Ratios & Proportions
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Lesson overview
Simplifying and sharing in ratios, combining ratios, direct and inverse proportion, rates, and shares and dividends.
Go to lesson quiz →Exam guidance · JAMB UTME Speed Lens
Pacing
~40 seconds per MCQ (180 questions in 120 minutes, no negative marking).
How JAMB marks this
Fast dimensional/sign elimination and standard formula shortcuts — skip multi-line algebra when substitution tests options faster.
Watch out for
UTME distractors routinely include the intermediate value (e.g., forgetting to convert cm³ to dm³ or °C to Kelvin).
Lesson content
Overview
Three friends who contribute ₦20,000, ₦30,000 and ₦40,000 to start a business expect the profit to be shared fairly. A recipe for 4 people must be scaled up for 10. A road project that takes 8 workers 15 days must be finished in 10. All of these are ratio and proportion problems. UTME questions on this topic are short but come in many disguises: sharing, combining two ratios into one, rates such as speed and exchange rates, and shares and dividends. The lesson builds one dependable method for each.
Key concept: ratio and proportion
A ratio a : b compares two quantities of the same kind. It has no units, and it is simplified like a fraction by dividing every term by the highest common factor: 20 : 30 : 40 = 2 : 3 : 4. Two quantities are in direct proportion if they increase in the same ratio: double one and the other doubles. Their quotient is constant, y/x = k, and the graph is a straight line through the origin. Two quantities are in inverse proportion if one increases in the same ratio as the other decreases: double one and the other halves. Their product is constant, xy = k. A rate compares quantities of different kinds, such as km per hour, naira per litre or naira per dollar. A share is a unit of ownership in a company with a nominal (face) value; its market price may differ. A dividend is a percentage of the nominal value paid to shareholders, and the yield is the dividend as a percentage of the money actually invested.
Methods for ratio problems
Sharing in a ratio. Add the parts to get the total number of shares, find the value of one share, then multiply by each part. Finding the whole from one part. If A's share of 2 parts is ₦8,000, one part is ₦4,000, and the whole is ₦4,000 × (total number of parts). Combining ratios. To join a : b and b : c, make the b terms equal by multiplying each ratio by a suitable number, then read off a : b : c. Direct proportion (unitary method). Find the value for one unit, then multiply. If 5 exercise books cost ₦1,750, one costs ₦350 and 8 cost ₦2,800. Inverse proportion. Find the constant product first. 8 workers taking 15 days is 8 × 15 = 120 worker-days of work, assuming they all work at the same rate. Rates. Use the rate as a multiplier, taking care with units: at ₦1,500 to $1, $250 costs 250 × 1,500 = ₦375,000. Shares and dividends. Money invested = number of shares × market price. Dividend = rate × nominal value × number of shares. Yield = dividend ÷ money invested × 100%.
Formulae to know
Share of a part = (part ÷ sum of parts) × total Direct proportion: y = kx (y/x constant) Inverse proportion: y = k/x (xy constant) Speed = distance ÷ time; average speed = total distance ÷ total time Dividend = rate × nominal value of shares held; yield = dividend ÷ amount invested × 100%
Worked example 1: sharing a profit
Ada, Bayo and Chidi contributed ₦20,000, ₦30,000 and ₦40,000 to a business. They share a profit of ₦45,000 in the ratio of their contributions. Find each person's share.
Active Worked Example · Predict Step 1
4 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Worked example 2: combining two ratios
If a : b = 2 : 3 and b : c = 4 : 5, find a : b : c. If c − a = 21, find a, b and c.
Active Worked Example · Predict Step 1
5 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Worked example 3 (UTME standard): inverse proportion
8 workers can tar a road in 15 days. Working at the same rate, how many workers are needed to finish it in 10 days? How long would 20 workers take?
Active Worked Example · Predict Step 1
5 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Common traps
• Dividing by one of the ratio numbers instead of by the sum of the parts. • Treating inverse proportion as direct: more workers means fewer days. • Combining ratios without first making the common term equal. • Comparing quantities in different units. Convert ₦ and kobo, or hours and minutes, before forming a ratio. • Calculating dividend on the market price. Dividends are on the nominal value.
Summary
• Simplify ratios by the HCF; share by finding the value of one part. • Join a : b and b : c by making b the same in both. • Direct proportion: y/x constant, a straight line through the origin. Inverse proportion: xy constant. • Rates are multipliers with units; check them carefully. • Dividend uses nominal value; yield = dividend ÷ amount invested × 100%.
Mastery Check · Lesson Quiz · pass 60%
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