Lessons / Quantitative aptitude
Quantitative aptitude · Day 2

Ratio and Proportion

A ratio is a recipe, not an amount. Write amounts as 'parts', find what one part is worth, then multiply.

Practise (9 questions) Present as slides

Why this topic matters

If a recipe uses 2 cups of rice for every 3 cups of water, the ratio is 2 : 3. The actual amounts can be 4 and 6 or 10 and 15. A ratio is a pattern of parts.

Ratios drive partnership, ages, mixtures, time and work and DI. The single best tool is the bar model: draw the parts as equal boxes and find what one box is worth.

What a ratio means: parts

a : b means for every a parts of the first there are b parts of the second. Multiply or divide both by the same number and the ratio is unchanged: 4 : 6 = 2 : 3.

To split an amount in the ratio a : b, the total is a + b parts. One part = amount / (a + b).

Ravi2 partsSita3 parts
Rs 1,000 in the ratio 2 : 3 is 5 parts, so one part = 200
Example 1Divide Rs 1,000 between Ravi and Sita in the ratio 2 : 3.
  1. Total parts = 2 + 3 = 5.Each box in the bar model is one part.
  2. One part = 1000 / 5 = 200.
  3. Ravi 400, Sita 600.

Combining ratios

If A : B = 2 : 3 and B : C = 4 : 5, make B the same in both. LCM of 3 and 4 is 12: A : B = 8 : 12 and B : C = 12 : 15, so A : B : C = 8 : 12 : 15.

Compound ratio of a : b and c : d is ac : bd.

Example 1A : B = 3 : 4 and B : C = 6 : 7. Find A : C.
  1. Make B equal: LCM(4, 6) = 12.B must be the same number of parts in both ratios.
  2. A : B = 9 : 12, B : C = 12 : 14.
  3. A : C = 9 : 14.

Changing a ratio by adding or removing

When something is added to both terms, the difference between them does not change. Use that: it is the easiest thing to keep track of.

Set up the new ratio and solve for the unknown, or try the options.

Example 1Two numbers are in the ratio 3 : 5. If 9 is added to each, the ratio becomes 3 : 4. Find the numbers.
  1. Numbers 3k and 5k. (3k + 9)/(5k + 9) = 3/4.
  2. 4(3k + 9) = 3(5k + 9) → 12k + 36 = 15k + 27 → k = 3.Cross-multiply.
  3. Numbers 9 and 15.

Proportion: direct and inverse

Four numbers are in proportion if a : b = c : d, i.e. a × d = b × c (product of extremes = product of means).

Direct proportion: both go up together (more items, more cost). Inverse proportion: one goes up as the other goes down (more workers, fewer days).

Mean proportional between a and b is √(ab). Third proportional to a and b is b2/a. Fourth proportional to a, b, c is bc/a.

Example 1Find the fourth proportional to 4, 9 and 12.
  1. 4 : 9 = 12 : x.The fourth term completes the proportion.
  2. 4x = 9 × 12 = 108.Extremes = means.
  3. x = 27.
Example 2Find the mean proportional between 9 and 16.
  1. √(9 × 16) = √(144).
  2. = 12.

Coins, ages and other ratio stories

A bag has 1-rupee, 50-paise and 25-paise coins in the ratio 5 : 6 : 8 and a total value of Rs 210. Value ratio = 5 × 1 : 6 × 0.5 : 8 × 0.25 = 5 : 3 : 2. So the value splits 105 : 63 : 42 and the 1-rupee coins number 105.

Example 1A bag has 1-rupee, 50-paise and 25-paise coins in the ratio 5 : 6 : 8, worth Rs 210 in all. How many 1-rupee coins?
  1. Value per 'set' of coins: 5 × 1 + 6 × 0.5 + 8 × 0.25 = 5 + 3 + 2 = 10 rupees.Turn the count ratio into money.
  2. Number of sets = 210 / 10 = 21.
  3. 1-rupee coins = 5 × 21 = 105.

Formula summary

Combine a:b and b:c
Make the common term equal: multiply across.
Mean proportional of a, b
√(ab).
Third proportional of a, b
b2 / a.
Fourth proportional of a, b, c
bc / a.
Divide in 1/a : 1/b : 1/c
Multiply by the LCM of a, b, c to get whole numbers first.

Shortcuts and tricks

Coins by value

Use it when: Bag of coins given in ratio by COUNT, total given in rupees.

  1. Convert count ratio to value ratio: multiply each by its coin value.
  2. Share the total by value ratio, then divide by coin value.
ExampleRs 1, 50p, 25p coins in count ratio 5:6:8, total Rs 210. How many 50p coins?
  1. Value ratio 5 : 3 : 2 = 10 parts, each Rs 21.
  2. 50p value = 63 → 126 coins.

Common mistakes

Adding the same number to both terms and keeping the ratio.
Adding changes the ratio; multiplying does not.
Combining A : B and B : C without making B equal.
Scale both ratios so B has the same number.

Before you move on, you should be able to...

  • split an amount in a given ratio with the bar model
  • combine two ratios through the common term
  • find mean, third and fourth proportionals

Practice questions (9)

Try each one before opening the solution. Or practise them one by one so your score is saved.

easy A:B = 2:3 and B:C = 4:5. A:B:C?
  1. Make B equal: 8:12 and 12:15.

Answer: 8:12:15

easy Divide Rs 1,300 in the ratio 1/2 : 1/3 : 1/4. Largest share?
  1. x12: 6:4:3, 13 parts of 100.

Answer: Rs 600

easy Mean proportional of 9 and 16?
  1. √(144).

Answer: 12

easy Third proportional to 4 and 6?
  1. 36/4.

Answer: 9

medium Two numbers are 3:5. Adding 10 to each makes it 5:7. The larger number?
  1. (3x+10)/(5x+10) = 5/7 → x = 5.

Answer: 25

medium What must be added to each of 6, 10, 14 and 22 so they become proportional?
  1. (6+x)(22+x) = (10+x)(14+x) → 132 + 28x = 140 + 24x → x = 2.

Answer: 2

hard Salaries of A and B are 2:3. Both get Rs 4,000 raises and the ratio becomes 3:4. B's new salary?
  1. (2x + 4000)/(3x + 4000) = 3/4 → 8x + 16000 = 9x + 12000 → x = 4000.
  2. B new = 12000 + 4000.

Answer: Rs 16,000

hard Incomes of A and B are 5:4, expenses 3:2, and each saves Rs 1,600. A's income?
  1. 5x - 3y = 1600 and 4x - 2y = 1600.
  2. y = 2x - 800 → 5x - 6x + 2400 = 1600 → x = 800.

Answer: Rs 4,000

hard Rs 5,625 is shared so that A gets half of what B and C together get, and B gets a quarter of what A and C together get. C's share?
  1. A = (B + C)/2 → A = 1/3 of total = 1875.
  2. B = (A + C)/4 → B = 1/5 of total = 1125.
  3. C = 5625 - 3000.

Answer: Rs 2,625

Class plan (2 hours, for trainers)

A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.

10 minWarm-up.
20 minRatio as parts with the bar model; splitting money.
15 minCombining ratios.
20 minAdding to both terms.
20 minProportion: direct, inverse, mean/third/fourth proportional.
10 minCoins.
20 minPractice set.
5 minRecap.
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