Foundation ratio and percentage problems

(a) £60

1. $2 + 5 = 7$ parts; one part is $84 \div 7 = £12$. Ben gets $5 \times 12 = £60$.

• M1 Finding one part: $84 \div 7 = 12$.
• A1 The correct answer, £60.

(b) £46

1. $40 \times 1.15 = 46$.

• M1 Writing $40 \times 1.15$ (or finding $15\%$ of 40, which is 6).
• A1 The correct answer, £46.

(a) £195

1. £156 is $80\%$ of the original price.
2. $$156 \div 0.8 = 195$$

• M1 Writing $156 \div 0.8$ (or $156 \div 80 \times 100$).
• A1 The correct answer, £195.

(b) The 20% was taken off the original price, which is bigger than £156, so 20% of £156 is too small.

1. The $20\%$ was a percentage of the original price, not of the sale price. £156 is only $80\%$ of the original, so adding $20\%$ of £156 (£31.20) gives £187.20, which is too little.

• C1 Saying the $20\%$ is of the original (bigger) price, so £156 is $80\%$ and must be divided by $0.8$.

(a) £$140$

1. Undo the final fixed reduction first.
   $$113+6$$
2. Divide the discounted price by 0.85.
   $$119/0.85$$
3. Therefore £$140$.

• P1 Undo the final fixed reduction first.
• P1 Divide the discounted price by 0.85.
• A1 Correct answer: £$140$

(b) £$27$

1. Compare the original price with the actual payment.
   $$140-(113)$$
2. Therefore £$27$.

• M1 Compare the original price with the actual payment.
• A1 Correct answer: £$27$

(a) £$3.60$

1. Find the travel share.
   $$96\times 3/8$$
2. Find ten per cent of only that share.
   $$36\times 0.1$$
3. Therefore £$3.60$.

• P1 Find the travel share.
• P1 Find ten per cent of only that share.
• A1 Correct answer: £$3.60$

(b) $3.75$%

1. Divide the extra money by the original total.
   $$(18/5)/96\times 100$$
2. Therefore $3.75$%.

• P1 Divide the extra money by the original total.
• A1 Correct answer: $3.75$%

(a) $17$ km/h

1. Convert 45 minutes to three quarters of an hour.
   $$45/60$$
2. Divide distance by time in hours.
   $$(51/4)/(3/4)$$
3. Therefore $17$ km/h.

• P1 Convert 45 minutes to three quarters of an hour.
• P1 Divide distance by time in hours.
• A1 Correct answer: $17$ km/h

(b) $22.7$ km

1. Convert 80 minutes to hours before multiplying by speed.
   $$17\times 80/60$$
2. Therefore $22.7$ km.

• M1 Convert 80 minutes to hours before multiplying by speed.
• A1 Correct answer: $22.7$ km

(a) £$10.08$

1. Multiply by the multiplier for a 5% increase.
   $$9.6\times 1.05$$
2. Therefore £$10.08$.

• M1 Multiply by the multiplier for a 5% increase.
• A1 Correct answer: £$10.08$

(b) £$16.80$

1. Multiply the hourly rise by the hours worked.
   $$0.48\times 35$$
2. Therefore £$16.80$.

• P1 Multiply the hourly rise by the hours worked.
• A1 Correct answer: £$16.80$

(a) The 1.5 litre bottle (£1.26 per litre against £1.32)

1. Compare the price per litre.
   $$1.89 \div 1.5 = 1.26, \qquad 0.99 \div 0.75 = 1.32$$
2. The 1.5 litre bottle costs less per litre, so it is better value.

• P1 Converting to the same unit (750 ml $= 0.75$ litres, or $1.5$ litres $= 1500$ ml).
• P1 Finding both unit prices (or two bottles of 750 ml cost £1.98 against £1.89).
• C1 Choosing the 1.5 litre bottle, supported by both comparisons.

(b) £2.13

1. $$1.89 \times 1.04^3 = 2.1259\ldots \approx 2.13$$

• M1 Writing $1.89 \times 1.04^3$.
• A1 The correct answer, £2.13.

(a) $7.5$%

1. $$960\times 3/8$$
2. $$360\times 0.2$$
3. $$72/960\times 100$$
4. Transport receives 3/8 £$\times$960 = £360.
5. Its increase is 20% of £360 = £72.
6. The budget must rise by 72/960 $\times$ 100 = 7.5%.

• P1 Establishing $960\times 3/8$ or an equivalent valid method.
• P1 Establishing $360\times 0.2$ or an equivalent valid method.
• P1 Establishing $72/960\times 100$ or an equivalent valid method.
• A1 Correct answer: $7.5$%