MENSURATION · TEXTBOOK LESSON
Ratio compares quantities multiplicatively. Proportion describes relationships in which ratios remain equal, while rates compare quantities measured in different units.
Learning objectives
By the end of this lesson, you should be able to:
- Write and simplify ratios.
- Share quantities in a given ratio.
- Solve direct proportion problems using scale factors or unit rates.
- Interpret rates such as speed, cost per item and density.
1. Key vocabulary and definition
Definition
The ratio a:b compares a with b. Equivalent ratios are formed by multiplying or dividing every part by the same non-zero number.
ratioproportionscale factorunit rateconstant of proportionality
2. Essential facts and rules
These are the facts you should know before attempting the worked example.
Key mathematical facts
- Ratios should usually be simplified by dividing all parts by their highest common factor.
- When sharing in a ratio, first find the total number of ratio parts.
- Direct proportion has the form y = kx for constant k.
3. Standard method
A reliable method helps prevent errors and makes your reasoning easy to follow.
1
Write the quantities in the same units where necessary.
2
Simplify or identify the scale factor.
3
For sharing, add the ratio parts and find the value of one part.
4
Scale each part or quantity consistently.
5
Check that the final parts add to the original total.
4. Worked example
Worked example
Share 84 in the ratio 2:5.
1
Total ratio parts = 2 + 5 = 7.
2
One part = 84 ÷ 7 = 12.
3
First share = 2 × 12 = 24.
4
Second share = 5 × 12 = 60. Check: 24 + 60 = 84.
5. Common mistakes
Watch out for these
- Starting calculations before deciding what the question is asking.
- Skipping important steps or changing notation midway through a solution.
- Accepting an answer without checking whether its size, sign or unit is sensible.
6. Video lesson
This topic is ready for a matching video lesson. The written textbook lesson is fully available now; additional videos can be added to the same lesson template.
7. Practice exercise
Complete these without looking at the answers first.
- Simplify 18:30.
- Share 96 in the ratio 5:3.
- If 4 notebooks cost 28 AED, find the cost of 7 at the same rate.
- A map scale is 1:50,000. What real distance is represented by 6 cm?
- If y is directly proportional to x and y=18 when x=6, find y when x=10.
Show answers
- 3:5
- 60 and 36
- 49 AED
- 3 km
- 30
8. Chapter summary
- Understand the underlying idea before memorising a procedure.
- Use precise mathematical notation and show a logical method.
- Always check the final result.