Ratios and Proportions
Ratios and proportions are fundamental quantitative concepts that express relationships between numbers. They appear extensively in Post-UTME questions, everyday life (recipes, maps, scale models), and are critical for solving word problems. A solid understanding of how to form, simplify, and apply ratios will give you an advantage in virtually every quantitative section of the exam.
Understanding Ratios
A ratio is a comparison of two or more quantities of the same kind, expressed in the same units.
How to Express a Ratio
- Using colon: a : b (read “a to b”)
- As a fraction: a/b
- In words: “a is to b”
Example: A class has 20 boys and 30 girls. Ratio of boys to girls = 20:30 = 2:3
Rules for Ratios
- Both quantities must be in the same unit
- The order matters — 3 : 5 is NOT the same as 5 : 3
- Ratios can be simplified by dividing by their HCF
- Multiplying or dividing both terms by the same non-zero number does NOT change the ratio
Simplifying: 20:30 → Divide both by 10 → 2:3 With decimals: 1.5:2.5 → Multiply by 10 → 15:25 → Divide by 5 → 3:5
Types of Ratios
Compound Ratio
The ratio formed by multiplying corresponding terms of two or more ratios. If the ratios are a:b and c:d, the compound ratio is (a×c):(b×d).
Example: Find the compound ratio of 2:3 and 4:5. Compound ratio = (2×4):(3×5) = 8:15
Dividing a Quantity in a Given Ratio
Example: Divide ₦50,000 in the ratio 3:7 between A and B.
- Total ratio parts = 3 + 7 = 10
- A’s share = (3/10) × 50,000 = ₦15,000
- B’s share = (7/10) × 50,000 = ₦35,000
Proportions
A proportion is an equality of two ratios. If a:b = c:d, we write a/b = c/d, or a:b::c:d.
The Terms of a Proportion
In a:b::c:d:
- Extremes: a and d (the first and last terms)
- Means: b and c (the middle terms)
- Property: The product of extremes = the product of means: a × d = b × c
Example: Find x in 3:x::12:20
- 3/ x = 12/20 → Cross-multiply: 3 × 20 = 12 × x → 60 = 12x → x = 5
Continued Proportion
Three quantities a, b, c are in continued proportion if a:b = b:c. This means b² = a × c, and b is called the mean proportional.
Example: Find the mean proportional between 4 and 16. b² = 4 × 16 = 64 → b = √64 = 8
Direct and Inverse Proportion
Direct Proportion (Direct Variation)
Two quantities are in direct proportion when an increase in one causes a proportional increase in the other.
- x ∝ y or x = ky
- As one increases, the other increases at the same rate
Example: Distance ∝ Time at constant speed. If a car travels at 60 km/h, 3 hours = 180 km.
Inverse Proportion (Indirect Variation)
Two quantities are in inverse proportion when an increase in one causes a proportional decrease in the other.
- x ∝ 1/y or xy = k (constant)
Example: If 5 men can build a wall in 20 days, how long will 8 men take?
- More men → Less days → Inverse proportion
- 5 × 20 = 8 × x → 100 = 8x → x = 12.5 days
The Unitary Method
The unitary method involves finding the value of one unit and then multiplying to find the value of the required number of units.
Example: If 8 books cost ₦2,400, how much will 12 books cost?
- Cost of 1 book = 2400/8 = ₦300
- Cost of 12 books = 300 × 12 = ₦3,600
Rates
A rate is a ratio between two different kinds of quantities.
- Speed = Distance/Time (e.g., 60 km/h)
- Population density = Population/Area
- Unit rate = Price per unit (e.g., ₦250/kg)
Unit Rate: Used for comparison shopping.
- Rice A: ₦5,000 for 20 kg = ₦250/kg
- Rice B: ₦3,600 for 12 kg = ₦300/kg
- Rice A is cheaper per kg.
Key Post-UTME Exam Facts
- Cross-multiplying: In a:b::c:d, product of extremes (a×d) = product of means (b×c)
- Direct: x/y = constant (x increases when y increases)
- Inverse: xy = constant (more of one means less of the other)
- Speed: Speed = Distance/Time; Distance = Speed × Time; Time = Distance/Speed
- ⚡ Exam tip: “More workers take less time” = INVERSE. “More distance takes more time” = DIRECT.
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision before your exam.
- Ratio: Comparison of same-kind quantities; express as a:b or a/b; simplify by dividing by HCF
- Compound ratio: (a×c):(b×d) for ratios a:b and c:d
- Proportion: Equality of two ratios; a:b::c:d → a×d = b×c
- Mean proportional: For a and c, mean proportional b = √(a×c)
- Direct proportion: x/y = k (same rate)
- Inverse proportion: xy = k (opposite rates)
- ⚡ Exam tip: The most common mistake is confusing direct with inverse proportion
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Working with Ratios in Word Problems
Age Problems
“The ratio of the ages of a father and son is 5:2. If the father is 40 years old, how old is the son?”
- 5 parts = 40 years → 1 part = 8 years → Son = 16 years
Scales and Maps
“A map uses a scale of 1:500,000. If two cities are 3.5 cm apart on the map, what is their actual distance?”
- 1 cm on map = 500,000 cm in reality
- Actual = 3.5 × 500,000 = 1,750,000 cm = 17.5 km
Work and Rate Problems
“If A can complete a job in 6 days and B can complete it in 9 days, how many days together?” A’s rate = 1/6; B’s rate = 1/9; Combined = 1/6 + 1/9 = (3+2)/18 = 5/18 Time = 1 ÷ (5/18) = 18/5 = 3.6 days
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer study timeline.
More Complex Ratio Problems
Sharing Profit in a Ratio
Three partners invest ₦20,000, ₦30,000, and ₦50,000. Profit = ₦200,000. Investment ratio = 20000:30000:50000 = 2:3:5 Total parts = 10 A: (2/10)×200,000 = ₦40,000; B: ₦60,000; C: ₦100,000
Mixing Alloys
“How much of 30% copper alloy must be mixed with 60% copper alloy to get 100 kg of 45% copper?” 0.30x + 0.60(100−x) = 0.45 × 100 0.30x + 60 − 0.60x = 45 −0.30x = −15 → x = 50 kg of each alloy
Content adapted based on your selected roadmap duration. Switch tiers using the selector above.