Ratio, Proportion & Rates of Change

A PE teacher measures out 3.6 m of rope for a skipping activity. How many centimetres of rope is this?

• A0.036 cm
• B36 cm
• C3600 cm
• D360 cm✓

Answer: One metre is equal to 100 centimetres, so a length in metres is multiplied by 100 to convert to centimetres. Multiplying 3.6 by 100 gives 360 centimetres. Multiplying by only 10 instead of 100 gives 36 centimetres, multiplying by 1000 (the metres-to-millimetres factor) gives 3600 centimetres, and dividing by 100 instead of multiplying gives 0.036 centimetres.

A school lesson lasts 45 minutes. How many seconds does the lesson last?

• A13500 seconds
• B27000 seconds
• C270 seconds
• D2700 seconds✓

Answer: One minute is equal to 60 seconds, so a time in minutes is multiplied by 60 to convert to seconds. Multiplying 45 by 60 gives 2700 seconds. Multiplying by only 6 instead of 60 gives 270 seconds, multiplying by 600 instead of 60 gives 27000 seconds, and mistakenly using 300 (the number of seconds in 5 minutes) as the conversion factor gives 13500 seconds.

Jamal saves 850 pence in a jar each week towards a new bike. How much money is this in pounds?

• A£850
• B£85.00
• C£8.50✓
• D£0.085

Answer: One pound is equal to 100 pence, so an amount in pence is divided by 100 to convert to pounds. Dividing 850 by 100 gives £8.50. Dividing by 10 instead of 100 gives £85.00, simply attaching a pound sign without converting at all gives £850, and dividing by an extra factor of 10 too many gives £0.085.

A map is drawn to a scale of 1 : 25 000. Two villages are 6 cm apart on the map. What is the real distance between the villages?

• A1.5 km✓
• B0.15 km
• C2 km
• D15 km

Answer: A scale of 1 : 25 000 means every centimetre on the map represents 25 000 centimetres in real life. Multiplying 6 by 25 000 gives 150 000 cm, which converts to 1.5 km. Placing the decimal point one too many places to the left gives 0.15 km, placing it one too many places to the right gives 15 km, and rounding to the nearest whole kilometre gives 2 km.

A model aeroplane kit uses a scale of 1 : 48. The real aeroplane has a wingspan of 14.4 m. What is the wingspan of the model, in centimetres?

• A1440 cm
• B144 cm
• C480 cm
• D30 cm✓

Answer: Converting 14.4 m to centimetres gives 1440 cm, and dividing by the scale factor of 48 gives the model's wingspan. Dividing 1440 by 48 gives 30 cm. Using a metres-to-centimetres factor of 10 instead of 100 gives 144 cm, forgetting to divide by the scale factor at all gives 1440 cm, and dividing by an incorrect scale factor of 3 instead of 48 gives 480 cm.

A drama club has 18 boys and 12 girls. What fraction of the club members are girls?

• A2/5✓
• B3/5
• C2/3
• D1/2

Answer: The total number of club members is 18 plus 12, which is 30, so the fraction who are girls is 12 out of 30. Simplifying 12/30 by dividing top and bottom by 6 gives 2/5. Finding the fraction of boys instead of girls gives 3/5, rounding to the nearest simple fraction gives 1/2, and comparing girls to boys instead of to the whole club gives 2/3.

Priya spent £15 of her £60 birthday money on a video game. What fraction of her birthday money did she spend?

• A3/4
• B1/4✓
• C3/20
• D1/3

Answer: The fraction spent is found by writing the amount spent over the total amount, which is 15 out of 60. Simplifying 15/60 by dividing top and bottom by 15 gives 1/4. Dividing by 15 incorrectly on only one side gives 1/3, finding the fraction remaining instead of spent gives 3/4, and mistakenly using 100 instead of 60 as the total gives 3/20.

A fruit bowl contains 8 apples and 12 oranges. Write the ratio of apples to oranges in its simplest form.

• A2 : 3✓
• B3 : 2
• C2 : 15
• D3 : 5

Answer: Both parts of the ratio 8 : 12 share a common factor of 4, so dividing each part by 4 gives the simplest form. Dividing 8 by 4 gives 2 and dividing 12 by 4 gives 3, so the simplified ratio is 2 : 3. Writing the ratio the wrong way round gives 3 : 2, comparing apples to an incorrect total of 15 gives 2 : 15, and comparing oranges to the whole bowl instead of to apples gives 3 : 5.

A recipe uses 300 g of butter and 1.2 kg of flour. Write the ratio of butter to flour in its simplest form.

• A4 : 1
• B1 : 3
• C1 : 5
• D1 : 4✓

Answer: Before simplifying, both quantities must be in the same unit, so 1.2 kg is converted to 1200 g. The ratio 300 : 1200 simplifies to 1 : 4 by dividing both parts by 300. Dividing by an incorrect factor gives 1 : 3, writing the ratio the wrong way round gives 4 : 1, and comparing butter to the whole mixture instead of to the flour gives 1 : 5.

£40 is shared between Alex and Ben in the ratio 3 : 5. How much money does Ben receive?

• A£25✓
• B£24
• C£15
• D£20

Answer: The ratio 3 : 5 has 8 equal parts in total, so £40 is divided by 8 to find the value of one part, which is £5. Ben's share is 5 parts, so 5 multiplied by £5 gives £25. Finding Alex's share instead of Ben's gives £15, using an incorrect part value close to £4.80 gives £24, and sharing the money equally instead of by the ratio gives £20.

A youth club splits a £180 grant between sports, art and music equipment in the ratio 2 : 3 : 4. How much does the art budget receive?

• A£60✓
• B£40
• C£90
• D£80

Answer: The ratio 2 : 3 : 4 has 9 equal parts in total, so £180 is divided by 9 to find the value of one part, which is £20. Art receives 3 parts, so 3 multiplied by £20 gives £60. Finding the sports share instead gives £40, finding the music share instead gives £80, and splitting the grant into only two equal shares gives £90.

A metal block has a mass of 540 g and a volume of 60 cm³. What is the density of the metal?

• A10 g/cm³
• B6 g/cm³
• C90 g/cm³
• D9 g/cm³✓

Answer: Density is found by dividing mass by volume, so 540 is divided by 60. This division gives a density of 9 g/cm³. Using an incorrect volume of 90 instead of 60 gives 6 g/cm³, dividing by 54 instead of 60 gives 10 g/cm³, and misplacing a decimal point gives 90 g/cm³.

In a class, the ratio of left-handed to right-handed pupils is 1 : 6. If there are 4 left-handed pupils, how many right-handed pupils are there?

• A28
• B20
• C24✓
• D10

Answer: The ratio 1 : 6 means there are 6 times as many right-handed pupils as left-handed pupils, so the number of left-handed pupils is multiplied by 6. Multiplying 4 by 6 gives 24 right-handed pupils. Multiplying by 5 instead of 6 gives 20, multiplying by 7 instead of 6 gives 28, and adding the two ratio parts to 4 instead of multiplying gives 10.

A pancake recipe for 4 people uses 200 g of flour. Using the same ratio of flour to people, how much flour is needed for 10 people?

• A500 g✓
• B250 g
• C1000 g
• D800 g

Answer: The recipe uses 200 g of flour for every 4 people, which is 50 g of flour per person. Multiplying 50 g by 10 people gives 500 g of flour. Multiplying the original 200 g by 4 instead of scaling by people gives 800 g, halving 500 g by mistake gives 250 g, and using an incorrect rate of 100 g per person gives 1000 g.

A jacket priced at £40 has its price increased by 15%. What is the new price?

• A£55
• B£42
• C£46✓
• D£600

Answer: A 15% increase on £40 is found by calculating 15% of £40, which is £6, and adding it to the original price. Adding £6 to £40 gives £46. Adding £15 directly instead of 15% gives £55, multiplying £40 by 15 instead of by 1.15 gives £600, and using a 5% increase instead of 15% gives £42.

A games console originally cost £250. In a sale it is reduced by 20%. What is the sale price?

• A£230
• B£200✓
• C£50
• D£225

Answer: A 20% reduction on £250 is found by calculating 20% of £250, which is £50, and subtracting it from the original price. Subtracting £50 from £250 gives £200. Subtracting a flat £20 instead of 20% gives £230, using a 10% reduction instead of 20% gives £225, and giving the size of the discount itself instead of the new price gives £50.

After a 10% price increase, a football now costs £33. What was the original price before the increase?

• A£30✓
• B£330
• C£143
• D£363

Answer: A 10% increase means the new price is 110% of the original price, so the original price is found by dividing £33 by 1.10. Dividing 33 by 1.1 gives an original price of £30. Multiplying 33 by 10 instead of recognising the 10% figure gives £330, adding the percentage number 110 directly to £33 instead of calculating it gives £143, and using a multiplier of 11 instead of 1.1 (forgetting the decimal point) gives £363.

Freya puts £300 into a savings account that pays 4% simple interest per year. How much interest will she have earned after 2 years?

• A£120
• B£324
• C£12
• D£24✓

Answer: Simple interest is calculated on the original amount each year, so 4% of £300, which is £12, is earned every year. Over 2 years, £12 is earned twice, giving a total of £24. Calculating the interest for only 1 year gives £12, confusing 4% with 40% for a single year gives £120, and giving the new account balance instead of the interest earned gives £324.

6 identical pens cost £4.50 in total. How much would 10 of the same pens cost?

• A£45.00
• B£5.25
• C£7.50✓
• D£15.00

Answer: Dividing the total cost by the number of pens gives the cost of one pen, which is £4.50 divided by 6, or £0.75. Multiplying £0.75 by 10 gives a cost of £7.50 for 10 pens. Multiplying the total cost of 6 pens by 10 instead of finding the unit price first gives £45.00, using an incorrect per-pen rate of £1.50 instead of £0.75 gives £15.00, and using an incorrect unit price of £0.525 gives £5.25.

The cost of hiring a bike is directly proportional to the number of hours it is hired. Hiring for 3 hours costs £12. What would hiring for 7 hours cost?

• A£140
• B£21
• C£24
• D£28✓

Answer: Direct proportion means the cost per hour stays constant, so dividing £12 by 3 gives a rate of £4 per hour. Multiplying £4 by 7 gives a cost of £28 for 7 hours. Multiplying the two given hours together instead of using the rate gives £21, using 6 hours instead of 7 gives £24, and using an incorrect rate of £20 per hour gives £140.

It takes 4 workers 9 hours to paint a fence. Working at the same rate, how long would it take 6 workers to paint the same fence?

• A6 hours✓
• B54 hours
• C216 hours
• D36 hrs

Answer: The total amount of work is constant, so 4 workers multiplied by 9 hours gives 36 worker-hours of work. Dividing 36 by 6 workers gives 6 hours to complete the job. Giving the total worker-hours instead of dividing by the new number of workers gives 36 hours, multiplying the new number of workers by the original time instead of dividing gives 54 hours, and multiplying the total work by the number of workers again gives 216 hours.

A cyclist travels 45 km in 3 hours at a constant speed. What is her average speed?

• A1.5 km/h
• B15 km/h✓
• C150 km/h
• D135 km/h

Answer: Average speed is found by dividing distance by time, so 45 km is divided by 3 hours. This division gives an average speed of 15 km/h. Dividing by 0.3 instead of 3 gives 150 km/h, multiplying distance by time instead of dividing gives 135 km/h, and misplacing a decimal point the other way gives 1.5 km/h.