Ratio notation and multiplicative comparison

(a) $30$ cm

1. $$12\times 2.5$$
2. A multiplicative comparison means multiply by the stated factor.
3. 12 $\times$ 2.5 = 30 cm.

• P1 Establishing $12\times 2.5$ or an equivalent valid method.
• A1 Correct answer: $30$ cm

(a) $2 : 3$

1. $12 : 18$. Divide both by the highest common factor, 6: $2 : 3$.

• M1 Writing $12 : 18$ or a partly simplified ratio such as $6 : 9$.
• A1 The correct answer, $2 : 3$.

(b) $\frac{2}{5}$

1. There are $12 + 18 = 30$ students, so the fraction of boys is $\frac{12}{30} = \frac{2}{5}$.

• B1 $\frac{2}{5}$ (or $\frac{12}{30}$ or $0.4$).

(a) $7.5 : 1$

1. Use the same units: $4.5$ m $= 450$ cm, so the ratio is $450 : 60$.
2. Divide both by 60: $7.5 : 1$.

• M1 Writing both in the same units: $450 : 60$ or $4.5 : 0.6$.
• A1 The correct answer, $7.5 : 1$.

(b) $1 : 7$

1. $245 \div 35 = 7$, so $35 : 245 = 1 : 7$.

• B1 The correct answer, $1 : 7$.

(a) $\frac{3}{10}$

1. There are $3 + 7 = 10$ parts, and 3 of them are red: $\frac{3}{10}$.

• B1 The correct answer, $\frac{3}{10}$.

(b) $70$

1. The difference between blue and red is $7 - 3 = 4$ parts.
   $$4 \text{ parts} = 28 \Rightarrow 1 \text{ part} = 7$$
2. There are 10 parts in total.
   $$10 \times 7 = 70$$

• P1 Finding that the difference is 4 parts.
• P1 Finding one part: $28 \div 4 = 7$.
• A1 The correct answer, $70$.

(a) $55$ cm

1. The total has one pin part and the rod parts.
   $$66/(5+1)$$
2. Multiply one part by the rod multiplier.
   $$11\times 5$$
3. Therefore $55$ cm.

• P1 The total has one pin part and the rod parts.
• P1 Multiply one part by the rod multiplier.
• A1 Correct answer: $55$ cm

(b) $\frac{1}{5}$

1. Therefore $\frac{1}{5}$.

• B1 Correct answer: $\frac{1}{5}$

(a) $156$

1. Find the size of one ratio part.
   $$72/6$$
2. Multiply by the total number of parts.
   $$12\times (6+7)$$
3. Therefore $156$.

• P1 Find the size of one ratio part.
• P1 Multiply by the total number of parts.
• A1 Correct answer: $156$

(b) The denominator must be 13, the total parts. 6/7 compares children with adults.

1. The denominator must be 13, the total parts. 6/7 compares children with adults.

• C1 Correct conclusion with supporting reasoning: The denominator must be 13, the total parts. 6/7 compares children with adults.

(a) $\frac{3}{2}$

1. Convert both lengths to centimetres.
   $$1.8\times 100=180$$
2. Divide the first length by the reference length.
   $$180/120$$
3. Therefore $\frac{3}{2}$.

• M1 Convert both lengths to centimetres.
• M1 Divide the first length by the reference length.
• A1 Correct answer: $\frac{3}{2}$

(b) $\frac{2}{3}$

1. Use the reciprocal ratio for the reverse comparison.
   $$120/180$$
2. Therefore $\frac{2}{3}$.

• M1 Use the reciprocal ratio for the reverse comparison.
• A1 Correct answer: $\frac{2}{3}$

(a) $8 : 15$

1. Make the $b$ parts equal. The lowest common multiple of 3 and 4 is 12.
   $$a : b = 8 : 12, \qquad b : c = 12 : 15$$
2. So $a : b : c = 8 : 12 : 15$ and $a : c = 8 : 15$.

• P1 Scaling both ratios so the $b$ parts match, such as $8 : 12$ and $12 : 15$.
• A1 The correct answer, $8 : 15$.

(b) $75$

1. $8 + 12 + 15 = 35$ parts, so one part is $175 \div 35 = 5$.
2. $c = 15 \times 5 = 75$.

• P1 Finding 35 parts and one part $= 5$.
• A1 The correct answer, $75$.