REAL NUMBERS AND RATIO

QUESTION 1 [5 marks] — Criterion A Medium

Order the fractions $\frac{3}{5}, \frac{7}{10}, \frac{2}{3}, \frac{5}{8}$ from smallest to largest.

a. Convert each fraction to an equivalent fraction with denominator 120.

[3]

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$\frac{3}{5}=\frac{72}{120}$, $\frac{7}{10}=\frac{84}{120}$, $\frac{2}{3}=\frac{80}{120}$, $\frac{5}{8}=\frac{75}{120}$

b. Using your converted fractions, write the original fractions in order from smallest to largest.

[2]

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Comparing numerators (72, 84, 80, 75): smallest to largest gives $$\frac{3}{5}, \frac{5}{8}, \frac{2}{3}, \frac{7}{10}$$

QUESTION 2 [3 marks] — Criterion A Medium

Complete the following conversions, showing full working:

a. Write $\frac{47}{6}$ as a mixed number.

[1]

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$47 \div 6 = 7$ remainder $5$, so $\frac{47}{6}=7\frac{5}{6}$

b. Write $5\frac{3}{8}$ as an improper fraction.

[1]

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$$5\frac{3}{8} = \frac{5\times8+3}{8} = \frac{43}{8}$$

c. Find the fraction equivalent to $\frac{3}{4}$ with denominator 32.

[1]

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$$\frac{3}{4} = \frac{3\times8}{4\times8} = \frac{24}{32}$$

QUESTION 3 [5 marks] — Criterion B Medium

Investigate the sequence of fractions $\frac{n}{n+1}$ as $n$ increases: $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots$

a. Write each of $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}$ as a decimal, correct to 3 decimal places.

[3]

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$\frac{1}{2}=0.500$, $\frac{2}{3}=0.667$, $\frac{3}{4}=0.750$, $\frac{4}{5}=0.800$, $\frac{5}{6}=0.833$

b. Describe what happens to the value of $\frac{n}{n+1}$ as $n$ gets larger.

[2]

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The value increases and gets closer and closer to 1, but never actually reaches 1 (since the numerator is always exactly 1 less than the denominator).

QUESTION 4 [5 marks] — Criterion C Medium

Explain, without using a calculator, how to determine which of $\frac{5}{7}$ and $\frac{7}{9}$ is larger.

a. Describe the cross-multiplication method for comparing two fractions.

[2]

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To compare $\frac{a}{b}$ and $\frac{c}{d}$ (both positive denominators), multiply $a\times d$ and $c\times b$. Whichever product is larger corresponds to the larger fraction.

b. Apply this method to determine which of $\frac{5}{7}$ and $\frac{7}{9}$ is larger, showing your working.

[3]

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$5\times9=45$ and $7\times7=49$. Since $45<49$, this means $\frac{5}{7}<\frac{7}{9}$, so $\frac{7}{9}$ is larger.

QUESTION 5 [5 marks] — Criterion D Medium

Two pizza deals are advertised: Deal A gives $\frac{3}{8}$ of a large pizza for \$6. Deal B gives $\frac{5}{12}$ of a large pizza for \$9.

a. Find the cost of a WHOLE pizza under each deal (i.e. scale each deal up to a full pizza).

[3]

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Deal A: $\$6 \div \frac{3}{8} = 6\times\frac{8}{3}=\$16$ per whole pizza. Deal B: $\$9 \div \frac{5}{12} = 9\times\frac{12}{5}=\$21.60$ per whole pizza.

b. Which deal offers better value, and by how much (per whole pizza)?

[2]

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Deal A is better value, since $\$16 < \$21.60$. The difference is $\$21.60-\$16=\$5.60$ per whole pizza.

Fractions