A seller mixes two varieties of rice costing Rs. 40/kg and Rs. 60/kg in a certain ratio and sells the mixture at Rs. 55/kg, gaining 10%. What is the ratio in which the two types are mixed?

Correct — A.
First find the cost price of the mixture.

Sold at Rs. 55 with a 10% gain: CP = 55 ÷ 1.1 = Rs. 50/kg

Alligation about the mean cost of 50:
Rs. 40 variety's share = 60 − 50 = 10
Rs. 60 variety's share = 50 − 40 = 10

Ratio = 10 : 10 = 1 : 1 → option (a)

Why the others are wrong

• (b)
  2:3 — 2 : 3 gives a cost of (2 × 40 + 3 × 60) ⁄ 5 = Rs. 52/kg. Selling at Rs. 55 would then earn about 5.8%, not 10%.
• (c)
  3:2 — 3 : 2 gives a cost of (3 × 40 + 2 × 60) ⁄ 5 = Rs. 48/kg. Rs. 55 on that is a gain of about 14.6%, not 10%.
• (d)
  4:1 — 4 : 1 gives a cost of (4 × 40 + 60) ⁄ 5 = Rs. 44/kg, and Rs. 55 on that is a 25% gain, not 10%.

Alligation needs every price on the same footing. Rs. 40 and Rs. 60 are cost prices, but Rs. 55 is a selling price, so strip the 10% gain first: 55 ÷ 1.1 = Rs. 50.

Then apply the rule: cheaper : dearer = (dearer − mean) : (mean − cheaper).

Here the mean, 50, sits exactly halfway between 40 and 60, so equal quantities balance it: 1 : 1.

Key facts

• Cost price = selling price ÷ (1 + gain%)
• Alligation: cheaper : dearer = (dearer − mean) : (mean − cheaper)
• The mean and the two prices must all be cost prices, or all selling prices
• A mean exactly midway between the two prices means a 1 : 1 mix

Common traps

• Using the selling price 55 as the mean, which gives (60 − 55) : (55 − 40) = 1 : 3
• Finding CP as 55 − 10% of 55 = 49.5 instead of 55 ÷ 1.1 = 50

No directly related past PYQ was found.