The average age of the boys in a class is 12 years. The average age of the girls in the class is 11 years. There are 50% more girls than boys in the class. Which one of the following is the average age of the class (in years)?
- (a)11.2 years
- (b)11.4 years
- (c)11.6 years
- (d)11.8 years
Correct — B, 11.4 years. Only the ratio of boys to girls matters, so take the smallest convenient numbers. Let there be 2 boys; fifty per cent more girls than boys means 3 girls. Total age of the boys is 2 × 12 = 24 years and of the girls 3 × 11 = 33 years, so the class totals 57 years across 5 children. The average is 57 ÷ 5 = 11.4 years. The general working gives the same thing: with b boys and 1.5b girls, total age is 12b + 11 × 1.5b = 12b + 16.5b = 28.5b over 2.5b students, and 28.5 ÷ 2.5 = 11.4. The answer must lie between 11 and 12 and closer to 11, because the girls outnumber the boys — which alone rules out two of the four options.
- (a)11.2 years — This is the weighted average for a boys-to-girls ratio of 1 : 4, not 2 : 3. It pulls the mean too far towards the girls' age of 11.
- (c)11.6 years — This is the mirror image of the correct answer — it is what you get if there are fifty per cent more boys than girls, that is 3 boys to 2 girls. Reversing the ratio is the trap the option is built for.
- (d)11.8 years — Too close to the boys' average of 12, which would need the boys to outnumber the girls four to one. Nothing in the stem supports that.
A combined average is a weighted average, and the weights are the group sizes. The result always lies between the two group averages, and it sits nearer the average of the larger group. Two habits make these items quick: pick concrete numbers that fit the ratio rather than carrying algebra, and check at the end that the answer falls on the correct side of the midpoint.
The midpoint of 11 and 12 is 11.5, and since the girls, with the lower average, are the larger group, the answer has to be below 11.5. That leaves only 11.2 and 11.4 in play before any arithmetic is done. The phrase 'fifty per cent more girls than boys' is the other place candidates lose the mark — it means girls are 1.5 times the boys, giving a 2 : 3 ratio, not a 3 : 2 one and not a 1 : 1.5 reading in the reverse direction.
- Fifty per cent more girls than boys means the ratio of boys to girls is 2 : 3.
- Taking 2 boys and 3 girls: total age 24 + 33 = 57 years over 5 students, giving 11.4.
- In general, average = (12b + 11 × 1.5b) ÷ 2.5b = 28.5 ÷ 2.5 = 11.4.
- A combined average always lies between the two group averages, nearer the larger group's.
- The midpoint 11.5 is the boundary; the answer must be below it because the girls are the larger group.
The side check alone removes 11.6 and 11.8 before any arithmetic is done.
- Taking the plain average of 12 and 11 as 11.5 and ignoring the group sizes.
- Reversing the ratio and answering 11.6.
- Reading 'fifty per cent more girls' as girls being half as many again as the total rather than as the boys.
Asked as a weighted-average item where the four options straddle the midpoint, so that a candidate who ignores the group sizes has no way to guess correctly.
Out of the three annual examinations, each with a total of 500 marks, a student secured average marks of 45 % and 55 % in the first and second annual examinations. To have an overall average of 60 %, how many marks does the student need to secure in the third annual examination?
- (a) 450
- (b) 400
- (c) 350
- (d) 300
Answer(b) 400
The same average machinery run backwards — the combined average is given and one component has to be recovered. Working in totals rather than in averages is the move in both.
The average age of father and elder son is 35 years, the average age of father and younger son is 32 years and the average age of the two sons is 17 years. What is the average age of the father and his two sons?
- (a) 30 years
- (b) 27 years
- (c) 28 years
- (d) 29 years
Answer(c) 28 years
CAPF's average item two years later, again solved by converting every stated average into a total before combining. The quantitative block sets one of these almost every year.
- practice — not a real PYQ
The average weight of 10 boys is 45 kg and of 15 girls is 40 kg. What is the average weight of the whole group?
- (a)41 kg
- (b)42 kg
- (c)42.5 kg
- (d)43 kg
Answer(b) 42 kg — total weight is 450 + 600 = 1050 kg over 25 children, and 1050 ÷ 25 = 42.
- practice — not a real PYQ
In a class there are 25% more boys than girls. If the average marks of boys is 60 and of girls is 69, what is the class average?
- (a)63
- (b)64
- (c)64.5
- (d)65
Answer(b) 64 — take 5 boys and 4 girls: (300 + 276) ÷ 9 = 576 ÷ 9 = 64.