A group of 30 students appeared in a test. The average score of 12 students is 62, and that of the rest is 76. What is the average score of the group?
- (a)71.4
- (b)70.4
- (c)69.4
- (d)68.4
Answer
Why
Correct — B. The two groups are different sizes, so combine the totals, not the averages.
12 students at 62 → 12 × 62 = 744
The rest = 30 − 12 = 18 students at 76 → 18 × 76 = 1,368
Group total = 744 + 1,368 = 2,112
Average = 2,112 ⁄ 30 = 70.4 → option (b).
Deviation check: take 62 as the base. The 18 students exceed it by 14 each, which is 252 marks, and 252 ⁄ 30 = 8.4, so the mean is 62 + 8.4 = 70.4.
Why the others are wrong
- (a)71.4 — 71.4 needs a group total of 71.4 × 30 = 2,142, which is 30 marks more than the 2,112 the two groups actually score between them.
- (c)69.4 — 69.4 implies a total of 2,082, thirty marks short. It also sits just above 69, the plain average of 62 and 76, which ignores the group sizes entirely.
- (d)68.4 — 68.4 implies a total of 2,052. It also falls below 69, the midpoint of the two averages — impossible when the larger group is the one scoring 76.
Concept
A weighted mean is the grand total divided by the head count, and the weights here are 12 and 18, not 1 and 1.
Two routes get there. Add the group totals and divide by 30. Or work from a base: take the lower average, count how far the other group sits above it, and spread that excess over everyone.
The second route is faster and worth practising, because it also tells you which side of the midpoint the answer must land on before you multiply anything.
The mean has to lie strictly between 62 and 76. Because 18 of the 30 students sit in the 76 group, it must also lie above the midpoint 69.
That one observation kills an option before any arithmetic is done.
Key facts
- Weighted mean = (n₁a₁ + n₂a₂) ⁄ (n₁ + n₂), which is not the same as (a₁ + a₂) ⁄ 2.
- A combined average always lies between the two group averages.
- It lies nearer the average of the larger group, so 18 students at 76 pull the mean above the midpoint 69.
- Deviation shortcut for this item: 62 + (18 × 14) ⁄ 30 = 62 + 8.4 = 70.4.
Study next
Common traps
- Averaging 62 and 76 to get 69 and ignoring that the groups differ in size
- Reading the rest as 30 students instead of 30 − 12 = 18
- Swapping the weights, which gives (18 × 62 + 12 × 76) ⁄ 30 = 67.6
SSC keeps the frame — 30 students split into a group of 12 and the rest — and moves the averages.
The rest score 74 at 11 Sep 2024, 16:00, Quant Q.9 (answer 69.2) and 72 at 13 Sep 2024, 16:00, Quant Q.8 (answer 68). A 12-at-60, rest-at-80 variant runs at 25 Sep 2024, 16:00, Quant Q.20.
Related PYQs
No directly related past PYQ was found.