Which one of the following is the average of first five multiples of each of the numbers from 11, 12, 13, …, 20 ?
40.5
42.5
44.5
46.5
Answer is Right!
Answer is Wrong!
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UPSC CAPF – 2024
For any number $N$, the first five multiples are $N, 2N, 3N, 4N, 5N$.
The sum of these multiples is $N + 2N + 3N + 4N + 5N = (1+2+3+4+5)N = 15N$.
The average of these five multiples is $\frac{15N}{5} = 3N$.
So, for each number $N$ from 11 to 20, the average of its first five multiples is $3N$.
We need to find the average of the values $3 \times 11, 3 \times 12, …, 3 \times 20$.
The average of these 10 numbers is $\frac{(3 \times 11) + (3 \times 12) + … + (3 \times 20)}{10}$.
We can factor out 3 from the numerator: $\frac{3 \times (11 + 12 + … + 20)}{10}$.
The numbers 11, 12, …, 20 form an arithmetic progression. The sum of an arithmetic progression is $\frac{\text{number of terms}}{2} \times (\text{first term} + \text{last term})$.
Sum of $11 + … + 20 = \frac{10}{2} \times (11 + 20) = 5 \times 31 = 155$.
The required average is $\frac{3 \times 155}{10} = \frac{465}{10} = 46.5$.
Alternatively, the average of $3N$ for $N=11, …, 20$ is $3 \times (\text{Average of } 11, …, 20)$. The average of $11, …, 20$ is $\frac{11+20}{2} = 15.5$. The required average is $3 \times 15.5 = 46.5$.