Shares, tax, commission and exchange · Commercial arithmetic

These questions use percentages and ratio in everyday settings. The skill is reading carefully what each percentage is of.

Partners usually share a profit in the ratio of what they invested. Take away any expenses first.

Sharing in a ratioAdd the ratio parts, then take each fraction of the total

Worked example · WAEC 2021

WAEC 2021 · Paper 2 · Q1 (a)

Kojo and Alidu invested $4,000.00 and $6,000.00 respectively in a business venture. After one year, a profit of $4,800.00 was realised. If after deduction of expenses of $1,280.00, the remainder is shared in proportion to their investment, find the amount received by each person.

What's left to share
$4800 - 1280 = 3520$ , so $3,520 is shared.
Think first. What must come off the profit first?
The ratio
$4000 : 6000 = 2 : 3$ , which is 5 parts, so one part is $\frac{3520}{5} = 704$ dollars.
Each share
Kojo: $2 \times 704 = 1408$ , so $1,408. Alidu: $3 \times 704 = 2112$ , so $2,112.

Income tax

Tax-free allowances come off the income first; only the rest, the taxable income, is taxed. When there are bands (the first ₦200 at one rate, the next at another), tax each band separately and add.

Income taxAllowances off first; tax only the taxable income

Worked example · WAEC 2014

WAEC 2014 · Paper 2 · Q2

A man has a wife and 6 children and his total income in a year was GH¢ 850.00. He was given the following tax-free allowances: personal GH¢ 120.00; wife GH¢ 30.00; children GH¢ 25.00 per child, for a maximum of 4 children; medical GH¢ 40.00. The rest was taxed as follows: first GH¢ 200.00 at $10\%$ ; next GH¢ 200.00 at $15\%$ ; next GH¢ 200.00 at $20\%$ ; remainder at $25\%$ . Calculate his: (i) taxable income; (ii) monthly tax.

Allowances
At most 4 children: $120 + 30 + 4 \times 25 + 40 =$ GH¢ 290.
Think first. How many children count for the allowance?
(i) Taxable income
$850 - 290 =$ GH¢ 560.
Tax in bands
First 200 at $10\%$ : 20. Next 200 at $15\%$ : 30. The remaining 160 at $20\%$ : 32. Total tax GH¢ 82 a year.
Think first. How much of the 560 is left for the third band?
(ii) Monthly tax
$\frac{82}{12} \approx$ GH¢ 6.83.

Commission and duty

A commission is a percentage of the selling price paid to an agent; import duty is a percentage of the cost of goods. “38 kobo in the naira” means $38\%$ .

CommissionA percentage of the selling price

Shares and dividends

Shares have a nominal value (printed on them, such as ₦1.00) and a market value (what they cost to buy now). The number of shares bought depends on the market price; the dividend is declared per share (or as a percentage of the nominal value).

Shares and dividendsNumber of shares = money ÷ market price

Foreign exchange

Write the rate as a conversion, such as ₦1.00 = Le 16.00, and decide whether to multiply or divide by asking which currency you want to end with. Leones to naira: divide by 16.

Exchange ratesMultiply or divide by the rate, towards the currency you want

Buying by instalments

The deposit is paid now; the balance is borrowed, and interest is charged on the balance only.

Buying by instalmentsPrice = deposit + balance; interest on the balance

Speed, distance and time

Speed is distance per unit of time:

\begin{aligned} \text{speed} &= \frac{\text{distance}}{\text{time}} \\ \text{distance} &= \text{speed} \times \text{time} \\ \text{time} &= \frac{\text{distance}}{\text{speed}} \end{aligned}

Keep the units matched: km with hours for km/h, and change minutes to hours (divide by 60) first. For a journey in parts, the average speed is the total distance ÷ total time. It is not the average of the speeds, because more time is spent on the slower part.

Speed, distance, timeCover the one you want: D = S × T, S = D ÷ T, T = D ÷ S

Average speedTotal distance ÷ total time

More: speed, distance and time

Your turn

WAEC 2016 · Paper 2 · Q6

(a)
A worker's annual income was $20\%$ tax-free and the remainder was taxed at a rate of 25 kobo in the naira. If he paid a tax of ₦14,000.00 in one year, how much was his income?
(b)
Every month a family spends $\frac25$ of its monthly income on education, $\frac16$ on clothes, $\frac38$ on food and saves the rest. If its monthly income is Le 36,000.00, how many years would it take to save Le 63,000.00?

Worked solution (try it first)

(a)

Let the income be ₦ $x$ .
The taxed part is $80\%$ of it, $0.8x$ .
Tax at 25 kobo in the naira is $25\%$ : $0.25 \times 0.8x = 0.2x = 14\,000$ , so $x = ₦70,000$ .

(b)

The fractions spent add up to $\frac25 + \frac16 + \frac38 = \frac{48 + 20 + 45}{120}$
$= \frac{113}{120}$ , so the family saves $\frac{7}{120}$ of its income: $\frac{7}{120} \times 36\,000 =$ Le 2,100 a month.
To save Le 63,000 takes $\frac{63\,000}{2100} = 30$ months $= 2\frac12$ years.

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