Sec 1 Maths: Ratio, Rate & Proportion — practice questions & worked solutions

(a)

Convert to the same unit (centimetres), then simplify:

$$\begin{aligned}1.6\text{ km} &= 160000\text{ cm}\\ 160000 : 6400 &= 25 : 1\end{aligned}$$

(b)

$$\begin{aligned}54\text{ m/s} &= 54 \times \frac{3600}{1000}\text{ km/h}\\ &= 194.4\text{ km/h}\end{aligned}$$

When rearranged equally, each box has:

$$\begin{aligned}\frac{180}{3} &= 60\text{ balls.}\end{aligned}$$

Box $A$ initially had:

$$\begin{aligned}60 – 30 &= 30\text{ balls.}\end{aligned}$$

Since $A$ corresponds to $3$ parts:

$$\begin{aligned}3\text{ parts} &= 30\\ 1\text{ part} &= 10\end{aligned}$$

Total number of parts:

$$\begin{aligned}\frac{180}{10} &= 18\\ 3 + 5 + n &= 18\\ n &= 10\end{aligned}$$

(a)

Convert via JPY:

$$\begin{aligned}1\text{ SGD} &= 114.72\text{ JPY}\\ 1\text{ KRW} &= 0.11\text{ JPY}\\ 1\text{ SGD} &= \frac{114.72}{0.11}\text{ KRW}\\ &= 1042.9\text{ KRW}\;(1\text{ d.p.})\end{aligned}$$

(b)

Convert both prices to JPY for comparison. Korean website:

$$\begin{aligned}9899 \times 0.11 &= 1088.89\text{ JPY}\end{aligned}$$

Japanese website $= 1284.86$ JPY. Since:

$$\begin{aligned}1088.89 &< 1284.86,\end{aligned}$$

the Korean website offered the better deal.

Make the $q$ terms equal (LCM of $7$ and $5$ is $35$):

$$\begin{aligned}p:q &= 4:7 = (4\times5):(7\times5) = 20:35 \\ q:r &= 5:2 = (5\times7):(2\times7) = 35:14 \\ p:q:r &= 20:35:14\end{aligned}$$

Difference between $p$ and $r$ in units $= 20 – 14 = 6$ units.

$$\begin{aligned}6 \text{ units} &= 12 \text{ cm} \\ 1 \text{ unit} &= 2 \text{ cm} \\ \text{Total units} &= 20+35+14 \\ &= 69 \\ \text{Total length} &= 69 \times 2 \\ &= 138 \text{ cm}\end{aligned}$$

(i)

Round $\$29.95$ to $\$30$ and $238$ to $240$:

$$\begin{aligned}\text{estimated cost} &\approx 30\times 240\\ &= 7200\end{aligned}$$

The estimated cost is about $\$7200$.

(ii)

The actual cost is:

$$\begin{aligned}29.95\times 238 &= 7128.10\end{aligned}$$

Since $\$7128.10 < \$8000$, Mr Goh has sufficient funds.

(i)

Divide each amount by $12\,500$:

$$\begin{aligned}50\,000 : 37\,500 : 87\,500 = 4 : 3 : 7\end{aligned}$$

(ii)

Total investment $= 50\,000 + 37\,500 + 87\,500 = 175\,000$.

$$\begin{aligned}\text{Profit} &= 345\,000 – 175\,000 \\ &= 170\,000\end{aligned}$$

Amy’s units $= 4$ out of $4 + 3 + 7 = 14$:

$$\begin{aligned}\text{Amy's profit} &= \frac{4}{14} \times 170\,000 \\ &= 48\,571.43\end{aligned}$$

Amy received $\$48\,571.43$.

(i)

$$\begin{aligned}\text{Jack} &= 5u, \quad \text{Jill} = 2u\end{aligned}$$

The total amount is unchanged, so $5u + 2u = 7u$ before and $11p + 5p = 16p$ after, giving $7u = 16p$.

$$\begin{aligned}\frac{5u – 0.3}{2u + 0.3} &= \frac{11}{5}\\ 5(5u – 0.3) &= 11(2u + 0.3)\\ 25u – 1.5 &= 22u + 3.3\\ 3u &= 4.8\\ u &= 1.6\end{aligned}$$
$$\begin{aligned}\text{Total} &= 7u = 7 \times 1.6 = 11.2 \text{ litres}\end{aligned}$$

(ii)

$$\begin{aligned}\text{Jack before} &= 5 \times 1.6 = 8 \text{ litres}\\ \text{Percentage change} &= \frac{0.3}{8} \times 100\% = 3.75\%\end{aligned}$$

The amount decreased by $3.75\%$.

Cross-multiply:

$$\begin{aligned}5(p-3q)&=2p-11q\\ 5p-15q&=2p-11q\\ 3p&=4q\\ \frac{p}{q}&=\frac{4}{3}\end{aligned}$$

(a)(i)

$$\begin{aligned}\frac{5}{7} \times 100\%\end{aligned}$$
$$= 71.4\%$$

(a)(ii)

Bob’s time $= 55$ min $40$ s $= 3340$ s.

$$\begin{aligned}\text{Mei's time} &= \frac{7}{4} \times 3340 \\ &= 5845 \text{ s}\end{aligned}$$
$$= 97 \text{ mins } 25 \text{ s}$$

(b)(i)

$$\begin{aligned}820 \times 1.28\end{aligned}$$
$$= \$1049.60$$

(b)(ii)

Celia is $60\%$ less than Mei, so Celia $= 40\%$ of Mei.

$$\begin{aligned}\text{Mei} &= \frac{820}{0.4} \\ &= 2050 \\ \text{Difference} &= 2050 – 820\end{aligned}$$
$$= \$1230$$

(c)(i)

Cost of $4$ hot dogs $= 29 – 12.80 – 4.90 – 3.50 = 7.80$.

$$\begin{aligned}\text{One hot dog} &= \frac{7.80}{4}\end{aligned}$$
$$= \$1.95$$

(c)(ii)

$$\begin{aligned}\text{GST} &= 0.07 \times (29 + 2.90)\end{aligned}$$
$$= \$2.23$$

(c)(iii)

$$\begin{aligned}\text{Total} &= 29 + 2.90 + 2.23\end{aligned}$$
$$= \$34.13$$

Cross-multiply and collect terms:

$$\begin{aligned}5(10x-6y) &= 2(9x-8y) \\ 50x-30y &= 18x-16y \\ 32x &= 14y \\ \frac{x}{y} &= \frac{14}{32}=\frac{7}{16}\end{aligned}$$

Then:

$$\begin{aligned}x:2y &= 7:(2\times 16) \\ &= 7:32\end{aligned}$$

From $2a = 3b$, $a:b = 3:2$. Given $b:c = 4:5$. Make the $b$ terms equal:

$$a:b = 6:4$$
$$b:c = 4:5$$
$$a:c = 6:5$$