Learn ratios & proportions
A ratio compares two quantities. A proportion states that two ratios are equivalent and can be solved with a scale factor or cross products.
You will compare quantities, find unit rates, and solve missing-value proportions.
The idea in plain English
A ratio compares two quantities. A unit rate tells the amount for exactly one unit.
A proportion says two ratios describe the same relationship, even when the numbers are scaled.
How to recognise this kind of question
- The problem compares two amounts or prices.
- Words such as per, for every, or at this rate appear.
- One value in two equal ratios is missing.
Quick decision: Asked for one? Divide for a unit rate. Asked for a matching amount? Use the same scale factor or a proportion.
The method
- Identify corresponding quantities in both ratios.
- Find the scale factor or form equal cross products.
- Solve the missing value and substitute it back to verify equivalent ratios.
Worked examples
3:5 = 12:n
- 3 is multiplied by 4 to make 12.
- Multiply 5 by 4 as well.
Answer: n = 20
5 notebooks cost $15. What do 8 cost?
- Find one notebook: $15 ÷ 5 = $3.
- Multiply by 8: $3 × 8.
Answer: $24
Your turn—with help
3 tickets cost $21. What is the cost per ticket?
Need a hint?
Per ticket means find the cost for one.
Show the answer and steps
- Divide total cost by number of tickets.
- $21 ÷ 3 = $7.
Answer: $7 per ticket
Watch out for this mistake
Incorrect: $12 for 3 items means $12 ÷ 4
Why it fails: The divisor must be the actual number of items, which is 3.
Do this instead: $12 ÷ 3 = $4 per item.
Quick check
Try these without looking. Then open each answer to check your thinking.
4 apples cost $6. Cost per apple?
Need a hint?
Divide cost by apples.
Show the answer and steps
- $6 ÷ 4 = $1.50.
Answer: $1.50
Need a hint?
What multiplies 5 to make 20?
Show the answer and steps
- 5 × 4 = 20.
- 2 × 4 = 8.
Answer: x = 8
A car goes 180 miles in 3 hours. Miles per hour?
Need a hint?
Find the distance for one hour.
Show the answer and steps
- 180 ÷ 3 = 60.
Answer: 60 miles per hour
Common mistakes
- Switching the order of only one ratio.
- Adding the scale factor instead of multiplying.
- Comparing quantities with different units before converting them.
Practice tips
- Label what each part of the ratio represents.
- Simplify both ratios to check whether they are equivalent.