Ratio & Proportion — Full Lesson
Lesson objectives
- Understand the meaning of ratio and how to write it.
- Use ratios to compare quantities and simplify them.
- Understand proportion and solve problems using cross-multiplication.
- Solve real-world word problems involving ratio and proportion.
Key definitions & notation
Ratio: a comparison of two quantities by division. If there are $a$ apples and $b$ oranges we write the ratio as $a:b$ or $\dfrac{a}{b}$. Example: if there are 6 apples and 4 oranges the ratio apples:oranges is $6:4 = 3:2$ after simplification.
Proportion: an equality between two ratios. For example, \frac{a}{b} = \frac{c}{d} . If this holds we say a:b::c:d or simply \dfrac{a}{b}=\dfrac{c}{d}.
Cross-multiplication: to test or solve a proportion \dfrac{a}{b}=\dfrac{c}{d} multiply across to get a\cdot d = b\cdot c.
Worked examples (with step-by-step solutions)
Example 1 — Simplify a ratio
Given the ratio $24:18$, simplify it.
Show steps
- Find the greatest common divisor (GCD) of 24 and 18, which is 6.
- Divide both parts by 6: $24\div6=4$, $18\div6=3$.
- So $24:18 = 4:3$.
Example 2 — Solve a proportion by cross-multiplication
Solve for $x$ in the proportion $\dfrac{3}{4} = \dfrac{x}{12}$.
Show steps
- Cross-multiply: $3\cdot12 = 4\cdot x$.
- Compute left side: $36 = 4x$.
- Divide both sides by 4: $x = 9$.
Example 3 — Word problem (scaling)
If a map scale is 1:50,000, a distance of 3 cm on the map represents how many km in reality?
Show steps
- Ratio is $1:50\,000$ which means 1 cm on map = 50,000 cm real.
- 3 cm on map = $3\times50,000 = 150,000$ cm real.
- Convert to kilometres: $150,000\text{ cm} = 1,500\text{ m} = 1.5\text{ km}$. (Because 100 cm = 1 m and 1000 m = 1 km.)
- Answer: $1.5\text{ km}$.
Quick rules & tips
- Always simplify ratios by dividing by the GCD.
- When solving proportions use cross-multiplication and isolate the unknown.
- Check units in word problems — convert everything to the same unit before forming ratios.
- Proportions can be used for scaling, recipes, mixture problems, speed/time/distance, and more.
Practice problems — try them, then reveal answers
Problem 1: Simplify the ratio $45:60$.
Problem 2: Find $x$ if $\dfrac{5}{x} = \dfrac{15}{12}$.
Problem 3 (word): A recipe calls for ingredients in the ratio sugar:flour = 2:5. If you want to use 300 g of flour, how much sugar do you need?
Problem 4 (real world): A car travels 180 km using 12 litres of petrol. How many litres are needed for 450 km, assuming constant consumption?
Problem 5 (challenge): If $\dfrac{x+1}{x-1} = \dfrac{6}{5}$, find $x$.
Interactive trials (mini-quiz)
Q1: Which of these ratios is equivalent to $8:12$?
Q2: Solve: $\dfrac{7}{x}=\dfrac{21}{9}$
Q3: A map uses scale 1:100,000. 5 cm on map corresponds to how many km?
Answers & Worked Solutions
Practice answers
- Problem 1: $45:60=3:4$.
- Problem 2: $x=4$.
- Problem 3: sugar = 120 g.
- Problem 4: 30 litres.
- Problem 5: $x=11$.
Quiz answers
- Q1: both (a) 4:6 and (c) 16:24 are equivalent to 8:12 (they simplify to 2:3). The best single answer in the options is (c) 16:24 if only one allowed.
- Q2: Solve $7/x = 21/9$: cross-multiply $7\cdot9 =21x\Rightarrow63=21x\Rightarrow x=3$. So (a).
- Q3: Scale 1:100,000 means 1 cm = 1 km? (No) — compute: 1 cm = 100,000 cm = 1,000 m = 1 km. So 5 cm = 5 km. Answer (b).
