“Concept of Average”-08

Question: The sum of 8 consecutive odd numbers is 656. Also the average of four consecutive even numbers is 87. What is the sum of the smallest odd number and second largest even number?   [Ans. (c) 163]

(a) 165 (b) 175 (c) 163 (d) can’t be determined (e) None of these

Solution:

Detailed Explanation:

Concept: 

1. “The average of two even numbers is an odd number.”

2. “The average of two odd numbers is an even number.”

3. “If number of terms of an A. P. series is odd, then the middle term is the average of the series.”

4. “If number of terms of an A. P. series is even, then no one is the middle term. In this case, there are two middle terms and the average of the two middle terms is the average of the whole series.”

Given that the sum of 8 consecutive odd numbers is 656, so their average is 656/8 = 82; which is the average the two middle numbers of the 8 given odd numbers.

The smallest odd number is 82 – 1 -2 -2 -2 = 75

[As there are four odd numbers less than 82; they are       75       77       79       81]   

Also, given that the average of four consecutive even numbers is 87, which is the average of the two middle numbers out of four consecutive even numbers.

The second largest even number is 87 +1 = 88 [As there are two even numbers greater than 87; they are       88       90]   

Therefore, the sum of the smallest odd number and second largest even number = 75 + 88 = 163 Ans.