QUESTION

CSAT

Easy

Reasoning

Prelims 2017

There are thirteen 2-digit consecutive odd numbers. If 39 is the mean of the first five such numbers, then what is the mean of all the thirteen numbers?

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Explanation

To solve this problem, let the first odd number be $x$ . Therefore, the consecutive odd numbers are $x$ , $x+2$ , $x+4$ , ...

Step 1: Find the value of $x$ We are given that the mean of the first five numbers is 39. The first five numbers are: $x$ , $x+2$ , $x+4$ , $x+6$ , $x+8$

The mean of these five numbers is given by: $(x + (x+2) + (x+4) + (x+6) + (x+8)) / 5 = 39$

Simplify the sum in the numerator: $(5x + 20) / 5 = 39$

Solve for $x$ : $5x + 20 = 195$ $5x = 175$ $x = 35$

Step 2: Find the mean of all thirteen numbers Now that we know the first number is $x = 35$ , the thirteen consecutive odd numbers are: 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59

The mean of these thirteen numbers is given by: $(35 + 37 + 39 + ... + 59) / 13$

Since these numbers form an arithmetic sequence with the first term 35 and the last term 59, the mean of an arithmetic sequence is the average of the first and last terms: Mean = $(35 + 59) / 2 = 94 / 2 = 47$