In what ratio, sugar costing ₹78 per kg be mixed with sugar costing ₹46 per kg such that by selling the mixture at ₹75 per kg there is a gain of 20%?

Correct — B. The 20% gain is on the mixture, so find the mixture's cost price first, then run alligation on it.

Cost price of the mixture = 75 ÷ 1.20 = ₹62.50 per kg

Alligation crosses the two gaps around 62.50:
62.50 − 46 = 16.50 → the share of the ₹78 sugar
78 − 62.50 = 15.50 → the share of the ₹46 sugar

₹78 sugar : ₹46 sugar = 16.50 : 15.50 = 33 : 31 → option (b).

Why the others are wrong

• (a)
  37 : 31 — 37 is not one of this mixture's gaps. The 31 is right — it is twice 15.50 — but its partner has to be twice 16.50, which is 33.
• (c)
  31 : 29 — 31 : 29 drops the 33 altogether. The only two numbers this alligation can produce are 16.50 and 15.50 (doubled, 33 and 31), and 29 is neither.
• (d)
  33 : 37 — 33 : 37 makes the ₹46 sugar the larger share. But ₹62.50 sits closer to 78 (a gap of 15.50) than to 46 (a gap of 16.50), so the dearer sugar has to dominate.

Alligation is the weighted-average rule written backwards. Mix quantities q₁ and q₂ at prices p₁ and p₂ and the mixture costs m, where m sits between them.

The rule: q₁ : q₂ = (p₂ − m) : (m − p₁). Each variety's quantity is proportional to the gap on the far side of the mean, which is why the differences cross.

Watch the first step. The ₹75 in the stem is a selling price, and alligation only works on cost prices, so the 20% gain has to be stripped out before any gap is measured.

Key facts

• Cost price of a mixture = selling price ÷ (1 + gain), so ₹75 at a 20% gain means ₹62.50 per kg.
• In alligation each quantity is proportional to the price gap on the opposite side of the mean.
• The variety priced nearer the mean is always present in the larger quantity.
• Here the gaps are 62.50 − 46 = 16.50 and 78 − 62.50 = 15.50, giving 33 : 31.

Common traps

• Applying the 20% gain to ₹78 or ₹46 instead of to the mixture.
• Answering 31 : 33 when the stem names the ₹78 sugar first.
• Writing the gaps without crossing them, which inverts the ratio.

No directly related past PYQ was found.