How much rice of ₹4 per kg should be mixed to 15 kg of rice of ₹10 per kg, so as to make a mixture worth ₹6.50 per kg?

Correct — B. Put the two prices and the required mean into alligation.

Cheaper ₹4, dearer ₹10, mean ₹6.50
Cheap share = 10 − 6.50 = 3.5, dear share = 6.50 − 4 = 2.5
Cheap : dear = 3.5 : 2.5 = 7 : 5
The dearer rice is 15 kg, so 5 parts = 15 kg and 1 part = 3 kg

Cheap rice = 7 × 3 = 21 kg → option (b).

Why the others are wrong

• (a)
  40 kg — 40 kg of ₹4 rice with 15 kg of ₹10 rice averages (160 + 150) ⁄ 55 = ₹5.64 per kg, well below the ₹6.50 the question demands.
• (c)
  30 kg — 30 kg gives (120 + 150) ⁄ 45 = ₹6.00 per kg. It answers a 2 : 1 split, which is what the alligation would give if the mean were ₹6 rather than ₹6.50.
• (d)
  15 kg — 15 kg against 15 kg gives (60 + 150) ⁄ 30 = ₹7.00 per kg. Equal quantities always land on the plain midpoint of ₹4 and ₹10, never on ₹6.50.

Alligation measures each price against the required mean and then crosses the gaps over.

The distance from the mean up to the dearer price, 10 − 6.50 = 3.50, becomes the share of the cheaper rice. The distance down to the cheaper price, 6.50 − 4 = 2.50, becomes the share of the dearer.

So cheap : dear = 3.50 : 2.50 = 7 : 5. The same answer falls out of one equation if you prefer algebra: 4q + 150 = 6.50(q + 15) gives q = 21.

Key facts

• The alligation ratio is (dearer − mean) : (mean − cheaper), and it reads as cheaper : dearer.
• Here that is (10 − 6.5) : (6.5 − 4) = 7 : 5, so 21 kg of ₹4 rice joins 15 kg of ₹10 rice.
• 21 kg at ₹4 and 15 kg at ₹10 cost ₹234 for 36 kg, which is exactly ₹6.50 per kg.

Common traps

• Crossing the gaps the wrong way and reporting the dearer share instead.
• Reading the 15 kg as the total mixture rather than as the dearer part alone.

No directly related past PYQ was found.