Fractions, percentages and financial arithmetic

Interpret fractions as quantities

A fraction a/b represents a divided by b, with b ≠ 0. It can describe part of a whole, a ratio or a point on the number line. Equivalent fractions have the same value because multiplying numerator and denominator by the same nonzero number multiplies the fraction by 1. Thus. Simplify by dividing both parts by a common factor. Before performing an operation, identify what the fraction refers to: one third of a class is not the same quantity as one third of a school.

For addition and subtraction, express fractions with a common denominator. Since and, their sum is 11/12. Adding numerator to numerator and denominator to denominator would give 3/7, which is incorrect. Multiplication gives, and division by a nonzero fraction means multiplying by its reciprocal. Therefore. Cancellation is valid between factors, never across an addition sign.

Connect decimals, percentages and ratios

A percentage is a fraction out of 100. Thus 0.35 = 35/100 = 35%, and 3/8 = 0.375 = 37.5%. To find 18% of ₦2,500, calculate. To find what percentage 18 is of 60, calculate (18/60) × 100% = 30%. The reference quantity belongs in the denominator. If a price rises from ₦800 to ₦1,000, the percentage increase is 200/800 × 100% = 25%, rather than 200/1,000 × 100%.

Repeated percentage changes multiply. A 20% increase followed by a 20% decrease changes a value by 1.20 × 0.80 = 0.96, an overall 4% decrease. For reverse percentages, divide by the multiplier. If a discounted price is ₦7,200 after a 10% reduction, it represents 90% of the original. The original is. Adding 10% of the discounted price would not undo the original reduction because the two percentage bases differ.

A ratio compares quantities using the same units. Sharing ₦12,000 in the ratio 2:3 means five equal parts, each worth ₦2,400. The shares are ₦4,800 and ₦7,200. A rate can compare different units, such as kilometres per hour. An average speed is total distance divided by total time, not automatically the mean of two speeds. Travelling 60 km at 30 km/h and another 60 km at 60 km/h takes three hours, so the overall speed is 40 km/h.

Work with stated financial rules

Simple interest is when r is an annual percentage and t is in years. For ₦40,000 at 6% per year for 18 months, t = 1.5 and interest is ₦3,600. Compound interest repeatedly adds interest to the balance: for n equal annual periods at a fixed annual rate. Interest earned is A − P. These are models with stated assumptions. Do not substitute a current bank rate, tax rate or fee unless the question explicitly supplies it.

Profit is selling price minus cost price. Profit percentage normally uses cost price: profit/cost × 100%. A ₦5,000 item sold for ₦6,000 has 20% profit on cost, while its profit margin on selling price is 1,000/6,000 × 100% ≈ 16.67%. Discount generally uses the marked price, commission uses the stated transaction base, and a tax calculation must identify whether a price excludes or includes the tax. With a stated 8% tax on a ₦2,000 pre-tax price, the total is ₦2,160.

Shares also require a clear base. If 100 shares have nominal value ₦2 each and a declared dividend is 15% of nominal value, the dividend is. It is not 15% of an unrelated market price. A model of straight-line depreciation subtracts the same amount each year; reducing-balance depreciation multiplies by the same remaining fraction. A machine costing ₦100,000 depreciating by 10% of its current value annually is worth ₦81,000 after two years, before other assumptions.

Report accuracy without hiding error

Decimal places count positions after the decimal point. Significant figures start at the first nonzero digit. To three significant figures, 0.004867 becomes 0.00487; to three decimal places, it becomes 0.005. Rounding a measured length to the nearest centimetre gives a possible true value within half a centimetre of the rounded value, using the stated rounding convention. The interval for a displayed 12 cm is typically 11.5 ≤ length < 12.5 cm. Exact counts do not automatically have the same uncertainty as measurements.

Absolute error is the magnitude of approximate minus reference value. Percentage error divides that absolute error by the magnitude of the reference value and multiplies by 100%. An approximation of 49 to a reference value of 50 has absolute error 1 and percentage error 2%. The denominator is the reference value, not the approximation. Keep several intermediate digits and round at the end, because repeated early rounding can distort a final answer, especially after subtraction or division.

Self-check: calculate 3/4 − 2/5, find the original price of an item sold for ₦4,500 after a 25% discount, and round 48.76 to three significant figures. The answers are 7/20, ₦6,000 and 48.8. Estimate before calculating: a 25% discount leaves three quarters of the original, so the original must exceed ₦4,500. In a written solution, include the unit and identify the percentage base. Those two habits prevent many otherwise convincing financial-arithmetic mistakes.

Compare payment schedules using the stated assumptions

A hire-purchase total is the deposit plus every instalment, including any separately stated fee. If a cash price is ₦80,000, a deposit is ₦20,000 and twelve monthly instalments are ₦6,000, the hire-purchase total is ₦92,000. The extra amount is ₦12,000, or 15% of the cash price. This does not automatically equal a bank-style annual interest rate because the outstanding balance changes as instalments are paid. Answer the arithmetic question actually specified rather than inventing an unstated finance convention.

Equal deposits with compound growth form a geometric series. If ₦1,000 is deposited at the end of each of three years and earns a stated 10% per year, the balance immediately after the third deposit is. The first deposit earns two years' interest, the second one year's, and the third none yet. Beginning-of-period deposits would have a different total, so timing is part of the model.

For an inclusive tax example, if a total of ₦1,200 includes a stated 20% tax, the pre-tax amount is and tax is ₦200. Taking 20% of the inclusive total gives ₦240 and uses the wrong base. Every rate in these examples is supplied for calculation; it does not assert a current Nigerian tax or lending rate.

When converting a monthly rate to an annual model, use the convention actually supplied. A monthly compound rate requires repeated multiplication, rather than assuming twelve additions produce the same result.

Extend the payment and capital-market models

An ordinary annuity consists of equal payments at the end of equal periods. If each payment is D, the effective rate per period is i ≠ 0 and there are n payments, its accumulated value immediately after the last payment is D[(1 + i)ⁿ − 1]/i. This is a finite GP sum: the earliest payment has n − 1 growth periods and the last has none. At i = 0 the value is simply nD. An annuity due pays at the start of each period, giving one extra growth period under otherwise identical assumptions.

Present value discounts future amounts to a chosen valuation date. For end-of-period equal payments, the elementary fixed-rate model is D[1 − (1 + i)⁻ⁿ]/i, with all timing and rate assumptions stated. These formulas describe mathematical models rather than guaranteed investment outcomes. For three end-of-year deposits of ₦1,000 at a stated 10% per year, future value is ₦3,310, while present value at the start is approximately ₦2,486.85. The dates differ, so the two values answer different questions.

A share represents an ownership interest, while a bond or debenture generally represents borrowing by its issuer. In a simple coupon model, annual interest is stated coupon rate × face value; a bond with face value ₦10,000 and coupon rate 6% pays ₦600 per year under the supplied terms. If purchased at ₦9,600, that coupon divided by purchase price is 6.25%, which is not by itself a full yield-to-maturity calculation. Do not substitute market price for face value when the coupon is defined on face value.

Straight-line depreciation allocates a fixed depreciable amount over a stated useful life. With cost ₦100,000, residual value ₦20,000 and life four years, yearly depreciation is. A corresponding classroom amortisation schedule may allocate a stated intangible-asset cost over its life; repaying a loan is a different use of the word amortisation and requires a payment/interest schedule. Self-check: reducing-balance value after two years at 10% yearly on ₦100,000 is ₦81,000, whereas the stated straight-line model above leaves ₦60,000. Read the model rather than treating the two methods as interchangeable.