Equivalent fractions and mixed numbers

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

1. The highest common factor of 18 and 24 is 6. $\frac{18 \div 6}{24 \div 6} = \frac{3}{4}$.

• B1 The correct answer, $\frac{3}{4}$.

(b) $\frac{13}{5}$

1. $2$ wholes are $\frac{10}{5}$. Add $\frac{3}{5}$: $\frac{10}{5} + \frac{3}{5} = \frac{13}{5}$.

• B1 The correct answer, $\frac{13}{5}$.

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

1. $17 \div 4 = 4$ remainder $1$, so $\frac{17}{4} = 4\frac{1}{4}$.

• B1 The correct answer, $4\frac{1}{4}$.

(d) $\frac{5}{8}$

1. $\frac{5}{8} = 5 \div 8 = 0.625$, and $0.625 > 0.600$.

• B1 $\frac{5}{8}$, with or without the conversion.

(a) $0.375$

1. $3 \div 8$: $8$ into $30$ is $3$ r $6$, into $60$ is $7$ r $4$, into $40$ is $5$. So $0.375$.

• M1 A correct division method, or $\frac{3}{8} = \frac{375}{1000}$.
• A1 The correct answer, $0.375$.

(b) $2.2$

1. $\frac{1}{5} = 0.2$, so $2\frac{1}{5} = 2.2$.

• B1 The correct answer, $2.2$.

(c) $\frac{9}{20}$

1. $0.45 = \frac{45}{100}$. Divide top and bottom by 5: $\frac{9}{20}$.

• M1 Writing $\frac{45}{100}$.
• A1 The correct answer, $\frac{9}{20}$.

(a) $10$

1. Convert the mixed number to an improper fraction.
   $$3+3/4=15/4$$
2. Divide by the ribbon required for one decoration.
   $$(15/4)/(3/8)$$
3. Therefore $10$.

• P1 Convert the mixed number to an improper fraction.
• P1 Divide by the ribbon required for one decoration.
• A1 Correct answer: $10$

(b) $0$ m

1. Subtract the length used by the complete decorations.
   $$15/4-10\times (3/8)$$
2. Therefore $0$ m.

• P1 Subtract the length used by the complete decorations.
• A1 Correct answer: $0$ m

(a) $0.85$

1. Write the fraction with denominator 100.
   $$17/20=85/100$$
2. Therefore $0.85$.

• M1 Write the fraction with denominator 100.
• A1 Correct answer: $0.85$

(b) Nia: 17/20 = 0.85, which is greater than 21/25 = 0.84.

1. Write Omar’s fraction with denominator 100 too.
   $$21/25=84/100$$
2. Nia: 17/20 = 0.85, which is greater than 21/25 = 0.84.

• M1 Write Omar’s fraction with denominator 100 too.
• C1 Correct conclusion with supporting reasoning: Nia: 17/20 = 0.85, which is greater than 21/25 = 0.84.

(a) $\frac{13}{5}$

1. Multiply the whole number by the denominator and add the numerator.
   $$2\times 5+3$$
2. Therefore $\frac{13}{5}$.

• M1 Multiply the whole number by the denominator and add the numerator.
• A1 Correct answer: $\frac{13}{5}$

(b) 10 2/5 m

1. Multiply the improper fraction by 4.
   $$4\times 13/5$$
2. Therefore 10 2/5 m.

• M1 Multiply the improper fraction by 4.
• A1 Correct answer: 10 2/5 m

(a) $0.875$

1. Divide 7 by 8, or write the fraction over 1000.
   $$7/8=875/1000$$
2. Therefore $0.875$.

• M1 Divide 7 by 8, or write the fraction over 1000.
• A1 Correct answer: $0.875$

(b) The 7/8 litre bottle: 7/8 = 0.875, 5/6 = 0.833... and 0.8 is smaller than both.

1. Compare 7/8 and 5/6 using a common denominator or decimals.
   $$7/8=21/24$$
2. The 7/8 litre bottle: 7/8 = 0.875, 5/6 = 0.833... and 0.8 is smaller than both.

• M1 Compare 7/8 and 5/6 using a common denominator or decimals.
• C1 Correct conclusion with supporting reasoning: The 7/8 litre bottle: 7/8 = 0.875, 5/6 = 0.833... and 0.8 is smaller than both.

(a) $504$ litres

1. $$3/4-5/7$$
2. $$18/(1/28)$$
3. After removal, the tank is 5/7 full. The added fraction is 3/4 $-$ 5/7 = 21/28 $-$ 20/28 = 1/28.
4. One twenty-eighth of the capacity is 18 litres, so the full capacity is 18 $\times$ 28 = 504 litres.

• P1 Establishing $3/4-5/7$ or an equivalent valid method.
• P1 Establishing $18/(1/28)$ or an equivalent valid method.
• A1 Correct answer: $504$ litres

(a) $7$

1. A fraction in its simplest form terminates exactly when its denominator has no prime factors other than 2 and 5.
2. From 2 to 20 these are $2, 4, 5, 8, 10, 16, 20$.
3. That is 7 values.

• P1 Using the rule that the denominator's only prime factors are 2 and 5.
• A1 The correct answer, $7$.

(b) $\frac{7}{35} = \frac{1}{5} = 0.2$

1. The rule applies to a fraction in its simplest form. $\frac{7}{35} = \frac{1}{5}$, and the denominator 5 has only the prime factor 5, so it terminates: $0.2$.

• C1 Simplifying to $\frac{1}{5}$ (or $0.2$) and saying the simplified denominator has only the prime factor 5.

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

1. $0.375 = \frac{375}{1000}$. Divide top and bottom by 125: $\frac{3}{8}$.

• M1 Writing $\frac{375}{1000}$ or an equivalent such as $\frac{75}{200}$.
• A1 The correct answer, $\frac{3}{8}$.