Decimals, fractions and percentages

(a) $56$

1. $$64/8$$
2. To find a fraction of an amount, divide by the denominator and multiply by the numerator.
3. 64 $\div$ 8 $\times$ 7 = 56.

• M1 Establishing $64/8$ or an equivalent valid method.
• A1 Correct answer: $56$

(a) $4\%$

1. Multiply by 100: $0.04 \times 100 = 4$, so $0.04 = 4\%$.

• B1 The correct answer, $4\%$.

(b) $\frac{7}{20}$

1. $35\% = \frac{35}{100}$. Divide top and bottom by 5: $\frac{7}{20}$.

• M1 Writing $\frac{35}{100}$.
• A1 The correct answer, $\frac{7}{20}$.

(c) $12$

1. $10\%$ of 80 is 8 and $5\%$ is 4, so $15\%$ is $8 + 4 = 12$.

• M1 A correct build-up such as $10\% = 8$ and $5\% = 4$, or $0.15 \times 80$.
• A1 The correct answer, $12$.

(a) $87.5\%$

1. $7 \div 8 = 0.875$, and $0.875 \times 100 = 87.5$, so $\frac{7}{8} = 87.5\%$.

• M1 Working out $7 \div 8 = 0.875$.
• A1 The correct answer, $87.5\%$.

(b) $\frac{3}{11}$, because $\frac{3}{11} = 27.27\ldots\%$

1. $3 \div 11 = 0.2727\ldots = 27.27\ldots\%$, which is greater than $27\%$.

• M1 Converting to the same form: $\frac{3}{11} = 0.2727\ldots$ or $27.27\ldots\%$, or $27\% = 0.27$.
• C1 Choosing $\frac{3}{11}$ with the comparison of $27.27\%$ (or $0.2727$) and $27\%$.

(a) Shop B (£42)

1. Shop A: $\frac{1}{4}$ of £60 is £15, so the price is $60 - 15 = £45$.
2. Shop B: $10\%$ of £60 is £6, so $30\%$ is £18 and the price is $60 - 18 = £42$.
3. Shop C: $60 - 16 = £44$.
4. The cheapest is Shop B at £42.

• P1 Finding one sale price or discount correctly, such as £15 off (Shop A) or £45.
• P1 Finding the second shop's price or discount, such as £18 off (Shop B) or £42.
• A1 All three prices: £45, £42 and £44 (or all three discounts: £15, £18, £16).
• C1 Shop B, supported by the three correct prices or discounts.

(a) $0.4375$

1. $$7/16$$
2. An equivalent fraction with denominator 10000 is 4375/10000 because 16 $\times$ 625 = 10000.
3. Therefore 7/16 = 0.4375.

• M1 Establishing $7/16$ or an equivalent valid method.
• A1 Correct answer: $0.4375$

(a) $37.5$%

1. Multiply the fraction by 100.
   $$(3/8)\times 100$$
2. Therefore $37.5$%.

• M1 Multiply the fraction by 100.
• A1 Correct answer: $37.5$%

(b) $225$

1. Use the fraction as an operator on the full response count.
   $$600\times (3/8)$$
2. Therefore $225$.

• P1 Use the fraction as an operator on the full response count.
• A1 Correct answer: $225$

(a) $37.5$%

1. Multiply the fraction by 100.
   $$3/8\times 100$$
2. Therefore $37.5$%.

• M1 Multiply the fraction by 100.
• A1 Correct answer: $37.5$%

(b) 2/5 = 40%, which is larger than 38%, 37.5% and 37%.

1. Convert 2/5 to a percentage.
   $$2/5\times 100$$
2. Therefore 2/5 = 40%, which is larger than 38%, 37.5% and 37%.

• M1 Convert 2/5 to a percentage.
• C1 Correct conclusion with supporting reasoning: 2/5 = 40%, which is larger than 38%, 37.5% and 37%.

(a) $400$

1. Write both fractions as decimals and add them.
   $$\frac{3}{8} = 0.375, \qquad 0.375 + 0.45 = 0.825$$
2. The cyclists are the rest.
   $$1 - 0.825 = 0.175$$
3. $$0.175 \text{ of the students} = 70 \Rightarrow \text{students} = 70 \div 0.175 = 400$$

• P1 Converting to one form and adding: $0.375 + 0.45 = 0.825$ (or $82.5\%$, or $\frac{33}{40}$).
• P1 Finding the cyclists' share, $0.175$ (or $17.5\%$ or $\frac{7}{40}$).
• A1 The correct answer, $400$.

(b) $18\%$

1. This is $\frac{2}{5}$ of $45\%$.
2. $$0.4 \times 45\% = 18\%$$

• M1 Writing $\frac{2}{5} \times 45\%$ or $0.4 \times 0.45$, or $\frac{72}{400}$.
• A1 The correct answer, $18\%$.