Of three numbers, the first is 4 times the second and 3 times the third. If the average of all the three numbers is 95, what is the third number ?

Options

A. 76
B. 60
C. 130
D. 57

Answer

To solve the problem step by step, we will define the three numbers based on the relationships provided and then calculate the third number.

### Step 1: Define the numbers
Let:
- The first number be \( A \)
- The second number be \( B \)
- The third number be \( C \)

According to the problem:
- The first number \( A \) is 4 times the second number \( B \):
\[ A = 4B \]
- The first number \( A \) is also 3 times the third number \( C \):
\[ A = 3C \]

### Step 2: Express \( B \) and \( C \) in terms of \( A \)
From the first equation \( A = 4B \), we can express \( B \):
\[ B = \frac{A}{4} \]

From the second equation \( A = 3C \), we can express \( C \):
\[ C = \frac{A}{3} \]

### Step 3: Calculate the average of the three numbers
The average of the three numbers is given by:
\[ \text{Average} = \frac{A + B + C}{3} \]

Substituting \( B \) and \( C \) in terms of \( A \):
\[ \text{Average} = \frac{A + \frac{A}{4} + \frac{A}{3}}{3} \]

### Step 4: Find a common denominator to simplify the average
The common denominator for 4 and 3 is 12. Thus, we rewrite \( B \) and \( C \):
\[ B = \frac{A}{4} = \frac{3A}{12}, \quad C = \frac{A}{3} = \frac{4A}{12} \]

Now substituting back:
\[ \text{Average} = \frac{A + \frac{3A}{12} + \frac{4A}{12}}{3} = \frac{A + \frac{7A}{12}}{3} \]

### Step 5: Combine the terms in the numerator
Convert \( A \) to have a common denominator:
\[ A = \frac{12A}{12} \]
So,
\[ \text{Average} = \frac{\frac{12A + 7A}{12}}{3} = \frac{\frac{19A}{12}}{3} = \frac{19A}{36} \]

### Step 6: Set the average equal to 95
According to the problem, the average is 95:
\[ \frac{19A}{36} = 95 \]

### Step 7: Solve for \( A \)
To find \( A \), multiply both sides by 36:
\[ 19A = 95 \times 36 \]
Calculating \( 95 \times 36 \):
\[ 95 \times 36 = 3420 \]
Now divide by 19:
\[ A = \frac{3420}{19} = 180 \]

### Step 8: Find the third number \( C \)
We already have \( C \) in terms of \( A \):
\[ C = \frac{A}{3} = \frac{180}{3} = 60 \]

Thus, the third number \( C \) is **60**.

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