The average of 5, 7, 12,15 and X is 10. What will be the value of Y if the average of 9, 12, 13, 17, X and Y is 14?

Options

A. 12
B. 13
C. 20
D. 22

Answer

To solve the problem step by step, we will first find the value of \( X \) using the information given about the average of the first set of numbers, and then we will use that value to find \( Y \) from the second set of numbers.

### Step 1: Find the value of \( X \)

We are given that the average of the numbers \( 5, 7, 12, 15, \) and \( X \) is \( 10 \).

The formula for average is:
\[ \text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}} \]

In this case, we have:
\[ 10 = \frac{5 + 7 + 12 + 15 + X}{5} \]

Now, let's calculate the sum of the known numbers:
\[ 5 + 7 + 12 + 15 = 39 \]

Substituting this back into the equation gives:
\[ 10 = \frac{39 + X}{5} \]

To eliminate the fraction, multiply both sides by \( 5 \):
\[ 50 = 39 + X \]

Now, solve for \( X \):
\[ X = 50 - 39 = 11 \]

### Step 2: Use the value of \( X \) to find \( Y \)

Now we need to find the average of the numbers \( 9, 12, 13, 17, X, \) and \( Y \), which is given to be \( 14 \).

Substituting \( X = 11 \) into the average formula gives:
\[ 14 = \frac{9 + 12 + 13 + 17 + 11 + Y}{6} \]

First, calculate the sum of the known numbers:
\[ 9 + 12 + 13 + 17 + 11 = 62 \]

Now substituting this back into the equation gives:
\[ 14 = \frac{62 + Y}{6} \]

Multiply both sides by \( 6 \) to eliminate the fraction:
\[ 84 = 62 + Y \]

Now, solve for \( Y \):
\[ Y = 84 - 62 = 22 \]

### Final Answer
The value of \( Y \) is \( 22 \).

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