Average, Mixture and Alligation Question 5 - 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
- Find the Initial Sum:
The average of three integers is 13, which means their combined sum is:
Initial Sum = 13 × 3 = 39
- Analyze the New State:
When a new natural number n is included, the dataset grows to 4 items. Let the new average be represented by K:
(39 + n) / 4 = K ⟹ 39 + n = 4K ⟹ n = 4K − 39
We are given two critical constraints: n is a natural number (n ≥ 1) and K must be an odd integer.
4K − 39 ≥ 1 ⟹ 4K ≥ 40 ⟹ K ≥ 10
The smallest odd integer that satisfies K ≥ 10 is 11. Substitute K = 11 back into our equation to find n:
n = 4 × 11 − 39 = 44 − 39 = 5
Correct Option: C
💡 FundaMakers Tip:
Setting up clear boundary inequalities prevents zeroing-in errors. By shifting the perspective to the new average variable (K), you instantly discover that it must cross a floor value of 10. Combining this floor value with the structural odd-number constraint lets you identify 11 as the key, giving you the value of n without any trial-and-error guesswork.