The average of three numbers is 58. First number is `3/4`th of the third number. If the third number is 24 more than the second number, what will be the difference between the first and second number?

Options

A. 24
B. 6
C. 16
D. 18

Answer

To solve the problem step by step, let's denote the three numbers as follows:

- Let the first number be \( A \).
- Let the second number be \( B \).
- Let the third number be \( C \).

### Step 1: Calculate the total sum of the three numbers using the average.
The average of the three numbers is given as 58. Therefore, the total sum of the three numbers can be calculated as follows:

\[ \text{Sum} = \text{Average} \times \text{Number of items} = 58 \times 3 = 174 \]

### Step 2: Set up the relationships based on the problem statement.
From the problem, we have the following relationships:

1. The first number \( A \) is \( \frac{3}{4} \) of the third number \( C \):
\[ A = \frac{3}{4}C \]

2. The third number \( C \) is 24 more than the second number \( B \):
\[ C = B + 24 \]

### Step 3: Substitute the expressions into the sum equation.
Now we can substitute the expressions for \( A \) and \( C \) into the total sum equation:

\[ A + B + C = 174 \]

Substituting \( A = \frac{3}{4}C \) and \( C = B + 24 \):

\[ \frac{3}{4}C + B + C = 174 \]

### Step 4: Replace \( C \) with \( B + 24 \).
Now we can replace \( C \) in the equation:

\[ \frac{3}{4}(B + 24) + B + (B + 24) = 174 \]

### Step 5: Simplify the equation.
Distributing \( \frac{3}{4} \):

\[ \frac{3}{4}B + 18 + B + B + 24 = 174 \]

Combining like terms:

\[ \frac{3}{4}B + 2B + 42 = 174 \]

Converting \( 2B \) to a fraction:

\[ \frac{3}{4}B + \frac{8}{4}B + 42 = 174 \]

This simplifies to:

\[ \frac{11}{4}B + 42 = 174 \]

### Step 6: Solve for \( B \).
Subtract 42 from both sides:

\[ \frac{11}{4}B = 174 - 42 \]
\[ \frac{11}{4}B = 132 \]

Multiply both sides by \( \frac{4}{11} \):

\[ B = 132 \times \frac{4}{11} = 48 \]

### Step 7: Find \( C \) using \( B \).
Now substitute \( B \) back to find \( C \):

\[ C = B + 24 = 48 + 24 = 72 \]

### Step 8: Find \( A \) using \( C \).
Now substitute \( C \) back to find \( A \):

\[ A = \frac{3}{4}C = \frac{3}{4} \times 72 = 54 \]

### Step 9: Calculate the difference between \( A \) and \( B \).
Now we can find the difference between the first number \( A \) and the second number \( B \):

\[ \text{Difference} = A - B = 54 - 48 = 6 \]

### Final Answer:
The difference between the first and second number is **6**.

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