Percentages, changes and multipliers

(a) $25\%$

1. $\frac{18}{72} = \frac{1}{4} = 25\%$.

• M1 Writing $\frac{18}{72}$.
• A1 The correct answer, $25\%$.

(b) $84$

1. $10\%$ of 240 is 24, so $30\%$ is 72; $5\%$ is 12.
2. $35\% = 72 + 12 = 84$.

• M1 A correct build-up, such as $30\% = 72$ and $5\% = 12$.
• A1 The correct answer, $84$.

(a) 63

1. An increase of 125% leaves 225% of the original.
2. 28 $\times$ 2.25 = 63 bookings.

• B1 Correct answer: 63

(a) £50.40

1. An increase of $12\%$ means multiplying by $1.12$.
2. $$45 \times 1.12 = 50.40$$

• M1 Writing $45 \times 1.12$, or finding $12\%$ of 45 ($5.40$) and adding it.
• A1 The correct answer, £50.40.

(b) $15\%$

1. The decrease is $80 - 68 = £12$.
2. Divide by the original price.
   $$\frac{12}{80} \times 100 = 15\%$$

• M1 Writing $\frac{12}{80}$ (or $\frac{68}{80} = 0.85$).
• A1 The correct answer, $15\%$.

(a) $90$

1. $$0.375\times 240$$
2. A percentage acts as a fraction out of 100: 37.5% = 0.375.
3. 0.375 $\times$ 240 = 90.

• M1 Establishing $0.375\times 240$ or an equivalent valid method.
• A1 Correct answer: $90$

(a) $15$%

1. $$(2.4-2.04)/2.4\times 100$$
2. Decrease = 2.4 $-$ 2.04 = 0.36 kg.
3. Percentage decrease = (0.36 $\div$ 2.4) $\times$ 100 = 15%.

• P1 Establishing $(2.4-2.04)/2.4\times 100$ or an equivalent valid method.
• A1 Correct answer: $15$%

(a) £$186.30$

1. Apply the increase multiplier first.
   $$180\times 1.15$$
2. Apply the discount to the increased amount.
   $$207\times 0.9$$
3. Therefore £$186.30$.

• P1 Apply the increase multiplier first.
• P1 Apply the discount to the increased amount.
• A1 Correct answer: £$186.30$

(b) $3.5$%

1. Compare the net increase with the original price.
   $$(186.3-180)/180\times 100$$
2. Therefore $3.5$%.

• M1 Compare the net increase with the original price.
• A1 Correct answer: $3.5$%

(a) $1.2 \times 0.8 = 0.96$, which is a $4\%$ decrease.

1. Multiply the multipliers.
   $$1.2 \times 0.8 = 0.96$$
2. $0.96$ is $96\%$ of the original, so the overall change is a decrease of $4\%$ (a change of $-4\%$).

• P1 Writing $1.2 \times 0.8$ (or using a starting value, such as $100 \to 120 \to 96$).
• A1 Reaching $0.96$ and stating that this is a $4\%$ decrease.

(b) $4.17\%$

1. The multiplier $m$ must take $0.96$ back to $1$.
   $$0.96 \times m = 1 \Rightarrow m = \frac{1}{0.96} = 1.041\overline{6}$$
2. So the increase is $4.1\overline{6}\% \approx 4.17\%$.

• P1 Writing $\frac{1}{0.96}$ (or $\frac{100}{96}$).
• A1 The correct answer, $4.17\%$.

(a) $28$%

1. $$0.8\times 0.9$$
2. $$100\times (1-0.72)$$
3. The combined value multiplier is 0.8 $\times$ 0.9 = 0.72.
4. 72% remains, so the overall decrease is 28%.

• P1 Establishing $0.8\times 0.9$ or an equivalent valid method.
• P1 Establishing $100\times (1-0.72)$ or an equivalent valid method.
• A1 Correct answer: $28$%