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KS3 Maths · Year 7Chapter 8 · Ratio, proportion and rates of change
Year 7 · Chapter 8

Ratio, proportion and rates of change

By the end of this chapter you will be able to

  • Change between metric units of length, mass, capacity, time, area and volume
  • Use a scale or a scale factor to find real lengths and enlarged lengths
  • Simplify a ratio, and share a quantity in a given ratio
  • Find a percentage change and simple interest using a multiplier
  • Recognise direct and inverse proportion, and solve problems with them
  • Use speed, density and unit price, and convert between speed units

You should already know

  • Fractions and percentages of an amount (Chapter 2)
  • Multiplication and division facts, and sharing equally (Year 6, Chapter 3)
  • Changing units of length and mass in Year 6 (Year 6, Chapter 9)

Where this leads

  • Ratio and proportion problems with algebra in Year 8
  • Rates of change on graphs and the gradient of a curve in Years 8 and 9

National Curriculum for England, Key Stage 3 mathematics, Year 7. Content: N12 and R1 change freely between related standard units: time, length, area, volume, capacity and mass, R2 scale factors, scale diagrams and maps, R3 express one quantity as a fraction of another, including fractions less than or greater than 1, R4 and R5 ratio notation, reduction to simplest form, and dividing a quantity into parts in a given ratio, R6 and R7 multiplicative relationships as a ratio or a fraction, and linking ratio to fraction arithmetic, R8 percentage change, reverse percentages and simple interest (reverse percentages are in Chapter 2), R9 direct and inverse proportion, in tables and algebraically, R10 compound units: speed, unit pricing and density.

Exercise P  Do you remember?

Non-calculator
1
FHow many millimetres are there in 1 centimetre?
2
FWrite 1/4 as a percentage.
3
FSimplify 6/9.
4
FShare 12 sweets equally between 3 people. How many does each person get?
5
FWork out 25% of 80.
8.1

Changing units

Year 7
N12, R1
Video lesson: Changing unitsComing soon
A builder needs 2.5 km of cable, but the cable company sells it in metres. A builder also needs a floor covered in tiles, measured in square metres, and buys tiles in square centimetres. Changing units correctly avoids costly mistakes.

Length: 1 m = 100 cm = 1000 mm, and 1 km = 1000 m. Mass: 1 kg = 1000 g. Capacity: 1 litre = 1000 ml. Time: 1 minute = 60 seconds, 1 hour = 60 minutes, 1 day = 24 hours.

Area and volume are harder. A square metre is 100 cm by 100 cm, so 1 m2 = 10,000 cm2. A cubic metre is 100 × 100 × 100 cm, so 1 m3 = 1,000,000 cm3.

Key ideas

  • Length: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm.
  • Mass: 1 kg = 1000 g. Capacity: 1 litre = 1000 ml.
  • Area: squared units. 1 m2 = 10,000 cm2, 1 cm2 = 100 mm2.
  • Volume: cubed units. 1 m3 = 1,000,000 cm3.
Worked example 8.1.1
Change 2.5 km to metres.

Method. Use 1 km = 1000 m.

  1. Write the conversion. 1 km = 1000 m.
    2.5 km = 2.5 × 1000 m
  2. Multiply by 1000. A kilometre is a thousand metres, so the number of metres is bigger.
    2.5 × 1000 = 2500
  3. Write the answer. Check it against the question and give the unit.
    2500 m

Check. 2500 ÷ 1000 = 2.5. ✓

Watch out. Bigger unit to smaller means multiply.

Worked example 8.1.2
Change 3 m2 to cm2.

Method. Area uses squared units: 1 m2 = 100 × 100 cm2.

  1. Find the factor for area. 1 m = 100 cm, so 1 m2 = 100 × 100 cm2.
    1 m2 = 10,000 cm2
  2. Multiply by 3. Three square metres is three lots of 10,000.
    3 × 10,000 = 30,000
  3. Write the answer. Check it against the question and give the unit.
    30,000 cm2

Check. 30,000 cm2 is only 3 m2 when divided by 10,000. ✓

Watch out. Do not multiply by 100 for area: that is only one of the two dimensions.

Worked example 8.1.3
Change 2 hours 15 minutes to minutes.

Method. Change hours to minutes, then add the minutes.

  1. Change hours to minutes. 1 hour = 60 minutes.
    2 × 60 = 120
  2. Add the extra minutes. Add the 15 minutes.
    120 + 15 = 135
  3. Write the answer. Check it against the question and give the unit.
    135 minutes

Check. 135 ÷ 60 = 2.25 hours. ✓

Watch out. Change hours to minutes, not to decimal hours, when you add minutes.

Watch out

Watch out: 1 m2 is NOT 100 cm2. Area needs 100 × 100 = 10,000 cm2, so squared units need squaring factors.

Exam tip

Grown-up tip. Ask the pupil to sketch a metre square with a 10 cm grid, to see that there are 100 squares of 10 cm, not 100 of 1 cm.

Exercise 8A  Changing units

Non-calculator
1
FChange 3.4 km to metres.
2
FChange 250 cm to metres.
3
FChange 4.2 kg to grams.
4
FChange 1750 g to kilograms.
5
FChange 6 litres to millilitres.
6
FChange 350 ml to litres.
7
FChange 2.5 hours to minutes.
8
FChange 3 weeks to days.
9
FChange 5 cm2 to mm2.
10
FChange 4 m2 to cm2.
11
FChange 2 m3 to cm3.
12
FChange: (a) 2 km to m (b) 500 ml to litres (c) 1 day to hours
(a)(a) 2 km to m
(b)(b) 500 ml to litres
(c)(c) 1 day to hours
13
PSA tiler needs 3 m2 of tiles. The shop sells tiles measured in cm2. How many cm2 does the tiler need?
14
PSA tap fills a jug at 150 ml per second. How many litres does it fill in 1 minute?
15
RSam says 1 m2 is 100 cm2. Is he right? Explain.
8.2

Scales and fractions of a whole

Year 7
R2, R3
Video lesson: Scales and fractions of a wholeComing soon
A map has a scale of 1 cm : 5 km. A line that is 3 cm long on the map is 15 km in real life. A scale drawing keeps all lengths in the same ratio to the real object.

A scale such as 1:50 means 1 unit on the drawing is 50 of the same unit in real life. A scale factor multiplies every length: a scale factor of 3 makes each length three times as big.

To write one quantity as a fraction of another, put the first on the top and the second on the bottom, then simplify. If the first is bigger than the second, the fraction is bigger than 1: 15 out of 10 is 15/10 = 3/2.

Key ideas

  • Scale 1 : n means 1 unit on the drawing is n of the same units in real life.
  • A scale factor multiplies every length in the same way.
  • A fraction of a whole: the part on top, the whole on the bottom, then simplify.
  • A fraction greater than 1 means the top is bigger than the bottom.
Worked example 8.2.1
On a plan with scale 1 cm : 5 m, a wall is 4 cm long. How long is the real wall?

Method. Multiply the drawing length by the scale.

  1. Read the scale. Each 1 cm on the plan stands for 5 m in real life.
    1 cm : 5 m
  2. Find the real length. Four lots of 5 m.
    4 × 5 = 20
  3. Write the answer. Check it against the question and give the unit.
    20 m

Check. 4 cm on the plan is 4 lots of 5 m: 20 m. ✓

Watch out. The scale tells you how many real units each drawn unit is.

Worked example 8.2.2
A rectangle 4 cm by 2 cm is enlarged with scale factor 3. What are the new sides?

Method. Multiply each side by 3.

  1. Multiply the length. The length 4 cm becomes 3 times as long.
    4 × 3 = 12
  2. Multiply the width. The width 2 cm becomes 3 times as wide.
    2 × 3 = 6
  3. Write the answer. Check it against the question and give the unit.
    12 cm and 6 cm

Check. 12 ÷ 3 = 4 and 6 ÷ 3 = 2. ✓

Watch out. Multiply every side by the same scale factor.

Worked example 8.2.3
Write 18 as a fraction of 24.

Method. Put 18 on the top and 24 on the bottom, then simplify.

  1. Write the fraction. 18 out of 24.
    18/24
  2. Simplify. Divide the top and bottom by 6.
    18/24 = 3/4
  3. Write the answer. Check it against the question and give the unit.
    3/4

Check. 3/4 × 24 = 18. ✓

Watch out. Simplify fully.

Watch out

Watch out: a scale of 1:50 does not mean 50 cm to 1 cm. It means 1 unit on the drawing is 50 units in real life, so the drawing is much smaller.

Exam tip

Grown-up tip. Ask the pupil to measure a room with a tape and draw it at 1:50 on squared paper, then check the real size.

Exercise 8B  Scales, scale factors and fractions

Non-calculator
1
FA plan has a scale of 1 cm : 5 m. A line on the plan is 4 cm long. What is the real length?
2
FA map has a scale of 1 : 20,000. A line on the map is 5 cm long. How far is it in real life, in km?
3
FA rectangle is 3 cm by 2 cm. It is enlarged by a scale factor of 3. What are the new lengths?
4
FA logo 10 cm wide is enlarged by a scale factor of 1.5. How wide is the new logo?
5
FWrite 18 as a fraction of 24 in its simplest form.
6
FWrite 12 as a fraction of 20 in its simplest form.
7
FWrite 15 as a fraction of 10 in its simplest form.
8
PSA class has 12 boys and 8 girls. What fraction of the class are girls? Give your answer in simplest form.
9
PSWrite 45 minutes as a fraction of 2 hours. Give your answer in simplest form.
10
PSA room is 5 m wide. It is drawn on a plan at a scale of 1 : 100. How wide is the room on the plan, in cm?
11
FWrite each as a fraction in simplest form: (a) 6 as a fraction of 8 (b) 25 as a fraction of 10
(a)(a) 6 as a fraction of 8
(b)(b) 25 as a fraction of 10
12
PSA model car is 1/24 the length of a real car. The real car is 3.6 m long. How long is the model, in cm?
13
RSam says a scale of 1 : 10 means 1 cm on the drawing is 10 cm in real life. Is he right? Explain why a scale of 1 : 100 is the same as 1 cm : 1 m.
8.3

Ratio and sharing in a given ratio

Year 7
R4, R5, R6, R7
Video lesson: Ratio and sharing in a given ratioComing soon
A fruit punch is made by mixing 2 parts orange juice with 3 parts lemonade. If the mix uses 10 litres in total, the amounts of each are found by sharing in the ratio 2 : 3.

A ratio compares two or more quantities of the same kind, written as 2 : 3. A ratio is simplified by dividing every part by their HCF: 8 : 12 = 2 : 3.

To share in a ratio, add the parts to find the total number of parts. Find one part by dividing the total by the number of parts, then multiply each ratio number by that amount. The fraction a part is of the whole is its ratio number over the total of the parts.

Key ideas

  • Simplify a ratio by dividing every part by the HCF.
  • Total the parts, then find the value of one part.
  • Each share is the number of parts times the value of one part.
  • A part of a ratio 2 : 3 is 2/5 of the total, because 2 + 3 = 5 parts.
Worked example 8.3.1
Simplify 12 : 18.

Method. Find the HCF of 12 and 18, and divide.

  1. Find the HCF. The HCF of 12 and 18 is 6.
    HCF = 6
  2. Divide both parts. Divide 12 and 18 by 6.
    12 ÷ 6 : 18 ÷ 6 = 2 : 3
  3. Write the answer. Check it against the question and give the unit.
    2 : 3

Check. 2 × 6 : 3 × 6 = 12 : 18. ✓

Watch out. Divide every part by the same number.

Worked example 8.3.2
Share £60 in the ratio 2 : 3.

Method. Find the total number of parts, then the value of one part.

  1. Find the total parts. 2 + 3 = 5 parts.
    5
  2. Find one part. Share £60 into 5 equal parts.
    £60 ÷ 5 = £12
  3. Find each share. Multiply each ratio number by £12.
    2 × £12 = £24 and 3 × £12 = £36
  4. Write the answer. Check it against the question and give the unit.
    £24 and £36

Check. £24 + £36 = £60. ✓

Watch out. Check the shares add up to the total.

Worked example 8.3.3
There are 24 pupils in a class, in the ratio boys : girls = 3 : 5. How many girls are there?

Method. Find the parts, then the girls.

  1. Find the total parts. 3 + 5 = 8 parts.
    8
  2. Find one part. 24 ÷ 8 = 3 pupils per part.
    3
  3. Find the girls. Girls are 5 parts.
    5 × 3 = 15
  4. Write the answer. Check it against the question and give the unit.
    15 girls

Check. 9 boys + 15 girls = 24. ✓

Watch out. Girls are the second number in the ratio.

Watch out

Watch out: a ratio 2 : 3 is not the same as a fraction 2/3 of the total. The fraction of the total is 2/5, because there are 5 parts altogether.

Exam tip

Grown-up tip. Ask the pupil to draw the bar: 5 equal boxes, 2 shaded for one share and 3 for the other, to see the parts clearly.

Exercise 8C  Ratio and sharing

Non-calculator
1
FSimplify 8 : 12.
2
FSimplify 15 : 25.
3
FSimplify 4 : 10 : 6.
4
FDivide £40 in the ratio 3 : 5.
5
FShare 56 sweets in the ratio 1 : 3.
6
PSThe ratio of red to blue counters is 4 : 9. There are 52 counters altogether. How many are red?
7
PSWrite the ratio 2 kg : 500 g in its simplest form.
8
RThe ratio of boys to girls is 3 : 5. What fraction of the pupils are boys?
9
FPriya is 12 and Ben is 18. Write their ages as a ratio in its simplest form.
10
FShare: (a) £30 in the ratio 2 : 1 (b) 42 km in the ratio 4 : 3
(a)(a) £30 in the ratio 2 : 1
(b)(b) 42 km in the ratio 4 : 3
11
PSCordial is mixed with water in the ratio 1 : 4. How much cordial is needed to make 500 ml of drink?
12
PSThe ratio of cats to dogs in a village is 2 : 5. There are 10 cats. How many dogs are there?
13
RSam says that in the ratio 2 : 3, two-fifths of the total is the first part. Is Sam right? Explain.
8.4

Percentage change and simple interest

Year 7
R8
Video lesson: Percentage change and simple interestComing soon
Sam puts £500 into a savings account that pays 3% simple interest each year. Each year he earns 3% of the original £500, so the interest does not grow.

A multiplier is a number that gives the new amount in one step. A 20% increase is multiplied by 1.2, and a 15% decrease is multiplied by 0.85. So £300 increased by 15% is 300 × 1.15 = £345.

Simple interest is worked out on the original amount only: I = P × r × t, where P is the principal, r is the rate as a decimal, and t is the time in years.

Key ideas

  • Increase by p%: multiply by (100 + p)/100. Decrease by p%: multiply by (100 − p)/100.
  • Simple interest: I = P × r × t, with r as a decimal.
  • Total amount = original + interest.
  • Repeated percentage changes are applied one at a time to the new amount.
Worked example 8.4.1
Work out the simple interest on £500 at 3% per year for 4 years.

Method. Use I = P × r × t.

  1. Write the values. P = 500, r = 0.03, t = 4.
    500 × 0.03 × 4
  2. Multiply. Work out 500 × 0.03 = 15, then 15 × 4.
    15 × 4 = 60
  3. Write the answer. Check it against the question and give the unit.
    £60

Check. £60 a year is £15 for 4 years: 15 × 4 = 60. ✓

Watch out. The rate must be a decimal: 3% is 0.03.

Worked example 8.4.2
Increase £300 by 15%.

Method. Use a multiplier.

  1. Find the multiplier. An increase of 15% gives 100% + 15% = 115%, so the multiplier is 1.15.
    1.15
  2. Multiply. The new amount is 300 times the multiplier.
    300 × 1.15 = 345
  3. Write the answer. Check it against the question and give the unit.
    £345

Check. 300 × 0.15 = 45, and 300 + 45 = 345. ✓

Watch out. Multiply by the new percentage, not the change.

Worked example 8.4.3
A price of £40 is decreased by 25%. Find the new price.

Method. Use the multiplier 0.75.

  1. Find the multiplier. A decrease of 25% gives 75%, so the multiplier is 0.75.
    0.75
  2. Multiply. New price = 40 × 0.75.
    40 × 0.75 = 30
  3. Write the answer. Check it against the question and give the unit.
    £30

Check. 25% of 40 is 10, and 40 − 10 = 30. ✓

Watch out. Use the decrease to find the multiplier.

Watch out

Watch out: a 10% increase and then a 10% decrease does not bring you back to the start. £100 goes to £110, then to £99.

Exam tip

Grown-up tip. Ask the pupil to write the multiplier first, before the calculation, so that the increase or decrease is clear.

Exercise 8D  Percentage change and simple interest

Non-calculator
1
FWork out the simple interest on £400 at 5% per year for 3 years.
2
FFind the total value of £400 after 3 years at 5% simple interest.
3
FWork out the simple interest on £250 at 4% per year for 2 years.
4
FWrite the multiplier for a 25% increase.
5
FWrite the multiplier for a 12% decrease.
6
FIncrease 300 by 15%.
7
FDecrease 40 by 25%.
8
PSA price of £100 increases by 10%, and then the new price increases by 10% again. What is the final price?
9
PSHow many years at 2% simple interest will £100 take to earn £10 interest?
10
FFind the multiplier for: (a) an 8% increase (b) a 35% decrease
(a)(a) 8% increase
(b)(b) 35% decrease
11
PSSam invests £800 at 2.5% simple interest per year. How much is the total after 4 years?
12
RAli says that a 10% increase followed by a 10% decrease brings £200 back to £200. Is Ali right? Show it with working.
8.5

Direct and inverse proportion

Year 7
R9
Video lesson: Direct and inverse proportionComing soon
Five pens cost £7.50. The cost goes up in the same way as the number of pens: double the pens, double the cost. This is direct proportion. If 4 people paint a fence in 6 hours, then twice as many people take half the time: that is inverse proportion.

Two quantities are in direct proportion when one is a fixed number times the other: y = kx, where k is the constant of proportionality. Their graph is a straight line through the origin.

Two quantities are in inverse proportion when their product is constant: y = k/x, so xy = k. Doubling one halves the other.

Key ideas

  • Direct proportion: y = kx. Doubling x doubles y.
  • Find k with k = y ÷ x, then use it.
  • Inverse proportion: y = k/x, so xy = k is constant. Doubling x halves y.
  • The graph of direct proportion is a straight line through the origin.
Worked example 8.5.1
y is directly proportional to x, and y = 12 when x = 4. Find y when x = 7.

Method. Find k, then use y = kx.

  1. Find k. k = y ÷ x = 12 ÷ 4 = 3.
    k = 3
  2. Use y = kx. Put x = 7 into y = 3x.
    y = 3 × 7 = 21
  3. Write the answer. Check it against the question and give the unit.
    y = 21

Check. 21 ÷ 7 = 3. ✓

Watch out. Find k first.

Worked example 8.5.2
6 workers take 12 days to build a wall. How long would 4 workers take, if the work is shared equally?

Method. Use inverse proportion: the product workers × days is constant.

  1. Find the constant. 6 × 12 = 72.
    72
  2. Find the days for 4 workers. Days = 72 ÷ 4.
    72 ÷ 4 = 18
  3. Write the answer. Check it against the question and give the unit.
    18 days

Check. 4 × 18 = 72. ✓

Watch out. Fewer workers take longer, so the answer is bigger.

Worked example 8.5.3
5 pens cost £7.50. How much do 8 pens cost?

Method. Find the cost of one pen, then multiply.

  1. Find one pen. £7.50 ÷ 5 = £1.50.
    £1.50
  2. Find eight pens. 8 × £1.50.
    8 × 1.50 = 12
  3. Write the answer. Check it against the question and give the unit.
    £12

Check. £12 ÷ 8 = £1.50. ✓

Watch out. Direct proportion means the price per pen stays the same.

Watch out

Watch out: y = 2x + 1 is not direct proportion, because it does not go through the origin: when x = 0, y = 1, not 0.

Exam tip

Grown-up tip. Ask the pupil to decide first whether the quantities go up together (direct) or one goes up while the other goes down (inverse).

Exercise 8E  Direct and inverse proportion

Non-calculator
1
Fy is directly proportional to x. y = 6 when x = 2. Find y when x = 5.
2
F4 pens cost £3.40. How much do 7 pens cost?
3
F5 kg of apples cost £7.50. How much do 2 kg cost?
4
PS4 people paint a wall in 6 hours. How long would 8 people take, working at the same rate?
5
Fy is inversely proportional to x. y = 10 when x = 2. Find y when x = 5.
6
FWhich of these is directly proportional? A: y = 5x B: y = 5/x C: y = x + 5
7
PSA car travels at a steady speed. Distance is directly proportional to time. It travels 60 km in 1 hour. How far does it travel in 2.5 hours?
8
PSA 60 km trip takes 2 hours at 30 km/h. How long does it take at 40 km/h?
9
PSA recipe for 4 people uses 300 g of flour. How much flour is needed for 10 people?
10
F(a) y is directly proportional to x, and y = 8 when x = 2. Find y when x = 3. (b) y is inversely proportional to x, and y = 3 when x = 4. Find y when x = 6.
(a)(a) direct, y = 8 at x = 2, find y at x = 3
(b)(b) inverse, y = 3 at x = 4, find y at x = 6
11
RExplain why y = 2x + 1 is not direct proportion. Use x = 0 to help.
8.6

Compound units: speed, density and unit price

Year 7
R10
Video lesson: Compound units: speed, density and unit priceComing soon
A runner covers 100 m in 12.5 seconds. A metal block has a mass of 540 g and a volume of 200 cm3. A pack of 6 pens costs £2.40. Each of these is a rate: a quantity divided by another, with its own unit.

A compound unit combines two units. Speed = distance ÷ time, in km/h or m/s. Density = mass ÷ volume, in g/cm3. Unit price = price ÷ amount, such as pence per gram.

To convert speed, use 1 km = 1000 m and 1 hour = 3600 seconds: 72 km/h = 72,000 m ÷ 3600 s = 20 m/s.

Key ideas

  • Speed = distance ÷ time. Distance = speed × time. Time = distance ÷ speed.
  • Density = mass ÷ volume.
  • Unit price = price ÷ amount: choose the cheapest by comparing unit prices.
  • Change units first, so that the units match before dividing.
Worked example 8.6.1
A car travels 210 km in 3 hours. What is its average speed in km/h?

Method. Speed = distance ÷ time.

  1. Write the formula. Speed = distance ÷ time.
    210 ÷ 3
  2. Divide. Divide the distance by the time.
    210 ÷ 3 = 70
  3. Write the answer. Check it against the question and give the unit.
    70 km/h

Check. 70 × 3 = 210. ✓

Watch out. Use the unit km/h.

Worked example 8.6.2
A block has mass 540 g and volume 200 cm3. Find its density.

Method. Density = mass ÷ volume.

  1. Write the formula. Density = mass ÷ volume.
    540 ÷ 200
  2. Divide. Divide the mass by the volume.
    540 ÷ 200 = 2.7
  3. Write the answer. Check it against the question and give the unit.
    2.7 g/cm3

Check. 2.7 × 200 = 540. ✓

Watch out. Put mass on top and volume on the bottom.

Worked example 8.6.3
Change 72 km/h to m/s.

Method. Change km to m, and hours to seconds.

  1. Change km to m. 72 km = 72,000 m.
    72,000 m
  2. Change hours to seconds. 1 hour = 3600 seconds.
    72,000 ÷ 3600 = 20
  3. Write the answer. Check it against the question and give the unit.
    20 m/s

Check. 20 × 3600 = 72,000. ✓

Watch out. Change both units.

Watch out

Watch out: 60 km/h is not 60 m/s. A metre per second is much slower than a kilometre per hour: 60 km/h is about 16.7 m/s.

Exam tip

Grown-up tip. Ask the pupil to say the units out loud when they divide: "kilometres divided by hours gives kilometres per hour".

Exercise 8F  Speed, density and unit price

Non-calculator
1
FA car travels 210 km in 3 hours. Find its average speed in km/h.
2
FA runner covers 100 m in 12.5 seconds. Find the speed in m/s.
3
FA coach travels 240 km at 80 km/h. How long does the journey take?
4
FA cyclist rides at 15 m/s for 4 seconds. How far does the cyclist travel?
5
FA block has mass 540 g and volume 200 cm3. Find the density in g/cm3.
6
FChange 72 km/h to metres per second.
7
FChange 5 m/s to km/h.
8
FSix pens cost £2.40. How much does one pen cost?
9
RA pack of 500 g of rice costs £2.25. Another pack of 750 g costs £3.00. Which pack is better value? Explain.
10
PSA tap fills a 60 litre bath at 15 litres per minute. How many minutes does it take?
11
FFind: (a) the speed of 4 km in 30 minutes, in km/h (b) the density of 3 kg in 500 cm3, in g/cm3 (c) the time to travel 90 miles at 60 miles per hour
(a)(a) 4 km in 30 minutes, in km/h
(b)(b) 3 kg in 500 cm3, in g/cm3
(c)(c) 90 miles at 60 mph
12
PSA walker travels 6 km in 1 hour 30 minutes. What is her average speed in km/h?

Chapter summary

SkillMethod
UnitsMetric: 1 km = 1000 m, 1 m = 100 cm, 1 kg = 1000 g, 1 l = 1000 ml; area squared, volume cubed
Scale and fractionScale 1 : n multiplies by n; a fraction is the part over the whole, simplified
RatioSimplify by the HCF; share by finding the total parts and the value of one part
Percentage changeMultiplier 1 + r or 1 − r; simple interest I = P × r × t
Direct and inverseDirect: y = kx; inverse: y = k/x, so xy is constant
Compound unitsSpeed = distance ÷ time; density = mass ÷ volume; unit price = price ÷ amount

Can you…?

  • I can change between metric units of length, mass, capacity, time, area and volume
  • I can use a scale or a scale factor, and write one quantity as a fraction of another
  • I can simplify a ratio, and share a quantity in a given ratio
  • I can use a multiplier for percentage change, and work out simple interest
  • I can recognise direct and inverse proportion, and solve problems with them
  • I can use speed, density and unit price, and convert speed units

Key terms

ratio
a comparison of two or more quantities of the same kind, such as 2 : 3
scale
the relationship between a drawing and the real object, such as 1 : 50
scale factor
the number every length is multiplied by when an object is enlarged or reduced
multiplier
a number that gives the new amount in one step, such as 1.15 for a 15% increase
simple interest
interest worked out on the original amount only
direct proportion
two quantities where one is a constant times the other, y = kx
inverse proportion
two quantities whose product is constant, y = k/x
density
mass divided by volume

Exercise 8Z  Check-up

Non-calculator
1
Change 3.6 km to metres.
2
Change 2.5 m2 to cm2.
3
Simplify 15 : 35.
4
Share £45 in the ratio 2 : 3.
5
Work out the simple interest on £600 at 4% per year for 2 years.
6
Find the multiplier for a 15% increase.
7
y is directly proportional to x. y = 12 when x = 3. Find y when x = 5.
8
It takes 5 people 8 hours to do a job. How long would 10 people take, working at the same rate?
9
A car travels 150 km in 2.5 hours. Find its average speed in km/h.
10
Change 3 hours 20 minutes to minutes.

Total 21 marks. Allow about 21 minutes.

Exercise 8X  Practice test questions

Non-calculator
1
FChange 4.5 kg to grams.[1 mark]
2
FSimplify 18 : 24.[1 mark]
3
FChange 5 m2 to cm2.[2 marks]
4
FShare £80 in the ratio 3 : 5.[2 marks]
5
FWork out the simple interest on £500 at 3% per year for 4 years.[2 marks]
6
FA price of £60 is decreased by 15%. Find the new price.[2 marks]
7
FA car travels 300 km in 4 hours. Find its average speed in km/h.[2 marks]
8
Fy is inversely proportional to x. y = 6 when x = 4. Find y when x = 3.[3 marks]
9
FA recipe for 4 people uses 250 g of pasta. How much pasta is needed for 6 people? Give your answer in grams.[3 marks]
10
RSam says that a 20% increase followed by a 20% decrease brings a price of £100 back to £100. Is Sam right? Show your working.[3 marks]
← Chapter 7: Sequences and graphsChapter 9: Angles, polygons and constructions →

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