REAL NUMBERS AND RATIO

QUESTION 1 [4 marks] — Criterion A Medium

For each number below, write it as a fraction $\frac{p}{q}$ (where $p, q$ are integers, $q\ne0$), showing that it is rational:

a. $0.75$

[1]

Show Solution

$$0.75 = \frac{75}{100} = \frac{3}{4}$$

b. $-3$

[1]

Show Solution

$$-3 = \frac{-3}{1}$$

c. $0.\dot{3}$ (i.e. $0.3333\ldots$)

[2]

Show Solution

Let $x=0.\dot{3}$. Then $10x=3.333\ldots$, so $10x-x=3$, giving $9x=3$, so $$x=\frac{3}{9}=\frac{1}{3}$$

QUESTION 2 [4 marks] — Criterion A Medium

Order the rational numbers $-1.5, \frac{3}{4}, -\frac{2}{3}, 0.6$ from smallest to largest.

a. Convert each number to a decimal (correct to 3 decimal places where needed).

[2]

Show Solution

$-1.5$, $\frac{3}{4}=0.750$, $-\frac{2}{3}=-0.667$, $0.6$

b. Write the four numbers in order from smallest to largest.

[2]

Show Solution

$$-1.5, \ -\frac{2}{3}, \ 0.6, \ \frac{3}{4}$$

QUESTION 3 [5 marks] — Criterion B Medium

Investigate whether adding or multiplying two rational numbers always gives a rational result (this is called 'closure').

a. Add $\frac{2}{3}$ and $\frac{1}{5}$, and multiply $\frac{2}{3}$ and $\frac{1}{5}$. Are both results rational?

[2]

Show Solution

$\frac{2}{3}+\frac{1}{5}=\frac{10}{15}+\frac{3}{15}=\frac{13}{15}$ (rational). $\frac{2}{3}\times\frac{1}{5}=\frac{2}{15}$ (rational). Both results are rational.

b. Using general fractions $\frac{a}{b}$ and $\frac{c}{d}$, explain why the sum $\frac{a}{b}+\frac{c}{d}$ will always be rational.

[3]

Show Solution

$$\frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{bd}$$ Since $a,b,c,d$ are integers, $ad+bc$ and $bd$ are also integers (sums and products of integers are integers), so the result is a ratio of two integers — by definition, rational.

QUESTION 4 [5 marks] — Criterion C Medium

Explain why every terminating decimal (a decimal that ends, like $0.125$) must be a rational number.

a. Using $0.125$ as an example, show how it can be written as a fraction with a power of 10 as the denominator.

[2]

Show Solution

$$0.125 = \frac{125}{1000}$$

b. Simplify this fraction, and then explain in general terms why ANY terminating decimal can always be written this way.

[3]

Show Solution

$\frac{125}{1000}=\frac{1}{8}$ (dividing by HCF 125). In general, a terminating decimal with $n$ digits after the decimal point can always be written as (the whole number formed by removing the decimal point) $\div 10^n$ — both of these are integers, so the result is always a ratio of integers, i.e. rational.

QUESTION 5 [6 marks] — Criterion D Medium

A recipe requires ingredients measured as $0.75$ cups of sugar, $\frac{1}{3}$ cup of oil, and $1.2$ cups of flour, for one batch.

a. A baker is making $2\frac{1}{2}$ batches. Find the total amount of each ingredient needed, giving each answer as a fraction in simplest form.

[4]

Show Solution

Sugar: $0.75\times2.5=\frac{3}{4}\times\frac{5}{2}=\frac{15}{8}=1\frac{7}{8}$ cups. Oil: $\frac{1}{3}\times\frac{5}{2}=\frac{5}{6}$ cups. Flour: $1.2\times2.5=\frac{6}{5}\times\frac{5}{2}=3$ cups.

b. The baker only has a $\frac{1}{4}$-cup measuring scoop. Explain whether the sugar amount can be measured exactly using only this scoop.

[2]

Show Solution

$1\frac{7}{8}$ cups $=\frac{15}{8}$ cups. Since $\frac{1}{4}=\frac{2}{8}$, and $15$ is not evenly divisible by $2$, the sugar amount cannot be measured using only whole $\frac{1}{4}$-cup scoops — a smaller measure would be needed for the extra $\frac{1}{8}$.

Rational numbers