If the mean of the numbers `27+x, 31+x, 89+x, 107+x, 156+x` is 82, then the mean of `130+x, 126+x, 68+x, 50+x, 1+x` is

Options

A. 75
B. 157
C. 82
D. 80

Explanation

To solve the problem step by step, we will first find the value of \( x \) using the information provided about the first set of numbers, and then we will calculate the mean of the second set of numbers.

### Step 1: Set up the equation for the first set of numbers

The first set of numbers is \( 27+x, 31+x, 89+x, 107+x, 156+x \). The mean of these numbers is given as 82.

The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of the numbers}}{\text{Total number of values}} \]

In this case, the mean can be expressed as:
\[ 82 = \frac{(27+x) + (31+x) + (89+x) + (107+x) + (156+x)}{5} \]

### Step 2: Simplify the equation

Now, we will simplify the sum in the numerator:
\[ (27 + 31 + 89 + 107 + 156) + 5x = 410 + 5x \]

So, we can rewrite the equation as:
\[ 82 = \frac{410 + 5x}{5} \]

### Step 3: Cross-multiply to solve for \( x \)

Cross-multiplying gives us:
\[ 82 \times 5 = 410 + 5x \]
\[ 410 = 410 + 5x \]

### Step 4: Solve for \( x \)

Subtract 410 from both sides:
\[ 410 - 410 = 5x \]
\[ 0 = 5x \]

Dividing both sides by 5 gives:
\[ x = 0 \]

### Step 5: Find the mean of the second set of numbers

Now that we have \( x = 0 \), we will calculate the mean of the second set of numbers: \( 130+x, 126+x, 68+x, 50+x, 1+x \).

Substituting \( x = 0 \) into the second set gives us:
\[ 130 + 0, 126 + 0, 68 + 0, 50 + 0, 1 + 0 \]
which simplifies to:
\[ 130, 126, 68, 50, 1 \]

### Step 6: Calculate the sum of the second set

Now we calculate the sum:
\[ 130 + 126 + 68 + 50 + 1 = 375 \]

### Step 7: Calculate the mean of the second set

Now, we can find the mean:
\[ \text{Mean} = \frac{375}{5} = 75 \]

### Final Answer

The mean of the numbers \( 130+x, 126+x, 68+x, 50+x, 1+x \) is **75**.

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