When 2 is subtracted from each of the given n numbers, the sum of the numbers so obtained is 102. When 5 is subtracted from each of them, the sum of the numbers so obained' is 12. What is the average of the given n numbers ?

Options

A. 5.8
B. 5.4
C. 6.6
D. 6.2

Answer

To solve the problem step by step, we will first set up equations based on the information provided in the question.

### Step 1: Set Up the Equations
Let the sum of the given n numbers be denoted as S.

1. When 2 is subtracted from each of the n numbers, the sum becomes:
\[ S - 2n = 102 \]
(Equation 1)

2. When 5 is subtracted from each of the n numbers, the sum becomes:
\[ S - 5n = 12 \]
(Equation 2)

### Step 2: Solve the Equations
Now we will solve these two equations to find S and n.

From Equation 1:
\[ S = 102 + 2n \]

From Equation 2:
\[ S = 12 + 5n \]

Since both expressions equal S, we can set them equal to each other:
\[ 102 + 2n = 12 + 5n \]

### Step 3: Rearranging the Equation
Rearranging the equation to isolate n:
\[ 102 - 12 = 5n - 2n \]
\[ 90 = 3n \]
\[ n = \frac{90}{3} = 30 \]

### Step 4: Substitute n Back to Find S
Now that we have n, we can substitute it back into either equation to find S. We will use Equation 1:
\[ S = 102 + 2n = 102 + 2(30) = 102 + 60 = 162 \]

### Step 5: Calculate the Average
The average of the n numbers is given by:
\[ \text{Average} = \frac{S}{n} = \frac{162}{30} \]

Calculating the average:
\[ \text{Average} = 5.4 \]

### Final Answer
The average of the given n numbers is **5.4**.

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