Y8 – Autumn – Block 1 – Step 1 – Understand the meaning and representation of ratio Answers

Question Answer

1

a) For every 2 squares there are 3 triangles.b) For every 5 circles there is 1 square.

For every 1 square there are 5 circles.

2

3Any three diagrams showing twice as much yellow as red. Ensure that students are drawing equal parts.

4

a) For every 1 pencil there are 2 rubbers. b) For every 2 tyres there are 3 tennis balls.c) Any two of:

For every 3 tyres there are 9 tennis balls.For every 2 tyres there are 6 tennis balls.For every 1 tyre there is 3 tennis balls.

5

a) 3 should be coloured red and 9 blue.b) 12 should be coloured green and 18 red.c) 20 squares should be shaded.

6

Possible answers:Eva £1, Dexter 50pEva £2, Dexter £1Eva £3, Dexter £1.50 etc.

£19.50

7 This representation will make the darkest purple as it has the highest proportion of blue.

Any representation where over half of the parts are shaded blue will make a darker purple.

8

2 in every 5 people wear glasses means if 2 people do, 3 people don’t.

For every £1 Whitney has, Mo has £1.50 means if Whitney has £2 then Mo has £3

For every 225g of flour there are 150g of sugar means for every 2g of sugar there is 3g of flour.

All are in the ratio 2 : 3

✓ ✓✓

Y8 – Autumn – Block 1 – Step 2 – Understand and use ratio notation Answers

Question Answer

1

a) The ratio of boys to girls is 3 : 4The ratio of girls to boys is 4 : 3

b)

The ratio of red to blue is 1 : 5The ratio of blue to red is 5 : 1

c) For every 2 adults, there are 5 children.For every 5 children, there are 2 adults.For every 4 adults, there are 10 children.

d) The ratio of blue to orange is 5 : 3The ratio of orange to blue is 3 : 5For every 5 blue there are 3 orange.For every 3 orange there are 5 blue.

e) For every 2 triangles there is 1 square.For every 1 square there are 2 triangles.The ratio of squares to triangles is 1 : 2The ratio of triangles to squares is 2 : 1The ratio of green shapes to yellow shapes is 2 : 1

2Any diagram showing four times as many tyres as cars. Ensure that students are drawing equal parts.

3

a) 5 : 3b) 2 : 3c) 2 : 5d) Any ratio such that the proportion of dark chocolate is increased e.g. 3 : 5 : 3. Children

could go for a ratio e.g. 3 : 5 : 6 so there is more dark chocolate than any other type.

4a) 3 : 2 b) 2 : 3 c) 1 : 1 d) 1 : 1Any diagram showing the ratios above. Ensure that students are drawing equal parts.

5

a) 1 : 3 2 : 2 or 1 : 1 8 : 4 or 2 : 1b) 1 : 4 2 : 4 or 1 : 2 8 : 4 or 2 : 1c) Observations relating to the number lines being identical but some of the ratios being

different.d) E F G E G F

Or a reflection of the above diagrams. Diagrams may have multiples of the gaps shownabove.

Y8 – Autumn – Block 1 – Step 3 – Solve problems involving ratios of the form 1 : n (or n : 1)

Question Answer

1

a) For every 1 tray, there are 6 muffins.b) The ratio of trays to muffins is 1 : 6c) For every 6 muffins, there is 1 tray.d) The ratio of muffins to trays is 6 : 1e) 12f) 18g) 5

2

a) 200 gb) 100 gc) The amounts of flour and sugar are the same in each question because the ratio is 1 : 1

3

a) CordialWater

b)

120 ml of juice can be made using 20 ml of cordial.c)

24 ml of juice can be made using 20 ml of water.d) The ratios are the same but in part b 20 ml represented one part and in part c 20 ml

represents 5 parts.

4

a) Method 1 – Each part should contain 13Method 2 – Multiply each side of the ratio by 13Method 3 – Multiply both numbers on the double number line by 13There are 13 adults in the school.

b) Students will need to discuss and justify their reasons verbally. They may comment on the efficiency of a method and when one method is more suitable than another.

5 75

6

a) 450b) 50c) 100

7a) The 1 because it is the smaller number in the ratio.b) 25 cm

20

20 20 20 20 20

4

4 4 4 4 4

Y8 – Autumn – Block 1 – Step 4 – Solve problems involving ratios of the form m : n

Question Answer

1

a) The ratio of apples to oranges is 2 : 3b) The ratio of oranges to apples is 3 : 2c) If there are 6 oranges, there will be 4 apples.d) If there are 6 apples, there will be 9 oranges.

2

TrueNot enough information. Discussion point – could potentially be true but not necessarily.TrueFalse

3

a) 18b) 32c) 4d) 3 : 32

4a) 400b) 144

5

a) 98b) 196c) 49d) If there are twice as many horses, there are twice as many sheep. If there are half as

many horses there are half as many sheep.e) 392

6 16 hours

7

a) The following ratios should be ticked.100 : 502 : 1Students may also tick 1 : 2 assuming the ratio Tommy to Annie.

b) £1.50 or 150 pc) £5d) £6e) £7.50

Y8 – Autumn – Block 1 – Step 5 – Divide in a given ratio

Question Answer

1

a) Each bar represents the amount of money each person has. The bracket labelled £50 represents the total amount of money.

b) Dora receives £15 and Ron receives £35

2

a)

Could also be set out as two bars..b) £7c) £35d) She will save £5 more per month.

3 50 ml

4

Mo receives £150Aisha receives £50Jack receives £200

5

a) 90 : 6090 : 60

b) 45 : 3030 : 45

c) 10 : 1540 : 60

d) In part a the answers are the same because the ratios are equivalent.In part b the the numbers in the answer are reversed because the numbers in the ratioare reversed.In part c each part of the second answer is four times larger because the ratios are the same and the amount shared was four times larger.

6a) Isosceles – The triangle has exactly two angles that are the same.b) 120°

7

If the mean is 16, the total of the four numbers is 64This means that one equal part in the ratio is equal to 4Therefore the range is 24

8

a) 16b) 36c) 20d) −20e) 364f) −28

Y8 – Autumn – Block 1 – Step 6 – Express ratios in their simplest integer form

Question Answer

1

a) Any three ratios equivalent to 2 : 3 e.g. 6 : 9, 4 : 6, 2 : 3b) Discussion about the ratios being equivalent and how they can be seen in the image.c) 2 : 3 as their only common factor is 1

2 2 : 5

3

a) For every 18 yellow squares, there are 6 white squares.For every 9 yellow squares, there are 3 white squares.For every 6 yellow squares, there are 2 white squares.For every 3 yellow squares, there is 1 white square.

b) The last statement.c) 3 : 1

4

For the left hand diagram we can divide each side of the ratio by the highest common factor which is 3For the right hand diagram we can factorise 15 and 18 to identify a common factor of 3

5

a) 3 : 1b) 3 : 2c) 1 : 4d) 6 : 7e) 2 : 1f) 3 : 5 : 2

67 : 8 8 : 7

7

13 : 15 – Mo has added 10 to each number in the ratio.7 : 9 – Mo has added 4 to each number in the ratio.5 : 3 – Mo has reversed the numbers in the ratio

8 Jack will need 24 blue cars.

9

a) 5 : 1b) 20 : 9c) 5 : 9d) 1 : 2e) 10 : 3f) 200 : 130 : 1g) 3 : 5

Y8 – Autumn – Block 1 – Step 7 – Express ratios in form 1 : n

Question Answer (accept fractional and decimal equivalents throughout)

1

a) Any three ratios equivalent to 1 : 4 e.g. 3 : 12, 2 : 8, 1 : 4b) Students explain verbally. They could group the rows to show this.c) 1 : 4 as the only common factor is 1d) 1 : 4

2 1 : 2

3

a) For every 12 white squares, there are 18 yellow squares.For every 6 white squares, there are 9 yellow squares.For every 2 white squares, there are 3 yellow squares.

b) For every 1 white square, there are 1.5 yellow squares.c) 1 : 1.5

4

a) 1 : 3b) 1 : 3c) 1 : 4d) 1 : 4.5e) 1 : 2.25f) 1 : 0.5

5

a) 4 : 1b) 2 : 1c) 2 : 1d) 0.5 : 1

e) 23 : 1

f) 0.25 : 1

6

For every 1 person who prefers red, 2 people prefer blue.For every 1 person who prefers red, 3 people prefer green.For every 1 person who prefers blue, 0.5 people prefer red.

For every 1 person who prefers green, 23 people prefer blue.

7

8

a) 1 : 0.6 !" : 1

b) 1 : 1.8 !# : 1

c) 1 : !$%%!$ : 1

d) 1 : &$'%'%&$ : 1

1 : n n : 1

men : women 1 : 1.5'" : 1

women : men 1 : '" 1.5 : 1

Y8 – Autumn – Block 1 – Step 8 – Compare ratios and fractions

Question Answer

1

a) !"

b) 3 : 1c) #

"d) 1 : 3

2 1 : 4

337

4

a)

b) 25

c) 35

5a) 4

11b) 7

11

6

74 : 1

8 4 : 5

9

a) 730

b) 17 : 6 : 7c) 6 : 7

Y8 – Autumn – Block 1 – Step 9 – Understand 𝝅 as a ratio

Question Answer

1

a) 6 cmb) 24 cmc) 6 : 24 or 1 : 4d) Yes - for a square.e) It will be the same for a rhombus as that also has four sides equal in length.

2

a) 10 cmb) 20 cmc) 2 mmd) 4 mm

315 cm – None of the other measurements show either the diameter or the radius.

4

a) 3.14b) 1 : 3.14c) 2 : 6.28 = 1 : 3.14d) 4 : 12.56 = 1 : 3.14e) They are all equivalent.f) For any circle, the ratio of the diameter : circumference can be written as 1 : 3.14 or, more

accurately, 1 : 𝜋

5

Multiply both by dd : 𝜋dThe circumference of a circle is equal to Pi multiplied by the diameter.C = 𝜋d

6

a) 12𝜋 cm or 37.70 cm to 2 d.p.b) 24𝜋 cm or 75.40 cm to 2 d.p.c) 8𝜋 cm or 25.13 cm to 2 d.p.d) 2𝑥𝜋 cm

Y8 – Autumn – Block 1 – Step 10 – Understand gradient as a ratio

Question Answer

1

a) 1 : 31 : 3

b) 4 : 41 : 1

c) 2 : 41 : 2

d) 4 : 61 : 1.5

e) 5 : 11 : 0.2

f) 12 : 361 : 3

g) The hypotenuse is steeper.

2

a) For every 1 square across, it goes 3 squares up. The gradient of line A is 3b) For every 1 square across, it goes 1 square up. The gradient of line B is 1c) For every 1 square across, it goes 2 squares up. The gradient of line C is 2d) For every 1 square across, it goes 1.5 squares up. The gradient of line D is 1.5e) For every 1 square across, it goes 0.2 squares up. The gradient of line E is 0.2

3

Gradient of A = 2Gradient of B = 1Gradient of C = 1Gradient of D = 3Gradient of E = 0.5

Gradient of F = 13

4

For example:

For every 1 square that each line goes across, it goes up 2. The lines are parallel.