A merchant mixes two varieties of sugar of 5 kg and 3 kg, which costs him ₹4 and ₹6 per kg, respectively. What is the cost price (in ₹) of the mixture per kg?
- (a)4.35
- (b)4.45
- (c)4.75
- (d)4.21
Answer
Why
Correct — C. The cost price of a mixture is the total money spent divided by the total weight, never the plain average of the two rates.
5 kg at ₹4 = ₹20
3 kg at ₹6 = ₹18
Total = ₹38 for 8 kg
38 ÷ 8 = ₹4.75 per kg
So the mixture costs ₹4.75 a kilogram → option (c). As a weighted average: ₹4 + (3⁄8) × ₹2 = ₹4.75, since only 3 of the 8 kg carry the extra ₹2.
Why the others are wrong
- (a)4.35 — At ₹4.35 a kg the 8 kg would have cost ₹34.80. The merchant actually laid out ₹20 + ₹18 = ₹38, and every rupee of that has to reappear in the rate.
- (b)4.45 — ₹4.45 × 8 = ₹35.60, which is ₹2.40 short of the ₹38 spent. The per-kg figure has to account for the whole outlay.
- (d)4.21 — ₹4.21 × 8 = ₹33.68, the furthest of the four from ₹38. It sits far nearer the ₹4 rate than a 5 : 3 split of the weights allows.
Concept
This is a weighted average, and the weights are the quantities. Total cost divided by total quantity always lands between the two rates, and leans towards whichever rate carries more weight.
Here 5 of the 8 kg cost ₹4 and 3 cost ₹6, so the mixture must sit below the midpoint of ₹5. ₹4.75 is 0.75 above the cheaper rate and 1.25 below the dearer, in the ratio 3 : 5.
That inversion is the rule of alligation: the distances from the mean stand in the reverse ratio of the quantities mixed.
The question asks for cost price, not selling price, so no profit or loss enters the working. The ₹4 and ₹6 are rates per kilogram, not totals.
Key facts
- Cost price per unit of a mixture = total cost ÷ total quantity.
- Alligation: the distances of the mean from the two rates are in the inverse ratio of the quantities mixed.
- 5 kg at ₹4 and 3 kg at ₹6 give ₹38 for 8 kg, so ₹4.75 per kg.
- A mixture rate always lies strictly between the two component rates.
Study next
Common traps
- Averaging ₹4 and ₹6 to get ₹5, which throws away the unequal weights.
- Swapping the weights and pricing 5 kg at ₹6 with 3 kg at ₹4, which gives ₹5.25 a kg.
- Reading 5 : 3 as a ratio of prices rather than of quantities.
The numbers are small enough that the direct total-cost route beats setting up alligation, which earns its keep only when the mean is given and a quantity is hidden.
The same combined-average idea runs the table item at 12 Sep 2024, 16:00, Quant Q.23, where two car types are averaged together across six entries.
Related PYQs
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