chapter 1 more about factorization of polynomials

1

Chapter 1 More about Factorization of Polynomials

1A p.2

1B p.9

1C p.17

1D p.25

1E p.32

Chapter 2 Laws of Indices

2A p.39

2B p.49

2C p.57

2D p.68

Chapter 3 Percentages (II)

3A p.74

3B p.83

3C p.92

3D p.99

3E p.107

3F p.119

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2

F3A: Chapter 1A

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Book Example 3


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Book Example 4


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Book Example 5


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Book Example 6


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Consolidation Exercise


(Full Solution)

Maths Corner Exercise 1A Level 1


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3




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Maths Corner Exercise 1A Multiple Choice

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E-Class Multiple Choice Self-Test

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4

Book 3A Lesson Worksheet 1A (Refer to §1.1A)

1.1 Factorization Using Identities

1.1A Using the Difference of Two Squares Identity

a2 – b2 ≡ (a + b)(a – b)

Example 1 Instant Drill 1

Factorize (a) x2 – 32, (b) 52 – y2.

Sol (a) x2 – 32 = (x + 3)(x – 3) (b) 52 – y2 = (5 + y)(5 – y)

Factorize (a) p2 – 42, (b) 82 – q2.

Sol (a) p2 – 42 = ( )( ) (b) 82 – q2 = ( )( ) ○○○○→→→→ Ex 1A 1, 2


Factorize (a) x2 – 4, (b) y2 – 36.

Sol (a) x2 – 4 = x2 – 22 = (x + 2)(x – 2) (b) y2 – 36 = y2 – 62 = (y + 6)(y – 6)

Factorize (a) h2 – 9, (b) k2 – 49.

Sol (a) h2 – 9 = ( )2 – ( )2 = ( )( )

(b) k2 – 49 = ( )2 ( )2 = ( )( )

1. Factorize (a) x2 – 64, (b) u2 – 100, (c) w2 – 121.

Do you remember the square numbers 1, 4, 9,

16, 25, ?

a = x, b = 3 a = ___, b = ___

a = 5, b = y

4 = 22 9 = ( )2

36 = 62 49 = ( )2

102 = ___ 112 = ___

a = ___, b = ___

Factorize: convert the polynomial into the product of its factors.

5

2. Factorize (a) 25 – y2, (b) 36 – p2, (c) 81 – n2.

○○○○→→→→ Ex 1A 3–5

Example 3 Instant Drill 3 Factorize (a) 4x2 – 1, (b) 4x2 – y2.

Sol (a) 4x2 – 1 = (2x)2 – 12 = (2x + 1)(2x – 1) (b) 4x2 – y2 = (2x)2 – y2 = (2x + y)(2x – y)

Factorize (a) 9x2 – 16, (b) 9x2 – 16y2.

Sol (a) 9x2 – 16 = ( )2 – ( )2 = (b) 9x2 – 16y2 = ( )2 – ( )2 =

3. Factorize (a) 1 – 36x2, (b) 25p2 – 49, (c) 64 – 81s2.

4. Factorize (a) 4x2 – 25y2, (b) 9h2 – 64k2, (c) 49m2 – 100n2.

○○○○→→→→ Ex 1A 8–14

5. Factorize (a) x2y2 – 16, (b) p2q2 – 81.

6. Factorize (a) 4x2 – y2z2, (b) s2 – 25p2q2.

a = 2x b = 1

x2y2 = (xy)2 4x2 = ( )2

y2z2 = ( )2

a = 2x b = y

a = ( )

b = ( )

4x2 = ( )2

25y2 = ( )2

a = ( )

b = ( )

6

○○○○→→→→ Ex 1A 15–18

Level Up Questions

Factorize the following polynomials. [Nos. 7−−−−8] 7. (a) 49b2 – 36a2c2 (b) –9x2y2 + 16w2 = = 16w2 – ( ) = (c) 121h2 – 144m2n2 = 8. (a) 8x2 – 8 (b) 6x2 – 6y2 = 8( ) = ( )( ) = =

B

Take out the common factor 8 first.

The common factor is _________.

7

New Century Mathematics (2nd Edition) 3A

1 More about Factorization of Polynomials

Level 1

Factorize the following polynomials. [Nos. 1–18]

1. x2 − 22 2. 42 − y2 3. z2 − 64

4. c2 − 100 5. 36 − x2 6. −y2 + 1

7. 1 − 16u2 8. 25m2 − 49 9. 16a2 − 9

10. −81x2 + 25 11. 36a2 − b2 12. 4p2 − 25q2

13. c2d2 − 9 14. −49 + h2k2 15. 16p2 − q2r2

16. x2 − 100y2z2 17. 11x2 − 11 18. 3m2 − 3n2 Level 2


19. (3 + x)2 − 1 20. (y − 3)2 − 25 21. 121 − (m − 2n)2

22. (3 + x)2 − (1 + 2x)2 23. (2x + 3y)2 − (x − 2y)2 24. (a − 2b)2 − (2a + b)2

25. 3 − 75x2 26. 18c2 − 72d2 27. 6ab2 − 24ac2

28. 5(p + q)2 − 45 29. 32h2 − 2(k − 3)2 30. 18(x + 2y)2 − 2(x − y)2

31. 16 − a2 + 4b − ab 32. 2x + 7y + 4x2 − 49y2 33. p2 + 3q − 3p − q2


1A

8

Answer

Consolidation Exercise 1A

1. (x + 2)(x − 2) 2. (4 + y)(4 − y)

3. (z + 8)(z − 8) 4. (c + 10)(c − 10)

5. (6 + x)(6 − x) 6. (1 + y)(1 − y)

7. (1 + 4u)(1 − 4u) 8. (5m + 7)(5m − 7)

9. (4a + 3)(4a − 3) 10. (5 + 9x)(5 − 9x)

11. (6a + b)(6a − b) 12. (2p + 5q)(2p − 5q)

13. (cd + 3)(cd − 3) 14. (hk + 7)(hk − 7)

15. (4p + qr)(4p − qr) 16. (x + 10yz)(x − 10yz)

17. 11(x + 1)(x − 1) 18. 3(m + n)(m − n)

19. (4 + x)(2 + x) 20. (y + 2)(y − 8)

21. (11 + m − 2n)(11 − m + 2n)

22. (4 + 3x)(2 − x) 23. (3x + y)(x + 5y)

24. −(3a − b)(a + 3b) 25. 3(1 + 5x)(1 − 5x)

26. 18(c + 2d)(c − 2d) 27. 6a(b + 2c)(b − 2c)

28. 5(p + q + 3)(p + q − 3)

29. 2(4h + k − 3)(4h − k + 3)

30. 2(4x + 5y)(2x + 7y)

31. (4 − a)(4 + a + b) 32. (2x + 7y)(1 + 2x − 7y)

33. (p − q)(p + q − 3)

9

F3A: Chapter 1B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 7


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Book Example 8


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Book Example 9


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Book Example 10


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Book Example 11


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(Full Solution)

Maths Corner Exercise 1B Level 1



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___________ ( )

Maths Corner Exercise 1B Multiple Choice


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10



_________

11

Book 3A Lesson Worksheet 1B (Refer to §1.1B)

1.1B Using the Perfect Square Identities

(I) Square of the Sum of Two Numbers

a2 + 2ab + b2 ≡ (a + b)2

Example 1 Instant Drill 1 Factorize x2 + 2x + 1.

Sol x2 + 2x + 1 = x2 + 2(x)(1) + 12 = (x + 1)2

Factorize x2 + 16x + 64.

Sol x2 + 16x + 64 = ( )2 + 2( )( ) + ( )2 = ( )2

1. Factorize (a) x2 + 14x + 49, (b) s2 + 20s + 100.

2. Factorize (a) 4 + 4m + m2, (b) 16 + 8x + x2.

○○○○→→→→ Ex 1B 1, 3–5, 11, 12


Factorize (a) 9x2 + 6x + 1, (b) 4y2 + 12y + 9.

Sol (a) 9x2 + 6x + 1 = (3x)2 + 2(3x)(1) + 12 = (3x + 1)2 (b) 4y2 + 12y + 9 = (2y)2 + 2(2y)(3) + 32 = (2y + 3)2

Factorize (a) 4h2 + 20h + 25, (b) 9k2 + 24k + 16.

Sol (a) 4h2 + 20h + 25 = ( )2 + 2( )( ) + ( )2 = ( )2 (b) 9k2 + 24k + 16 =

3. Factorize (a) 81x2 + 18x + 1, (b) 9n2 + 42n + 49.

4. Factorize (a) 25y2 + 60y + 36, (b) 49m2 + 4 + 28m.

a = x, b = 1

a = ___, b = ___

9x2 = (3x)2

Rearrange the terms.

12

○○○○→→→→ Ex 1B 6, 13

(II) Square of the Difference of Two Numbers

a2 – 2ab + b2 ≡ (a – b)2


Factorize x2 – 8x + 16.

Sol x2 – 8x + 16 = x2 – 2(x)(4) + 42 = (x – 4)2

Factorize x2 – 14x + 49.

Sol x2 – 14x + 49 = ( )2 – 2( )( ) + ( )2 = ( )2

5. Factorize (a) x2 – 4x + 4, (b) s2 – 16s + 64.

6. Factorize (a) 1 – 2y + y2, (b) 36 – 12k + k2.

○○○○→→→→ Ex 1B 2, 7–9


Factorize (a) 16x2 – 8x + 1, (b) 9y2 – 12y + 4.

Sol (a) 16x2 – 8x + 1 = (4x)2 – 2(4x)(1) + 12 = (4x – 1)2 (b) 9y2 – 12y + 4 = (3y)2 – 2(3y)(2) + 22 = (3y – 2)2

Factorize (a) 25h2 – 30h + 9, (b) 36k2 – 60k + 25.

Sol (a) 25h2 – 30h + 9 = ( )2 – 2( )( ) + ( )2 = ( )2 (b) 36k2 – 60k + 25 =

7. Factorize (a) 81x2 – 36x + 4, (b) 16p2 – 56p + 49.

8. Factorize (a) 64h2 – 48h + 9, (b) 49y2 + 36 – 84y.

a = ___, b = ___

Rearrange the terms.

a = x, b = 4

13

○○○○→→→→ Ex 1B 10, 14

9. Factorize (a) x2 + 8xy + 16y2, (b) h2 – 6hk + 9k2.

10. Factorize (a) x2 + 4y2 + 4xy, (b) –10pq + 25p2 + q2.

○○○○→→→→ Ex 1B 15–18

Level Up Questions

Factorize the following polynomials. [Nos.11–14] 11. 25a2 – 40ab + 16b2 12. 100x2 + 140xy + 49y2 13. (a) 49x2 + 4y2 + 28xy (b) – 48ab + 9b2 + 64a2

a2 + 2ab + b2

≡ ( a2 − 2ab + b2

≡ (

14

14. 3x2 – 18x + 27

Take out the common factor

of all the terms first.

15



Level 1


1. x2 + 2(x)(2) + 22 2. x2 − 2(x)(7) + 72 3. k2 + 2k + 1

4. r2 + 16r + 64 5. 49c2 + 14c + 1 6. m2 − 6m + 9

7. u2 − 18u + 81 8. 64y2 − 16y + 1 9. 100 + 20t + t2

10. p2 + 16 − 8p 11. 9k2 + 42k + 49 12. 25 − 40x + 16x2

13. 36x2 + 12xy + y2 14. u2 − 22uv + 121v2 15. 144a2 + 24ab + b2

16. 25p2 − 110pq + 121q2 17. 56cd + 49c2 + 16d2 18. 4m2 + 81n2 − 36mn Level 2


19. 3x2 + 18x + 27 20. −4k2 − 28k − 49

21. −100t2 + 120t − 36 22. −a2 + 12ab − 36b2

23. 2m2 − 28mn + 98n2 24. −112x2 − 168xy − 63y2

25. x3 − 4x2 + 4x 26. −p3 + 10p2q − 25pq2

27. 12y + 12xy + 3x2y 28. (x − 2)2 + 6(x − 2) + 9

29. 25(m + n)2 + 10(m + n) + 1 30. 16(a + b)2 − 8c(a + b) + c2

31. (a) m2 − 8m + 16 (b) m2 − 8m + 16 − n2

32. (a) p2 + 18pq + 81q2 (b) p2 + 18pq + 81q2 − 25

33. (a) 36a2 − 12ab + b2 (b) 36a2 − 12ab + b2 − 66a + 11b


1B

16

Answer

Consolidation Exercise 1B

1. (x + 2)2 2. (x − 7)2

3. (k + 1)2 4. (r + 8)2

5. (7c + 1)2 6. (m − 3)2

7. (u − 9)2 8. (8y − 1)2

9. (10 + t)2 10. (p − 4)2

11. (3k + 7)2 12. (5 − 4x)2

13. (6x + y)2 14. (u − 11v)2

15. (12a + b)2 16. (5p − 11q)2

17. (7c + 4d)2 18. (2m − 9n)2

19. 3(x + 3)2 20. −(2k + 7)2

21. −4(5t − 3)2 22. −(a − 6b)2

23. 2(m − 7n)2 24. −7(4x + 3y)2

25. x(x − 2)2 26. −p(p − 5q)2

27. 3y(2 + x)2 28. (x + 1)2

29. (5m + 5n + 1)2 30. (4a + 4b − c)2

31. (a) (m − 4)2

(b) (m − 4 + n)(m − 4 − n)

32. (a) (p + 9q)2

(b) (p + 9q + 5)(p + 9q − 5)

33. (a) (6a − b)2

(b) (6a − b)(6a − b − 11)

17

F3A: Chapter 1C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 12


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Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)

Book Example 15


(Video Teaching)

Book Example 16


(Video Teaching)



(Full Solution)

Maths Corner Exercise 1C Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 1C Multiple Choice


___________ ( )

18



_________

19

Book 3A Lesson Worksheet 1C (Refer to §1.2A)

1.2 Factorization Using the Cross-method

1.2A Factorization of Polynomials in the Form of x2 + bx + c Cross-method Using the fact that (x + m)(x + n) = x2 + (m + n)x + mn to factorize polynomials.



Sol [Step 1111: Write the constant term +3 as a product of two factors. +3 = (+1)(+3) +3 = (–1)(–3)

Step 2222: Test each possible pair of factors by the cross-method.

x +1 x –1 x +3 x –3 +x + 3x = +4x –x – 3x = –4x ]

x2 + 4x + 3 = (x + 1)(x + 3)


Sol [Step 1111: Write the constant term +7 as a product of two factors. +7 = (+1)( ) +7 = (–1)( )


x +1 x –1 x x +x + ___ = ___ –x _____ = ___ ]

x2 + 8x + 7 = (x )(x )

1. Factorize x2 – 3x + 2.

x2 – 3x + 2 =


○○○○→→→→ Ex 1C 2, 3


Factorize x2 + 2x – 3.

Sol [Step 1111: Write the constant term –3 as a product of two factors.

Factorize x2 + 6x – 7.

Sol [Step 1111: Write the constant term ( ) as a product of two factors.

1111: +2 = ( )( ) +2 = ( )( )

2222: x ( ) x ( ) x ( ) x ( )

Find the x term. Can it give

+4x?

Which can give +8x?

� x + m

×) x + n x2 + mx +) nx + mn x2 + (m + n)x + mn

1111: +11 = ( )( ) +11 = ( )( )

2222: x ( ) x ( ) x ( ) x ( )

20

–3 = (–3)(+1) –3 = (+3)(–1)


x –3 x +3 x +1 x –1 –3x + x = –2x +3x – x = +2x ]

x2 + 2x – 3 = (x + 3)(x – 1)

( ) = ( )( ) ( ) = ( )( )


x ( ) x ( ) x ( ) x ( )

]

x2 + 6x – 7 = (x )(x )

3. Factorize the following polynomials. (a) x2 + 12x – 13

(b) x2 – 12x – 13

4. Factorize the following polynomials. (a) x2 – 16x – 17

(b) x2 + 18x – 19

○○○○→→→→ Ex 1C 4



Sol [Step 1111: Write the constant term +8 as a product of two factors:

+8: (+1)(+8), (+2)(+4), (–1)(–8), (–2)(– 4)


x +1 +2 x +8 +4 +x + 8x +2x + 4x = +9x = +6x ] x2 + 6x + 8 = (x + 2)(x + 4)


Sol [Step 1111: +6: (+1)(+6), (+2)(+3)


x ( ) ( ) x ( ) ( ) ]

x2 + 7x + 6 = (x )(x )

This pair can give +2x.

x ( )

x ( )

Since the coefficient of x is +6, do we need to test (–1)(–8), (–2)(– 4)? Why?

1111: –13 = ( )( ) –13 = ( )( )

2222: x ( ) x ( ) x ( ) x ( )

� We can skip writing (–1)(–6) and (–2)(–3).

21



7. Factorize x2 + 8x + 15.

8. Factorize x2 + 10x + 24.

○○○○→→→→ Ex 1C 5–9

9. Factorize x2 + 8x – 9.

10. Factorize –15 + x2 + 2x.

11. Factorize x2 – 4x – 21.

12. Factorize x2 – 22 – 9x.

○○○○→→→→ Ex 1C 10–12, 18, 19

Level Up Questions

Arrange the terms in descending order first.

x ( )

x ( ) 1111: +10: ( )( ), ( )( ) 2222: x ( ) ( ) x ( ) ( )

22

13. Factorize 15x + x2 + 26. 14. Factorize x2 – 3x – 18.

23



Level 1

1. (a) List out all the possible ways of writing −10 as a product of two factors. (The first one is already done as an example for you.)

−10 = (+1)(−10), −10 = ( )( ), −10 = ( )( ), −10 = ( )( )

(b) Using the result of (a), factorize the following polynomials.

(i) x2 + 9x − 10 (ii) x2 − 9x − 10 (iii) x2 − 3x − 10


2. x2 + 4x + 3 3. x2 − 3x + 2 4. x2 + 6x − 7

5. r2 + 5r + 4 6. k2 + 13k + 22 7. a2 − 10a + 9

8. m2 − 12m + 35 9. h2 − 7h + 12 10. w2 + 8w − 9

11. b2 + 3b − 10 12. p2 + 7p − 18 13. c2 − c − 20

14. y2 − 12y − 28 15. q2 + 13q + 40 16. −2v + v2 − 15

17. 4n − 21 + n2 18. −10s + 24 + s2 19. 42 + z2 − 13z Level 2


20. −x2 + 10x + 11 21. −x2 − 14x − 13 22. −x2 + 5x − 4

23. −x2 − 4x + 32 24. −2a + 35 − a2 25. 11y − y2 + 12

26. −20 + 12z − z2 27. b2 + 24b + 128 28. m2 − 27m − 90

29. u2 + 8u − 84 30. −14q − q2 + 72 31. x2 − 12xy + 11y2

32. r2 + 4rs − 21s2 33. −p2 − 11pq + 26q2 34. −b2 + 48c2 + 8bc

35. (a) Factorize k2 + 10k − 39.

(b) Hence, factorize hk − 3h − k2 − 10k + 39.


1C

24

Answer

Consolidation Exercise 1C 1. (a) −10 = (−1)(+10), −10 = (+2)(−5),

−10 = (−2)(+5)

(b) (i) (x − 1)(x + 10)

(ii) (x + 1)(x − 10)

(iii) (x + 2)(x − 5)

2. (x + 1)(x + 3) 3. (x − 1)(x − 2)

4. (x + 7)(x − 1) 5. (r + 1)(r + 4)

6. (k + 11)(k + 2) 7. (a − 9)(a − 1)

8. (m − 7)(m − 5) 9. (h − 3)(h − 4)

10. (w + 9)(w − 1) 11. (b + 5)(b − 2)

12. (p + 9)(p − 2) 13. (c + 4)(c − 5)

14. (y − 14)(y + 2) 15. (q + 5)(q + 8)

16. (v + 3)(v − 5) 17. (n + 7)(n − 3)

18. (s − 4)(s − 6) 19. (z − 6)(z − 7)

20. −(x + 1)(x − 11) 21. −(x + 1)(x + 13)

22. −(x − 1)(x − 4) 23. −(x + 8)(x − 4)

24. −(a + 7)(a − 5) 25. −(y − 12)(y + 1)

26. −(z − 10)(z − 2) 27. (b + 16)(b + 8)

28. (m + 3)(m − 30) 29. (u + 14)(u − 6)

30. −(q + 18)(q − 4) 31. (x − 11y)(x − y)

32. (r + 7s)(r − 3s) 33. −(p + 13q)(p − 2q)

34. −(b + 4c)(b − 12c)

35. (a) (k − 3)(k + 13)

(b) (k − 3)(h − k − 13)

25

F3A: Chapter 1D

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 17


(Video Teaching)

Book Example 18


(Video Teaching)

Book Example 19


(Video Teaching)



(Full Solution)

Maths Corner Exercise 1D Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 1D Multiple Choice


___________ ( )



_________

26

Book 3A Lesson Worksheet 1D (Refer to §1.2B)

1.2B Factorization of Polynomials in the Form of ax2 + bx + c


Factorize 2x2 + 7x + 5.

Sol [Step 1111: Write 2x2 as a product of two factors. 2x2 = (x)(2x)

Step 2222: The constant term +5 can be written as: (+1)(+5), (+5)(+1)


x +1 +5 2x +5 +1 +2x + 5x +10x + x = +7x = +11x ] 2x2 + 7x + 5 = (x + 1)(2x + 5)


Sol [Step 1111: Write 3x2 as a product of two factors. 3x2 = ( )( )

Step 2222: The constant term ( ) can be written as: ( )( ), ( )( )


( ) ( ) ( ) ( ) ( ) ( ) ] 3x2 + 5x + 2 = ( )( )

1. Factorize 5x2 – 34x – 7.

5x2 – 34x – 7 =


3. Factorize 2x2 + 3x – 5.

4. Factorize 3x2 – 10x + 3.

○○○○→→→→ Ex 1D 1–7, 18 Example 2 Instant Drill 2

Factorize 6x2 – 11x + 3.

Sol [Step 1111: The term 6x2 can be written as: (x)(6x), (2x)(3x)

Step 2222: The constant term +3 can be written as: (–1)(–3), (–3)(–1)



Sol [Step 1111: The term 8x2 can be written as: ( )( ), ( )( )

Step 2222: The constant term ( ) can be written as: ( )( ), ( )( )


Do not skip anyone of

1111: 5x2: ( )( ) 2222: –7: ( )( ),

( )( ) 3333: ( ) ( )

( ) ( ) ( ) ( )

27

x –1 –3 6x –3 –1 –6x – 3x –18x – x = –9x = –19x 2x –1 –3 3x –3 –1 –3x – 6x –9x – 2x = –9x = –11x ] 6x2 – 11x + 3 = (2x – 3)(3x – 1)

] 8x2 + 14x + 5 = ( )( )

5. Factorize 4x2 + 4x – 3.

6. Factorize 10x2 – 11x + 3.

7. Factorize 14x2 – 19x – 3. ○○○○→→→→ Ex 1D 8–11, 17, 19



Sol [Step 1111: The term 6x2 can be written as: (x)(6x), (2x)(3x)

Step 2222: The constant term +14 can be written as: (+1)(+14), (+14)(+1), (+2)(+7), (+7)(+2)


x +1 +14 +2 +7 6x +14 +1 +7 +2 +6x + 14x +84x + x +12x + 7x +42x + 2x = +20x = +85x = +19x = +44x 2x +1 +14 +2 +7 3x +14 +1 +7 +2 +3x + 28x

= +31x ]

6x2 + 31x + 14 = (2x + 1)(3x + 14)

Factorize 10x2 + 17x – 6.

Sol [Step 1111: The term 10x2 can be written as: ( )( ), ( )( )

Step 2222: The constant term ( ) can be written as: ( )( ), ( )( ), ( )( ), ( )( )


]

1111: 4x2: ( )( ), ( )( )

2222: –3: ( )( ), ( )( ) 3333:

28


9. Factorize 15x2 – 22x + 8.

10. Factorize 21x2 + 41x + 10.

11. Factorize 13x + 6x2 – 28.

○○○○→→→→ Ex 1D 12–16, 20, 21

Level Up Questions

12. Factorize 12x2 – 37x + 21. 13. Factorize –7x2 – 6x + 13. –7x2 – 6x + 13

= –( )

1111: 9x2: ( )( ), ( )( )

2222: – 4: ( )( ), ( )( ), ( )( ), ( )( )

3333:

30



Level 1


1. 2x2 + 3x + 1 2. 3y2 + 7y + 2 3. 5z2 + 16z + 3

4. 3a2 + a − 2 5. 2b2 − 13b − 7 6. 5n2 − 13n + 6

7. 8y2 + 25y + 3 8. 7u2 − 12u − 4 9. 10t2 + 3t − 1

10. 28d2 − 11d + 1 11. 21x2 − 5x − 6 12. 8m2 + 22m + 15

13. 15r2 − 23r + 4 14. 18y2 + 9y − 14 15. 11 + 14c + 3c2

16. 10k2 − 7 − 9k 17. −2 + 25x2 − 5x 18. −z − 35 + 6z2

Level 2


19. −2x2 + 5x − 2 20. −3y2 − 40y − 13 21. −11t + 3 − 20t2

22. 8x − 4x2 + 21 23. 2p2 − 24p + 22 24. 5k2 + 90 − 45k

25. −35c + 14c2 − 126 26. −12a2 + 9a + 30 27. 58u − 14 − 48u2

28. 11m2 − 32mn − 3n2 29. 5x2 − 18xy − 8y2 30. 18r2 + 45rs − 38s2

31. ab − 63a2 + 12b2 32. 3p2 + 108q2 − 39pq 33. 4h2 + 42k2 − 34hk

34. (a) Factorize 8x2 + 2xy − 3y2.

(b) Hence, factorize 3y2 − 8x2 + 5y − 10x − 2xy.


1D

31

Answer

Consolidation Exercise 1D

1. (2x + 1)(x + 1) 2. (3y + 1)(y + 2)

3. (5z + 1)(z + 3) 4. (3a − 2)(a + 1)

5. (2b + 1)(b − 7) 6. (5n − 3)(n − 2)

7. (8y + 1)(y + 3) 8. (7u + 2)(u − 2)

9. (5t − 1)(2t + 1) 10. (7d − 1)(4d − 1)

11. (7x + 3)(3x − 2) 12. (4m + 5)(2m + 3)

13. (5r − 1)(3r − 4) 14. (3y − 2)(6y + 7)

15. (3c + 11)(c + 1) 16. (5k − 7)(2k + 1)

17. (5x + 1)(5x − 2) 18. (3z + 7)(2z − 5)

19. −(2x − 1)(x − 2) 20. −(3y + 1)(y + 13)

21. −(4t + 3)(5t − 1) 22. −(2x + 3)(2x − 7)

23. 2(p − 11)(p − 1) 24. 5(k − 6)(k − 3)

25. 7(2c − 9)(c + 2) 26. −3(4a + 5)(a − 2)

27. −2(8u − 7)(3u − 1) 28. (11m + n)(m − 3n)

29. (5x + 2y)(x − 4y) 30. (3r − 2s)(6r + 19s)

31. −(7a + 3b)(9a − 4b)

32. 3(p − 9q)(p − 4q) 33. 2(h − 7k)(2h − 3k)

34. (a) (2x − y)(4x + 3y)

(b) (y − 2x)(5 + 4x + 3y)

32

F3A: Chapter 1E

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 20


(Video Teaching)

Book Example 21


(Video Teaching)

Book Example 22


(Video Teaching)

Book Example 23


(Video Teaching)

Book Example 24


(Video Teaching)



(Full Solution)

Maths Corner Exercise 1E Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 1E Multiple Choice


___________ ( )

33



_________

34

Book 3A Lesson Worksheet 1E (Refer to §1.3)

1.3 Factorization Using the Difference and Sum of Two Cubes Identities

(A) Difference of the Two Cubes Identity

a3 −−−− b3 ≡ (a −−−− b)(a2 + ab + b2)


Factorize (a) x3 − 1, (b) y3 − 27.

Sol (a) x3 − 1 = x3 − 13 = (x − 1)[x2 + x(1) + 12] = (x − 1)(x2 + x + 1)

(b) y3 − 27 = y3 − 33 = (y − 3)[y2 + y(3) + 32] = (y − 3)(y2 + 3y + 9)

Factorize (a) x3 − 64, (b) y3 − 125.

Sol (a) x3 − 64 = x3 − ( )3 = ( − )[( )2 + ( )( ) + ( )2] =

(b) y3 − 125 = y3 − ( )3 =

1. Factorize (a) p3 – 216, (b) h3 – 1 000. (a) p3 – 216 = ( )3 − ( )3 =

(b) h3 – 1 000 =

2. Factorize (a) 8 – k3, (b) 343 – n3.

3. Factorize (a) 27x3 – 1, (b) 8h3 – 125.

(a) 27x3 – 1 = ( )3 – 13 =

4. Factorize (a) x3 – 64y3, (b) 512p3 – q3.

43 = ______

53 = ______

Try to memorize the following cube numbers:

13 = 1 63 = 216 23 = 8 73 = 343 33 = 27 83 = 512 43 = 64 93 = 729 53 = 125 103 = 1 000

35

(b) 8h3 – 125 = ○○○○→→→→ Ex 1E 2, 4, 6, 8, 10, 11, 13

(B) Sum of the Two Cubes Identity

a3 + b3 ≡ (a + b)(a2 −−−− ab + b2)


Factorize (a) x3 + 1, (b) y3 + 8.

Sol (a) x3 + 1 = x3 + 13 = (x + 1)[x2 − x(1) + 12] = (x + 1)(x2 − x + 1)

(b) y3 + 8 = y3 + 23

= (y + 2)[y2 − y(2) + 22] = (y + 2)(y2 − 2y + 4)

Factorize (a) x3 + 27, (b) y3 + 125.

Sol (a) x3 + 27 = x3 + ( )3 = ( + )[( )2 − ( )( ) + ( )2] =

(b) y3 + 125 = ( )3 + ( )3 =

5. Factorize (a) p3 + 216, (b) h3 + 729.

6. Factorize (a) 64 + k3, (b) 343 + n3.

7. Factorize (a) 64p3 + 1, (b) 343h3 + 27.

8. Factorize (a) m3 + 512n3, (b) 1 000h3 + k3.

○○○○→→→→ Ex 1E 1, 3, 5, 7, 9, 12, 14

9. Factorize

Pay attention to the sign!

36

(a) 27x3 – 125y3, (b) 8p3 + 729q3.

○○○○→→→→ Ex 1E 15–18

Level Up Questions

10. Factorize x3 +8

1.

11. Factorize x3y3 − 512z3.

3

32

1

8

1

==

x3y3 = (xy)3

37



Level 1


1. (3k)3 + 1 2. 1 − (5r)3 3. (11x)3 + 1

4. y3 − 27 5. 64 + z3 6. 1 − 8w3

7. 216c3 + 1 8. 125 − x3y3 9. a3 + 343b3

10. 125m3 − 8 11. 343 + 27s3 12. 512x3 − 729

13. 27x3 − 64y3 14. 125a3 + 729b3 15. 1 000p3 − 343q3 Level 2


16. x3 −

64

1 17. 8y3 +

27

1 18. 27x3 −

125

3y

19. 4k3 + 108 20. −448r3 + 7 21. 686a3 − 54b3

22. 500x3 − 32y3 23. ab4 − a4b 24. 135xy3 − 40x4

25. (x − 2)3 − 729 26. 64x3 + (x − 1)3 27. (1 − 3x)3 − (1 + 3x)3

28. (a) Factorize 9x2 − 4.

(b) Hence, factorize 729x6 − 64.


1E

38

Answer

Consolidation Exercise 1E

1. (3k + 1)(9k2 − 3k + 1)

2. (1 − 5r)(1 + 5r + 25r2)

3. (11x + 1)(121x2 − 11x + 1)

4. (y − 3)(y2 + 3y + 9)

5. (4 + z)(16 − 4z + z2)

6. (1 − 2w)(1 + 2w + 4w2)

7. (6c + 1)(36c2 − 6c + 1)

8. (5 − xy)(25 + 5xy + x2y2)

9. (a + 7b)(a2 − 7ab + 49b2)

10. (5m − 2)(25m2 + 10m + 4)

11. (7 + 3s)(49 − 21s + 9s2)

12. (8x − 9)(64x2 + 72x + 81)

13. (3x − 4y)(9x2 + 12xy + 16y2)

14. (5a + 9b)(25a2 − 45ab + 81b2)

15. (10p − 7q)(100p2 + 70pq + 49q2)

16.

++

−

16

1

44

1 2 xxx

17.

+−

+

9

1

3

24

3

12 2 y

yy

18.

++

−

255

39

53

22 yxy

xy

x

19. 4(k + 3)(k2 − 3k + 9)

20. 7(1 − 4r)(1 + 4r + 16r2)

21. 2(7a − 3b)(49a2 + 21ab + 9b2)

22. 4(5x − 2y)(25x2 + 10xy + 4y2)

23. ab(b − a)(b2 + ab + a2)

24. 5x(3y − 2x)(9y2 + 6xy + 4x2)

25. (x − 11)(x2 + 5x + 67)

26. (5x − 1)(13x2 + 2x + 1)

27. −18x(3x2 + 1)

28. (a) (3x + 2)(3x − 2)

(b) (3x + 2)(3x − 2)(81x4 + 36x2 + 16)

39

F3A: Chapter 2A

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 1


(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



_________

40

Book 3A Lesson Worksheet 2A (Refer to §2.1)

2.1A Zero Index

a0 = 1, where a ≠ 0. � e.g. 30 = 1, (–5)0 = 1, (2x)0 = 1 1. (a) Evaluate the following without using a calculator. (i) 70 (ii) (–8)0 (iii) –90

(b) Simplify the following expressions, where b, c, d, e ≠ 0.

(i)

0

4

b (ii) –5c

0 (iii) (–d 3e4)0

○○○○→→→→ Ex 2A 1, 2, 16, 17

2.1B Negative Integral Indices

a–n =na

1, where a ≠ 0 and n is a positive integer. � e.g. 3–1 =

3

1, 4–2 =

24

1=

16

1, (6x)–1 =

x6

1

2. (a) Evaluate the following without using a calculator. (i) 7–1 (ii) 2–3 (iii) (–8)–2

(b) Simplify the following expressions (where r, s, t ≠ 0) and express the answers with positive indices. (i) r–

4 (ii) (–s)–1 (iii) (8t) –2

○○○○→→→→ Ex 2A 3, 4, 21

Without using a calculator, evaluate the following and give the answers in fractions. [Nos. 3–4] 3. 4–1 + 2–1

4. 3–2 × (–9)0

○○○○→→→→ Ex 2A 5–9

2.1C Laws of Integral Indices

If m and n are integers and a, b ≠ 0, then

(a) am × an = am + n

(b) am ÷ an = am – n

41

(c) (am)n = am × n (d) (ab)n = anbn

(e)

n

b

a

=

n

n

b

a


Without using a calculator, evaluate

(a) 5–

4 × 53, (b) 7–2 ÷ 7–

4.

Sol (a) 5–

4 × 53

= 5–

4 + 3 � am × an = am + n = 5–1

=5

1 � a–1 =

a

1


(a) 65 × 6–3, (b) 9–7 ÷ 9–8.

Sol (a) 65 × 6–3 = 6( ) ( ) =

(b) 7–2 ÷ 7–

4

= 7–2 – (–

4) � am ÷ an = am – n = 7–2 + 4 = 72 = 49

(b) 9–7 ÷ 9–8 = 9( ) ( ) = ○○○○→→→→ Ex 2A 7–11

Simplify the following expressions (where p, q, r, s ≠ 0) and express the answers with positive indices. [Nos. 5–6]

5. (a) p–8 × p 2

(b) 3r7 × r–5

6. (a) q5 ÷ q–3

(b) 16s–

4 ÷ 2s3

○○○○→→→→ Ex 2A 22–24


Without using a calculator, evaluate (a) (2–3)–2,

(b) 1

7

4−

.

Sol (a) (2–3)–2

= 2–3 × (–2) � (am)n = am × n

= 26 = 64

Without using a calculator, evaluate (a) (4–1)3,

(b) 1

3

5−

.

Sol (a) (4–1)3

= ( )( ) ( ) =

Remember to express the answers with positive indices!

am × an = am + n am ÷ an = am – n

42

(b)

1

7

4−

= 1

1

7

4−

−

�

n

b

a

=

n

n

b

a

= 4

7

(b)

1

3

5−

=)(

)(

)(

)(

=

○○○○→→→→ Ex 2A 12–15

Simplify the following expressions and express the answers with positive indices. [Nos. 7–8] (All the letters in the expressions represent non-zero numbers.)

7. (a) (k2)–5 (b) (y–3)–6

8. (a)

12

−

h

(b)

2−

q

p

○○○○→→→→ Ex 2A 25–27 ○○○○→→→→ Ex 2A 30


Simplify the following expressions (where

x, y ≠ 0) and express the answers with positive indices. (a) (4x)–2 (b) (–5y–1)3

Sol (a) (4x)–2

= 4–2x–2 � (ab)n = anbn

= 216

1

x � a–1 =

a

1

Simplify the following expressions (where

r, s ≠ 0) and express the answers with positive indices. (a) (–3r–1)

4 (b) (2s2)–5

Sol (a) (–3r–1)

4

= ( )( ) ( )( )

=

1

1

−

−

7

4 =

1

1

4

7

(am)n = am × n n

b

a

=

n

n

b

a

43

(b) (–5y–1)3

= (–5)3y–1 × 3

= (–5)3y–3

= 3

125

y−

(b) (2s2)–5 =

Simplify the following expressions and express the answers with positive indices. [Nos. 9–14] (All the letters in the expressions represent non-zero numbers.) 9. (a) (7g4)–3 (b) (–6t

–5)2

10. (a) (–hk)–9 (b) (xy–2)–4

○○○○→→→→ Ex 2A 31

11. (a) 3)2(

24−− s

(b) 6

51 )(

e

e−−

12. (a) d

d

9

)3( 23 −

(b) 24

3

)5(

5−f

f

(ab)n = anbn

44

○○○○→→→→ Ex 2A 28, 29

13. (a) (4h–1k0)3

(b)

2

4

03−

−

−

q

p

14. (a) (6rt –2)2

(b) (–5ab4)–3

○○○○→→→→ Ex 2A 32, 33

Level Up Questions

45

15. Evaluate the following without using a calculator.

(a) 4–1 ÷ 2–3 = (2( ))–1 ÷ 2–3 = (b) 272 × (–3)–

4

16. Simplify the following expressions (where x, y ≠ 0) and express the answers with positive indices.

(a) (–x–3y2)–

4 (b) 6

145 )(−

−−

x

yx

(c) 512

70

)( −−−

−

− yx

yx

Convert 27 into the powers of 3. i.e. 27 = 3( )

46


2 Laws of Indices

Level 1

Without using a calculator, find the values of the following expressions and give the answers in

integers or fractions. [Nos. 1−−−−12]

1. −70 2. (−7)0 3. (190)−2

4. 18

1−

5. 6−1 + (3 + 1)0 6. 3−2 × (−2−1)

7. 80 ÷ (−5)−2 8. 97 × 9–5 9. 7−3 ÷ 7−2

10. 1

3

2−

− 11.

2

5

3−

− 12. (2−2)−2

Simplify the following expressions and express the answers with positive indices. [Nos. 13−−−−27] (All the letters in the expressions represent non-zero numbers.)

13. (8p)0 14. 5q0 15. −(r−9)0

16. (a3b−3)0 17. (−x0)−6 18. (−y)−9

19. c−4 × c5 20. g ÷ g−4 21. (s−3)5

22. (−k −3)−4 23. 2)5(

4−b

24. 2

5

1−

z

25.

34

2

−

u 26. (3c−2d)2 27.

3

2

1−

−

mn


2A

47

Level 2 Without using a calculator, find the values of the following expressions and give the answers in

integers or fractions. [Nos. 28−−−−33]

28. 25−2 ÷ 5−3 29. 4

1

2

1)4(

−

−

−×− 30. 5−4 ÷ 125−4 × 25−5

31. 24 ÷ 6−2 × 2−4 32. (2−4 × 1250) ÷ 12−2 33. 62 − 6–11 ÷ 6−12

Simplify the following expressions and express the answers with positive indices. [Nos. 34−−−−45] (All the letters in the expressions represent non-zero numbers.)

34. (x −3y −2)−3 35. (7−1a−5b3)−1 36. (−5−1r3s−2)−2

37. 4

132 )(−

−−

n

nm 38.

2

6

05−

−

d

c 39.

1

42

20−

−

−

ts

ts

40. 72

95

3

3−

−−

x

x 41. (mn3)−2(nm−4)−1 42.

31

212

)(

)(

ba

ba−

−

43. 215

243

)(4

)2(−−−−

−

yx

yx 44.

6

2633

)2(

)()6(

−

−×

a

aa 45.

3

12

51

15

48

−×

−

−−

−−

ba

b

a

ba

48

Answer


1. −1

2. 1

3. 1

4. 8

5. 6

7

6. 18

1−

7. 25

8. 81

9. 7

1

10. 2

3−

11. 9

25

12. 16

13. 1

14. 5

15. −1

16. 1

17. 1

18. 9

1

y−

19. c

20. g5

21. 15

1

s

22. k12

23. 100b2

24. z10

25. 12

8

u

26. 4

29

c

d

27. −8m3n3

28. 5

1

29. −4

30. 25

1

31. 36

32. 9

33. 30

34. x9y6

35. 3

57

b

a

36. 6

425

r

s

37. 2

7

m

n

38. 25

12d

39. 2

6

s

t

40. 2

27

x−

41. 7

2

n

m

42. 5

7

b

a

43. 4

6

x

y−

44. 27

8 9a

45. 118

1

b−

49

F3A: Chapter 2B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



_________

50

Book 3A Lesson Worksheet 2B (Refer to §2.2)

2.2 Scientific Notation

2.2A Introduction

A positive number expressed in scientific notation is in the form of

a × 10n,

where 1 ≤ a < 10, and n is an integer.


Express each of the following numbers in scientific notation. (a) 5 000 (b) 10 500 000

Sol (a) 5 000

= 5 × 103

Express each of the following numbers in scientific notation. (a) 60 000 (b) 4 020 000

Sol (a) 60 000

= 6 × 10( )

(b) 10 500 000

= 1.05 × 107

(b) 4 020 000 =

1. Express each of the following numbers in scientific notation.

(a) 300 000 (b) 87 000 000 (c) 923.1

○○○○→→→→ Ex 2B 2(a)–(c)


Express each of the following numbers in scientific notation. (a) 0.004 (b) 0.000 062

Sol (a) 0.004

= 4 × 10–3

Express each of the following numbers in scientific notation. (a) 0.000 3 (b) 0.000 008 4

Sol (a) 0.000 3

= 3 × 10( )

(b) 0.000 062

= 6.2 × 10–5

(b) 0.000 008 4 =


(a) 0.000 07 (b) 0.001 34

5 000.

Move to the left

for 3 digits

10 500 000.

7 digits

60 000.

___ digits

4 020 000.

___ digits

0.000 062

5 digits 0.000 008 4

___ digits

0.004

Move to the right

for 3 digits 0.000 3

___ digits

51

(c) 0.000 000 269 (d) 0.000 000 049

○○○○→→→→ Ex 2B 2(d)–(f)


Express the following numbers as integers or decimals.

(a) 7 × 103

(b) 6 × 10–5

(c) 2.1 × 10–6

Sol (a) 7 × 103

= 7 × 1 000 = 7 000

Express the following numbers as integers or decimals.

(a) 9.2 × 105

(b) 2 × 10–7

(c) 3.8 × 10–4

Sol (a) 9.2 × 105

= 9.2 × ( ) =

(b) 6 × 10–5

= 6 × 0.000 01 = 0.000 06

(b) 2 × 10–7 =

(c) 2.1 × 10–6

= 2.1 × 0.000 001 = 0.000 002 1

(c) 3.8 × 10–4 =

3. Express the following numbers as integers or decimals.

(a) 8 × 104 (b) 7.3 × 106

(c) 4 × 10–8 (d) 5.06 × 10–5

○○○○→→→→ Ex 2B 4

2.2B Applications of Scientific Notation

I. Simplifying Operations


7.000

Move to the right

for 3 digits

0000 06.

Move to the left

for 5 digits __________2.

___ digits

0000 002.1

6 digits

9.2________

___ digits

________3.8

___ digits

52


5 × 106 + 3.8 × 107 and express the answer in scientific notation.

Sol 5 × 106 + 3.8 × 107

= 0.5 × 107 + 3.8 × 107

= (0.5 + 3.8) × 107

= 4.3 × 107


6.3 × 108 + 2 × 107 and express the answer in scientific notation.

Sol 6.3 × 108 + 2 × 107

=



(4 × 102) × (3 × 106) and express the answer in scientific notation.

Sol (4 × 102) × (3 × 106)

= (4 × 3) × 102 + 6

= 12 × 108

= 1.2 × 109


(6 × 10–5) × (7 × 109) and express the answer in scientific notation.

Sol (6 × 10–5) × (7 × 109) =

Without using a calculator, evaluate the following expressions and express the answers in scientific

notation. [Nos. 4–7]

4. 7.2 × 106 – 4.2 × 105

5. 2 000 000 + 40 000 000

○○○○→→→→ Ex 2B 5(a), (b)

6. (6.2 × 109) × (2 × 10–13)

7. (8.1 × 10–4) ÷ (3 × 105)

○○○○→→→→ Ex 2B 5(c), (d)

II. Practical Applications

8. Express the following data in scientific notation.

(a) The radius of moon is about 17 381 000 m.

(b) The length of an Amoeba is about 0.000 22 m.

(c) The world population in 2017 is about 7 510 000 000.

(d) The diameter of a human red blood cell is about 0.000 006 2 m.

○○○○→→→→ Ex 2B 7

�� ‘Explain Your Answer’ Question

5 × 106

= (0.5 × 101) × 106

= 0.5 × 107

Express each term in scientific notation

am ÷ an = am –

n

2 × 107

= [( ) × 101] × 107

= ( ) × 108

am × an = am +

n

10–5 × 109 = 10( ) +

( )

12 = 1.2 × 101

53

9. The diameter of the Earth is about 1.27 × 104 km. The distance between the Jupiter and the Earth is about 629 000 000 km. Susan claims that the distance between the Jupiter and the Earth is more than 50 000 times the diameter of the Earth. Do you agree? Explain your answer.

Distance between the Jupiter and the Earth = 629 000 000 km = ( ) × 10( ) km The required number of times =

∵ ______________ ( > / = / < ) 50 000

∴ The claim is (agreed / disagreed).

Level Up Questions

10. Round off the following numbers to 3 significant figures and express the results in scientific notation.

(a) 2 468.3 = ( ), cor. to 3 sig. fig. i.e. 2 468.3 = , cor. to 3 sig. fig.

(b) 0.000 517 29

11. Without using a calculator, evaluate 7.4 × 106 – 3.62 × 105 + 8 × 104 and express the answer in scientific notation.

Express this data in scientific notation

Round off the number first.

54


2 Laws of Indices

Level 1

1. Determine whether each of the following is expressed in scientific notation. If not, express the

number in scientific notation.

(a) −3.14 × 100 (b) 4.27 × (−10)5

(c) 8.73 × 10−8 (d) −65.3 × 10−5


(a) 8 000 000 000 (b) 2 296.03

(c) 9 580 000 (d) 0.003 109

(e) 0.000 401 (f) 0.000 098

3. Round off the following numbers to 3 significant figures and express the results in scientific

notation.

(a) 9 753.1 (b) 907 684.27

(c) 0.040 742 (d) 0.000 246 89

(e) 51

27360 (f)

11

573

4. Express each of the following numbers as an integer or a decimal.

(a) 5 × 103 (b) 9.53 × 106

(c) −6.19 × 105 (d) −10−3

(e) 6.1 × 10–1 (f) 3.93 × 10−5

5. Without using a calculator, evaluate the following expressions and express the answers in

scientific notation.

(a) 256 000 000 000 + 8 300 000 000 (b) 0.000 000 045 − 0.000 000 144

(c) (5 × 1036) × (2 × 10−24) (d) (6 × 10–4) ÷ (3 × 102)

6. Use a calculator to evaluate the following expressions and express the answers in scientific

notation.

(a) 8.5 × 106 + 7.3 × 107 (b) 2.3 × 1015 − 6.9 × 1014

(c) (5.4 × 103) × (9.5 × 10−6) (d) (2.43 × 10−12) ÷ (9 × 10−7)


2B

55

7. Express the following data in scientific notation.

(a) The surface area of the moon is about 38 000 000 km2.

(b) Hong Kong citizens dispose about 15 000 000 000 g of garbage every day.

(c) The average diameter of human hair is about 0.000 05 m.

(d) The diameter of a water molecule is about 0.000 000 29 mm.

Level 2

8. Without using a calculator, evaluate the following expressions and express the answers in scientific notation.

(a) 4 × 102 + 0.7 × 104 − 2 × 103

(b) 6.6 × 10−6 + 20 × 10−8 + 1 × 10−7

(c) 34 × 103 + 5.8 × 104 + 0.72 × 105

(d) –840 × 10−2 + 98 × 10−1 + 7 × 101 9. Without using a calculator, find the values of the following expressions and express the answers

in scientific notation.

(a) 5000000000.0

000000500000025 ×

(b) 1

2

0000008002

00000000030080000003−

−

10. Use a calculator to find the values of the following expressions correct to 2 significant figures, and express the answers in scientific notation.

(a) 3

9

00032

7

00013−

+ (b) 83 ÷ (4 001−3 × 4 0078)

(c) 6106.4

3312−×

(d)

2

2

32

102

104.610

×

×−

−−

11. A jet has travelled for 39.7 hours. If the average speed of the jet is 2.8 × 102 m/s, find the distance travelled in m.

(Give the answer correct to 3 significant figures and express the result in scientific notation.)

12. △ABC is a right-angled triangle with ∠ABC = 90°. If AB = 1.3 × 1010 m and

BC = 9.6 × 109 m, find the length of AC in m. (Give the answer correct to 2 significant figures and express the result in scientific notation.)

56

Answer


1. (a) yes (b) no, −4.27 × 105

(c) yes (d) no, −6.53 × 10−4

2. (a) 8 × 109 (b) 2.296 03 × 103

(c) 9.58 × 106 (d) 3.109 × 10−3

(e) 4.01 × 10−4 (f) 9.8 × 10−5

3. (a) 9.75 × 103 (b) 9.08 × 105

(c) 4.07 × 10−2 (d) 2.47 × 10−4

(e) 3.61 × 102 (f) 7.35 × 101

4. (a) 5 000 (b) 9 530 000

(c) −619 000 (d) −0.001

(e) 0.61 (f) 0.000 039 3

5. (a) 2.643 × 1011 (b) −9.9 × 10−8

(c) 1 × 1013 (d) 2 × 10−6

6. (a) 8.15 × 107 (b) 1.61 × 1015

(c) 5.13 × 10−2 (d) 2.7 × 10−6

7. (a) 3.8 × 107 km2 (b) 1.5 × 1010 g

(c) 5 × 10−5 m (d) 2.9 × 10−7 mm

8. (a) 5.4 × 103 (b) 6.9 × 10−6

(c) 1.64 × 105 (d) 7.14 × 101

9. (a) 2.5 × 1023

(b) 1.96 × 1021

10. (a) 6.3 × 10−12 (b) 8.0 × 10−17

(c) 2.3 × 104 (d) 9.9 × 10−5

11. 4.00 × 107 m

12. 1.6 × 1010 m

57

F3A: Chapter 2C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 9


(Video Teaching)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)

Book Example 12


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



_________

58

Book 3A Lesson Worksheet 2C (Refer to §2.3)

2.3A Denary System and Denary Numbers (a) Denary system consists of ten numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 e.g. 2310 and 48910 are denary numbers. (b) Taking 8 54210 as an example,

Digit 8 5 4 2

Place value 103 102 10 1

∴ 8 54210 = 8 ×××× 103 + 5 ×××× 102 + 4 ×××× 10 + 2 ×××× 1

The expression is called the expanded form of 8 54210.


Consider the denary number 7 05810. (a) Write down the place value of each digit in

the number. (b) Hence, express 7 05810 in the expanded

form.

Sol (a) The place value of each digit is listed as follows:

Digit 7 0 5 8


(b) 7 05810

= 7 ×××× 103 + 0 ×××× 102 + 5 ×××× 10 + 8 ×××× 1

Consider the denary number 6 90410. (a) Write down the place value of each digit in

the number. (b) Hence, express 6 90410 in the expanded

form.


Digit 6 9 0 4

Place value 10 1

(b) 6 90410

= 6 × 10( ) + 9 × ( ) +

1. Consider the denary number 29810. (a) Write down the place value of each

digit in the number. (b) Hence, express 29810 in the expanded

form. (a) The place value of each digit is listed

as follows:

Digit

Place value

(b)

2. Express the denary number 18 30710 in the expanded form.

○○○○→→→→ Ex 2C 1, 6, 9

Represent each of the following expressions as a denary number. [Nos. 3–4]

× 10 × 10 × 10

Digit 1 8 3 0 7

Place

v

a

� ‘10’ indicates that they are denary numbers.

59

3. 5 × 102 + 6 × 10 + 7 × 1

4. 8 × 103 + 0 × 102 + 5 × 10 + 9 × 1

○○○○→→→→ Ex 2C 12

2.3B Binary System and Binary Numbers (a) Binary system consists of two numerals: 0 and 1 e.g. 1012 and 1001102 are binary numbers.

(b) Taking 11102 as an example,

Digit 1 1 1 0


∴ 11102 = 1 ×××× 23 + 1 ×××× 22 + 1 ×××× 2 + 0 ×××× 1

The expression is called the expanded form of 11102.


Consider the binary number 10112. (a) Write down the place value of each digit in

the number. (b) Hence, express 10112 in the expanded

form.


Digit 1 0 1 1


(b) 10112

= 1 ×××× 23 + 0 ×××× 22 + 1 ×××× 2 + 1 ×××× 1

Consider the binary number 11112. (a) Write down the place value of each digit in

the number. (b) Hence, express 11112 in the expanded

form.


Digit 1 1 1 1

Place value 2 1

(b) 11112

= 1 × ( ) + 1 × ( ) +

5. Consider the binary number 1002. (a) Write down the place value of each

digit in the number. (b) Hence, express 1002 in the expanded


as follows:

Digit

6. Consider the binary number 110012. (a) Write down the place value of each

digit in the number. (b) Hence, express 110012 in the

expanded form.

× 2 × 2 × 2

Digit

Place value 102 10 1

� ‘2’ indicates that they are binary numbers.

60

Place value

(b)

○○○○→→→→ Ex 2C 2, 7

Express the following binary numbers in the expanded form. [Nos. 7–8] 7. 10012

8. 101012

○○○○→→→→ Ex 2C 10

Represent each of the following expressions as a binary number. [Nos. 9–10]

9. 1 × 22 + 1 × 2 + 0 × 1

10. 1 × 24 + 0 × 23 + 0 × 22 + 1 × 2 + 1 × 1

○○○○→→→→ Ex 2C 13

2.3C Hexadecimal System and Hexadecimal Numbers (a) Hexadecimal system consists of sixteen numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F, where the values of A to F are as follows:

A B C D E F

10 11 12 13 14 15

e.g. 1A2B16 and 56CD8916 are hexadecimal numbers.

(b) Taking C50D16 as an example,

Digit C 5 0 D


∴ C50D16 = 12 ×××× 163 + 5 ×××× 162 + 0 ×××× 16 + 13 ×××× 1

The expression is called the expanded form of C50D16.

Digit 1 0 1 0 1

Place

v

a

Digit 1 0 0 1

Place

v

a

Digit

Place value 22 2 1

× 16 × 16 × 16

� ‘16’ indicates that they are hexadecimal

numbers.

61

Example 3 Instant Drill 3 Consider the hexadecimal number 34AB16. (a) Write down the place value of each digit in

the number. (b) Hence, express 34AB16 in the expanded

form.


Digit 3 4 A B


(b) 34AB16

= 3 ×××× 163 + 4 ×××× 162 + 10 ×××× 16 + 11 ×××× 1

Consider the hexadecimal number 2CE16. (a) Write down the place value of each digit in

the number. (b) Hence, express 2CE16 in the expanded

form.


Digit 2 C E

Place value 1

(b) 2CE16

= 2 × ( ) + ( ) × ( ) +

11. Consider the hexadecimal number F0516. (a) Write down the place value of each

digit in the number. (b) Hence, express F0516 in the expanded


as follows:

Digit

Place value

(b)

12. Consider the hexadecimal number D7A816. (a) Write down the place value of each

digit in the number. (b) Hence, express D7A816 in the

expanded form.

○○○○→→→→ Ex 2C 3, 8

Express the following hexadecimal numbers in the expanded form. [Nos. 13–14] 13. 36916

14. B657016

Digit B 6 5 7 0

Place

v

a

Digit 3 6 9

Place value

62

○○○○→→→→ Ex 2C 11

Represent each of the following expressions as a hexadecimal number. [Nos. 15–16]

15. 8 × 162 + 14 × 16 + 12 × 1

16. 13 × 163 + 0 × 162 + 15 × 16 + 11 × 1

○○○○→→→→ Ex 2C 14

Level Up Questions

17. Represent 5 × 104 + 3 × 102 + 2 × 10 as a denary number.

5 × 104 + 3 × 102 + 2 × 10

= 5 × 104 + ( ) × 103 + 3 × 102 + 2 × 10 + ( ) × ( ) = 18. Represent 25 + 23 + 1 as a binary number.

19. Represent 165 + 162 + 16 + 1 as a hexadecimal number.

Digit


Digit 5 0

Place value 104 103 102 10 1

� Add the place holder ‘0’

in suitable places.

64


2 Laws of Indices

Level 1

1. Write down the place value of each digit in 3 04510.

Digit 3 0 4 5

Place value

2. Write down the place value of each digit in 101102.

Digit 1 0 1 1 0

Place value

3. Write down the place value of each digit in A4CF16. Digit A 4 C F

Place value

4. Write down the place value of the underlined digit in each of the following numbers.

Number Place value

(a) 4610

(b) 100112

(c) 357916

(d) 110001012

(e) D24B16

5. Consider the denary number 3 57910. (a) Write down the place value of each digit in the number. (b) Hence, express 3 57910 in the expanded form. 6. Consider the binary number 10112. (a) Write down the place value of each digit in the number. (b) Hence, express 10112 in the expanded form. 7. Consider the hexadecimal number E4B216. (a) Write down the place value of each digit in the number. (b) Hence, express E4B216 in the expanded form.


2C

65

8. Express the following denary numbers in the expanded form. (a) 2310 (b) 15310 (c) 2 34510

9. Express the following binary numbers in the expanded form. (a) 1102 (b) 11012 (c) 110012

10. Express the following hexadecimal numbers in the expanded form. (a) B516 (b) C3F16 (c) A5DF16

11. Represent each of the following expressions as a denary number.

(a) 4 × 10 + 7 × 1

(b) 7 × 102 + 6 × 10 + 0 × 1

(c) 8 × 102 + 0 × 10 + 1 × 1 12. Represent each of the following expressions as a binary number.

(a) 1 × 22 + 0 × 2 + 1 × 1

(b) 1 × 23 + 0 × 22 + 1 × 2 + 1 × 1 13. Represent each of the following expressions as a hexadecimal number.

(a) 9 × 162 + 5 × 16 + 3 × 1

(b) 11 × 162 + 10 × 16 + 2 × 1 Level 2

14. Represent each of the following expressions as a denary number.

(a) 5 × 102 + 3 × 10

(b) 2 × 105 + 5 × 102 + 4 × 10

(c) 8 × 100 + 9 × 100 000 + 3 × 1 000 + 4 15. Represent each of the following expressions as a binary number. (a) 24 + 23

(b) 1 × 16 + 0 × 8 + 1 × 2 + 0 × 1 + 0 × 4

(c) 1 × 32 + 3 × 16 + 1 × 8 + 1 × 4 + 0 × 2 + 2 × 1 16. Represent each of the following expressions as a hexadecimal number.

(a) 2 × 165 + 10 × 164 + 256

(b) 15 × 16 + 12 × 163 + 2 × 164 + 4 × 162 + 13

(c) 18 × 16 + 14 × 163 + 3 × 164 + 2 × 162 + 19

66

17. Use each of the digits 0, 2, 3 and 6 once only to form the smallest 4-digit denary number and the

largest 4-digit denary number respectively. Write these two numbers in the expanded form. 18. (a) Find the smallest 4-digit binary number in which only one digit is 0. Write the number in the expanded form. (b) Find the largest 4-digit binary number in which only one digit is 0. Write the number in the expanded form.

19. Evaluate each of the following expressions and express the answer as a denary number. (a) 5216 + 1610 + 102

(b) (34516 − 67810) ÷ 112

20. If x is a digit between 0 and 9 inclusive such that x5A16 = 2 39410, find the value of x. 21. If y is a digit between 0 and 9 inclusive such that 1yAB16 + 1012 = 5 04010, find the value of y.

67

Answer

Consolidation Exercise 2C

1.

2.

3.

4. (a) 1 (b) 22 (c) 16 (d) 25 (e) 163

5. (a)

(b) 3 × 103 + 5 × 102 + 7 × 10 + 9 × 1

6. (a)

(b) 1 × 23 + 0 × 22 + 1 × 2 + 1 × 1

7. (a)

(b) 14 × 163 + 4 × 162 + 11 × 16 + 2 × 1

8. (a) 2 × 10 + 3 × 1

(b) 1 × 102 + 5 × 10 + 3 × 1

(c) 2 × 103 + 3 × 102 + 4 × 10 + 5 × 1

9. (a) 1 × 22 + 1 × 2 + 0 × 1

(b) 1 × 23 + 1 × 22 + 0 × 2 + 1 × 1

(c) 1 × 24 + 1 × 23 + 0 × 22 + 0 × 2 + 1 × 1

10. (a) 11 × 16 + 5 × 1

(b) 12 × 162 + 3 × 16 + 15 × 1

(c) 10 × 163 + 5 × 162 + 13 × 16 + 15 × 1

11. (a) 47 (b) 760 (c) 801

12. (a) 1012 (b) 10112

13. (a) 95316 (b) BA216

14. (a) 530 (b) 200 540 (c) 903 804

15. (a) 110002 (b) 100102 (c) 10111102

16. (a) 2A010016 (b) 2C4FD16 (c) 3E33316

17. smallest: 2 036,

2 × 103 + 0 × 102 + 3 × 10 + 6 × 1

largest: 6 320,

6 × 103 + 3 × 102 + 2 × 10 + 0 × 1

18. (a) 10112, 1 × 23 + 0 × 22 + 1 × 2 + 1 × 1

(b) 11102, 1 × 23 + 1 × 22 + 1 × 2 + 0 × 1

19. (a) 10010 (b) 5310

20. 9 21. 3

Digit 3 0 4 5


Digit 1 0 1 1 0

Place value 24 23 22 2 1

Digit A 4 C F


Digit in 3 57910 3 5 7 9


Digit in 10112 1 0 1 1


Digit in E4B216 E 4 B 2


68

F3A: Chapter 2D

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)

Book Example 15


(Video Teaching)

Book Example 16


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



_________

69

Book 3A Lesson Worksheet 2D (Refer to §2.4A)

2.4A Conversion of Binary or Hexadecimal Numbers into Denary Numbers


Convert 1102 into a denary number.

Sol 1102

= 1 × 22 + 1 × 2 + 0 × 1 = 4 + 2 + 0 = 6

Convert 10112 into a denary number.

Sol 10112

= 1 × ( ) + 0 × ( ) + __________ _______________________________ =

1. Convert 100012 into a denary number.

2. Convert 110012 into a denary number.

○○○○→→→→ Ex 2D 1–6


Convert 12B16 into a denary number.

Sol 12B16

= 1 × 162 + 2 × 16 + 11 × 1 = 256 + 32 + 11 = 299

Convert 1A516 into a denary number.

Sol 1A516

= 1 × ( ) + ( ) × ( ) + _________________________ =

3. Convert CD16 into a denary number.

4. Convert 1F016 into a denary number.

Express 1102 in the expanded form first.

Digit 1 1 0

Place value 22 2 1

Digit 1 0 1 1

Place

val

ue

Digit 1 1 0 0 1

Place

v

a

Express 12B16 in the expanded form first.

Digit 1 2 B


Digit 1 A 5

Place

value

Digit 1 0 0 0 1

Place

v

a

Digit C D

Place value

Digit 1 F 0

Place value

A = 10, B = 11, …

70

○○○○→→→→ Ex 2D 7–15


5. Paul claims that 1111102 must be greater than BE16 because 1111102 has more digits than BE16. Do you agree? Explain your answer.

1111102 = 1 × ( ) + ___________________________________

______________________________________________

= BE16 =

∵

i.e. 1111102 (> / = / <) BE16


Level Up Questions

6. Convert 11101112 into a denary number. 7. Convert DEC16 into a denary number.

Digit 1 1 1 0 1 1 1

Place

v

a

First, convert 1111102 and BE16 into denary numbers. Then, do comparison.

71


2 Laws of Indices

Level 1

Convert the following binary numbers into denary numbers. [Nos. 1−−−−5]

1. (a) 102 (b) 1002 2. (a) 1112 (b) 10112 3. (a) 110002 (b) 110012 4. (a) 100112 (b) 111112 5. (a) 1100102 (b) 1001112

Convert the following hexadecimal numbers into denary numbers. [Nos. 6−−−−12]

6. (a) 2916 (b) D716 7. (a) 2F16 (b) E016 8. (a) 35716 (b) 60E16 9. (a) B3016 (b) C0F16 10. (a) 201716 (b) 6C5B16 11. (a) DBA16 (b) DFAC16 12. (a) A3BD16 (b) BDFC16


2D

72

Level 2 Convert the following denary numbers into binary numbers. [Nos. 13−−−−15]

13. (a) 3010 (b) 3210 14. (a) 11210 (b) 23710 15. (a) 37710 (b) 39310

Convert the following denary numbers into hexadecimal numbers. [Nos. 16−−−−18]

16. (a) 6410 (b) 24310 17. (a) 42610 (b) 62510 18. (a) 2 57510 (b) 3 66410 19. Convert the following binary numbers into hexadecimal numbers.

(a) 11001002 (b) 100110112 20. Convert the following hexadecimal numbers into binary numbers.

(a) 4216 (b) CD16 21. Consider the following three numbers:

111111112, 24010, F416

Arrange them in descending order.

73

Answer


1. (a) 2 (b) 4

2. (a) 7 (b) 11

3. (a) 24 (b) 25

4. (a) 19 (b) 31

5. (a) 50 (b) 39

6. (a) 41 (b) 215

7. (a) 47 (b) 224

8. (a) 855 (b) 1 550

9. (a) 2 864 (b) 3 087

10. (a) 8 215 (b) 27 739

11. (a) 3 514 (b) 57 260

12. (a) 41 917 (b) 48 636

13. (a) 111102 (b) 1000002

14. (a) 11100002 (b) 111011012

15. (a) 1011110012 (b) 1100010012

16. (a) 4016 (b) F316

17. (a) 1AA16 (b) 27116

18. (a) A0F16 (b) E5016

19. (a) 6416 (b) 9B16

20. (a) 10000102 (b) 110011012

21. 111111112 > F416 > 24010

74

F3A: Chapter 3A

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 1


(Video Teaching)

Book Example 2


(Video Teaching)

Book Example 3


(Video Teaching)

Book Example 4


(Video Teaching)

Book Example 5


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )

75



_________

76

Book 3A Lesson Worksheet 3A (Refer to §3.1)

3.1 Simple Interest

If the principal is $P and the annual interest rate is R%, then the simple interest ($I) received after T years is given by:

I = P × R% × T


$4 000 is deposited in a bank at an annual interest rate of 3%. Find the simple interest received after (a) 2 years, (b) 5 years.

Sol (a) Simple interest received

= $4 000 × 3% × 2 = $240 (b) Simple interest received

= $4 000 × 3% × 5 = $600

$6 000 is deposited in a bank at an annual interest rate of 4%. Find the simple interest received after (a) 3 years, (b) 8 years.

Sol (a) Simple interest received

= $( ) × ( )% × ( ) = $ (b) Simple interest received

= $( ) × ( )% × ( ) = $

1. $30 000 is deposited in a bank at an

interest rate of 10% p.a. Find the simple interest received after

(a) 7 years, (b) 2.5 years.

(a) Simple interest received

= $( ) × ( ) × ( ) =

(b)

2. $15 000 is deposited in a bank at an interest rate of 8% p.a. Find the simple interest received after

(a) 6 months, (b) 9 months.

(a) Simple interest received

= $( ) × ( ) ×

) (

) (

=

(b)

○○○○→→→→ Ex 3A 1

P = 4 000, R = 3,

T = 2. Find I.

P = ? R = ? T

6 months

=

) (

) ( years

9 months

=

) (

) ( years

‘p.a.’ means ‘per year’.

77


Mr Chan deposits $2 000 in a bank at an interest rate of 5% p.a. Find the time required to receive a simple interest of $600.

Sol Let T years be the time required.

600 = 2 000 × 5% × T 600 = 100T T = 6

∴ The time required is 6 years.

Miss Wong deposits $5 000 in a bank at an interest rate of 6% p.a. Find the time required to receive a simple interest of $1 200.

Sol Let T years be the time required.

( ) = ( ) × ( ) × T =

∴ The time required is years.

3. Gloria deposits $9 000 in a bank at an interest rate of 3% p.a. How long will it take to receive a simple interest of $945?

○○○○→→→→ Ex 3A 4, 5

4. A sum of money is deposited in a bank at an interest rate of 2% p.a. The simple interest received after 4 years will be $260. Find the sum of money deposited.

Let $P be the sum of money deposited.

( ) = ( ) × ( ) × ( ) =

∴ The sum of money deposited is

.

5. Ben borrows a sum of money from a bank. The interest rate is 9% p.a. If he has to pay a simple interest of $10 800 after 5 years, find the sum of money borrowed.

○○○○→→→→ Ex 3A 6

6. $75 000 is deposited in a bank. The simple interest received after 8 years will be $42 000. Find the interest rate per annum.

○○○○→→→→ Ex 3A 7, 8

Set up an equation

to find T.

78

(a) Amount ($A) = principal ($P) + interest ($I)

(b) Since I = P × R% × T, we have

A = P(1 + R% × T)


Edward deposits $700 in a bank at a simple interest rate of 4% p.a. Find the amount received after 2 years.

Sol Interest = $700 × 4% × 2 = $56 Amount = $(700 + 56) = $756

Alternative

Amount = $700 × (1 + 4% × 2) = $756

Joey deposits $3 000 in a bank at a simple interest rate of 6% p.a. Find the amount received after 5 years.

Sol Interest = $( ) × ( ) × ( ) = Amount = $[( ) + ( )] =

Alternative

Amount = $( ) × [1 + ( ) × ( )] =

7. Mr Poon deposits $5 000 in a bank at a

simple interest rate of 3% p.a. Find the amount received after 4.5 years.

8. Teresa borrows $32 000 from a bank at a simple interest rate of 10% p.a. How much will she repay after 3 months?

○○○○→→→→ Ex 3A 2, 3

9. $1 800 is deposited in a bank at a simple

interest rate of 5% p.a. How long will it take to receive an amount of $2 160?

○○○○→→→→ Ex 3A 10, 11

10. Samuel deposits a sum of money in a bank at a simple interest rate of 2% p.a. If he receives an amount of $51 300 after 7 years, find the sum of money deposited.

○○○○→→→→ Ex 3A 12

I = P × R% × T

A = P + I

A = P(1 + R% × T)

A = P + I or

A = P(1 + R% × T)

� i.e. A = P + I A = P + I

= P + P × R% × T

= P(1 + R% × T)

79

11. Flora deposits $6 000 in a bank at a certain simple interest rate. If she receives an amount of $9 240 after 9 years, what is the interest rate per annum?

Method 1 Interest = $[( ) – ( )] = Let R% be the interest rate per annum.

( ) = ( ) × R% × ( )

=

Method 2 Let R% be the interest rate per annum.

( ) = ( ) × ( )

=

○○○○→→→→ Ex 3A 13


12. Mr Hung invests $20 000 in a bond which offers simple interest at 4.7% p.a. Will the simple interest received after 20 years be greater than his original principal? Explain your answer.

Simple interest received =

∵ $ ________ ( < / = / > ) $20 000

∴ The simple interest received (will be / will not be) greater than his original principal.

Level Up Question

13. A sum of money $P is deposited in a bank at a simple interest rate of 5% p.a. How long will it take to receive an amount which is 3 times the original principal?

Amount = $______

Let T years be the time required.

( ) = ( ) × [( ) + ( ) × ( )]

=

A = P(1 + R% × T)

Express the amount in terms of P.

Then set up an equation to find

80


3 Percentages (II)

Level 1

1. Complete the following table.

Principal Interest rate (p.a.) Time Simple interest

(a) $2 000 5% 4 years

(b) $3 000 6.25% 8 years

(c) $47 000 3% 2.5 years

(d) $96 000 8% 13 months

2. $175 000 is deposited in a bank at a simple interest rate of 4% p.a. Find the amount received after 9 months. 3. $8 000 is deposited in a bank at an interest rate of 3.5% p.a. How long will it take to receive a simple interest of $560? 4. $96 000 is deposited in a bank at an interest rate of 1% p.a. How many months will it take to receive a simple interest of $4 320? 5. A sum of money is deposited in a bank at an interest rate of 2% p.a. The simple interest received after 7.5 years will be $4 119. Find the sum of money deposited. 6. $30 000 is deposited in a bank. The simple interest received after 6 years will be $14 400. Find the interest rate per annum. 7. A sum of money is deposited in a bank at an interest rate of 3% p.a. The simple interest received after 9 months will be $1 800. Find the amount received. 8. $24 000 is deposited in a bank at a simple interest rate of 7% p.a. How long will it take to receive an amount of $29 040? 9. Sam invests $60 000 at a simple interest rate of 4.8% p.a. How long will it take to receive an amount of $75 840?

10. Mandy deposits a sum of money in a bank at a simple interest rate of 4% p.a. If she receives an amount of $7 410 after 42 months, find the sum of money deposited. 11. Samson plans to deposit $50 000 in either bank H or bank K. The table below shows the simple

interest rates per annum offered by the two banks.

Bank H Bank K

Interest rate 5.4% p.a. 3.7% p.a.

Samson will take out the amount after 4 years. (a) Which bank should Samson choose in order to earn more interest? Explain your answer. (b) If Samson deposits the money in the bank in (a), find the interest he will earn.


3A

81

Level 2

12. A railway company plans to borrow $5 000 000 from a bank. The amount will be repaid after 6 years. Bank M charges simple interest at 4% p.a. while bank N charges simple interest at 5.5% p.a. How much more interest will the company pay if it borrows the money from bank N rather than bank M? 13. Susan deposits $36 000 in bank A at a simple interest rate of 2% p.a. and $5 000 in bank B at a simple interest rate of 1.8% p.a. Find the total amount she will receive after 7 years.

14. Annie deposits $380 000 in a bank at a simple interest rate of 6% p.a. (a) Find the amount received after 9 months. (b) If she deposits the amount in (a) in another bank which offers a simple interest rate of 4% p.a., how many months will it take to receive an amount of $401 071?

15. When a sum of money is deposited in a bank at a simple interest rate of 5% p.a., how long will

it take to triple the original sum of money?

16. Jessica borrows $30 000 from a bank at a simple interest rate of R% p.a. Find the value of R in each of the following. (a) The interest is 21% of the original principal after 3.5 years. (b) The amount is 2 times the original principal after 20 years. 17. William deposits $18 000 in a bank at a simple interest rate of R% p.a. for T years. If he

deposits the money for further 6 months, he will receive $360 more. Find the value of R. 18. Rick wants to borrow $41 000 from a bank for 3 years. The bank offers the following loan scheme. (a) Find the interest that Rick will pay after 3 years. (b) If Rick saves a fixed amount of money every month during these 3 years to repay the loan and the interest found in (a), can the amount of money saved each month be less than $1 300? Explain your answer.

19. Mrs Chan deposits $100 000 in a bank at a simple interest rate of 3% p.a. for 2 months. Meanwhile, she makes an investment of $30 000 and loses 1.25%. On the whole, does she make a profit or a loss? Explain your answer.

For a loan term of more than 4 years, the interest rate is 9% p.a. Otherwise, the interest rate is 7% p.a.

Note: Simple interest is charged for all loans.

82

Answer


1. (a) $400 (b) $1 500

(c) $3 525 (d) $8 320

2. $180 250 3. 2 years 4. 54

5. $27 460 6. 8% 7. $81 800

8. 3 years 9. 5.5 years 10. $6 500

11. (a) bank H (b) $10 800

12. $450 000 13. $46 670

14. (a) $397 100 (b) 3 months

15. 40 years

16. (a) 6 (b) 5

17. 4

18. (a) $8 610 (b) no

19. a profit

83

F3A: Chapter 3B

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 6


(Video Teaching)

Book Example 7


(Video Teaching)

Book Example 8


(Video Teaching)

Book Example 9


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



_________

84

Book 3A Lesson Worksheet 3B (Refer to §3.2)

[In this worksheet, give the answers correct to the nearest dollar if necessary.]

3.2A Formula for Compound Interest

If the principal is $P and the interest rate per period is R%, then the amount ($A) received after n periods is given by: A = P(1 + R%)n

Example 1 Instant Drill 1 If $500 is deposited in a bank at an interest rate of 4% p.a. compounded yearly, find the amount received after (a) 5 years, (b) 10 years.

Sol (a) Amount

= $500 × (1 + 4%)5 = $608, cor. to the nearest dollar (b) Amount

= $500 × (1 + 4%)10 = $740, cor. to the nearest dollar

If $1 000 is deposited in a bank at an interest rate of 7% p.a. compounded yearly, find the amount received after (a) 4 years, (b) 8 years.

Sol (a) Amount

= $( ) × [1 + ( )%]( )

= $ , cor. to the nearest dollar (b) Amount

= $( ) × [1 + ( )%]( )

= $ , cor. to the nearest dollar

1. If $20 000 is deposited in a bank at an

interest rate of 5% p.a. compounded yearly, find the amount received after

(a) 6 years, (b) 4.5 years.

(a) Amount

= $( ) × ( )( ) =

(b)

2. Grace deposits $P in a bank at an interest rate of 3% p.a. compounded yearly. If the amount received after 2 years is $63 654, find the value of P.

( ) = P × ( )( )

=

○○○○→→→→ Ex 3B 8

The compound interest ($I) is given by: I = P[(1 + R%)n – 1]

P = 500, R = 4, n = 5. Find A.

P = ? R = ? n

Set up an equation to

∵ A = P + I ∴ I= A – P

= P(1 + R%)n – P = P[(1 + R%)n – 1]

�

85


If $2 000 is deposited in a bank at an interest rate of 6% p.a. compounded yearly, find the compound interest received after 8 years.

Sol Amount

= $2 000 × (1 + 6%)8

= $3 187.70, cor. to the nearest $0.01 Compound interest = $(3 187.70 – 2 000) = $1 188, cor. to the nearest dollar

Alternative Compound interest

= $2 000 × [(1 + 6%)8 – 1] = $1 188, cor. to the nearest dollar

If $8 000 is deposited in a bank at an interest rate of 2% p.a. compounded yearly, find the compound interest received after 5 years.

Sol Amount

= $( ) × ( )( )

= $( ), cor. to the nearest $0.01 Compound interest = $[( ) – ( )] = $ , cor. to the nearest dollar

Alternative Compound interest

= $( ) × [( )( ) – 1] = $ , cor. to the nearest dollar

3. If $50 000 is deposited in a bank at an interest rate of 8% p.a. compounded yearly, find the compound interest received after 6 years.

○○○○→→→→ Ex 3B 1–3

4. Leo deposits a sum of money in a bank at an interest rate of 10% p.a. compounded yearly. If he receives a compound interest of $5 296 after 3 years, find the sum of money deposited.

Let $P be the sum of money deposited.

( ) = P × [( )( ) – 1]

=

∴ The sum of money deposited is

. ○○○○→→→→ Ex 3B 9

5. Complete the following table. (The interest is compounded yearly.)

Principal

Interest rate

(p.a.) Time Amount

Compound

interest

(a) $7 000 3% 4 years

(b) $15 000 7% 5 years

(c) 5% 3 years $46 305

(d) 4% 2 years $16 320

A = P(1 + R%)n

I = A – P

I = P[(1 + R%)n – 1]

86

3.2B Comparison between Compound Interest and Simple Interest


$4 000 is deposited in a bank at an interest rate of 3% p.a. for 10 years. Find the amount received in each of the following situations. (a) Simple interest is calculated. (b) The interest is compounded yearly.

Sol (a) Amount

= $4 000 × (1 + 3% × 10) = $5 200 (b) Amount

= $4 000 × (1 + 3%)10 = $5 376, cor. to the nearest dollar

$6 500 is deposited in a bank at an interest rate of 4% p.a. for 15 years. Find the amount received in each of the following situations. (a) Simple interest is calculated. (b) The interest is compounded yearly.

Sol (a) Amount = (b) Amount =

6. Nancy deposits $30 000 in a bank at an interest rate of 2.5% p.a. for 4 years. Find the interest

received in each of the following situations. (a) Simple interest is calculated.

(b) The interest is compounded yearly.

○○○○→→→→ Ex 3B 10, 11

3.2C Interest Compounded at Different Periods

Suppose $P is deposited in a bank at an interest rate of R% per period, and the

compound interest is calculated once per period. Then the amount ($A) after n

periods is given by: A = P(1 + R%)n

For simple interest,

A = P(1 + R% × T)

For compound interest,

A = P(1 + R%)n

I = P × R% × T I = P[(1 + R%)n – 1]

interest rate per period

total number of periods

87


$1 000 is deposited in a bank at an interest rate of 6% p.a. for 5 years. Find the amount received if the interest is compounded half-yearly.

Sol Interest rate for half a year

=

2

%6

= 3% Taking half a year as a period, number of periods in 5 years

= 5 × 2 = 10 amount

= $1 000 × (1 + 3%)10

= $1 344, cor. to the nearest dollar

$5 000 is deposited in a bank at an interest rate of 8% p.a. for 3 years. Find the amount received if the interest is compounded quarterly.

Sol Interest rate per quarter

=

Taking a quarter as a period, number of periods in ( ) years = amount =

7. $9 000 is deposited in a bank at an interest

rate of 10% p.a. compounded half-yearly. Find the amount after 4 years.

8. Hubert borrows $25 000 from a bank at an interest rate of 14% p.a compounded quarterly. Find the amount to be repaid after 6 years.

○○○○→→→→ Ex 3B 12, 13

P = 1 000 ∵ Number of periods per year = 2 ∴ Interest rate per period (R%) =

2

%6

Total number of periods (n) = 5 × 2

P = 5 000 ∵ Number of periods per year = ( ) ∴ R% =

) (

%8

n = 3 × ( )

88


9. Anna wants to deposit $100 000 in either bank A or bank B for 3 years. Bank A offers an interest rate of 4.5% p.a. compounded yearly while bank B offers an interest rate of 4% p.a. compounded half-yearly. Which bank pays a higher interest? Explain your answer.

Level Up Questions

10. At the beginning of 2017, Miss Lau deposits $4 000 in a bank at an interest rate of 8% p.a. compounded yearly. Find the amount she will receive at the beginning of 2024.

Time for deposit (in years) = 2024 – ( ) = Amount = 11. Winnie borrows a sum of money at an interest rate of 12% p.a. compounded monthly. If she

has to pay an interest of $482.4 after 2 months, find the sum of money borrowed.

Find the time for

deposit first.

89


3 Percentages (II)

[In this exercise, give the answers correct to the nearest dollar if necessary.]

Level 1

Complete the following table. [Nos. 1–3] (The interest is compounded yearly.)

Principal

Interest rate

(p.a.) Time Amount

Compound

interest

1. $2 000 3% 4 years

2. $8 000 5% 6 years

3. $15 000 7% 8 years

Complete the following table, given that the interests are compounded half-yearly. [Nos. 4–6]

Principal

Interest rate

(p.a.)

Interest rate

per period Time

Number of

periods Amount

4. $1 000 2% 1% 3 years

5. $300 000 4% 5 years 10

6. $60 000 8% 42 months

7. Mr Wong deposits $P in a bank at an interest rate of 6% p.a. compounded yearly. If the amount received after 5 years is $70 000, find the value of P, correct to the nearest integer.

8. $4 000 is deposited in a bank at an interest rate of 3% p.a. for 6 years. Find the interest received in each of the following situations. (a) Simple interest is calculated. (b) The interest is compounded yearly. 9. $90 000 is deposited in a bank at an interest rate of 7% p.a. for 5 years. Find the difference between the interests calculated on the bases of simple interest and compound interest (compounded yearly).

10. $36 000 is borrowed from a bank at an interest rate of 4% p.a. compounded quarterly. Find the amount to be repaid after 2 years.

11. $68 000 is deposited in a bank at an interest rate of 5% p.a. compounded monthly. Find the amount after 36 months. 12. Jackie borrows $10 000 from a bank at an interest rate of 16% p.a. compounded every 4 months. Find the interest he will pay after 5 years.


3B

90

Level 2

13. Ann deposits $14 000 in a bank for 2 years. The interest is compounded yearly. (a) Find the compound interest received if the interest rate is (i) 3% p.a., (ii) 6% p.a. (b) Is the compound interest found in (a)(ii) twice the interest found in (a)(i)? Explain your answer. 14. Judie plans to borrow $60 000 for 4 years. She can borrow the money from bank A at a simple

interest rate of 8% p.a. or from bank B at an interest rate of 7% p.a. compounded yearly. Which bank should she choose in order to pay less interest? Explain your answer.

15. David borrows a sum of money from a bank at an interest rate of 9% p.a. compounded quarterly.

If the interest he pays after 5 years is $22 000, find the sum of money borrowed. 16. At the beginning of 2014, Mr Tam deposited a sum of money in a bank at an interest rate of 6% p.a. compounded yearly. If he received an interest of $13 000 at the beginning of 2017, find (a) the sum of money deposited, (b) the amount received at the beginning of 2019.

17. 3 years ago, Mary deposited $400 000 in a bank, and a simple interest of $96 000 is just received. Now, she deposits the amount obtained in another bank at the same annual interest rate, but the interest will be compounded monthly. Find the amount she can obtain after another 3 years.

18. Roy wants to deposit $20 000 in a bank for 3 years. He can deposit the money in bank A at an

interest rate of 4% p.a. compounded half-yearly, or in bank B at 3% p.a. compounded daily. Which bank should he choose in order to earn more interest? Explain your answer.

(Assume there are 365 days in a year.)

19. Sandy plans to deposit $96 000 in a bank for 4 years. The saving schemes offered by bank H and bank K are as follows: Bank H: interest rate of 4% p.a. compounded monthly with a cash reward of $200 per year. Bank K: interest rate of 4.5% p.a. compounded quarterly. Which bank should she choose in order to earn more interest? Explain your answer. 20. Anderson is going to borrow $300 000 from a bank for 8 years. Bank A charges interest at 9% p.a. compounded yearly. Bank B charges interest at 7% p.a. compounded monthly.

(a) Which bank should he choose in order to pay less interest? Explain your answer. (b) Suppose the interest rate charged by bank B changes to 8.5% p.a. (i) Which bank should he choose in order to pay less interest? Explain your answer. (ii) When compared to the original interest rate of 7% p.a., how much more interest will be charged by bank B now?

91

Answer


1. amount = $2 251, interest = $251

2. amount = $10 721, interest = $2 721

3. amount = $25 773, interest = $10 773

4. number of periods = 6, amount = $1 062

5. interest rate per period = 2%,

amount = $365 698

6. interest rate per period = 4%,

number of periods = 7, amount = $78 956

7. 52 308

8. (a) $720 (b) $776

9. $4 730 10. $38 983 11. $78 980

12. $11 802

13. (a) (i) $852.6 (ii) $1 730.4 (b) no

14. bank B 15. $39 250

16. (a) $68 057 (b) $91 076

17. $630 038 18. bank A 19. bank K

20. (a) bank B (b) (i) bank B (ii) $66 398

92

F3A: Chapter 3C

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 10


(Video Teaching)

Book Example 11


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



_________

93

Book 3A Lesson Worksheet 3C (Refer to §3.3A)

3.3A Increasing at a Constant Rate

If a value P increases at a constant rate of R% per period, its new value A after n periods is given by: A = P(1 + R%)n where (1 + R%) is called the growth factor.


John weighs 50 kg this year. If his weight increases at a constant rate of 9% per year, find his weight after 4 years. (Give the answer correct to the nearest 0.1 kg.)

Sol John’s weight after 4 years

= 50 × (1 + 9%)4 kg = 70.6 kg, cor. to the nearest 0.1 kg

A metal rod is at 78°C now. If its temperature increases at a constant rate of 5% per hour, find its temperature after 3 hours.

(Give the answer correct to the nearest 0.1°C.)

Sol Temperature after 3 hours

= ( ) × ( )( ) °C

= ,

1. Peter buys a gold coin at $4 000. If its value increases steadily at a rate of 8% per year, what will its value be after 10 years?

(Give the answer correct to the nearest $10.)

2. In a city, the number of hotels was 600 in 2012. If the number of hotels in the city increases at a steady rate of 1.5% per year, find the number of hotels in 2017, correct to the nearest integer.

Number of periods = ( ) – ( ) = Number of hotels in 2017 =

○○○○→→→→ Ex 3C 1–6

P = 50 R = 9 n =

P = ___ R = ___ n =

94

3. The height of a tree increases at a constant rate of 20% per year. If the present height of the tree is 16 m, find its height 2 years ago.

(Give the answer correct to the nearest 0.01 m.)

Let P m be its height 2 years ago.

( ) = P × ( )( )

=

4. After a promotion campaign, the number of blood donors increases at a constant rate of 4% per day.

(a) Find the growth factor. (b) If there are 1 200 blood donors today,

how many blood donors were there one week ago?

(Give the answer correct to the nearest integer.)

○○○○→→→→ Ex 3C 10–13

5. The profit of a company is $100 000 this year. It is estimated that the profit of the company will increase by 16% every 2 years. Estimate the profit after 10 years.

(Give the answer correct to the 2 significant figures.)

Taking _____ years as a period, number of periods in 10 years

=

) (

10

= Estimated profit after 10 years =

6. In an experiment, the number of bacteria increases by 3% every half an hour. If there are 5 000 bacteria now, how many bacteria will there be after 12 hours?

(Give the answer correct to the nearest integer.)

○○○○→→→→ Ex 3C 7, 8

95


7. The population of a town is 75 000 this year. If its population grows at a steady rate of 2.3% per year, will the population of the town exceed 90 000 after 8 years? Explain your answer.

Level Up Questions

8. Due to technological improvement, the rice production of a farm increases at a constant rate of 10% every 5 years. Suppose its rice production is 65 tonnes this year. Find the rice production of the farm 20 years ago, correct to the nearest 0.1 tonnes.

9. In 2010, the monthly salary of Fred was $20 000. His monthly salary increased at a constant

rate of 3% per year from 2010 to 2016. (a) Find the growth factor. (b) Find the increase in his monthly salary from 2010 to 2016. (Give the answer correct to 3 significant figures.)

1 tonne = 1 000 kg

96


3 Percentages (II)

Level 1

Complete the following table. [Nos. 1–3]

Original value Growth rate

per year Growth factor Time New value

1. 300 50% 1.5 1 year

2. 6 000 40% 3 years

3. $20 000 30% 5 years

4. The water consumption of a village is 30 units in a certain week, and it increases steadily at a rate of 6% per week. Find the water consumption after 4 weeks, correct to 3 significant figures.

5. The population of a city was 540 000 in 2015 and it increases steadily at a rate of 3.8% per year. Find the population in 2021. (Give the answer correct to the nearest integer.)

6. The present value of an oil painting is $256 000. If its value increases by 30% every 4 years, what will its value be after 12 years? 7. A computer virus spreads through a certain network. The number of infected computers increases at a steady rate of 300% every 10 minutes. If 2 computers are infected initially, find the number of infected computers after 1 hour.

8. The average temperature of a town in January is 20°C. It increases steadily at a rate of 4% every

month until August. Will the average temperature in June be higher than 25°C? Explain your answer.


Original value Growth rate

per year Growth factor Time New value

9. 10% 1.1 1 year 1 210

10. 60% 3 years 49 152


3C

97

Level 2 11. The weight of a dog was 115 g at the beginning of 2013. Then, its weight increased steadily at a

rate of 2% per year from 2013 to 2016. (a) Find the weight of the dog at the end of 2016.

(b) If the weight of the dog increases by 4% every year from 2017 onwards, find its weight at the end of 2018.

(Give the answers correct to the nearest g.)

12. A tree was 210 cm tall 4 years ago. Then, its height increases steadily at a rate of 3% per year. (a) Find its height at present. (b) Find the percentage increase in its height over these 4 years. (Give the answers correct to 3 significant figures.)

13. The average stock price of a company increased by 1% every month over the past 9 months. It is known that the average stock price is $78 this month.

(a) Find the growth factor. (b) Find the increase in the average stock price over the past 9 months. (Give the answer correct to 3 significant figures.) 14. Over the past 3 years, the monthly income of Joe increased at a constant rate of 25% per year. It

is known that the monthly income of Joe at present is $50 000. (a) What was the monthly income of Joe 3 years ago? (b) Find the increase in the monthly income of Joe over the past 3 years. 15. In 2015, the number of fish in a pond was 260. In 2016, the number of fish was 273. (a) Find the growth factor. (b) Suppose the growth factor remains unchanged. Find the number of fish in the pond in

2020. (Give the answer correct to the nearest integer.) 16. From 2014 to 2016, company A’s profit increased by 3% per year and company B’s profit

increased by 4% per year. It is given that both companies made the same profit in 2016. (a) Which company’s profit was higher in 2014? Explain your answer.

(b) If the difference of profits of companies A and B in 2014 was $40 000, find the profit of each company in 2016.

(Give the answer correct to the nearest dollar.)

98

Answer

Consolidation Exercise 3C

1. 450

2. growth factor = 1.4, new value = 16 464

3. growth factor = 1.3, new value = $74 258.6

4. 37.9 units 5. 675 426 6. $562 432

7. 8 192 8. no 9. 1 100

10. original value = 12 000, growth factor = 1.6

11. (a) 124 g (b) 135 g

12. (a) 236 cm (b) 12.6%

13. (a) 1.01 (b) $6.68

14. (a) $25 600 (b) $24 400

15. (a) 1.05 (b) 332 16. (a) company A (b) $2 217 332

99

F3A: Chapter 3D

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 12


(Video Teaching)

Book Example 13


(Video Teaching)

Book Example 14


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )



___________ ( )



_________

100

Book 3A Lesson Worksheet 3D (Refer to §3.3B)

3.3B Decreasing at a Constant Rate

If a value P decreases at a constant rate of R% per period, its new value A after n periods is given by: A = P(1 – R%)n where (1 – R%) is called the decay factor.


A pool contains 300 m3 of water originally. If the volume of water decreases at a constant rate of 10% per hour, find the volume of water in the pool after 6 hours. (Give the answer correct to the nearest 0.1 m3.)

Sol Volume of water in the pool after 6 hours

= 300 × (1 – 10%)6 m3 = 159.4 m3, cor. to the nearest 0.1 m3

The monthly charge of a mobile data plan is $180. If the monthly charge decreases at a constant rate of 5% per year, find the monthly charge after 3 years. (Give the answer correct to the nearest dollar.)

Sol Monthly charge after 3 years

= $( ) × ( )( )

= ,

1. The number of tigers in a country is 1 400 at present. It is known that the number of tigers decreases by 4% per year.

(a) Find the decay factor. (b) Find the number of tigers after 5

years. (Give the answer correct to the nearest integer.)

2. In a shop, the profit from selling a model of calculator was $50 000 in 2013. The profit from selling this model decreases at a rate of 15% per year. Find the profit from selling this model in 2017.

(Give the answer correct to the nearest $100.)

Number of periods = ( ) – ( ) = Profit from selling this model in 2017 =

○○○○→→→→ Ex 3D 1, 2, 5–7

P = 300 R = 10 n = 6

P = _____ R = _____ n =

101

3. Suppose the weight of a block of dry ice decreases steadily at a rate of 2% per minute. If its present weight is 200 g, find its weight 5 minutes ago.

(Give the answer correct to the nearest 0.01 g.)

Let P g be its weight 5 minutes ago.

( ) = P × ( )( )

=

4. The number of newborn babies in a city decreases steadily at 4% per year. If the number of newborn babies in the city is 18 000 this year, find the number of newborn babies 9 years ago.

(Give the answer correct to the nearest thousand.)

○○○○→→→→ Ex 3D 3, 4

(a) Depreciation is the decrease in value of a product after it

has been used for a period of time, where depreciation = original value – new value

depreciation rate =

valueoriginal

ondepreciati × 100%

(b) For a product with the original value $P and depreciation rate R% per period, its new value $A after n periods is given by:

A = P(1 – R%)n


The original price of a watch is $4 000. If its value depreciates by 7% per year, what will the value of the watch be after 2 years?

Sol Value of the watch after 2 years

= $4 000 × (1 – 7%)2 = $3 459.6

The original price of a machine is $30 000. If its value depreciates by 10% per year, what will the value of the machine be after 3 years?

Sol Value of the machine after years

= $( ) × ( )( )

=

� Depreciation is a kind of decrease

at a constant rate.

102

5. The present value of a sofa is $5 600. If its

depreciation rate is 11% per year, find its value after 5 years, correct to the nearest dollar.

6. The value of a smartphone depreciates by 20% every year. If its present value is $3 072, find its value 3 years ago.

○○○○→→→→ Ex 3D 10

7. The original price of a piano is $47 000. If its value depreciates at 18% every 2 years, find the value of the piano after 6 years, correct to the nearest $100.

○○○○→→→→ Ex 3D 8, 9

8. The value of a camera depreciates by 5% every 6 months. Its value was $7 500 in 2015. Find the depreciation of the camera from 2015 to 2017, correct to the nearest $10.

○○○○→→→→ Ex 3D 11


9. Steven bought a printer 2 years ago. The depreciation rate is 25% per year and its present value is $810. Steven claims that he spent less than $1 500 for buying the printer. Do you agree? Explain your answer.

Level Up Questions

10. The expenditure of a family in March is $35 200. The family decreases their expenditure at a rate of 5% every 2 months since March. Find the expenditure of the family in

(a) July, (b) September in the same year.

depreciation = original value –

new value

103

11. Four years ago, the number of traffic accidents in a district was 400. Suppose the number of

traffic accidents decreases steadily at a rate of 3.7% per year. (a) Find the decay factor. (b) Find the percentage decrease in the number of traffic accidents over the past four years. (Give the answer correct to 2 significant figures.)

Find the number of traffic accidents this year first.

104


3 Percentages (II)

Level 1


Original value Rate of decrease

per year Decay factor Time New value

1. 800 30% 4 years

2. $7 500 0.75 2 years

3. 20% 3 years 1 600 cm3

4. A tank contains 20 L of water. Owing to an accident, the volume of water in the tank decreases

at a rate of 2% per minute. Find the volume of water in the tank after 4 minutes. (Give the answer correct to the nearest L.) 5. In February, Ron joins a slimming programme and his weight decreases by 3% per month. If

his weight in February is 120 kg, find his weight in July, correct to the nearest kg. 6. The original price of a mobile phone is $6 000. If its value depreciates by 4% every 3 months, find the value of the mobile phone after 1 year. (Give the answer correct to the nearest dollar.) 7. The value of a book decreases by 20% every year. If its present value is $400, find its value 2 years ago. 8. The value of a bike depreciates by 10% every 3 years. Its value was $8 748 in 2016. Find its value in 2007. Level 2 9. The distance between a car and a building is 80 km. The car is now moving towards the

building so that the distance between them decreases by 6% every 10 minutes. Find the distance between them after

(a) 20 minutes, (b) 1 hour. (Give the answers correct to 3 significant figures.)


3D

105

10. Thomas bought a car 5 years ago. Its value depreciates by 7% every year. Its present value is $142 000. (a) Find the original price of the car, correct to the nearest dollar. (b) When the depreciation of the car is greater than $60 000, Thomas will buy a new car. Will he buy a new car now? Explain your answer. 11. The value of a computer was $4 000 in 2015. Its value then decreases by 13% every year. (a) Find the value of the computer in 2017. (b) What is the percentage decrease in the value of the computer from 2015 to 2017? 12. Kate bought a piano for $64 000 in 2015. (a) If the value of the piano decreases by 6% every 6 months, find its value in 2017. (b) If the value of the piano decreases at a rate of 12% every year, will its value in 2017 be the same as that found in (a)? Explain your answer. If not, what is the difference? (Give the answers correct to the nearest dollar if necessary.) 13. The population of a city was 350 000 in 1961. It decreased to 346 150 in 1962. (a) Find the percentage change in the population over that year.

(b) If the population continues to change at the same rate as in (a) per year, find the population in

(i) 1980, (ii) 2000. (Give the answers correct to the nearest integer.) 14. The value of an ebook reader depreciates by x% every 8 months. It is known that the present value of the reader is $1 100 and its value will be $770 after 8 months. (a) Find the value of x. (b) Find the depreciation of the ebook reader after 2 years as compared to the present value.

106

Answer


1. decay factor = 0.7, new value = 192.08

2. rate of decrease per year = 25%,

new value = $4 218.75

3. original value = 3 125 cm3, decay factor = 0.8

4. 18 L 5. 103 kg 6. $5 096

7. $625 8. $12 000

9. (a) 70.7 km (b) 55.2 km

10. (a) $204 114 (b) yes

11. (a) $3 027.6 (b) 24.31%

12. (a) $49 968 (b) no, $406

13. (a) −1.1% (b) (i) 283 660 (ii) 227 365

14. (a) 30 (b) $722.7

107

F3A: Chapter 3E

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 15


(Video Teaching)

Book Example 16


(Video Teaching)

Book Example 17


(Video Teaching)

Book Example 18


(Video Teaching)

Book Example 19


(Video Teaching)



(Full Solution)




___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 3E Multiple Choice


___________ ( )

108



_________

109

Book 3A Lesson Worksheet 3E (Refer to §3.4)

3.4A Successive Percentage Changes

If a value N increases by x% and then decreases by y%, then

new value = N(1 + x%)(1 – y%)


75 increases by 20% and then decreases by 30%. Find the new value.

Sol New value

= 75 × (1 + 20%) × (1 – 30%)

= 75 × 1.2 × 0.7 = 63

120 increases by 50% and then decreases by 15%. Find the new value.

Sol New value

= 120 × ( ) × ( ) =

1. $300 decreases by 10% and then increases by 40%. Find the new value.

2. 5 kg increases by 30% and then increases by a further 70%. Find the new value.

○○○○→→→→ Ex 3E 1

3. In a tutorial school, there were 1 250 students in 2013. The number of students increased by 10% in 2014 and then decreased by 4% in 2015. Find the number of students in 2015.

4. On Monday, Helen practised on violin for 80 minutes. Her practising time decreased by 20% on Tuesday, and then decreased by a further 25% on Wednesday. Find her practising time on Wednesday.

○○○○→→→→ Ex 3E 3, 4

increase by 50% : (_________)

decrease by 15% : increase by 20% : (1 + 20%)

decrease by 30% : (1 – 30%)

110

5. A shop sold 200 TVs in May. The number of TVs sold decreased by 50% in June and then increased by 80% in July.

(a) Find the number of TVs sold in July. (b) What was the increase or decrease in

the number of TVs sold from May to July?

6. The original price of a suit is $5 600. The price of the suit first increases by 20% and then decreases by 10%.

(a) Find the final price of the suit. (b) Find the increase or decrease in the

price of the suit as compared to the original price.

○○○○→→→→ Ex 3E 5, 6

7. The production cost of a tennis racket is $P. If the production cost is first increased by 16%

and then decreased by 25%, the new production cost will be $609. Find the value of P.

P( )( ) = ( ) =

○○○○→→→→ Ex 3E 7–9

111

3.4B Percentage Changes of Different Components

To find the overall percentage change in a quantity with different components: Step1111: Find the original value of the quantity. Step2222: Find the new value of each component and hence the new value of the quantity.

Step3333: Use the formula ‘percentage change =

valueoriginal

change× 100%’ to find

the overall percentage change.


Last year, there were 10 boys and 30 girls in a choir. This year, the number of boys increases by 20% and the number of girls decreases by 40%. Find the percentage change in the total number of children in the choir.

Sol Original total number of children �Step 1111 = 10 + 30 = 40 New number of boys �Step 2222

= 10 × (1 + 20%) = 12 New number of girls

= 30 × (1 – 40%) = 18 New total number of children = 12 + 18 = 30

∴ Percentage change in the �Step 3333

total number of children

=

40

4030 − × 100%

= –25%

There are 40 large tables and 20 small tables in a hall. Now, the number of large tables increases by 25% while the number of small tables decreases by 5%. Find the percentage change in the total number of tables in the hall.

Sol Original total number of tables �Step 1111 = ( ) + ( ) = New number of large tables �Step 2222

= ( ) × ( ) = New number of small tables

= ( ) × ( ) = New total number of tables = ( ) + ( ) =

∴ Percentage change in the �Step 3333

total number of tables

=

) (

) () ( − × 100%

=

112

8. Last month, the income and the expenditure of a flower shop were $24 000 and $18 000

respectively. This month, the income decreases by 10% while the expenditure increases by 10%. Find the percentage change in the profit of the shop.

Profit of the shop last month = $[( ) – ( )] = Income of the shop this month = Expenditure of the shop this month = Profit of the shop this month =

∴ Percentage change in the profit of the shop =

○○○○→→→→ Ex 3E 10, 11

9. In the figure, the length and the width of a rectangle are 25 cm and 18 cm respectively. (a) Find the area of the rectangle. (b) If the length of the rectangle increases by 80% while the width decreases by 40%, find the percentage change in the area of the rectangle. ○○○○→→→→ Ex 3E 14, 15

25 cm

18 cm

113

10. A professional examination consists of papers I, II and III. In the first attempt, Calvin scored 50,

70 and 80 in papers I, II and III respectively. (a) Find the total score of Calvin in the first attempt. (b) In the second attempt, Calvin’s scores in papers I and II both increase by 20% and his score in paper III remains unchanged. Find the percentage change in his total score as compared to the first attempt. ○○○○→→→→ Ex 3E 12, 13


11. A bowl contains 300 mL of water originally. The volume of water in the bowl increases by 30% and then decreases by 40%. Sandy claims that the overall percentage change of the volume of water in the bowl is –10%. Do you agree? Explain your answer.

New volume of water = 300 × ( ) × ( ) mL =

Overall percentage change =

∵ _______ ( = / ≠ ) –10%


Level Up Questions

12. A value decreases by 30% and then increases by 50%. Find the overall percentage change. Let P be the original value. New value =

∴ Overall percentage change

Paper I Paper II Paper III

1st

atte

mpt

50 70 80

2nd

atte

114

=

13. After the fare of a bus route increases from $6 to $6.3, the number of passengers reduces by

14%. Find the percentage change in the revenue obtained from the bus route. Let x be the original number of passengers. Original revenue = $6x

New number of passengers = x × ( ) =

New revenue = $( ) × ( ) = $( )

∴ The required percentage change

=

115


3 Percentages (II)

Level 1

1. Find the new value in each of the following situations. (a) 200 increases by 30% and then increases by a further 40%. (b) 8 m decreases by 50% and then decreases by a further 10%. (c) $1 600 decreases by 25% and then increases by 60%. 2. Find the original value in each of the following situations. (a) After increasing by 6% and then increasing by a further 7%, the new value is 90 736. (b) After increasing by 30% and then decreasing by 20%, the new value is 208 g. (c) After decreasing by 40% and then increasing by 60%, the new value is 76.8 cm3. 3. In January, the number of visitors of a theme park is 50 000. The number decreases by 24% from January to February, and then increases by 7% from February to March. Find the number of visitors in March. 4. The income of a taxi driver on Friday is 40% more than that on Saturday. The income on

Saturday is 8% less than that on Sunday. If the income on Sunday is $2 000, find the income on Friday.

5. Ron’s first examination result is 60 marks. His second examination result is 20% higher than his

first one, while his third examination result is 25% lower than his second one. (a) What is Ron’s third examination result? (b) What is the increase or decrease in Ron’s third examination result as compared with his first examination result? 6. A hawker sells 200 apples on Thursday. The number of apples sold increases by 40% from Thursday to Friday and then decreases by 30% from Friday to Saturday. Is the number of apples sold on Saturday less than that on Thursday? Explain your answer. 7. The cost of a product decreases by 20% and then increases by 25%. The final cost of the product is $77. Find the original cost of the product.


3E

116

8. Last year, there were 800 male students and 900 female students in a university. This year, the number of male students increases by 30% and the number of female students decreases by 15%. (a) (i) Find the number of male students this year. (ii) Find the number of female students this year. (b) Find the percentage change in the total number of students, correct to 3 significant figures. 9. Last year, 6 000 candidates took an accountant examination and 1 500 of them failed the

examination. This year, the number of candidates who pass the examination increases by 18%, and the number of candidates who fail the examination decreases by 16%. Find the percentage change in the total number of candidates.

10. Last week, the number of books sold in bookstores P, Q and R were 150, 240 and 320

respectively. As compared to last week, the number of books sold this week in bookstore P decreases by 2%, that in bookstore Q increases by 5% and that in bookstore R remains unchanged. Find the percentage change in the total number of books sold in the three bookstores, correct to 3 significant figures.

11. The length of a rectangle is 10 cm and the width is 7 cm. (a) Find the area of the rectangle. (b) If the length of the rectangle decreases by 10% and the width increases by 10%, find (i) the new area, (ii) the percentage change in the area. 12. Last month, there were 500 workers in a factory and the wage of each worker was $12 000. This

month, the number of workers decreases by 12% and the wage of each worker increases by 6%. Find the percentage change in the sum of wages of all workers over these two months.

Level 2 13. Find the percentage change in each of the following situations. (a) A value increases by 80% and then decreases by 25%. (b) A value decreases by 17% and then increases by 17%. 14. The profit of a company increased by 40% from 2014 to 2015, and then decreased by 35% from 2015 to 2016. What was the percentage change in the profit of the company from 2014 to 2016? 15. In a shop, the price of a product is decreased by 20% and then increased by 30%. Find the

percentage change in the price of the product.

117

16. The tax revenue of a government increased by 20%, 10%, 35% respectively in the first three

quarters of 2016, and then decreased by 45% in the fourth quarter of the year. What was the percentage change in the quarterly tax revenue over the whole year 2016?

17. If a number is decreased by 90% and then increased by x%, the overall percentage change is

−83%. Find x. 18. The operating cost ($F) of a yoga club can be calculated by the following formula: F = 150 000 + 20 000N + R, where N is the number of yoga instructors and $R is the monthly rent of the yoga club.

(a) Last month, there were 20 yoga instructors and the monthly rent was $300 000. Find the operating cost.

(b) This month, the number of yoga instructors decreases by 25% and the monthly rent increases by 15%. Does the operating cost decrease by 10% as compared to last month? Explain your answer.

19. The numbers of tourists of three cities A, B and C in 2015 are listed below:

City A B C

Number of tourists 2 500 900 1 600

In 2016, the numbers of tourists of cities B and C increased by 40% and 20% respectively. If the total number of tourists of the three cities increased by 4%, find the percentage change in the number of tourists of city A.

20. If the base of a parallelogram increases by 18% and its height decreases by 30%, find the percentage change in its area. 21. Last week, the numbers of male customers and the female customers in a shopping mall were in

the ratio 1 : 3. This week, the number of male customers increases by 16% while the number of female customers decreases by 12%. Is there an increase in the total number of customers as compared to last week? Explain your answer.

118

Answer

Consolidation Exercise 3E

1. (a) 364 (b) 3.6 m (c) $1 920

2. (a) 80 000 (b) 200 g (c) 80 cm3

3. 40 660 4. $2 576

5. (a) 54 marks (b) a decrease of 6 marks

6. yes 7. $77

8. (a) (i) 1 040 (ii) 765 (b) +6.18%

9. +9.5% 10. +1.27%

11. (a) 70 cm2 (b) (i) 69.3 cm2 (ii) −1%

12. −6.72%

13. (a) +35% (b) −2.89%

14. −9% 15. +4% 16. −1.99%

17. 70

18. (a) $850 000 (b) no

19. −19.2% 20. −17.4% 21. no

119

F3A: Chapter 3F

Date Task Progress

Lesson Worksheet


(Full Solution)

Book Example 20


(Video Teaching)

Book Example 21


(Video Teaching)

Book Example 22


(Video Teaching)

Book Example 23


(Video Teaching)



(Full Solution)

Maths Corner Exercise 3F Level 1



___________ ( )




___________ ( )




___________ ( )

Maths Corner Exercise 3F Multiple Choice


___________ ( )



_________

120

Book 3A Lesson Worksheet 3F (Refer to §3.5)

3.5A Rates

Rates for the year = rateable value × rates percentage charge

Rates for a quarter =

4

year for the rates

(In this worksheet, the rates percentage charge is set as 5%.)


The rateable value of a flat is $300 000. Find the rates payable in a year.

Sol Rates payable in a year

= $300 000 × 5% = $15 000

The rateable value of a flat is $85 000. Find the rates payable in a year.

Sol Rates payable in a year

= ( ) × ( ) = ○○○○→→→→ Ex 3F 1, 2

1. The rateable value of a building is $1 600 000. Find the rates for a quarter of a year.

Rates for a quarter of a year

=

2. The rateable value of a piece of land is $2 304 000. Find the rates for a quarter of a year.

○○○○→→→→ Ex 3F 3, 4

3. Mr Luk pays $9 000 for the annual rates on his flat. Find the rateable value of the flat.

Let $P be the rateable value of the flat.

P × ( ) = ( ) =

4. The owner of a building pays $57 100 of rates quarterly. Find the rateable value of the building.

○○○○→→→→ Ex 3F 5–7

3.5B Salaries Tax

To calculate the salaries tax payable: Step1111: Split up the net chargeable income into several parts, which are called ‘tax bands’. (see Table 1) Step2222: Calculate the tax for each part by

Rateable value

↓ × 5%

Rates for a year

↓ ÷ 4

Rates for a quarter

Rates for a quarter

=4

year for the rates

Net chargeable income Tax rates On the first $40 000 2%

On the next $40 000 7%

On the next $40 000 12%

Remainder 17%

Table 1

121

multiplying the corresponding tax rates. Step3333: Salaries tax payable = sum of the taxes in Step2222

Note: The government may adjust the tax bands and the tax rates.


The net chargeable income of May is $125 000. Find her salaries tax payable.

Sol Step1111: Net chargeable income = $125 000 = $(40 000 + 40 000 + 40 000 + 5 000)

Step2222:

Net chargeable

income Rate Tax

On the first $40 000

2% $40 000 × 2% = $800

On the next $40 000

7% $40 000 × 7% = $2 800

On the next $40 000

12% $40 000 × 12% = $4 800

Remainder $5 000

17% $5 000 × 17% = $850

Step3333: Her salaries tax payable = $(800 + 2 800 + 4 800 + 850) = $9 250

The net chargeable income of Alex is $130 000. Find his salaries tax payable.

Sol Step1111: Net chargeable income = $130 000 = $(40 000 + 40 000 + ______________ )

Step2222:

Net chargeable

income Rate Tax


2% $40 000 × 2% = $800

On the next $40 000

7% $40 000 × 7% = _________

On the next _________

____

Remainder _________

____

Step3333: His salaries tax payable = $(800 + ________________________ ) =

� In this worksheet, refer to Table 1 when calculating salaries tax.

122

5. The net chargeable income of Ann is $150 000. Find her salaries tax payable.

Net chargeable income =

Net chargeable income Rate Tax

On the first $40 000 2% $40 000 × ______ = _________

Her salaries tax payable = ○○○○→→→→ Ex 3F 8–10

Net chargeable income = annual income – allowances

Example 3 Instant Drill 3 The annual income of Kary is $220 000. If she has a total allowance of $120 000, find (a) her net chargeable income, (b) her salaries tax payable.

Sol (a) Net chargeable income = $(220 000 – 120 000) = $100 000 (b) Net chargeable income = $100 000 = $(40 000 + 40 000 + 20 000)

Net chargeable

income Rate Tax


2% $40 000 × 2% = $800

On the next $40 000

7% $40 000 × 7% = $2 800

Remainder $20 000

12% $20 000 × 12% = $2 400

Her salaries tax payable = $(800 + 2 800 + 2 400) = $6 000

The annual income of Tim is $240 000. If he has a total allowance of $150 000, find (a) his net chargeable income, (b) his salaries tax payable.

Sol (a) Net chargeable income = $[( ) – ( )] = $ (b) Net chargeable income = $( ) = $(40 000 + 40 000 + ________ )

Net chargeable

income Rate Tax


2% $40 000 × 2% = _________

His salaries tax payable =

6. The annual income of Charles is $408 000. If he has a total allowance of $132 000, find his salaries tax payable.

Net chargeable income =

Net chargeable income Rate Tax

On the first $40 000 2% $40 000 × ______ = _________

123

His salaries tax payable = ○○○○→→→→ Ex 3F 12, 13

Level Up Question

7. The average monthly income of Vanessa is $13 000 and her total allowance is $120 000. Find her salaries tax payable.

124


3 Percentages (II)

[In this exercise, when calculating salaries tax, refer to the tax rates as shown in Table 1 on P.3.44 of the textbook.] Level 1

1. The rateable value of a flat is $360 000. Find the rates payable in a year.

2. The rateable value of a shopping centre is $3 000 000. Find the rates payable in a quarter of a year. 3. The rateable value of a piece of land is $76 900 000. Find the rates payable in a quarter of a year. 4. Mr Kan pays $18 000 for the annual rates on his property. Find the rateable value of the property. 5. David pays $7 200 for the quarterly rates on his flat. Find the rateable value of the flat. 6. The owner of an apartment pays $22 000 of rates quarterly. What is the rateable value of the apartment? Find the salaries tax payable for each of the persons below. [Nos. 7–9]

7. A taxi driver has a net chargeable income of $32 000. 8. Carrie is a dancer with a net chargeable income of $79 000. 9. Jon is an artist with a net chargeable income of $101 000. 10. The net chargeable incomes of Ramsey and Paul are $8 000 and $24 000 respectively. Is the salaries tax paid by Paul 3 times that paid by Ramsey? Explain your answer. 11. The annual income of Ellen is $186 000. If she has a total allowance of $110 000, find her salaries tax payable. Level 2 12. Priscilla paid $6 500 for the rates last year. At the beginning of this year, she moves to a new flat and the rates payable decreases by 10% as compared to last year. What is the rateable value of the new flat?


3F

125

13. Sally paid $9 400 for the rates last year. At the beginning of this year, she moves to a new flat and its rateable value is $142 800. What is the percentage change in the rates payable this year as compared to last year? (Give the answer correct to the nearest 1%.)

14. The average monthly income of Gigi is $25 000. If she has a total allowance of $120 000, find her salaries tax payable.

15. (a) Find the salaries tax payable in each of the following situations: (i) Net chargeable income = $40 000

(ii) Net chargeable income = $80 000

(iii) Net chargeable income = $120 000

(b) Joe has to pay the salaries tax of $10 100. Using the results of (a), calculate his net chargeable income. 16. Benny’s salaries tax payable is $13 500. Find his net chargeable income. 17. Glen has a total allowance of $130 000. He has to pay $1 500 in salaries tax. (a) Find his net chargeable income. (b) Find his average monthly income. 18. (a) The net chargeable income of Gary is $123 000. Find his salaries tax payable. (b) The net chargeable income of Raymond is one third of that of Gary. Is the salaries tax payable of Raymond one third of that of Gary? Explain your answer. If not, find the difference in their salaries tax payable.

126

Consolidation Exercise 3F

1. $18 000 2. $37 500 3. $961 250

4. $360 000 5. $576 000 6. $1 760 000

7. $640 8. $3 530 9. $6 120

10. yes 11. $3 320 12. $117 000

13. −24% 14. $18 600

15. (a) (i) $800 (ii) $3 600 (iii) $8 400

(b) $130 000

16. $150 000

17. (a) $50 000 (b) $15 000

18. (a) $8 910 (b) no, $8 040

chapter 1 more about factorization of polynomials

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