Sec 1 Maths: Percentage — practice questions & worked solutions

(i)

$88\%$ of the original price equals 148 000 dollars:

$$\begin{aligned}\text{Original price}&=\frac{148\,000}{0.88}\\ &=168\,182 \text{ dollars (nearest dollar)}\end{aligned}$$

(ii)

$$\begin{aligned}\text{Profit}&=148\,000-90\,000=58\,000\\ \text{Percentage profit}&=\frac{58\,000}{90\,000}\times 100\%\\ &=64.4\%\end{aligned}$$

(iii)

$$\begin{aligned}\text{Downpayment}&=0.30\times 148\,000=44\,400\\ \text{Monthly payments}&=2000\times 12\times 5=120\,000\\ \text{Total under hire-purchase}&=44\,400+120\,000=164\,400\\ \text{Additional cost}&=164\,400-148\,000=16\,400\end{aligned}$$

(i)

Down payment:

$$\begin{aligned}0.20\times 97000 &= 19400\end{aligned}$$

Loan amount:

$$\begin{aligned}97000-19400 &= 77600\end{aligned}$$

Simple interest over $5$ years:

$$\begin{aligned}77600\times \frac{2.78}{100}\times 5 &= 10786.40\end{aligned}$$

Total to repay:

$$\begin{aligned}77600+10786.40 &= 88386.40\end{aligned}$$

Monthly payment over $5\times 12=60$ months:

$$\begin{aligned}\frac{88386.40}{60} &= 1473.106\ldots\\ &\approx 1473.11 \text{ (nearest cent)}\end{aligned}$$

(ii)

First $59$ months paid at 1473.11 dollars:

$$\begin{aligned}59\times 1473.11 &= 86913.49\end{aligned}$$

Final month payment:

$$\begin{aligned}88386.40-86913.49 &= 1472.91\end{aligned}$$

Plan A: compounded quarterly, rate per period $=\dfrac{4\%}{4}=1\%$, number of periods $=2\times 4=8$.

$$\begin{aligned}\text{amount} &= 40000\times(1.01)^{8}\\ &= 43314.27\end{aligned}$$

Interest earned:

$$\begin{aligned}43314.27-40000 &= 3314.27\end{aligned}$$

Total gain (including the 100 dollar voucher):

$$\begin{aligned}3314.27+100 &= 3414.27\end{aligned}$$

Plan B: simple interest.

$$\begin{aligned}40000\times \frac{4.25}{100}\times 2 &= 3400\end{aligned}$$

Since the gain from Plan A (3414.27 dollars) exceeds the gain from Plan B (3400 dollars), Katherine should choose Plan A.

(a)

$$\begin{aligned}\text{Percentage increase} &= \frac{1240 – 800}{800} \times 100\% \\ &= \frac{440}{800} \times 100\% \\ &= 55\%\end{aligned}$$

(b)

A decrease of $67\%$ leaves $33\%$ of the 2019 figure:

$$\begin{aligned}33\% \text{ of (2019 flights)} &= 126\,060 \\ \text{2019 flights} &= \frac{126\,060}{0.33} \\ &= 382\,000\end{aligned}$$

(i)

$$\begin{aligned}\text{Deposit} &= 40\% \times 105\,600 = 42\,240 \\ \text{Balance} &= 105\,600 – 42\,240 = 63\,360\end{aligned}$$

(ii)(a)

Simple interest on the balance:

$$\begin{aligned}I &= 63\,360 \times \frac{3}{100} \times 5 \\ &= 9\,504\end{aligned}$$

Total amount paid for the car:

$$\begin{aligned}&= 42\,240 + 63\,360 + 9\,504 \\ &= 115\,104\end{aligned}$$

(ii)(b)

Loan amount repaid $= 63\,360 + 9\,504 = 72\,864$ over $60$ months:

$$\begin{aligned}\text{Monthly instalment} &= \frac{72\,864}{60} \\ &= 1214.40\end{aligned}$$

(iii)

Finance Company XYZ amount owed (compound interest):

$$\begin{aligned}63\,360 \times \left(1 + \frac{2.8}{100}\right)^5 &= 72\,741.25\end{aligned}$$

Bank ABC requires repayment of 72 864 dollars while Finance Company XYZ requires 72 741.25 dollars. Since 72 741.25 dollars is less than 72 864 dollars, Jessica should take the loan from Finance Company XYZ.

Interest required to reach 34 000 dollars:

$$\begin{aligned}34000-30000&=4000\end{aligned}$$

Simple interest per year:

$$\begin{aligned}30000\times\frac{2.2}{100}&=660\end{aligned}$$

Number of years for interest to reach 4000 dollars:

$$\begin{aligned}\frac{4000}{660}&=6.06\ldots\end{aligned}$$

Since the amount must exceed 34 000 dollars, the number of years required is $7$.

(a)

Each year the value becomes $89\%$ of the previous year. Let $V$ be the value 2 years ago:

$$\begin{aligned}V \times 0.89^2 &= 158\,000 \\ V &= \frac{158\,000}{0.89^2} \\ V &= 199\,470 \quad (\text{nearest dollar})\end{aligned}$$

(b)

Breadth $= x$, length $= 5x$:

$$\begin{aligned}\text{original perimeter} &= 2(5x + x) = 12x \\ \text{new length} &= 5x \times 0.6 = 3x \\ \text{new breadth} &= x \times 1.35 = 1.35x \\ \text{new perimeter} &= 2(3x + 1.35x) = 8.7x \\ \text{decrease} &= 12x – 8.7x = 3.3x \\ \text{percentage decrease} &= \frac{3.3x}{12x} \times 100\% \\ &= 27.5\%\end{aligned}$$

Original daily earnings:

$$\begin{aligned}300\times2000 &= 600000\end{aligned}$$

New price per computer:

$$\begin{aligned}2000\times1.15 &= 2300\end{aligned}$$

New number sold:

$$\begin{aligned}300\times0.80 &= 240\end{aligned}$$

New earnings:

$$\begin{aligned}240\times2300 &= 552000\end{aligned}$$

Percentage change:

$$\begin{aligned}\frac{552000-600000}{600000}\times100\% &= \frac{-48000}{600000}\times100\%\\ &= -8\%\end{aligned}$$

The earnings decrease by $8\%$.

After the $5\%$ discount:

$$\begin{aligned}\text{After discount} &= \frac{95}{100} \times 268.35 \\ &= 254.9325\end{aligned}$$

Apply the $10\%$ service charge and $8\%$ GST:

$$\begin{aligned}\text{Total amount} &= 254.9325 \times 1.1 \times 1.08 \\ &= 302.86\end{aligned}$$

Time $=4$ years. Using $I = \dfrac{PRT}{100}$:

$$\begin{aligned}I &= \frac{PRT}{100} \\ 5376 &= \frac{P\times 2.8\times 4}{100} \\ 5376 &= 0.112\,P \\ P &= 48000\end{aligned}$$

James put in 48 000 dollars.

(a)

Meya total (with 10-year warranty):

$$\begin{aligned}\text{Total} &= 3350+235\times 48+100 \\ &= 3350+11280+100 \\ &= \text{S\$}14730\end{aligned}$$

(b)

UK website: watch price after discounts (discounts not on shipping):

$$\begin{aligned}\text{Discounted price} &= 9600\times 0.90\times 0.95 \\ &= 9600\times 0.855 \\ &= \text{£}8208\end{aligned}$$

Add shipping:

$$\begin{aligned}\text{Total (£)} &= 8208+120 \\ &= \text{£}8328\end{aligned}$$

Convert to Singapore dollars:

$$\begin{aligned}8328\times 1.75 &= \text{S\$}14574\end{aligned}$$

Since SGD 14 574 (UK website) is less than SGD 14 730 (Meya), Mrs Sim should buy from the UK website as it is cheaper.

Original area:

$$A = 2x \times x = 2x^2$$

New length $=110\%$ of $2x = 2.2x$; new width $=80\%$ of $x = 0.8x$:

$$A_{\text{new}} = (2.2x)(0.8x)$$
$$A_{\text{new}} = 1.76x^2$$

Change:

$$\frac{1.76x^2 – 2x^2}{2x^2} \times 100\% = -12\%$$

There is a percentage decrease of $12\%$.

Total reliefs:

$$3500 + 2000 + 2(4200) + 23000 + 1500 = 38400$$

Chargeable income:

$$110400 – 38400 = 72000$$

Tax on first 40 000 dollars $= 550$ dollars; remaining 32 000 dollars taxed at $7\%$:

$$550 + 0.07(32000)$$
$$= 550 + 2240$$
$$= 2790$$

(a)

Simple interest for $3$ years on SGD 5600 at $2.5\%$ per annum:

$$\text{interest} = 5600 \times \tfrac{2.5}{100} \times 3$$
$$= 420$$
$$\text{total} = 5600 + 420$$
$$= \text{S\$}6020$$

(b)

US website price after $15\%$ discount:

$$4800 \times 0.85 = \text{US\$}4080$$

Add $8\%$ Value Added Tax on the product:

$$4080 \times 1.08 = \text{US\$}4406.40$$

Add shipping (no VAT):

$$4406.40 + 130 = \text{US\$}4536.40$$

Convert to Singapore dollars:

$$4536.40 \times 1.34 = \text{S\$}6078.78 \text{ (to 2 d.p.)}$$

Since SGD 6020 is less than SGD 6078.78, Eric should buy from Isten Shopping Centre as it is cheaper.