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FRACTIONS EXERCISES Mathematics for Academic Studies 3rd ESO

ALFONSO GONZÁLEZ. I.E.S. FERNANDO DE MENA. MATHEMATICS DEPARTMENT

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SECTION 1: Equivalent and irreducible fraction. Comparing fractions

NOTE: In each exercise, it is recommended checking your answers by means of your calculator.

1. Check whether the following fractions are equivalent:

a) 2 30and3 45 (Sol: YES)

b) 25 5and16 4 (Sol: NO)

c) 7 84and5 60 (Sol: YES)

d) 2 26and5 65

− (Sol: NO)

2. In each section, simplify –if possible– and amplify to find an equivalent fraction:

a) 23

b) 1625

c) 3624

d) 85

−

3. Find out the fractions with a denominator of 100 equivalent to the following ones:

a) 1325

b) 3950

c) 1120

4. Complete the missing terms: 5 157 84

= =

5. Express the following fractions in the simplest form (if some of them are irreducible, explain why):

a) 9018

(Sol: 1/5)

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b) 108252

− (Sol: -7/3)

c) 1625

(Sol: Irreducible)

d) 1751

(Sol: 3)

e) 99937

(Sol: 1/27)

f) 1404900 (Sol: 39/25)

g) 252420

(Sol: 3/5)

h) 495330

(Sol: 3/2)

i) 231770

(Sol: 3/10)

j) 147231

(Sol: 7/11)

k) 63110

(Sol: Irreducible)

l) 969361

(Sol: 51/19)

m) 999296

(Sol: 8/27)

6. Find out which of the following fractions are equivalent (do not use the decimal form!): 3 12 6 2, , and15 60 20 10

7. Reduce 3/4 and 4/5 to a common denominator and put them in order (from smallest to largest).






8. In each section, reduce the fractions to a common denominator and put them in order from smallest to largest:

a) 65

43

21

b)

157

53

21

(Sol: 7/15<1/2<3/5)

c) 1 3 2 9 6 55 4 7 8 5 6

− (Sol: -2/7<1/5<3/4<5/6<9/8<6/5)

9. In each case, find any fraction between the two given fractions (do not use the decimal form!). Then, check your answer using your calculator:

a) 4 2and5 3

b) 3 5and2 3






c) 5 4and4 3

10. Mental calculation: Put the following fractions in order: 2 3 7, and3 4 5

−

11. Given the following fractions: 3 4 5, and5 3 2

a) Reduce them to a common denominator and put them in order from smallest to largest. (Sol: 3/5<4/3<5/2)

b) Represent each one on a number line:

12. Do the same here: 5 15 12 2, , and

3 4 5 5−

a) (Sol: -2/5<5/3<12/5<15/4)

b)






13. a) Mark the following rational numbers on a number line:

2 7 16 5 18 5 9 3

3 6 3 7 5 4 2− − −

b) According to the previous result, arrange them from smallest to largest. (Sol: -9/2<-18/5<-5/7<2/3<7/6<5/4<3<16/3)

c) Use your calculator to check your answers.

14. a) Write the (irreducible) fraction pointed out by the arrow inside each box:

b) In each case, calculate the corresponding decimal expression (indicating the corresponding division). What kind of decimal are they?

c) Obtain the fraction with a denominator of 18 equivalent to the first one. Explain your answer.

0 1

0 1

0 1






d) Find out an integer number and five (irreducible) fractions between the two last ones. Explain your answer.

15. Given the following fractions: 56

and 12

a) Reduce them to a common denominator and put them in order (from least to greatest): (Sol: 1/2<5/6)

b) Using the previous result find the simplest fraction between the given ones.

(Sol: 2/3)

c) Show on a number line the two given fractions and the fraction obtained before, and check graphically your previous answers.

16. Given 116

and 2110

,

a) In each case, calculate its decimal expression (indicating the corresponding division) What type of decimal is it?

b) According to the previous result, write an integer number between the two given fractions. c) Find a (irreducible) fraction between the former integer and the first fraction.

0

0

0






d) Represent the fractions of the problem statement on a number line:

17. a) Write the (irreducible) fraction pointed out by the arrow inside the box:

Approximately represent 5/3:

b) Find out the corresponding decimal expression (indicating the division) of the two previous fractions. In both cases write what type of decimal it is:

53=

c) Obtain the fraction with a denominator of 42 equivalent to the second one. Explain your process:

53=

d) Find an integer and a (irreducible) fraction with a denominator of 2 between the two former fractions:

18. a) Reduce, if possible, the following fractions, explaining your process. If some of them are irreducible, explain

why:

147231

0 1 2 3

0 1 2 3

0 1

0 1






63

110

b) Find a fraction between the two numbers given (do not use the decimal form!). Then, check your answer using your calculator:

c) Reduce the following fractions to a common denominator and put them in order from smallest to largest

63110

711

1110

d) Mark the last fraction on a number line:

19. Consider the fractions 5

A2

= and 17

B6

=

a) Find a fraction with a denominator of 18 that is equivalent to B:

b) Reduce A and B to a common denominator and put them in order from smallest to largest:

c) Find a fraction C between A and B (do not use the decimal form!). Then, check your answer using your calculator:

0 1






d) Show A and B on a number line:

e) Calculate the corresponding decimal expression of A and B (and write the divisions). What kind of decimal are they?

20. Juan has spent 18/45 of his savings, María 21/35 and Pedro 4/5. Who has spent more? (Do not use decimals!) (Sol: Pedro)

21. Juan has eaten 2/5 of a pie and María has eaten 4/7 of the other one. Who has eaten more? (Do not use

decimals!) (Sol: María)

A MATHEMATICAL CURIOSITY: The Italian mathematician Leonardo de Pisa (first half of the 13th century), better known as Fibonacci, was the first in using the current notation for fractions, that is to say, two numbers separated by an horizontal bar.






SECTION 2: Adding and subtracting fractions

1. Work out these fractions, giving your answer in the simplest form (as in the examples):

a) 3 1 45 5 5+ =

b) 5 23 3+ =

c) 5 16 6− =

d) 7 25 5− =

e) 2 3 4 9 133 2 6 6

++ = =

f) 2 35 2+ =

(Sol: 19/10)

g) 3 14 2+ =

(Sol: 5/4)

h) 7 2 35 6 293 5 15 15

−− = =

i) 4 13 2− =

(Sol: 5/6)

j) 4 13 2+ =

(Sol: 11/6)

k) 3 22 3− =

(Sol: 5/6)

l) 2 33 2− =

(Sol: -5/6)

m) 1 55 2+ =

(Sol: 27/10)

n) 1 24 7− =

(Sol: -1/28)

o) 7 33 2− =

(Sol: 5/6)

p) 2 15 2+ =

(Sol: 9/10)

q) 8 75 2− =

(Sol: -19/10)

r) 4 13 8+ =

(Sol: 35/24)

s) 1 6 1 723 3 3

++ = =

t) 715

+ = (Sol: 12/5)

u) 233

− = (Sol: 7/3)

v) 5 23+ =

(Sol: 11/3)

w) 1 33− =

(Sol: -8/3)

x) 2 43 5

− − = (Sol: -22/15)

y) 6 33 2+ =

(Sol: 7/2)

z) 9 14 2

− − = (Sol: -11/4)

α) 3 15 3

− − = (Sol: -14/15)

β) 235

− = (Sol: 13/5)

γ) 10 499 45+ =

(Sol: 11/5)






δ) 1 13 15+ =

(Sol: 2/5) ε) 3 31

8 63− =

(Sol: -59/504)

2. Do the same here:

a) 3 2 1 18 20 15 535 3 2 30 30

+ ++ + = =

b) 3 1 22 4 3+ + =

(Sol: 29/12)

c) 3 1 35 3 2− + =

(Sol: 53/30)

d) 1 2 56 3 2+ − =

(Sol: -5/3)

e) 1 513 2

+ + =

(Sol: 23/6)

f) 7 1 23 3 5+ + =

(Sol: 46/15)

g) 8 2 25 3+ + =

(Sol: 64/15)

h) 7 112 3+ + =

(Sol: 29/6)

i) 5 3 16 4 3+ + =

(Sol: 23/12)

j) 3 1 22 4 3− − =

(Sol: 7/12)

k) 3 1 22 4 3

− − + =

(Sol: -13/12)

l) 2 1 37 3 2+ + =

(Sol: 89/42)

m) 1 1 13 6 2− + =

(Sol: 2/3)

n) 1 423 5

+ − =

(Sol: 23/15)

o) 1 314 4

+ + =

(Sol: 2)

p) 1 2 13 5 6+ − =

(Sol: 17/30)

q) 1 1 3 72 4 5 3− + + =

(Sol: 191/60)

r) 1 1 15 29 145+ + =

(Sol: 7/29)






s) 1 1 1 12 3 15 50+ + + =

(Sol: 23/25)

t) 25 6 4 19 81 3 27

− + − =

(Sol: 4)

u) 25 6 14 16 8

− + =

(Sol: 6)

v) (*) 1 1 1 1 16 24 58 87 232+ + + + =

(Sol: 7/29)

w) 8 28 103 4

− − + + =

(Sol: -1/6)

x) 3 3 121 84 28

+ − =

(Sol: 1/7)

3. Work out, alternating in each section between supressing conveniently brackets or calculating expressions inside of brackets (as in the examples):

a) supressing brackets1 3 2 1 3 2 15 18 20 232 5 3 2 5 3 30 30

− − − − + = − − = = ←

b) 7 4 1 7 8 3 7 5 42 20 22 114 3 2 4 6 4 6 24 24 12

− − − − = − = − = = = ←

calculating expressions inside of brackets

c) 2 1 45 2 3

− − = (Sol: 37/30)

d) 5 1 1 28 6 2 3

+ − − = (Sol: 23/24)

e) 5 1 412 3 5

− + − = (Sol: 59/30)

f) 2 4 1 323 5 3 4

+ + − − = (Sol: 233/60)

g) 2 1 319 3 4

− − + = (Sol: 67/36)

h) 1 5 1 42 2 3 5

− − − = (Sol: -37/15)

i) 2 1 317 3 2

− − + = (Sol: -19/42)






SECTION 3: Products and divisions of fractions

1. Calculate the following products, simplifying every step (not only at the end!) (See the examples):

a) 3 7 3·7 21·5 2 5·2 10

= =

b) 5 2 5·2 5·2 5·4 3 4·3 2·2 ·3 6

= = =

c) 5 3·6 4

= (Sol: 5/8)

d) 7 2·5 5

= (Sol: 14/25)

e) 2 3·3 2

= (Sol: 1)

f) 23 3·5 23

= (Sol: 3/5)

g) 3 1·4 2

= (Sol: 3/8)

h) 7 2·8 14

= (Sol: 1/8)

i) 4 1·3 5 − =

(Sol: -4/15)

j) 10 11·3 2

− =

(Sol: -55/3)

k) 3 7·2 12

− − =

(Sol: 7/8)

l) 1316·8

= (Sol: 26)

m) 15 21·14 5

= (Sol: 9/2)

n) 744·11

= (Sol: 28)

o) 7 6 1 7·6 7·3 ·2 7· ·3 5 4 3·5·4 3 ·5·2 ·2 10

= = =

p) 2 1 7· ·5 2 8

= (Sol: 7/40)

q) 2 5 3· ·9 4 2

= (Sol: 5/12)

r) 4 8 1· ·3 5 3

= (Sol: 32/45)

s) 1 12 7· ·3 5 3 − =

(Sol: -28/15)

t) 1 7·4·8 5

= (Sol: 7/10)

u) 2 7 25· ·3 5 21

− − =

(Sol: 10/9)

v) 5 7 5· ·3 2 4

= (Sol: 175/24)

w) 1 63· ·27 5

= (Sol: 2/15)

x) 6 3 4· ·3 2 13 − − =

(Sol: 12/13)

y) 9 1 8· ·4 2 3

−=

(Sol: -3)

z) 4 3 7· ·9 5 6− −

= (Sol: 14/45)






α) 108372

=· (Sol: 9/2)

β) 2 7 3 53 15 4 2

=· · · (Sol: 7/12)

γ) 3 3 1487 5 9

− =

· · · (Sol: -16/5)

d) 22 21 2025 24 23

=· · (Sol: 77/115)

2. Do the same:

a) 4 5 4·2 8:3 2 3·5 15

= =

b) 5 7 5·2 5·2 5:4 2 4·7 2·2 ·7 14

= = =

c) 5 3:6 4

= (Sol: 10/9)

d) 7 5:5 2

= (Sol: 14/25)

e) 7 2:5 5

= (Sol: 7/2)

f) 100 50:3 7

= (Sol: 14/3)

g) 3 1:4 2

= (Sol: 3/2)

h) 7 2:8 14

= (Sol: 49/8)

i) 4 1:3 5 − =

(Sol: -20/3)

j) 10 11:3 2

− =

(Sol: -20/33)

k) 3 7:2 12

− − =

(Sol: 18/7)

l) 525:4=

(Sol: 20)

m) 15 21:14 5

= (Sol: 25/98)

n) 990:7=

(Sol: 70)

o) 7 :143

= (Sol: 1/6)

p) 2 7:5 8

− = (Sol: -16/35)

q) 5 3:4 2

= (Sol: 5/6)

r) 4 8:3 5

−=

(Sol: -5/6)

s) 1 7:3 3−

= (Sol: -1/7)

t) 1 7:8 5− −

= (Sol: 5/56)

u) 2 10:3 21

− − =

(Sol: 7/5)

v) 5 5:3 4

= (Sol: 4/3)

w) 63:5=

(Sol: 5/2)

x) 1 1:2 3

− − =

(Sol: 3/2)






y) 9 1:4 2

−=

(Sol: -9/2)

z) ( )4 : 29−

− = (Sol: 2/9)

α) 4 :13

=

β) 31:4=

γ) 121:18

= (Sol: 3/2)

δ) 41:5

− =

ε) 372 :5=

(Sol: 120)

ξ) 5 1:6 12

= (Sol: 10)

η) 3 1:108 72

= (Sol: 2)

3. Do the same here:

a) 3 2 7 3 ·2·2 4· :5 3 2 5·3 ·7 35

= =

b) 3 1 2: ·2 4 3

= (Sol: 4)

c) 3 1 3· :5 3 2

= (Sol: 2/15)

d) 1 2 5 3 5 3 5 5: ·6 3 2 6 2 2 3 2 2 2 8

/= = =

/· ·

· · · · ·

e) 1 51: ·3 2

= (Sol: 15/2)

f) 7 1 2· :3 3 5

− =

(Sol: -35/18)

g) 8 2· : 25 3

= (Sol: 8/15)

h) 7 1:12 ·2 3

= (Sol: 7/72)

i) 5 3 1: :6 4 3

= (Sol: 10/3)

j) 3 1 2: ·2 4 3

− − =

(Sol: 4)

k) 3 1 2· :2 4 3

− − =

(Sol: 9/16)

l) 4 4 2· :3 5 3

=

m) 3 1 2: :2 4 3

− − =

(Sol: 9)

n) 3 1 2: :2 4 3

− − =

(Sol: 4)

4. a) How much is the half of 300 m3?






b) What is a third of 90 kg?

c) Work out two thirds of 90 kg.

d) Work out 1/5 of 1 000 euros.

e) Calculate 4/5 of 1 000 euros.

f) What is half of a half of a dozen?

g) How many days are in a third of a half of September?

h) Calculate the 5% of 1000 €.

i) Calculate 5% of 20% of an amount. (Sol: 1%)

5. The reverse problem:

a) The half of my age is 20. How old am I?

b) A third of my savings is 150 €. How much money do I have?

c) Two fifths of Ana’s savings are 6 $. How much money does she have? (Sol: 15 €)

d) 10% of an amount is 15. What is this amount?

e) 3/8 of a total country population is 6 000 000 inhabitants. What is its population? (Sol: 16000 000 inhabitants)






f) 15 % of a the total price of a consumer good is 9 €. What is its price? (Sol: 60 €)

g) 20 percent of the total price of a book is 12 $. How much is the book? (Sol: 60 $)

h) 60 % of a product is 21 €. How much does it cost? (Sol: 35 €)

i) 78 is 15% of what number? (Sol: 520)






SECTION 4: Combined operations with fractions (I)

1. Work out the following mixed operations, simplifying every step (not only at the end!). Remember hierarchy rules (BEDMAS):

1) 1 3 2·2 2 3

+ = (Sol: 13/12)

2) 1 3 2·2 2 3

+ = (Sol: 17/12)

3) 1 3 14·2 2 5+ =

(Sol: 47/10)

4) 2 1 4 1·5 2 3 6

+ − = (Sol: 41/30)

5) 2 1 4 1·5 2 3 6+ − =

(Sol: 9/10)

6) 2 1 4 1: :5 2 3 6

− = (Sol: -36/5)

7) 5 1 1 2·8 6 2 3

− − = (Sol: 47/72)

8) 5 1 1 2·8 6 2 3− + =

(Sol: 29/24)

9) 17 1 4·15 5 3

+ = (Sol: 39/25)

10) 5 1 41: ·2 3 5− =

(Sol: 1/10)

11) 2 4 12 :3 5 2

− + = (Sol: -7/3)

12) 3 2 1 21 :4 9 3 3

− − + = (Sol: -49/24)






13) ( )( )( )1 2 2 31 2 2 3+ +

=+ + (Sol: 11/15)

14) 343 16 45 7 174 3 :64 49 4 4 16

+ − + =· · (Sol: 12)

15) 3 2 1 21 :4 9 3 3 − − + =

(Sol: 85/12)

16) 1 5 1 4·2 2 3 5

− − = (Sol: 5/3)

17) 1 5 1 4·2 2 3 5

− − = (Sol: -22/15)

18) 1 5 1 4·2 2 3 5

− − = (Sol: 14/15)

19) 2 3 1 41 :3 2 4 3

− + − = (Sol: -2/13)

20) 2 3 1 41:3 2 4 3

− + − = (Sol: -37/6)

21) 2 3 1 41:3 2 4 3

− + − = (Sol: 7/78)

22) 1 2 1 3·5 7 3 2

− − = (Sol: -317/210)

23) 3 1 6 3· 32 3 5 2

+ − + = (Sol: -11/5)

24) 1 8 5 1· : 2 12 3 3 5

− − + = (Sol: 38/27)






25) 2 1 2 1 63 : ·5 3 3 3 5

− + − = (Sol: -19/5)

26) 64 25 45 5 174 : 3125 16 4 4 16

− + + + =· · (Sol: 12)

27) 65 258 723 23

− =· · (Sol: 15)

28) 2 1 5 1: · 1 53 3 3 2

− + + = (Sol: 12/89)

29) 3 6 4 24 · : 48 5 3 3

− + − = (Sol: 283/60)

30) 3 6 4 24 · : 48 5 3 3 − + − = (Sol: 249/80)

31) 2 1 31: ·7 3 2

− = (Sol: -14)

32) 1 1 1 1: :35 35 7 35

− = (Sol: -4)

33) 5 7 2 15 25 : · ·6 5 5 4 5

+ − − = (Sol: 4/3)

34) 1 3 2 4 1 1 1 =35 7 5 13 3 4

− − − −

: ·

(Sol: 2/3)

35) 2 3 4 3 11 13 2 9 5 6

− + −

· : · (Sol: -31/33)






36) 37 7 233 2 47 4 56− −

− + =· · · (Sol: -75/7)

37) 23 256 1165 23

− =: · (Sol: 5)

38) 37 7 232 3 27 4 56− −

− − =· · · (Sol: -43/7)

39) 1 1 5 1 1 12 1 12 3 2 3 7 3

+ − − + − − =

· · (Sol: 1/2)

40) 301 109 52 3 265 65 13− −

− − =· · · (Sol: -5)

41) 51 515 2 3 8 4 922 22

− − − =

· · · · (Sol: 6)

42) 301 109 53 2 465 65 13− −

− + =· · · (Sol: -9)

2. ¿True or False? If false, correct the mistake:

a) A +B A B2 2

= +

b) A +BC A BC2 2 2

= +






SECTION 5: Combined operations with fractions (II)

1. Calculate the following mixed operations, simplifying every step (not only at the end!) and taking hierarchy

rules (BEDMAS) into account!: a) =−

42

45 (Sol: 3/4)

b) =−44

55 (Sol: 0)

c) =−4

1655 (Sol: -3)

d) =− 4 −32 (Sol: -14/3)

e) =

−−−

−+ 2

23164

2132 (Sol: 16)

f) =+56 ·

31

41 (Sol: 13/20)

g) =

+56 ·

31

41 (Sol: 7/10)

h) =−51 ·

321 (Sol: 13/15)

i) =

−51 ·

321 (Sol: 1/15)

j) =+−21 ·

34

32 (Sol: 0)

k) =−−312 (Sol: -7/3)

l) =

−+−

56 ·

31

211 (Sol: -1)






m) =−+−56 ·

31

54 ·

31

52 (Sol: -8/15)

n) =

+−52 ·

31

211 (Sol: 1/3)

o) =+−52 ·

31

211 (Sol: 19/30)

p) =−

+−56 ·

31

54 ·

31

52 (Sol: -34/75)

q) + − + =1 1 4 1 5 8 · · 2 3 3 12 4 3

(Sol: 151/36)

r) + ⋅ − + ⋅ =

1 1 4 1 5 8 2 3 3 12 4 3

(Sol: 157/36)

s) 1 4 2 1 5 · · 2 7 14 2 7

− − + = (Sol: -1/14)

t) 1 4 2 1 5 · · 2 7 14 2 7

− − + =

(Sol: 1/7)

u) 21 19 1 2 15 9 3 : · : 2 2 5 5 8 2 4

− + − =

(Sol: -11/2)

v) − + + − + =

17 15 4 1 2 1 14 169 5 3 5 3 15 3 8

: : (Sol: 26/9)

w) + ⋅ − ⋅ + =

1 4 5 1 3 10: 43 3 6 2 2 9

(Sol: 73/15)

x) : − + ⋅ =

21 19 1 2 152 2 5 5 8

(Sol: 1/2)

y)

2

2 1

5 4

+ − − − =

3 3 324 4 2 (Sol: -3/4)






SECTION 6: Combined operations with fractions (III)

1. Work out the following mixed operations, simplifying every step (not only at the end!). Remember

hierarchy rules (BEDMAS)!:

a) + − − =

2 3 113 4 6

(Sol: 13/12)

b) 4 7 3 1 1 7 6 · 2 4:5 3 7 5 2 3 5

− + + − + =

(Sol: 13/10)

c) 2 5 3 4 5 3 12 : 43 4 5 10 4 5 5

+ + − + + =

(Sol: 193/60)

d) 1 7 2 52 : 25 3 4 3

+ + − + =

(Sol: 112/55)

e) 2 4 2 3 7 4 · :7 5 8 2 5 7

− + − =

(Sol: -797/280)

f) 33 7 235 3 27 4 56− −

+ + =· · · (Sol: -28)

g) 21 15 16 15 12 5 · : 35 4 3 30 4 4

+ − + + = (Sol: 291/10)

h) − − − − + − − + − =

2 3 1 2 1 6 1 3 1 13 2 5 5 3 5 2 4 2 3

(Sol: -37/20)






i) − − + − − + − =

4 1 2 1 4 12 23 2 5 3 3 5

(Sol: -49/30)

j) + − − − + =

5 7 2 12 32 10 5 4

(Sol: 29/20)

k) − + − − − + − =

3 1 5 7 14 28 2 4 2 8

(Sol: -1)

l) − − + − − + − =

4 1 5 2 723 9 4 3 2

(Sol: 19/36)

m) 4 1 7 4 5 1 1: · 6 2 3 12 6 15

+ − + =

(Sol: 7/3)

n) 1 4 1 1 2 5 · 1 · 33 5 3 3

+ − − − =

(Sol: 259/225)

o) + − − − =

4 12 1 2 3 1 2: 3 : 15 16 6 3 8 6 5

(Sol: 71/30)

p) − ⋅ − ⋅ + =

3 1 4 4 2 15: 12 2 3 3 3 8

(Sol: 23/26)

q) 1 256 35

−=

− (Sol: 3)






SECTION 7: Operations with complex fractions (I)

1. Calculate the following operations using complex fractions, simplifying every step (not only at the end!) and taking hierarchy rules (BEDMAS) into account! (See the example):

a) 11

11·6 11·2·3 33101 10 2·5 56

+/

= = = =/

−

3 1

5 22 1

3 2

b) +

=⋅

14 235

3

13

(Sol: 25/4)

c) :

−=

112 312

5

56

(Sol: 5/36)

d) =⋅

+−

56

2

32

31

21

5

(Sol: 7/24)

e) =

+

⋅+

61:

23

21

321

61

2

(Sol: 1/16)

f) =

+

−+

⋅

31

52

53

21

3

432:

(Sol: -39/17)

g) =+

−⋅+

12:1

324

312

345

(Sol: 35/27)

h) =−−

−+

312

51

94

94:2

625

5 531

(Sol: 27/17)






i) =

++−

−+−

3

51

31 ·

211

351 ·

31

211

(Sol: -73/98)

j) 1 1 1 1 12 7 14 112 2241 1 1 1 12 8 16 32 112

=+ + + +

+ + + +

(Sol: 1)

k) 1 2 312 :3 5 5

2 3 4 1· :3 2 9 2

=

− +

+ −

(Sol: -1/2)

l) 3 2 6 73:2 5 5 20

3 4 6 43 · :2 10 5 5

=

+ −

+ −

(Sol: 20/11)

m) 4 :3

2 4:5 7

−=

−

1 7 · 56 4

7 9 · 3 2

(Sol: -2600/931)

n) 5 1: :33 8

7 1 1:3 4 3 8

−

=

−

+

1 73 4

1 : +3

1

(Sol: -245/51)

o) 20 1 27 5 3

3 274 5

− = − −

·

:

(Sol: 1/15)






p)

+ −=

+ +

1 1 2 9: 2 · 2 3 5 5

1 2 1 5 2: · 3 3 3 2 3

(Sol: -9/20)

q) 1 7 5 13 :4 8 2 2

3 1 1 1 19:4 2 3 8 12

− − − =

− + −

(Sol: -5/8)

r)

110

110

− −=

− − −

3 1 3+ 2 + 5 5 53 1 3 25 5 5

:

: :

(Sol: -104/29)

s) =

++

−−

3311

1

2211

1 (Sol: 9/4)

t) =

+

−⋅+

⋅

−⋅

72

313

52

21

72

31

213:

52

(Sol: -47/606)

u) =

−

+⋅−

+

⋅

21:

25

31

71

72

349

321:

53

(Sol: 1323/3665)

v) =

++−

++−

5

41:

23

52 ·

31

21

541:

23

52 ·

31

21

(Sol: -31/9)






w) =

++

−+

31

25 ·

31

32:

31

21

52 · 2

31:

21

(Sol: 81/50)

x) =+

+−

−−

+−

56

32

31

21

1

32

46

5236

52

(Sol: 893/1512)

y) =

−−−+

−+

412

31

2 · 1

2

411

62532

1

(Sol: -49/130)

z) =+

+−+

5215

43

35

3

317

451:

(Sol: 205/162)

α) − + + = − +

2 1 3 8 33 :5 2 2 27 2

2 3 8 1 33 :5 2 27 2 2

·

· ·

(Sol: 59/32)

β) + +

=

+ − ⋅

1 2 3 24 4 4 9

1 1 62 23 3 5

⋅

⋅

(Sol: 55/152)

γ) 5 2 2 1 3: 33 3 5 2 11

14 13 2 1: 33 3 5 2

− − +

= − − +

·

(Sol: 13/41)






SECTION 8: Operations with complex fractions (II)

1. Work out the following operations using complex fractions, simplifying every step (not only at the end!). Take hierarchy rules (BEDMAS) into account!:

a) =

+−

+

825·2:

31

52

825·

522:

31

(Sol: -64/455)

b) =−+

−+−

2:13

:2

2511:2

515

35

(Sol: -125/189)

c) =

+

−+

+

910

53

51

52:

51

53

153

910 ·

53:

51

52

2:

(Sol: 585/347)

d) =+

++

211

11

11

(Sol: 8/5)

e) =−−

−+−

352

31:

32

21

71

352

31

32

21

71

(Sol: 311/342)

f) =+

++

323

23

23 (Sol: 139/39)

g) ( )

+ −=

+ +

1 8 3 9 : 12 3 5 251 98 3 : 12 253 5

(Sol: 108/299)

h) =

−

++

+−

361

42

31

32

32

83 23:

53

(Sol: -21/380)






i) =

−

+

−

+

23:3

52

278

23

21

23:3

52

278

23

21

(Sol: 59/32)

j) =+

++

765

43

21 (Sol: 233/151)

k) =

−+

+

+

−−+

19282:

543

21

5:6293

53

32

21

23

(Sol: 1/2)

l) =

−−

−

+−

512

31

32

109

32

32

815

32

23

(Sol: -125/28)

m) =−

+−−

−

+−−

51

52:

51

322:

32

34

51

52:

51

322

32

34

(Sol: 128/33)

n) =

−−

−

+−−

+−

353

164

35:

105

31

93

41

302

242

61

53

(Sol: -11/13)

o) =⋅

+

+

−+

23

51:

31−2

51

23

51:

312

51

(Sol: 325/21)

p) + − =

+ −

3 2 6 73 :2 5 5 20

3 4 6 43 :2 10 5 5⋅

(Sol: 20/11)

q) − + + =

− +

2 1 2 14 :3 5 3 32 1 2 14 : · 3 5 3 3

(Sol: 131/23)






SECTION 9: Decimal expression of a fraction 1. Work out the decimal expression of the following fractions, by dividing numerator and denominator without

using your calculator, and indicate the type of decimal obtained in each case:

a) 53

(Sol: Pure repeating decimal)

b) 76

(Sol: Mixed repeating decimal)

c) 95

− (Sol: Exact decimal)

d) 176


e) 513

(Sol: Integer)

f) 84210


g) 111240

(Sol: Exact decimal)






h) 203 (Sol: Exact decimal)

i) 125 (Sol: Mixed repeating decimal)

j) 5051 (Sol: Exact decimal)

k) 1825 (Sol: Mixed repeating decimal)

l) 111


m) 83


n) 38


o) 415







p) 125


q) 231175


r) 21440


s) 84120


t) 1120


A MATHEMATICAL CURIOSITY: Some examples of long repetend numbers:

Examples Number of repeating figures

6

6

6

16

22

PRACTICAL RULE TO FIND OUT IF A IRREDUCIBLE FRACTION IS EQUAL TO AN EXACT OR REPEATING DECIMAL (without doing the division): "If the only prime factors of its denominator are 2 and/or 5 then its decimal expression is exact; otherwise, is repeating"

1/23=0 0434782608695652173913.

1/17=0 0588235294117647.

1/21=0 047619.

1/13=0 076923.

1/7=0 142857.






2. Apply the previous rule to find, in each case, three fractions equivalent to:

a) Integers numbers:

b) Exact decimals:

c) Repeating decimals:

3. (In your notebook) Repeat exercise 1, but this time use the practical rule.

4. (In your notebook) Apply the practical rule to find out what type of decimal (exact or repeating) the following

fractions are equivalent to, and check your answers by doing the divisions:

a) 9

16

357

181

1823

123

211

71

1223

507

203

21

(Sol: E, E, E, R, R, R, R, R, R, E, R)

b) 6

7

21

132

9

23

7

3

3

2

25

13

20

23

5

7

4

3 (Sol: E, E, E, E, R, R, R, R, R)

4. MENTAL CALCULATIONS: Write the decimal expression of the following most common proper fractions:

12

= 13

= 14

= 15

= 23

= 34

= 25

=

And then, convert the following proper fractions to a decimal number by decomposing (rounding off to 2 decimal places), as in the examples:

a) 16 15 1 15 5 0 3 5 333 3 3 3

= + = + = + =. .

b) 94=

c) 43=

d) 4 5 4 0 5 4 0 5 2 0 25 2 252 2 2 2

+= = + = + =

. . . . .

e) 3 753

=.

f) 17·5=






SECTION 10: Decimals and their corresponding fraction

1. Express as a (irreducible) fraction and check your answers by dividing numerator and denominator (Do not use your calculator!):

1) 0.25 (Sol: 1/4)

2) 60,

(Sol: 2/3)

3) 30,2

(Sol: 7/30)

4) 0.12 (Sol: 3/25)

5) 20,1

(Sol: 11/90)

6) ∩350,12 (Sol: 1223/9900)

7) 1.125 (Sol: 9/8)

8) ∩1260, (Sol: 14/111)

9) 50,34

(Sol: 311/900)

10) 81,1

(Sol: 107/90)

11) 31,2

(Sol: 37/30)

12) 25.372 (Sol: 6343/250)






13) ∩2012, (Sol: 1208/99)

14) 55,13

(Sol: 2311/450)

15) ∩4012,13 (Sol: 120127/9900)

16) 124,12

(Sol: 21709/900)

17) 0 012∩

. (Sol: 2/165)

18) 0 012∩

. (Sol: 4/333)

19) 3 09∩

. (Sol: 34/11)

20) 1 5 6∩

. (Sol: 47/30)

21) 2 56. (Sol: 64/25)

22) 1 012. (Sol: 253/250)

23) 1 012∩

. (Sol: 167/165)

24) 1 012∩

. (Sol: 337/333)






25) 2 21∩

. (Sol: 73/33)

26) 2 0 3∩

. (Sol: 61/30)

27) 20 5. (Sol: 41/2)

28) 1 12∩

. (Sol: 37/33)

29) 1 12∩

. (Sol: 101/90)

30) 1 12. (Sol: 28/25)

31) 3 09∩

. (Sol: 34/11)

32) 1 5 6∩

. (Sol: 47/30)

33) 1 012. (Sol: 253/250)

34) 0 04772∩

. (Sol: 21/440)

35) 9 32. (Sol: 233/25)

36) 47 72∩

. (Sol: 525/11)

37) 2 6∩

. (Sol: 8/3)






38) 0 375. (Sol: 3/8)

39) 0 2 6∩

. (Sol: 4/15)

2. Work out the following operations in two different ways (and verify that the same result is obtained):

1º Doing the decimal operation, directly (without calculator).

2º Converting each decimal into its corresponding fraction, like in the example:

a) 0 3 0 6 3 6 9 19 9 9

+ = + = =. .

Sol : 1( )

b) =∩

− 150,0,3

∩Sol : 49 / 330 = 0 148( ).

c) =⋅0,1 40,

Sol : 2 / 45 = 0 04( ).

d) =+ 32,013,

Sol : 463 / 90 = 5 14( ).

e) =52, · 4

Sol : 92 / 9 = 10 2( ).

f) =∩

−∩

783,894,

Sol : 10 / 9 = 1 1( ).

g) =72,-8

Sol : 47 / 9 = 5 2( ).






h) 1,5 · 3,3 =

Sol : 5( )

i) =+− 11,61,11,25

Sol : 43 / 36 = 1 194( ).

3. Convert each decimal into a fraction and calculate the result of the following operations:

a) =+∩∩∩130,1:262,·1,872, (Sol: 25)

b) =++ )50,5(0,2 0,2521,9

(Sol: 17/8=2.125)

c) =72,

(Sol: 5/3= 61,

)

d) =− 6,0:8,038,0

(Sol: -11/30= 60,3

)

e) =− 3,0:51,01,11·308,4

(Sol: 1211/27=∩85144, )

f) =+ 72,0·83,16,0

(Sol: 5/3= 61,

)

g) 2 5 1 1 2 2+ =. · . .

(Sol: 5)



· fractions exercises mathematics for academic studies 3rd eso alfonso gonzÁlez. i.e.s. fernando...

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