GCSE Maths: Rates of pay, unit prices and best buys (Higher)
Part of the GCSE Maths units and compound measures chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Rates of pay, unit prices and best buys
A rate compares two different quantities by saying how much of one there is for one of the other. £9.50 per hour, 45p per litre, 12 litres per minute and 3000 people per km² are all rates. The word “per” means “for each” (or “divided by”). Rates are compound measures, like speed and density, and you use them with multiplication and division.
Rates of pay
Pay = rate of pay (per hour) × hours worked. Overtime is extra hours beyond the normal hours and is often paid at a higher rate: “time and a half” means 1.5 times the normal rate and “double time” means 2 times. Work out the normal pay and the overtime pay separately, then add them.
Unit pricing and best buys
To find the best value, compare the price of the same amount each time. The unit price is the cost of one unit (one gram, 100 g, one litre, one egg). Divide the price by the quantity; if the answer is a tiny number (0.4p per gram) scale up to a friendlier amount such as 100 g. The smallest unit price is the best buy. You can also compare how much you get for the same money: more grams per pound is better.
The unitary method
If you know the cost of 5 pens you can find the cost of 8: first work out the cost of one (divide) then multiply. This is the same unitary method from Chapter 15. It works whenever two quantities are in a fixed rate, such as price and number of items.
Changing the units of a rate
You can change the units of a rate in the same way as for speed. For example £2.40 per kg is 240p per 1000 g, which is 24p per 100 g. Work in two stages: change the money unit, then the quantity unit. Watch out when the quantity is on the bottom: if you change 1 kg into 1000 g, the price for each gram becomes 1000 times smaller.
Key ideas
- Pay = rate × hours. Overtime: work out the normal and overtime pay separately and add.
- Unit price = price ÷ quantity. Lowest unit price is the best buy.
- Compare the same quantity (per 100 g, per litre, per item) and use the same units.
- Unitary method: find the value for 1, then multiply.
Key words. A rate compares two different quantities, such as pounds per hour. Overtime is extra hours worked, often paid at a higher rate.
Method. Total pay = rate of pay per hour × number of hours (work out overtime separately).
- Pay for the normal hours. Rate of pay means pay per hour. Multiply the rate by the hours.£9.20 × 35 = £322
- Find the overtime rate. Overtime is paid at 1.5 times the normal rate (this is "time and a half").£9.20 × 1.5 = £13.80 per hour
- Pay for the overtime hours. £13.80 × 6 = £82.80
- Add the two amounts. £322 + £82.80 = £404.80
Check. Round numbers: about £9 × 41 = £369, close to £404.80. ✓
Watch out. Overtime hours are paid at the higher rate only for the extra hours, not for all hours.
Key words. Unit price is the cost of one unit (1 g, 100 g, 1 kg, 1 litre). A best buy is the option with the lowest unit price.
Method. Best value means the lowest cost for the same amount. Work out the cost of 100 g for each, then compare.
- Find the cost per 100 g for Pack A. To compare fairly we need the same amount each time. Divide the price by the quantity, then scale to 100 g.189p ÷ 450 × 100 = 42p per 100 g
- Find the cost per 100 g for Pack B. To compare fairly we need the same amount each time. Divide the price by the quantity, then scale to 100 g.260p ÷ 750 × 100 = 34.6667p per 100 g ≈ 34.67p
- Compare the unit costs. The smallest cost per unit is the best value.Pack A: 42p, Pack B: 34.67p
The lowest is Pack B.
Check. Another way: work out how much you get for 1p (or £1) each time; the biggest amount is the best buy. The same item wins. ✓
Watch out. The biggest pack is not always the cheapest per unit, and the lowest price is not always the best value.
Method. Unitary method: first find the value for ONE, then multiply to get the amount you need.
- Write down the rate you know. The same rate applies to every amount, so the amount for one can be found from any known amount.5 pens goes with £3.50
- Find the cost for one pen. Dividing by 5 gives the amount for 1.£3.50 ÷ 5 = £0.70
- Multiply by 8. We want 8 lots of one pen.£0.70 × 8 = £5.60
Check. Compare with the original: 8 pens is 1.6 times 5 pens, so the cost is 1.6 times £3.50 = £5.60. ✓
Watch out. Divide by the quantity you were given first, then multiply by the quantity you want.
Method. Find how many litres are used, then multiply by the cost per litre.
- Find how many lots of 100 km there are. The fuel use is given per 100 km.260 ÷ 100 = 2.6
- Find the litres used. Each 100 km uses 6.5 litres.2.6 × 6.5 = 16.9 litres
- Find the cost. Cost = litres × price per litre.16.9 × 1.50 = £25.35
Check. Rough check: 260 km is about 17 litres; 17 × £1.50 = £25.50. ✓
Watch out. Check which quantity the rate is “per”. Here it is per 100 km, not per km.
Method. Change the money to pence and the mass to 100 g units.
- Change pounds to pence. £1 = 100p.£2.40 = 240p, so 240p per kg
- Change 1 kg to 100 g units. 1 kg = 1000 g = 10 lots of 100 g. The same 240p now buys 10 lots, so each 100 g costs 10 times less.240 ÷ 10 = 24p per 100 g
- Write the answer. Give the new units.24p per 100 g
Check. 10 lots of 100 g at 24p each is 240p = £2.40 for 1 kg. ✓
Watch out. When the quantity on the bottom gets bigger, the unit price gets smaller. Divide, do not multiply.
Watch out
The biggest pack is not always the cheapest per unit, and the cheapest pack is not always the best value for you. Always compare the price for the same quantity. Make sure both quantities are in the same units (grams with grams) before you divide.Exercise 16H Rates of pay, unit prices and best buys
Calculator allowedKey words. A rate compares two different quantities, such as pounds per hour. Overtime is extra hours worked, often paid at a higher rate.
Method. Total pay = rate of pay per hour × number of hours (work out overtime separately).
- Multiply the rate by the hours worked. A rate of pay is the money earned in 1 hour. 1 hour earns the rate, so 2 hours earns 2 times as much, and so on.£8.40 × 25 = £210
- Write the answer. Pay = £210
Check. Round numbers: about £8 × 25 = £200, close to £210. ✓
Watch out. Overtime hours are paid at the higher rate only for the extra hours, not for all hours.
Key words. A rate compares two different quantities, such as pounds per hour. Overtime is extra hours worked, often paid at a higher rate.
Method. Total pay = rate of pay per hour × number of hours (work out overtime separately).
- Multiply the rate by the hours worked. A rate of pay is the money earned in 1 hour. 1 hour earns the rate, so 2 hours earns 2 times as much, and so on.£11.50 × 36 = £414
- Write the answer. Pay = £414
Check. Round numbers: about £12 × 36 = £432, close to £414. ✓
Watch out. Overtime hours are paid at the higher rate only for the extra hours, not for all hours.
Key words. A rate compares two different quantities, such as pounds per hour. Overtime is extra hours worked, often paid at a higher rate.
Method. Total pay = rate of pay per hour × number of hours (work out overtime separately).
- Multiply the rate by the hours worked. A rate of pay is the money earned in 1 hour. 1 hour earns the rate, so 2 hours earns 2 times as much, and so on.£10.25 × 40 = £410
- Write the answer. Pay = £410
Check. Round numbers: about £10 × 40 = £400, close to £410. ✓
Watch out. Overtime hours are paid at the higher rate only for the extra hours, not for all hours.
Key words. A rate compares two different quantities, such as pounds per hour. Overtime is extra hours worked, often paid at a higher rate.
Method. Total pay = rate of pay per hour × number of hours (work out overtime separately).
- Pay for the normal hours. Rate of pay means pay per hour. Multiply the rate by the hours.£9 × 35 = £315
- Find the overtime rate. Overtime is paid at 1.5 times the normal rate (this is "time and a half").£9 × 1.5 = £13.50 per hour
- Pay for the overtime hours. £13.50 × 5 = £67.50
- Add the two amounts. £315 + £67.50 = £382.50
Check. Round numbers: about £9 × 40 = £360, close to £382.50. ✓
Watch out. Overtime hours are paid at the higher rate only for the extra hours, not for all hours.
Method. Unitary method: first find the value for ONE, then multiply to get the amount you need.
- Write down the rate you know. The same rate applies to every amount, so the amount for one can be found from any known amount.6 apples goes with £2.70
- Find the cost for one apple. Dividing by 6 gives the amount for 1.£2.70 ÷ 6 = £0.45
- Multiply by 10. We want 10 lots of one apple.£0.45 × 10 = £4.50
Check. Compare with the original: 10 apples is 1.667 times 6 apples, so the cost is 1.667 times £2.70 = £4.50. ✓
Watch out. Divide by the quantity you were given first, then multiply by the quantity you want.
Method. Unitary method: first find the value for ONE, then multiply to get the amount you need.
- Write down the rate you know. The same rate applies to every amount, so the amount for one can be found from any known amount.8 pens goes with £5.20
- Find the cost for one pen. Dividing by 8 gives the amount for 1.£5.20 ÷ 8 = £0.65
- Multiply by 5. We want 5 lots of one pen.£0.65 × 5 = £3.25
Check. Compare with the original: 5 pens is 0.625 times 8 pens, so the cost is 0.625 times £5.20 = £3.25. ✓
Watch out. Divide by the quantity you were given first, then multiply by the quantity you want.
Method. Unitary method: first find the value for ONE, then multiply to get the amount you need.
- Write down the rate you know. The same rate applies to every amount, so the amount for one can be found from any known amount.12 eggs goes with £3
- Find the cost for one egg. Dividing by 12 gives the amount for 1.£3 ÷ 12 = £0.25
- Multiply by 30. We want 30 lots of one egg.£0.25 × 30 = £7.50
Check. Compare with the original: 30 eggs is 2.5 times 12 eggs, so the cost is 2.5 times £3 = £7.50. ✓
Watch out. Divide by the quantity you were given first, then multiply by the quantity you want.
Method. Unitary method: first find the value for ONE, then multiply to get the amount you need.
- Write down the rate you know. The same rate applies to every amount, so the amount for one can be found from any known amount.15 litres goes with £22.50
- Find the cost for one litre. Dividing by 15 gives the amount for 1.£22.50 ÷ 15 = £1.50
- Multiply by 40. We want 40 lots of one litre.£1.50 × 40 = £60
Check. Compare with the original: 40 litres is 2.667 times 15 litres, so the cost is 2.667 times £22.50 = £60. ✓
Watch out. Divide by the quantity you were given first, then multiply by the quantity you want.
Key words. Unit price is the cost of one unit (1 g, 100 g, 1 kg, 1 litre). A best buy is the option with the lowest unit price.
Method. Best value means the lowest cost for the same amount. Work out the cost of 100 g for each, then compare.
- Find the cost per 100 g for A. To compare fairly we need the same amount each time. Divide the price by the quantity, then scale to 100 g.150p ÷ 375 × 100 = 40p per 100 g
- Find the cost per 100 g for B. To compare fairly we need the same amount each time. Divide the price by the quantity, then scale to 100 g.285p ÷ 750 × 100 = 38p per 100 g
- Compare the unit costs. The smallest cost per unit is the best value.A: 40p, B: 38p
The lowest is B.
Check. Another way: work out how much you get for 1p (or £1) each time; the biggest amount is the best buy. The same item wins. ✓
Watch out. The biggest pack is not always the cheapest per unit, and the lowest price is not always the best value.
Key words. Unit price is the cost of one unit (1 g, 100 g, 1 kg, 1 litre). A best buy is the option with the lowest unit price.
Method. Best value means the lowest cost for the same amount. Work out the cost of 100 ml for each, then compare.
- Find the cost per 100 ml for A. To compare fairly we need the same amount each time. Divide the price by the quantity, then scale to 100 ml.180p ÷ 250 × 100 = 72p per 100 ml
- Find the cost per 100 ml for B. To compare fairly we need the same amount each time. Divide the price by the quantity, then scale to 100 ml.260p ÷ 400 × 100 = 65p per 100 ml
- Find the cost per 100 ml for C. To compare fairly we need the same amount each time. Divide the price by the quantity, then scale to 100 ml.350p ÷ 500 × 100 = 70p per 100 ml
- Compare the unit costs. The smallest cost per unit is the best value.A: 72p, B: 65p, C: 70p
The lowest is B.
Check. Another way: work out how much you get for 1p (or £1) each time; the biggest amount is the best buy. The same item wins. ✓
Watch out. The biggest pack is not always the cheapest per unit, and the lowest price is not always the best value.
Key words. Unit price is the cost of one unit (1 g, 100 g, 1 kg, 1 litre). A best buy is the option with the lowest unit price.
Method. Best value means the lowest cost for the same amount. Work out the cost of 1 egg for each, then compare.
- Find the cost per 1 egg for the box of 6. To compare fairly we need the same amount each time. Divide the price by the quantity.174p ÷ 6 = 29p per egg
- Find the cost per 1 egg for the box of 12. To compare fairly we need the same amount each time. Divide the price by the quantity.324p ÷ 12 = 27p per egg
- Find the cost per 1 egg for the box of 15. To compare fairly we need the same amount each time. Divide the price by the quantity.420p ÷ 15 = 28p per egg
- Compare the unit costs. The smallest cost per unit is the best value.the box of 6: 29p, the box of 12: 27p, the box of 15: 28p
The lowest is the box of 12.
Check. Another way: work out how much you get for 1p (or £1) each time; the biggest amount is the best buy. The same item wins. ✓
Watch out. The biggest pack is not always the cheapest per unit, and the lowest price is not always the best value.
Method. Find the litres used, then multiply by the cost per litre.
- Litres used. The rate is per 100 km, and 260 km is 2.6 lots of 100 km.2.6 × 6.5 = 16.9 litres
- Cost. Cost = litres × price per litre.16.9 × 1.50 = £25.35
Check. About 17 litres at about £1.50 is about £25. ✓
Method. Divide the amount by the rate to get the time.
- Divide. Time = amount ÷ rate.90 ÷ 12 = 7.5 minutes
- Change 0.5 minutes to seconds. 1 minute = 60 seconds.0.5 × 60 = 30 seconds
- Write the answer. 7 minutes 30 seconds
Check. 12 litres per minute for 7.5 minutes is 12 × 7.5 = 90 litres. ✓
Method. Change pounds to pence, then 1 kg to lots of 100 g.
- Change to pence. £1 = 100p.£3.60 = 360p per kg
- Change the quantity. 1 kg = 10 lots of 100 g, so the price for 100 g is 10 times smaller.360 ÷ 10 = 36p per 100 g
Check. 36p × 10 = 360p = £3.60. ✓
Method. Find the number of hours worked in a year, then divide the pay by the hours.
- Hours worked in a year. 36 hours a week for 52 weeks.36 × 52 = 1872 hours
- Pay per hour. Rate = pay ÷ hours.18 720 ÷ 1872 = 10
- Write the answer. £10 per hour
Check. 1872 × 10 = 18 720. ✓
Method. Make up a small pack and a big pack where the big pack costs more per gram, and show the unit prices.
- Small pack. Price ÷ quantity, scaled to 100 g.120p ÷ 500 × 100 = 24.00p per 100 g
- Big pack. The big pack is 1 kg = 1000 g.260p ÷ 1000 × 100 = 26p per 100 g
- Compare. 24p is less than 26p.The smaller pack is cheaper per 100 g, so Tom is wrong.
Check. Per kg: 500 g costs £1.20 so 1 kg would cost £2.40, less than £2.60. ✓
Method. Population density = number of people ÷ area.
- Divide. This is a rate: people per km².54 000 ÷ 18 = 3000
- Write the answer with units. 3000 people per km²
Check. 3000 × 18 = 54 000. ✓
Method. Multiply the density by the area.
- Use density × area. The rate is per km², so multiply by the number of km².3000 × 45 = 135000
- Write the answer. Give the number of people.135000 people
Check. 135 000 ÷ 45 = 3000. ✓
Method. Find the area to paint, then divide by the area one litre covers, and round up.
- Area of the wall. Area = length × height.5.5 × 2.4 = 13.2 m²
- Area for two coats. Two coats means painting the area twice.13.2 × 2 = 26.4 m²
- Litres needed. Divide by the coverage per litre.26.4 ÷ 12 = 2.2 litres
- Round up to whole tins. You cannot buy part of a tin, and 2.2 litres is not enough with only 2 tins.3 tins
Check. 3 tins cover 3 × 12 = 36 m², which is more than 26.4 m²; 2 tins cover only 24 m². ✓
Watch out. Round up, not to the nearest whole number, when you must have enough.
Method. Find the number of kilowatt-hours, then multiply by the price per kWh.
- Energy used in kWh. kWh = kW × hours.2.5 × 3 = 7.5 kWh
- Cost in pence. Cost = kWh × price per kWh.7.5 × 28 = 210p
- Change to pounds. 100p = £1.210 ÷ 100 = £2.10
Check. 7.5 kWh is about 8; 8 × 28p = 224p. ✓
Method. Find how much juice you really get from Brand X, then compare the cost per 100 ml.
- Find the extra amount. 20% of 500 ml is 500 ÷ 5.20% of 500 = 100 ml
- Find the total amount in Brand X. Add the extra to the normal amount.500 + 100 = 600 ml
- Cost per 100 ml of X. Price ÷ quantity, scaled to 100 ml.110 ÷ 600 × 100 = 18.33p
- Cost per 100 ml of Y. Same method.90 ÷ 450 × 100 = 20p
- Compare. 18.33p is less than 20p.Brand X is better value.
Check. 600 ml of Y would cost 6 × 20p = 120p, more than the 110p for 600 ml of X. ✓
Method. The first hour has a different price, so deal with it separately.
- Cost of the first hour. This is charged at £48.£48
- Remaining time. Take the first hour off the total time.4.5 − 1 = 3.5 hours
- Cost of the remaining time. Rate × time.31 × 3.5 = £108.50
- Total. Add the two costs.48 + 108.5 = £156.50
Check. Rough check: 3.5 hours at about £30 is £105, plus £48 is about £153. ✓
Watch out. Because of the higher first-hour charge, cost is not directly proportional to time.
Method. Change pounds to pence, then divide by the number of pencils.
- Change to pence. £1 = 100p.£p = 100p pence
- Divide by the number of pencils. Unit price = price ÷ quantity.100pn pence
Check. n = 10, p = 2: 100 × 2 ÷ 10 = 20p each. ✓
Method. Work out the cost per pencil for the second pack and compare with part (a).
- Cost in pence. £2.4p = 240p pence.240p pence for 3n pencils
- Cost of one pencil. Divide by 3n.240p3n = 80pn pence
- Compare. Same bottom, so compare the tops: 80p < 100p.The second pack costs less per pencil, so it is better value.
Check. n = 10, p = 2: 20p each compared with 16p each. ✓
Method. Take off the normal pay, then divide the rest by the overtime rate.
- Pay for the first 35 hours. Normal rate × normal hours.12 × 35 = 420
- Find the overtime rate. Time and a half is 1.5 times the rate.12 × 1.5 = 18
- Pay left for overtime. Total minus normal pay.528 − 420 = 108
- Overtime hours. Divide by the overtime rate.108 ÷ 18 = 6 hours
- Total hours. Add the normal and overtime hours.35 + 6 = 41 hours
Check. 35 hours × £12 = £420 and 6 hours × £18 = £108; 420 + 108 = 528. ✓