Mathematics · Lesson 4

Interest: Mama Nkechi's savings

JAMBWAECNECO

What the syllabus asks

JAMB: Number and numeration: simple and compound interest, depreciation, percentage error, ratio and proportion

WAEC: Commercial arithmetic: percentages, simple interest, compound interest, depreciation, profit and loss, discount, commission, hire purchase, VAT, shares and dividends

NECO: Commercial arithmetic: percentages, simple interest, compound interest, depreciation, profit and loss, discount, commission, hire purchase, VAT, shares and dividends

What you will learn

Calculate percentages, percentage increase and decrease, and percentage error.
Solve simple interest problems for I, P, R or T.
Calculate compound interest year by year and by formula.
Solve depreciation and appreciation problems.
Work out profit, loss, discount, commission, VAT and hire purchase.
Choose between simple and compound interest in real decisions.

Key points

x% of Q = xQ/100; percentage change = change/original × 100%.
Increase by R%: × (1 + R/100); decrease: × (1 − R/100); reverse by dividing.
SI: I = PRT/100, A = P + I; T in years.
CI: A = P(1 + R/100)ⁿ; CI = A − P; for k periods a year use R/k and nk.
Depreciation: V = P(1 − R/100)ⁿ; straight-line loses a fixed amount yearly.
Profit and loss % are on cost price.
Discount from marked price; VAT added on; commission is a % of sales.
HP price = deposit + instalments; dividend is on nominal value; yield on cost.
Share in a ratio: add the parts, find one part, multiply; check the total.
Direct proportion: both rise together; inverse: product (e.g. man-days) stays fixed.
Taxable income = gross − allowances; apply the rates band by band.

Start here: Percentages: the foundation

Almost every money question in WAEC rests on percentages, so we begin there. "Per cent" means "per hundred". So 20% means 20 out of every 100, or 20/100 = 0.2.

To find x% of a quantity Q: (x/100) × Q.
To express A as a percentage of B: (A/B) × 100%.

Mama Nkechi sells tomatoes. If 15% of a ₦12,000 basket spoils, the spoilt value is 15/100 × 12,000 = ₦1,800. If she sold ₦9,000 worth out of ₦12,000, she sold 9,000/12,000 × 100% = 75%.

Percentage increase and decrease

Percentage change = (change ÷ original) × 100%.

A bag of rice rose from ₦40,000 to ₦50,000. The change is ₦10,000, and as a percentage of the original price that is 10,000/40,000 × 100% = 25%. Notice we always divide by the original, not the new value.

A quick way to increase by R% is to multiply by (1 + R/100). To decrease by R%, multiply by (1 − R/100). So a 20% rise on ₦2,500 is 2,500 × 1.2 = ₦3,000. This "multiplier" idea is the key to compound interest later.

Worked example 4.1

A phone priced ₦80,000 is reduced by 12.5%. Find the new price.

Multiplier = 1 − 0.125 = 0.875. We use a multiplier because it does the subtraction in one step.
New price = 80,000 × 0.875 = ₦70,000.

₦70,000

Worked example 4.2 (reverse percentage)

After a 25% increase, a bus fare is ₦1,500. What was the old fare?

New = old × 1.25, so old = 1,500 ÷ 1.25. We divide because we are undoing the multiplier.
Old = ₦1,200.

₦1,200

⚠️ Common mistake: in reverse questions, taking 25% of the NEW price and subtracting it (1,500 − 375 = 1,125). This is wrong, because the 25% was of the old price.

Percentage error

If a length of 50 cm is measured as 48 cm, the error is 2 cm and the percentage error is 2/50 × 100% = 4%. JAMB asks this often.

Try it yourself. A student's mark rose from 48 to 60. Find the percentage increase.

Show solution

12/48 × 100% = 25%.

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Practice questions

1. Find the SI on ₦45,000 at 4% for 5 years.

Show worked answer

₦9,000.

2. At what rate does ₦12,000 earn ₦2,160 SI in 3 years?

Show worked answer

6%.

3. Find the CI on ₦30,000 at 10% for 3 years.

Show worked answer

30,000 × 1.331 = 39,930; CI = ₦9,930.

4. A ₦2,500,000 car depreciates by 15% yearly. Find its value after 2 years.

Show worked answer

2,500,000 × 0.7225 = ₦1,806,250.

5. By selling a goat for ₦54,000 a farmer gains 20%. Find the cost price.

Show worked answer

54,000/1.2 = ₦45,000.

Key words

Principal (P): The amount of money borrowed or invested.
Rate (R): The percentage charged or earned per year (per annum).
Time (T or n): The period in years.
Interest (I): The money paid for the use of money.
Amount (A): Principal plus interest.
Simple interest: Interest on the original principal only.
Compound interest: Interest on principal plus accumulated interest.
Per annum (p.a.): Per year.
Depreciation: Loss in value of an asset over time.
Appreciation: Gain in value over time.
Cost price (CP): The price at which an article is bought.
Selling price (SP): The price at which it is sold.

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