Quantitative Aptitude · 20 min read

Ratio & Proportion

Ratio and Proportion is the structural topic of the arithmetic section. Partnership, alligation, mixtures, ages, time-and-work and speed problems are all ratio questions underneath, which is why fluency here quietly raises your score on topics that never mention the word "ratio".

The skills that matter are manipulation, not definition: chaining two ratios into one, replacing a ratio with variables so an equation can be written, and reading alligation as a ratio of distances. All three are worked in full below.

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Current affairs · 19 September 2026

Today’s current affairs, checked at the source

Every item is dated, read on the conducting body’s or ministry’s own site, and written with the question it becomes. Read today’s items, take the quiz, or download the month as a PDF.

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19September 2026

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What to note today

  1. 01

    No charges on UPI payments up to ₹2,000 and on RuPay debit cards, by notification

    Economy and banking14 SeptemberMinistry of Finance notification of 14 September 2026

  2. 02

    Retail inflation rose to 4.82 per cent in August 2026

    Economy and banking14 SeptemberMoSPI CPI press release of 14 September 2026

  3. 03

    SEMICON India 2026 opened at Yashobhoomi on the theme “Silicon to Systems”

    Science and technology17–19 SeptemberPrime Minister’s Office note of 16 September 2026; inauguration confirmed by agreeing reports of 17 September

  4. 04

Why it matters

Ratio & Proportion in the exam

Direct question counts move between cycles, so treat these as ranges rather than promises. Check the notification for the pattern you are sitting.
ExamExpected questionsHow it usually appears
SSC CGL / CHSL Tier 12–4 direct questionsPlus partnership and mixture questions built on the same skill.
Banking Prelims & Mains3–5 questionsHeavily used in Data Interpretation, where quantities are compared as ratios.
RRB NTPC / Group D2–4 questionsUsually direct: simplify, divide a sum, find a proportional.
CAT / MBA entrances3–5 questionsMulti-step chaining and mixture-replacement questions.

Start here

What a ratio is

A ratio compares two quantities of the same kind by division. Writing a : b is another way of writing the fraction a/b, and every property follows from that.

Terms
In a : b, a is the antecedent (first term) and b the consequent (second term). Order matters — 3 : 4 and 4 : 3 are different ratios.
Same units only
A ratio compares like with like, so both quantities must be in the same unit before you write it. 500 g to 2 kg is 500 : 2000 = 1 : 4, not 500 : 2.
A ratio has no unit
Because the units cancel, a ratio is a pure number. That is exactly what makes it comparable across different contexts.
Multiplying or dividing both terms changes nothing
a : b = ma : mb for any non-zero m. This is why every ratio has a simplest form — divide both terms by their HCF.
Inverse ratio
The inverse of a : b is b : a. It appears whenever two quantities are inversely proportional, such as speed and time over a fixed distance.
Compounded ratio
The compounded ratio of a : b and c : d is ac : bd — multiply the antecedents and the consequents separately.
Duplicate and sub-duplicate
The duplicate ratio of a : b is a² : b², the triplicate is a³ : b³, and the sub-duplicate is √a : √b. Sub-duplicate of 16 : 25 is 4 : 5.

A ratio is not a quantity

Knowing two salaries are in the ratio 3 : 5 tells you nothing about their size — they might be ₹3 000 and ₹5 000 or ₹30 000 and ₹50 000. You always need one absolute value, or a sum, before a ratio can produce a number.

Core skill

Chaining and combining ratios

Given a : b and b : c, producing a : b : c is the single most-used manipulation in the topic. The trick is to make the shared term identical in both ratios.

Take the LCM of the two values of the shared term and scale each ratio up to match. Once b is the same number in both, the three terms can simply be read off side by side.

If a : b = 2 : 3 and b : c = 4 : 5, find a : b : c.

  1. b appears as 3 in the first ratio and as 4 in the second. Their LCM is 12.
  2. Scale the first by 4: a : b = 8 : 12
  3. Scale the second by 3: b : c = 12 : 15
  4. b is now 12 in both, so the terms line up directly.

Answer: a : b : c = 8 : 12 : 15.

Related manipulations

  • If 2A = 3B = 4C, then A : B : C = 1/2 : 1/3 : 1/4Take the LCM of the denominators (12) and multiply through: 6 : 4 : 3. Verify: 2×6 = 3×4 = 4×3 = 12.
  • If A : B = a : b and C is unrelated, you cannot combine themA shared term is required. Without one, the two ratios carry no common scale.
  • Compounded ratio of a : b and c : d = ac : bdCompound 2 : 3 with 4 : 5 to get 8 : 15.

The big one

The k-method

Whenever a question gives a ratio and then changes it, replace the ratio with variables and write an equation. This one habit solves most of the harder questions in the topic.

If two numbers are in the ratio 5 : 8, write them as 5x and 8x. The single unknown x carries the scale, and any further condition in the question becomes an equation in x that you can solve directly.

Two numbers are in the ratio 5 : 8. If each is increased by 9, the ratio becomes 8 : 11. Find the numbers.

  1. Write the numbers as 5x and 8x.
  2. After the increase: (5x + 9)/(8x + 9) = 8/11
  3. Cross-multiply: 11(5x + 9) = 8(8x + 9)
  4. 55x + 99 = 64x + 72, so 9x = 27 and x = 3.
  5. Check: 24 and 33 are in the ratio 8 : 11.

Answer: The numbers are 15 and 24.

Use one variable, not two

Writing the numbers as 5x and 8y throws away the information the ratio gave you and leaves an unsolvable pair. The whole point of the ratio is that a single scale factor governs both.

Definitions worth knowing

Proportion and proportionals

Four quantities are in proportion when a : b = c : d. The three named "proportionals" are just this equation with one term missing, and each has a one-line formula.

The proportionals

  • a : b = c : d ⟺ ad = bcThe product of the extremes equals the product of the means. a and d are the extremes; b and c the means.
  • Mean proportional between a and b = √(ab)Between 9 and 25 it is √225 = 15. This is the b in a : b = b : c.
  • Third proportional to a and b = b² ÷ aTo 4 and 12 it is 144/4 = 36, because 4 : 12 = 12 : 36.
  • Fourth proportional to a, b and c = (b × c) ÷ aTo 3, 6 and 9 it is 6 × 9 / 3 = 18, because 3 : 6 = 9 : 18.
  • Continued proportion a : b = b : c ⟹ b² = acThe middle term is the mean proportional of the outer two.

Third and mean are not the same

The mean proportional sits between the two given numbers; the third proportional sits after them. For 4 and 12 the mean proportional is √48 ≈ 6.93 while the third proportional is 36. Questions often offer both as options.

Shortcut

Componendo and dividendo

A family of transformations that turn an awkward ratio equation into a clean one. Worth recognising, because a question built for them collapses in one step.

The transformations

  • Componendo: if a/b = c/d then (a + b)/b = (c + d)/dAdd 1 to both sides and simplify.
  • Dividendo: if a/b = c/d then (a − b)/b = (c − d)/dSubtract 1 from both sides.
  • Componendo and dividendo: (a + b)/(a − b) = (c + d)/(c − d)The combined form, and by far the most useful. It removes the need to cross-multiply and expand.
  • Invertendo: a/b = c/d ⟹ b/a = d/cInvert both sides.
  • Alternendo: a/b = c/d ⟹ a/c = b/dSwap the means.

If (3x + 2y) : (3x − 2y) = 5 : 1, find x : y.

  1. Apply componendo and dividendo in reverse, treating a = 3x and b = 2y.
  2. Sum over difference is 5/1, so (a + b)/(a − b) = 5/1.
  3. This gives a/b = (5 + 1)/(5 − 1) = 6/4 = 3/2.
  4. So 3x/2y = 3/2, which means x/y = 1/1.

Answer: x : y = 1 : 1.

Exam classic

Dividing a quantity, and partnership

Splitting an amount in a given ratio is the most common direct question. Partnership is the same operation, with each partner's share weighted by both capital and time.

The formulas

  • Dividing N in the ratio a : b : c gives Na/(a+b+c), Nb/(a+b+c), Nc/(a+b+c)Add the parts first; that total is the denominator for every share.
  • Simple partnership (same duration): profit ratio = capital ratioInvest ₹4 000 and ₹6 000 for the same year → profits split 2 : 3.
  • Compound partnership: profit ratio = capital × timeThe product is often called the monthly equivalent of the investment.
  • Working vs sleeping partnerA working partner's salary or commission is taken out of the profit FIRST; only the remainder is split in the capital ratio.

A invests ₹10 000 for 6 months and B invests ₹15 000 for 8 months. In what ratio should a profit of ₹9 000 be divided?

  1. A's weight = 10 000 × 6 = 60 000
  2. B's weight = 15 000 × 8 = 120 000
  3. Ratio = 60 000 : 120 000 = 1 : 2
  4. Three parts of ₹9 000 gives ₹3 000 per part.

Answer: A receives ₹3 000 and B receives ₹6 000.

High yield

Alligation and mixtures

Alligation finds the ratio in which two things at different rates must be mixed to reach a given average. It is a ratio of distances from the mean, and it works for prices, speeds, percentages and concentrations alike.

The rule of alligation

  • Cheaper quantity : Dearer quantity = (Dearer rate − Mean) : (Mean − Cheaper rate)Each side gets the distance from the mean on the OPPOSITE side. This crossing over is the whole rule.
  • Repeated replacement: pure left = V × (1 − x/V)ⁿV is the vessel, x the amount drawn off and replaced each time, n the number of operations.
  • Mean of a mixture = (q₁r₁ + q₂r₂) ÷ (q₁ + q₂)The weighted average — alligation is just this equation rearranged.

In what ratio must rice at ₹30 a kg be mixed with rice at ₹45 a kg to obtain a mixture worth ₹36 a kg?

  1. Distance of the dearer rate from the mean: 45 − 36 = 9
  2. Distance of the cheaper rate from the mean: 36 − 30 = 6
  3. Cross over: cheaper : dearer = 9 : 6
  4. Simplify by 3.

Answer: 3 : 2 — three parts of the ₹30 rice to two of the ₹45 rice.

A vessel holds 40 litres of pure milk. Four litres are drawn off and replaced with water, and this is done a second time. How much milk remains?

  1. Each operation leaves a fraction (1 − 4/40) = 9/10 of the milk.
  2. After two operations: 40 × (9/10)²
  3. 40 × 81/100

Answer: 32.4 litres of milk (and 7.6 litres of water).

Read this twice

Traps that cost marks

Four errors account for most lost marks here, and every one is a misreading rather than a miscalculation.

Adding ratios term by term
If A : B = 2 : 3 and C : D = 4 : 5, then (A + C) : (B + D) is NOT 6 : 8. Ratios only add when they share a common scale, which two unrelated ratios do not.
Forgetting that order matters
3 : 4 and 4 : 3 are different answers, and both usually appear in the options. Check which quantity the question named first.
Mixing units
A ratio needs both quantities in the same unit. 45 minutes to 2 hours is 45 : 120 = 3 : 8, not 45 : 2.
Alligation the wrong way round
The distance from the mean on one side gives the quantity on the OTHER side. If you do not cross over, you get the reciprocal — which is also in the options.

Solved examples

Worked line by line

Read the steps rather than the answer. The method is what transfers to the next question.

Divide ₹1 200 among three people in the ratio 3 : 4 : 5.

  1. Total parts = 3 + 4 + 5 = 12
  2. One part = 1200 ÷ 12 = ₹100
  3. Multiply out each share.

Answer: ₹300, ₹400 and ₹500.

Find the mean proportional between 9 and 25.

  1. Mean proportional = √(a × b)
  2. √(9 × 25) = √225

Answer: 15, because 9 : 15 = 15 : 25.

Find the third proportional to 4 and 12.

  1. Third proportional = b² ÷ a
  2. 144 ÷ 4

Answer: 36, because 4 : 12 = 12 : 36.

Find the fourth proportional to 3, 6 and 9.

  1. Fourth proportional = (b × c) ÷ a
  2. (6 × 9) ÷ 3

Answer: 18, because 3 : 6 = 9 : 18.

If 2A = 3B = 4C, find A : B : C.

  1. Set all three equal to a constant k, so A = k/2, B = k/3, C = k/4.
  2. A : B : C = 1/2 : 1/3 : 1/4
  3. Multiply through by the LCM of the denominators, 12.
  4. Check: 2 × 6 = 3 × 4 = 4 × 3 = 12.

Answer: A : B : C = 6 : 4 : 3.

If x : y = 3 : 4, find (2x + 3y) : (3x − 2y).

  1. Take x = 3 and y = 4.
  2. 2x + 3y = 6 + 12 = 18
  3. 3x − 2y = 9 − 8 = 1

Answer: 18 : 1.

A mixture of 60 litres has milk and water in the ratio 2 : 1. How much water must be added to make the ratio 1 : 1?

  1. Milk = 60 × 2/3 = 40 litres and water = 20 litres.
  2. Milk does not change, so the final water must also be 40 litres.
  3. Water to add = 40 − 20

Answer: 20 litres.

In what ratio must tea at ₹20 a kg be mixed with tea at ₹32 a kg to get a mixture at ₹26 a kg?

  1. Dearer − mean = 32 − 26 = 6
  2. Mean − cheaper = 26 − 20 = 6
  3. Cross over: cheaper : dearer = 6 : 6

Answer: 1 : 1 — equal quantities, since ₹26 is exactly halfway.

The ratio of two numbers is 7 : 9. If each is increased by 6, the ratio becomes 5 : 6. Find the larger number.

  1. Write the numbers as 7x and 9x.
  2. (7x + 6)/(9x + 6) = 5/6
  3. 6(7x + 6) = 5(9x + 6) gives 42x + 36 = 45x + 30.
  4. 3x = 6, so x = 2 and the numbers are 14 and 18.

Answer: The larger number is 18. (Check: 20 : 24 = 5 : 6.)

From 20 litres of pure milk, 5 litres are removed and replaced by water. This is repeated once more. How much milk is left?

  1. Each operation leaves (1 − 5/20) = 3/4 of the milk.
  2. After two operations: 20 × (3/4)²
  3. 20 × 9/16

Answer: 11.25 litres.

Practice

16 questions on Ratio & Proportion

Work each one out before you reveal the answer — the explanation is where the marks are.

  1. Q1The ratio 48 : 60 in its simplest form is:

    • A4 : 5
    • B8 : 10
    • C3 : 4
    • D6 : 7

  2. Q2₹640 is divided in the ratio 3 : 5. The smaller share is:

    • A₹320
    • B₹240
    • C₹280
    • D₹200

  3. Q3If a : b = 2 : 3 and b : c = 4 : 5, then a : c is:

    • A4 : 5
    • B2 : 5
    • C8 : 15
    • D3 : 4

  4. Q4The mean proportional between 4 and 49 is:

    • A18
    • B26.5
    • C28
    • D14

  5. Q5The third proportional to 6 and 12 is:

    • A24
    • B30
    • C18
    • D36

  6. Q6The fourth proportional to 5, 8 and 15 is:

    • A18
    • B24
    • C20
    • D30

  7. Q7Two numbers are in the ratio 7 : 9. Increasing each by 6 makes the ratio 5 : 6. The larger number is:

    • A14
    • B16
    • C18
    • D20

  8. Q8If 2A = 3B = 4C, then A : B : C equals:

    • A3 : 4 : 6
    • B4 : 3 : 2
    • C2 : 3 : 4
    • D6 : 4 : 3

  9. Q9A and B invest capitals in the ratio 5 : 6 for times in the ratio 6 : 5. Their profits are in the ratio:

    • A1 : 1
    • B5 : 6
    • C6 : 5
    • D25 : 36

  10. Q10In what ratio must rice at ₹20 a kg be mixed with rice at ₹32 a kg to get a mixture at ₹26 a kg?

    • A3 : 2
    • B1 : 1
    • C1 : 2
    • D2 : 1

  11. Q11A 60-litre mixture has milk and water in the ratio 2 : 1. Water to be added to make it 1 : 1 is:

    • A30 litres
    • B10 litres
    • C20 litres
    • D15 litres

  12. Q12The duplicate ratio of 3 : 4 is:

    • A6 : 8
    • B3 : 8
    • C27 : 64
    • D9 : 16

  13. Q13The sub-duplicate ratio of 16 : 25 is:

    • A4 : 5
    • B32 : 50
    • C8 : 12.5
    • D256 : 625

  14. Q14If x : y = 5 : 2, then (3x + 2y) : (2x − y) equals:

    • A17 : 8
    • B19 : 8
    • C15 : 7
    • D19 : 6

  15. Q15From 20 litres of pure milk, 5 litres are removed and replaced with water twice. Milk left is:

    • A12.5 litres
    • B15 litres
    • C11.25 litres
    • D10 litres

  16. Q16Three numbers in the ratio 2 : 3 : 5 add up to 200. The largest is:

    • A120
    • B80
    • C60
    • D100

Questions

Ratio & Proportion — FAQs

What is the difference between ratio and proportion?

A ratio compares two quantities of the same kind — 3 : 4. A proportion states that two ratios are equal — 3 : 4 = 9 : 12. Every proportion contains ratios, but a single ratio on its own is not a proportion.

How do I combine a : b and b : c into a : b : c?

Make the shared term identical in both. If a : b = 2 : 3 and b : c = 4 : 5, the LCM of 3 and 4 is 12, so scale the first ratio by 4 and the second by 3. That gives 8 : 12 and 12 : 15, so a : b : c = 8 : 12 : 15.

What is the k-method in ratio problems?

Replace the ratio with variables sharing one scale factor. Two numbers in the ratio 5 : 8 become 5x and 8x. Any further condition in the question then becomes a single equation in x. Using two different variables discards the ratio and leaves the problem unsolvable.

What is the difference between mean proportional and third proportional?

The mean proportional between a and b is √(ab) and sits between them. The third proportional to a and b is b²/a and comes after them. For 4 and 12 the mean proportional is √48 while the third proportional is 36 — questions usually offer both.

How does the rule of alligation work?

Write the two rates and the required mean, then take each rate's distance from the mean and cross them over. The quantity of the cheaper item is proportional to (dearer rate − mean), and the quantity of the dearer item to (mean − cheaper rate). Forgetting to cross over gives the reciprocal, which is always in the options.

How is profit shared in a partnership?

In the ratio of capital × time for each partner. If A invests ₹10 000 for 6 months and B ₹15 000 for 8 months, the weights are 60 000 and 120 000, so profits split 1 : 2. If one partner draws a salary, deduct it from the profit first and split only the remainder.

What is the formula for repeated replacement in a mixture?

If x litres are drawn from a vessel of V litres and replaced with water, n times, the original liquid remaining is V × (1 − x/V)ⁿ. Drawing 4 litres twice from 40 litres of milk leaves 40 × (0.9)² = 32.4 litres.

Can I add two ratios together?

Not term by term. If A : B = 2 : 3 and C : D = 4 : 5, then (A + C) : (B + D) is not 6 : 8 — the two ratios have no common scale. You can only combine ratios that share a term, and then only by chaining them.

Why do units matter in a ratio?

A ratio compares like with like, so both quantities must be converted to the same unit first. 45 minutes to 2 hours is 45 : 120 = 3 : 8. Writing 45 : 2 compares minutes with hours and is meaningless.

Which topics depend on Ratio and Proportion?

Partnership, alligation and mixtures directly; and less obviously ages, time and work, time-speed-distance, and most Data Interpretation. Speed is inversely proportional to time over a fixed distance, and work rates combine as ratios — so fluency here pays off well beyond the questions labelled "ratio".

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