Lesson 2 of 3 | Foundations: Percentages and Ratios
Ratio and Proportion: Parts, Partnerships, and Applications
Learning outcomes
After this lesson, you will be able to simplify ratios, divide a quantity in a given ratio, use direct and inverse proportion, combine linked ratios, and recognise when a word problem should be translated into proportional parts.
1. Ratio as a comparison
The ratio (a:b) compares two quantities measured in the same units. It is equivalent to the fraction (a/b), but the notation emphasises comparison.
Before forming a ratio, make the units identical. For example:
- 2.5 kg : 750 g
- 2500 g : 750 g
- 10 : 3 after dividing by 250
Simplification rule
Divide every term by the greatest common divisor. The ratio 84:126 becomes 2:3 because both terms are divided by 42.
If a ratio contains fractions, multiply every term by the least common multiple of the denominators. For (1/2:2/3:3/4), multiply by 12:
[ 6:8:9 ]
2. Dividing a quantity in a ratio
If ₹8,400 is divided between A and B in the ratio 3:4, total parts = 7.
- A = 8400 × 3/7 = ₹3,600
- B = 8400 × 4/7 = ₹4,800
Always verify that the shares add back to the original total.
Difference-based shortcut
A and B are in the ratio 5:8 and their difference is ₹1,350. The difference of 3 parts equals ₹1,350, so one part equals ₹450. Therefore A = ₹2,250 and B = ₹3,600.
This is faster than introducing two algebraic variables.
3. Combining two ratios
Suppose A:B = 2:3 and B:C = 4:5. The value representing B must match in both ratios.
- A:B = 2:3; multiply by 4 → 8:12
- B:C = 4:5; multiply by 3 → 12:15
Therefore A:B:C = 8:12:15.
Do not simply write 2:3:5; the middle quantities in the original ratios are not equal.
4. Direct proportion
Two quantities are directly proportional when their ratio remains constant. More workers producing more items in the same time, at the same rate, is a direct relationship.
Worked example
Eight identical machines make 1,200 components in 5 hours. How many components will 12 machines make in 8 hours?
Production is directly proportional to both machines and time:
[ 1200 \times \frac{12}{8} \times \frac{8}{5} = 2,880 ]
Write the direction of each factor before multiplying: more machines → more output; more time → more output.
5. Inverse proportion
Two quantities are inversely proportional when their product stays constant. For a fixed amount of work, more workers generally mean fewer days.
Worked example
Fifteen workers complete a job in 24 days. At the same efficiency, 18 workers need:
[ 15 \times 24 = 18 \times d ]
[ d = 20 \text{ days} ]
The answer must be less than 24 because the workforce increased. This direction check catches many errors.
6. Partnership
Profit is divided in the ratio of capital × time.
A invests ₹60,000 for 12 months. B invests ₹80,000 for 9 months.
- A's weighted capital = 60,000 × 12 = 720,000
- B's weighted capital = 80,000 × 9 = 720,000
They receive equal profit even though B invested more money, because B invested for less time.
If the total profit is ₹54,000, each receives ₹27,000.
7. Mixture and replacement through ratios
A 40-litre mixture has milk and water in the ratio 7:3.
- Milk = 40 × 7/10 = 28 litres
- Water = 40 × 3/10 = 12 litres
If 5 litres of water are added, the new ratio is 28:17. Only the changed component should be adjusted.
For repeated replacement, use the standard remaining-quantity model:
[ \text{Quantity remaining after }n\text{ operations} = V\left(1-\frac{x}{V}\right)^n ]
where (V) is container volume and (x) is replaced each time.
8. Age problems
Age ratios change because the same number of years is added to every person's age, not because the ratio stays fixed.
Present ages of A and B are in the ratio 4:5. After 8 years they will be in the ratio 6:7.
Let present ages be 4x and 5x:
[ \frac{4x+8}{5x+8}=\frac{6}{7} ]
[ 28x+56=30x+48 ]
[ x=4 ]
Present ages are 16 and 20 years.
9. Common exam traps
- Comparing quantities before converting their units.
- Treating inverse proportion as direct proportion.
- Combining ratios without equalising the common term.
- Dividing profit by capital alone when investment durations differ.
- Assuming an age ratio remains constant over time.
- Confusing “ratio increases by 2” with “each term increases by 2”.
10. Timed approach
- Use parts when a total or difference is given.
- Use a common middle term for linked ratios.
- Write D for direct and I for inverse above each changing factor.
- For partnership, build a two-column capital × time table.
- Estimate the direction before exact arithmetic.
Check your understanding
- Divide 1,560 in the ratio 5:7: total parts 12, so shares are 650 and 910.
- A:B = 3:5 and B:C = 10:7. Therefore A:B:C = 6:10:7.
- Twelve workers take 15 days. Twenty workers take 9 days.
- ₹40,000 for 12 months and ₹60,000 for 8 months produce the ratio 480,000:480,000, or 1:1.
- In 32 litres at 5:3, the first component is 20 litres.
Revision recap
Ratios compare equal-unit quantities. Proportional parts turn many word problems into one-step arithmetic. Decide whether a relationship is direct or inverse, equalise shared terms when combining ratios, and use capital × time for partnership.