The average of a ,b and c is 8 less than d .If the average of a ,b,c and d is 42 ,then what is the average of (3d-2) and (d+5)?

Options

A. `96.5`
B. `97.5`
C. `99.5`
D. `98.5`

Answer

To solve the problem step by step, we will follow the given information and derive the necessary values.

### Step 1: Set up the equations based on the problem statement.

We know that:
1. The average of \( a, b, c \) is 8 less than \( d \).
\[ \frac{a + b + c}{3} = d - 8 \]
This implies:
\[ a + b + c = 3(d - 8) = 3d - 24 \]

2. The average of \( a, b, c, d \) is 42.
\[ \frac{a + b + c + d}{4} = 42 \]
This implies:
\[ a + b + c + d = 42 \times 4 = 168 \]

### Step 2: Substitute the expression for \( a + b + c \) into the second equation.

From the first equation, we have:
\[ a + b + c = 3d - 24 \]
Substituting this into the second equation:
\[ (3d - 24) + d = 168 \]

### Step 3: Simplify the equation.

Combine like terms:
\[ 4d - 24 = 168 \]

### Step 4: Solve for \( d \).

Add 24 to both sides:
\[ 4d = 168 + 24 \]
\[ 4d = 192 \]
Now, divide by 4:
\[ d = \frac{192}{4} = 48 \]

### Step 5: Find the average of \( 3d - 2 \) and \( d + 5 \).

Now that we have \( d = 48 \), we can calculate:
1. \( 3d - 2 = 3(48) - 2 = 144 - 2 = 142 \)
2. \( d + 5 = 48 + 5 = 53 \)

### Step 6: Calculate the average of \( 142 \) and \( 53 \).

The average is given by:
\[ \text{Average} = \frac{(3d - 2) + (d + 5)}{2} = \frac{142 + 53}{2} = \frac{195}{2} = 97.5 \]

### Final Answer:
The average of \( (3d - 2) \) and \( (d + 5) \) is \( 97.5 \).

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