🧮 Aptitude & Reasoning · Chapter 2

Ratios, proportion and mixtures

A ratio never tells you how much — it tells you how many parts, and every ratio question is solved the moment you find what one part is worth.

Study this lesson free in the app →

The basics

A ratio compares two quantities as parts: milk to water 3:1 means for every 3 cups of milk there is 1 cup of water — whether the pot holds 4 cups or 400.

The problem it solves: splitting, scaling and mixing things when you know the shares but not the amounts.

The mental model: think in parts. 3:1 is 4 parts in total. Once the question gives you any real amount, divide to find what one part is worth, then multiply back. A proportion is just two ratios set equal (a:b = c:d, so a×d = b×c), which is how you scale a recipe or a map.

Mixtures add one idea. When you blend a cheap thing with a dear thing, the blend's price (or strength) lands in between, closer to whichever you used more of. Alligation turns that into a ratio in one line: cheap : dear = (dear − mean) : (mean − cheap).

Explained

₹1,200 split between Ravi and Sana in the ratio 3:5.

Don't think "3 out of what?". Think 8 parts. One part = 1,200 ÷ 8 = ₹150. Ravi gets 3 parts = ₹450, Sana gets 5 parts = ₹750.

That one move — find the value of one part — solves most ratio questions you will meet in a placement test.

The chai stall that never measures

Raju's chai stall near Dadar station makes tea in a 2-litre pan in the morning and a 6-litre pan at 5 pm when the office crowd arrives.

He never measures litres. He measures parts: 2 of milk to 1 of water, whatever the pan. The bigger pan just has a bigger part.

One evening his helper tops up a half-used pan with only water. The tea tastes thin — not because there is less tea, but because the ratio changed. Pouring out a glass never thins the tea; adding just one ingredient always does.

That is the whole topic: ratios are about shares, and only adding or replacing changes the shares.

The method, then the shortcut

Standard method (always works). Rice at ₹40/kg and ₹60/kg is mixed to sell at ₹52/kg. Ratio?

Let x kg at 40, y kg at 60.
Cost balances:   40x + 60y = 52(x + y)
                 60y − 52y = 52x − 40x
                       8y  = 12x
                       x:y = 8:12 = 2:3

Shortcut (alligation). Distances to the mean, crossed over:

cheap : dear = (60 − 52) : (52 − 40) = 8 : 12 = 2 : 3

Same answer in five seconds, because each cheap kilo gains ₹12 toward the mean and each dear kilo loses ₹8 — gains must equal losses.

When the shortcut fails:

• Three or more ingredients — infinitely many ratios work; you need more information.
• Wrong quantity in the cross — mixing 3:1 and 5:3 vessels? Use the milk fraction (3/4, 5/8), not 3 and 5.
• Mean outside the range — ₹65 from ₹40 and ₹60 is impossible; the question is lying or you misread it.

Picture it

Ratio bar — ₹1,200 in 3:5:

Ravi │■■■│              3 parts
Sana │■■■■■│            5 parts
      └─ 8 parts = ₹1,200 → 1 part = ₹150
Ravi = 3 × 150 = ₹450      Sana = 5 × 150 = ₹750

Alligation cross — rice at ₹40 and ₹60, mix at ₹52:

   cheap 40            dear 60
          \            /
           \          /
             mean 52
           /          \
          /            \
   60 − 52 = 8      52 − 40 = 12
   (cheap qty)      (dear qty)

   cheap : dear = 8 : 12 = 2 : 3

Read the cross diagonally: each number lands under the opposite ingredient.

Solving any ratio or mixture question

The accent step is where marks are won: the quantity that does not change is your anchor.

The milkman's two rounds

A can holds 40 litres of pure milk. The milkman draws off 4 litres and tops it up with water. Then he does it again: draws 4 litres of the mixture and tops up with water.

TCS NQT, question 14, 70 seconds left

Vessel A has milk and water in the ratio 3:1. Vessel B has them in the ratio 5:3. In what ratio should they be mixed to get milk and water in the ratio 2:1?

Step 1 — convert to one per-unit quantity: the fraction that is milk.

A: 3/(3+1) = 3/4      B: 5/(5+3) = 5/8      target: 2/(2+1) = 2/3

Step 2 — use 24ths so nothing is fiddly: A = 18/24, B = 15/24, target = 16/24.

Step 3 — alligation:

A : B = (16 − 15) : (18 − 16) = 1 : 2

Step 4 — check in five seconds: 1 part of A + 2 parts of B → milk = 3/4 + 2 × 5/8 = 3/4 + 5/4 = 2 out of 3 parts → 2:1 ✓

The candidate who ran out of time crossed 3 and 5 against 2. Ratios are not per-unit values; fractions are.

The traps examiners set

• Chaining without a common term — A:B = 2:3 and B:C = 4:5 does not give A:C = 2:5. Make B = 12: A:B:C = 8:12:15, so A:C = 8:15
• Adding ratios across vessels — 3:1 plus 5:3 is not 8:4. Different vessel sizes, different-sized parts. Convert to actual litres or fractions first
• Alligation on the wrong number — cross per-unit values (₹/kg, % strength, milk fraction), never raw ratio terms
• Reversing the answer — the cross gives the distance under the opposite ingredient. Check: the mix should sit nearer the one you used more of
• Treating replacement as subtraction — 4 L out twice from 40 L leaves 32.4 L of milk, not 32
• Dividing by value instead of count — ₹180 in coins of ₹1, 50p, 25p with number ratio 2:3:4 is not 180 split 2:3:4

Words to know

In two minutes

• A ratio counts parts, not amounts: 3:5 means 8 parts, and any real figure tells you what one part is worth.
• To split ₹1,200 in 3:5, one part is 1,200/8 = 150, so the shares are ₹450 and ₹750.
• To chain A:B and B:C, make B the same number in both, then read A:B:C straight across.
• Taking some mixture out never changes its ratio; adding only one ingredient always does.
• Alligation: cheap : dear = (dear − mean) : (mean − cheap) — the quantity ratio is the inverse of the distances.
• Alligation works on per-unit values of the same unit — price per kg, or the fraction of milk in each vessel, never the milk:water ratio itself.
• Replacing x litres of a V-litre vessel n times leaves V × (1 − x/V)ⁿ of the original liquid, never V − n·x.

Test yourself

There are 10 questions, 3 puzzles and 5 recall cards for this lesson in the app — the recall cards come back on later days, just before you would forget.