Commercial Arithmetic: JAMB Mathematics Notes

A step-by-step guide to simple interest, compound interest and depreciation, profit and loss, discounts, tax, commission, and percentage applications.

Simple Interest

Simple interest grows by the same amount each year because every year's interest is calculated from the original principal.

Use

$I = \frac{P R T}{100}$ ,

where $P$ is the principal, $R$ is the annual rate percent, $T$ is the time in years, and $I$ is the interest.

The final amount is

$A = P + I$ .

Before substituting, identify exactly what the question gives. If time is in months, change it to years first. For example, 9 months is $9/12 = 3/4$ year.

Example 1: Find the simple interest on ₦50,000 for 3 years at 4% per annum.

$I = \frac{50000 \times 4 \times 3}{100} = 6000$ , so the interest is ₦6,000.

Example 2: Find the amount on ₦80,000 for 2 years at 5% simple interest.

First, $I = \frac{80000 \times 5 \times 2}{100} = 8000$ , so the interest is ₦8,000.

Then $A = 80000 + 8000 = 88000$ , so the amount is ₦88,000.

Example 3: A sum earns ₦9,000 simple interest in 3 years at 6% per annum. Find the principal.

$9000 = \frac{P \times 6 \times 3}{100}$ .

So $P = \frac{9000 \times 100}{18} = 50000$ , giving a principal of ₦50,000.

Example 4: ₦40,000 earns ₦7,200 simple interest in 3 years. Find the annual rate.

$7200 = \frac{40000 \times R \times 3}{100}$ .

Thus $R = 6%$ .

Example 5: ₦30,000 earns ₦6,000 at 5% simple interest. Find the time.

$6000 = \frac{30000 \times 5 \times T}{100}$ .

Hence $T = 4$ years.

Compound Interest & Depreciation

Compound interest changes the base from one period to the next. Interest is added to the balance and the next period's interest is calculated on the new balance.

For annual compounding,

$A = P (1 + \frac{R}{100})^{n}$ ,

where $n$ is the number of years. Compound interest itself is

$C I = A - P$ .

For depreciation at $R %$ per year, the retained fraction is $1 - R /100$ , so

$V = P (1 - \frac{R}{100})^{n}$ .

Do not use the simple-interest formula for repeated percentage growth or depreciation.

Example 6: Find the amount on ₦100,000 for 2 years at 10% compound interest.

$A = 100000 (1.10)^{2} = 121000$ , so the amount is ₦121,000.

Example 7: Find the compound interest in Example 6.

$C I = A - P = 121000 - 100000 = 21000$ , so the compound interest is ₦21,000.

Example 8: A machine worth ₦200,000 depreciates by 10% per year. Find its value after 2 years.

Each year 90% remains, so $V = 200000 (0.9)^{2} = 162000$ . The value is ₦162,000.

Example 9: A sum becomes ₦133,100 after 3 years at 10% compound interest. Find the principal.

$133100 = P (1.1)^{3} = 1.331 P$ .

Therefore $P = 100000$ , so the principal is ₦100,000.

Example 10: An asset falls from ₦500,000 to ₦405,000 in 2 years at a constant annual depreciation rate.

$(1 - r)^{2} = 405000/500000 = 0.81$ , where $r$ is the rate as a decimal.

So $1 - r = 0.9$ , giving $r = 0.1 = 10%$ .

Profit, Loss & Markup

Start by identifying the cost price $C P$ and the selling price $S P$ .

If $S P > C P$ , there is a profit:

$profit = S P - C P$ .

If $S P < C P$ , there is a loss:

$loss = C P - S P$ .

The denominator for profit or loss percentage is the cost price:

$profit % = \frac{profit}{C P} \times 100%$ ,

and similarly for loss percentage.

Markup is an increase added to cost price before selling. A markup of $m %$ gives

$S P = C P (1 + m /100)$ .

Example 11: An item costs ₦40,000 and sells for ₦50,000. Find the profit percentage.

Profit is ₦10,000. Therefore $\frac{10000}{40000} \times 100% = 25%$ .

Example 12: A phone costs ₦120,000 and is sold for ₦102,000. Find the loss percentage.

Loss is ₦18,000. Thus $\frac{18000}{120000} \times 100% = 15%$ .

Example 13: An article is sold for ₦69,000 at a 15% profit. Find the cost price.

The selling price is 115% of cost: $1.15 C P = 69000$ .

So $C P = 60000$ , giving a cost price of ₦60,000.

Example 14: A trader wants a 20% markup on an item costing ₦75,000.

$S P = 75000 (1.20) = 90000$ , so the selling price is ₦90,000.

Example 15: Goods cost ₦30,000 and an extra ₦5,000 is spent on repairs before sale. If they sell for ₦42,000, use the total effective cost ₦35,000.

Profit is ₦7,000, so profit percentage is $\frac{7000}{35000} \times 100% = 20%$ .

Discount, Tax & Percentage Applications

A discount reduces the marked price. If the discount is $d %$ , then

$selling price = marked price (1 - \frac{d}{100})$ .

Tax or VAT usually increases a stated pre-tax amount. A tax rate of $t %$ gives

$total = base amount (1 + \frac{t}{100})$ .

Commission is normally a percentage of sales or another specified base. Always identify the correct base before multiplying by the percentage.

For any percentage application, translate the words carefully: “ $p %$ of $Q$ ” means $pQ /100$ .

Example 16: An item marked ₦80,000 receives a 15% discount.

The customer pays 85%: $0.85 \times 80000 = 68000$ . The selling price is ₦68,000.

Example 17: After a 20% discount an item costs ₦72,000. Find the marked price.

The sale price is 80% of the marked price: $0.8 P = 72000$ .

Hence $P = 90000$ , so the marked price is ₦90,000.

Example 18: A pre-tax bill is ₦120,000 and VAT is 7.5%.

VAT is $0.075 \times 120000 = 9000$ , or ₦9,000.

The total bill is ₦129,000.

Example 19: A salesperson receives 4% commission on ₦350,000 sales.

Commission is $0.04 \times 350000 = 14000$ , or ₦14,000.

Example 20: A worker gives 10% of a monthly income to charity. If the contribution is ₦12,000, find the income.

$0.10 I = 12000$ , so $I = 120000$ . The income is ₦120,000.

A simple exam plan

Write down the quantity the percentage is based on before calculating.
For simple interest, use the original principal throughout.
For compound growth or depreciation, use a repeated multiplier and a power.
For profit or loss percentage, divide by cost price.
For discount, subtract the discount from 100% to find the fraction paid.
For tax or commission, identify the stated base amount first.
If the question gives the final amount and asks for the original, use reverse percentage rather than subtracting the same percentage.

Estimate first when possible, then check that your final answer is sensible.

Commercial Arithmetic

Simple Interest

Compound Interest & Depreciation

Profit, Loss & Markup

Discount, Tax & Percentage Applications

A simple exam plan

Key terms

Practise Commercial Arithmetic questions

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