Averages , Ratio and Proportion Question 6 - CAT 2022
The average of three integers is 13. When a natural number n is included, the average of these four integers remains an odd integer. The minimum possible value of n is:
Step-by-Step Explanation
- Set up the aggregate sum equation:
The average of three integers is 13, so their sum is:
13 × 3 = 39
When a natural number n is added, the new average across all 4 numbers becomes:
New Average = (39 + n)/4
- Apply the odd integer constraint:
Let this new average be an odd integer k:
(39 + n)/4 = k
39 + n = 4k
n = 4k − 39
- Find the minimum value:
Since n must be a natural number (n ≥ 1):
4k − 39 ≥ 1
4k ≥ 40
k ≥ 10
The smallest odd integer that satisfies k ≥ 10 is k = 11.
n minimum = 4(11) − 39 = 44 − 39 = 5
Correct Option: C
💡 FundaMakers Tip:
Work forward from the constraints on the average. Since the sum of the first three numbers is 39, the next total value must be a multiple of 4 (40, 44, 48, …). Dividing 40 by 4 gives an even average (10), while dividing 44 by 4 gives an odd average (11). This reveals that 39 + n = 44, so n = 5 almost instantly.