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
Take off the expenses: ${4800 - 1280 = 3520}$ , so $3,520 is shared.
Think first. What must come off the profit first?
The ratio
- ${4000 : 6000 = 2 : 3}$ , which is ${2 + 3 = 5}$ parts.
- 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. Sometimes the tax comes in bands: the first ₦200 at one rate, the next part at another, and so on. Tax each band separately, then add.

Income taxAllowances off first; tax only the taxable income

Slide the income and watch the bands fill one after another. Each band’s rate applies only to the part of the income inside that band.

Income tax in bandsSlide the yearly income

₦1,300,000taxable income₦140,000tax a year₦11,667tax a month (÷ 12)

yearly income = ₦1,500,000

Allowance first: ₦1,500,000 − ₦200,000 = ₦1,300,000 taxable. Then band by band: ₦300,000 at 5% = ₦15,000; ₦500,000 at 10% = ₦50,000; ₦500,000 at 15% = ₦75,000. Total ₦140,000.

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
Only 4 children count, the maximum: ${120 + 30 + 4 \times 25 + 40 = 290}$ , that is GH¢ 290.
Think first. How many children count for the allowance?
(i) Taxable income
Income minus allowances: ${850 - 290 = 560}$ , that is GH¢ 560.
Tax in bands
- First 200 at $10\%$ : 20.
- Next 200 at $15\%$ : 30.
- The remaining ${560 - 400 = 160}$ at $20\%$ : 32.
- Total tax: ${20 + 30 + 32 = 82}$ , that is GH¢ 82 a year.
Think first. How much of the 560 is left for the third band?
(ii) Monthly tax
${\frac{82}{12} \approx 6.83}$ , that is GH¢ 6.83 a month.

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 brought into the country. “38 kobo in the naira” means $38\%$ , because there are 100 kobo in a naira.

CommissionA percentage of the selling price

Shares and dividends

Shares have two values:

the nominal value, printed on them, such as ₦1.00;
the market value, what they cost to buy now.

The number of shares you can buy depends on the market price. The dividend, the share of the profit paid out, is given 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. Then ask which currency you want to end with, to decide whether to multiply or divide. Naira to leones: multiply by 16. 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 rest, 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. Change minutes to hours first, by dividing by 60.

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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