Fraction arithmetic and exact answers

(a) $24$

1. $\frac{1}{8}$ of $64$ is $8$, so $\frac{3}{8}$ is $3 \times 8 = 24$.

• M1 Finding $64 \div 8 = 8$.
• A1 The correct answer, $24$.

(b) $10\pi\ \text{cm}^2$

1. $\frac{2}{5} \times 25\pi = \frac{50\pi}{5} = 10\pi$.

• M1 Writing $\frac{2}{5} \times 25\pi$.
• A1 The correct answer, $10\pi$.

(a) $\frac{11}{15}$

1. Use the common denominator 15.
   $$\frac{2}{5} + \frac{1}{3} = \frac{6}{15} + \frac{5}{15} = \frac{11}{15}$$

• M1 Writing both fractions over a common denominator, such as $\frac{6}{15} + \frac{5}{15}$.
• A1 $\frac{11}{15}$.

(b) $\frac{1}{6}$

1. Multiply tops and bottoms, then simplify.
   $$\frac{3 \times 2}{4 \times 9} = \frac{6}{36} = \frac{1}{6}$$

• M1 Writing $\frac{6}{36}$ or cancelling before multiplying.
• A1 The correct answer, $\frac{1}{6}$.

(c) $\frac{3}{4}$

1. Change both to quarters.
   $$2\tfrac{1}{2} = \frac{10}{4}, \quad 1\tfrac{3}{4} = \frac{7}{4}$$
2. $$\frac{10}{4} - \frac{7}{4} = \frac{3}{4}$$

• M1 Writing both as improper fractions over 4, $\frac{10}{4} - \frac{7}{4}$.
• A1 The correct answer, $\frac{3}{4}$.

(a) $\frac{11}{24}$

1. $$20/24-9/24$$
2. Use the common denominator 24: 5/6 = 20/24 and 3/8 = 9/24.
3. Subtract the numerators: 20/24 $-$ 9/24 = 11/24.

• M1 Establishing $20/24-9/24$ or an equivalent valid method.
• A1 Correct answer: $\frac{11}{24}$

(a) $60$

1. $$150\times 3/5$$
2. $$90\times 2/3$$
3. The morning sale is 2/5 $\times$ 150 = 60, leaving 90.
4. The afternoon fraction acts on the remainder: 1/3 $\times$ 90 = 30.
5. There are 90 $-$ 30 = 60 postcards left.

• P1 Establishing $150\times 3/5$ or an equivalent valid method.
• P1 Establishing $90\times 2/3$ or an equivalent valid method.
• A1 Correct answer: $60$

(a) $18$

1. Find the quantity left after the first sale.
   $$72\times 3/4$$
2. One third of that remainder survives the second sale.
   $$54/3$$
3. Therefore $18$.

• P1 Find the quantity left after the first sale.
• P1 One third of that remainder survives the second sale.
• A1 Correct answer: $18$

(b) $\frac{3}{4}$

1. Compare the total sold with the original quantity.
   $$\frac{72-18}{72}$$
2. Therefore $\frac{3}{4}$.

• P1 Compare the total sold with the original quantity.
• A1 Correct answer: $\frac{3}{4}$

(a) $3$ cups

1. Write the mixed number as an improper fraction and multiply.
   $$2/3\times 9/2$$
2. Therefore $3$ cups.

• P1 Write the mixed number as an improper fraction and multiply.
• A1 Correct answer: $3$ cups

(b) $\frac{13}{12}$ cups

1. Use a common denominator of 12.
   $$21/12-8/12$$
2. Therefore $\frac{13}{12}$ cups.

• P1 Use a common denominator of 12.
• A1 Correct answer: $\frac{13}{12}$ cups

(a) $\frac{8}{3} \times \frac{9}{8} = \frac{72}{24} = 3$

1. Write each mixed number as an improper fraction.
   $$2\tfrac{2}{3} = \frac{8}{3}, \qquad 1\tfrac{1}{8} = \frac{9}{8}$$
2. Multiply.
   $$\frac{8}{3} \times \frac{9}{8} = \frac{72}{24} = 3$$

• M1 Both mixed numbers correctly changed to $\frac{8}{3}$ and $\frac{9}{8}$.
• A1 Reaching $\frac{72}{24} = 3$, or cancelling to $\frac{1}{1} \times \frac{3}{1} = 3$, with the working shown.

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

1. Change to improper fractions and multiply by the reciprocal.
   $$\frac{16}{5} \div \frac{4}{3} = \frac{16}{5} \times \frac{3}{4} = \frac{48}{20}$$
2. $\frac{48}{20} = \frac{12}{5} = 2\frac{2}{5}$.

• M1 Writing $\frac{16}{5}$ and $\frac{4}{3}$.
• M1 Multiplying by the reciprocal: $\frac{16}{5} \times \frac{3}{4}$.
• A1 The correct answer, $2\frac{2}{5}$.

(a) $120$ litres

1. The 42 litres are the difference between the two fractions of the tank.
   $$\frac{3}{5} - \frac{1}{4} = \frac{12}{20} - \frac{5}{20} = \frac{7}{20}$$
2. $$\frac{7}{20} \text{ of the tank} = 42 \Rightarrow \frac{1}{20} = 6 \Rightarrow \text{capacity} = 120$$

• P1 Finding the fraction used, $\frac{7}{20}$.
• P1 Setting $\frac{7}{20}$ of the capacity equal to 42, or finding $\frac{1}{20} = 6$.
• A1 The correct answer, $120$ litres.

(b) $\frac{1}{12}$

1. One third of the water remains.
   $$\frac{1}{3} \times \frac{1}{4} = \frac{1}{12}$$

• M1 Finding $\frac{1}{3}$ of $\frac{1}{4}$, or finding the $30 - 20 = 10$ litres left out of $120$.
• A1 The correct answer, $\frac{1}{12}$.