Fractions within ratio problems

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

1. $$18/45$$
2. Use the same units, then form 18/45.
3. Divide the numerator and denominator by 9: 2/5.

• M1 Establishing $18/45$ or an equivalent valid method.
• A1 Correct answer: $\frac{2}{5}$

(a) $\frac{3}{8}$

1. $$3+5$$
2. The whole bag has 3 + 5 = 8 equal ratio parts.
3. Green counters make up 3 of these 8 parts, so the fraction is 3/8.

• P1 Establishing $3+5$ or an equivalent valid method.
• A1 Correct answer: $\frac{3}{8}$

(a) $\frac{2}{7}$

1. There are $5 + 2 = 7$ parts; sugar is 2 of them: $\frac{2}{7}$.

• B1 The correct answer, $\frac{2}{7}$.

(b) $250$ g

1. One part is $350 \div 7 = 50$ g.
2. Flour is $5 \times 50 = 250$ g.

• M1 Finding one part: $350 \div 7 = 50$.
• A1 The correct answer, $250$ g.

(a) $\frac{5}{32}$

1. $$5/8\times 1/4$$
2. Blue counters form 5/(5 + 3) = 5/8 of the total.
3. A quarter of 5/8 is 5/32.

• P1 Establishing $5/8\times 1/4$ or an equivalent valid method.
• A1 Correct answer: $\frac{5}{32}$

(a) $\frac{1}{4}$

1. Use the total budget as the denominator and simplify.
   $$48/192$$
2. Therefore $\frac{1}{4}$.

• M1 Use the total budget as the denominator and simplify.
• A1 Correct answer: $\frac{1}{4}$

(b) $75$%

1. Subtract the spending, divide by the original total and convert to a percentage.
   $$(192-48)/192\times 100$$
2. Therefore $75$%.

• M1 Subtract the spending, divide by the original total and convert to a percentage.
• A1 Correct answer: $75$%

(a) $\frac{7}{19}$

1. The reference group is the 19 walkers.
   $$7/19$$
2. Therefore $\frac{7}{19}$.

• M1 The reference group is the 19 walkers.
• A1 Correct answer: $\frac{7}{19}$

(b) $\frac{3}{19}$

1. Subtract bridge users from walkers, then divide by all students.
   $$\frac{19-7}{76}$$
2. Therefore $\frac{3}{19}$.

• M1 Subtract bridge users from walkers, then divide by all students.
• A1 Correct answer: $\frac{3}{19}$

(a) $3 : 5$

1. Ben gets $1 - \frac{3}{8} = \frac{5}{8}$. So Ali : Ben $= 3 : 5$.

• B1 The correct answer, $3 : 5$.

(b) £168

1. The difference is $5 - 3 = 2$ parts out of 8.
   $$2 \text{ parts} = 42 \Rightarrow 1 \text{ part} = 21$$
2. $$8 \times 21 = 168$$

• P1 Recognising that the £42 difference is 2 parts (or $\frac{2}{8}$ of the total).
• P1 Finding one part £$=$21 (or $\frac{1}{8}$ of the total £$=$21).
• A1 The correct answer, £168.

(a) 1/5

1. Write the original amounts as 5k and 3k. After moving x beads, 5k $-$ x = 3k + x.
2. So x = k, which is k/(5k) = 1/5 of the original amount in A.

• B1 Correct answer: 1/5