There are three numbers. If the average of them is added to the three numbers, the sums obtained are 167, 153 and 148. What is the average of the given numbers?

Options

A. 76
B. 78
C. 81
D. 67

Answer

To solve the problem step by step, let's denote the three numbers as \( a \), \( b \), and \( c \).

### Step 1: Define the average of the numbers
The average of the three numbers \( a \), \( b \), and \( c \) can be expressed as:
\[ \text{Average} = \frac{a + b + c}{3} \]
Let's denote the average as \( A \). Therefore, we have:
\[ A = \frac{a + b + c}{3} \]

### Step 2: Express the sums after adding the average
According to the problem, when the average \( A \) is added to each of the three numbers, the resulting sums are given as follows:
- \( a + A = 167 \)
- \( b + A = 153 \)
- \( c + A = 148 \)

### Step 3: Rewrite the equations
From the equations above, we can express \( a \), \( b \), and \( c \) in terms of \( A \):
\[ a = 167 - A \quad (1) \]
\[ b = 153 - A \quad (2) \]
\[ c = 148 - A \quad (3) \]

### Step 4: Substitute into the average equation
Now, we can substitute equations (1), (2), and (3) into the average equation:
\[ A = \frac{(167 - A) + (153 - A) + (148 - A)}{3} \]

### Step 5: Simplify the equation
Combining the terms in the numerator:
\[ A = \frac{167 + 153 + 148 - 3A}{3} \]
Calculating the sum:
\[ 167 + 153 + 148 = 468 \]
So, we have:
\[ A = \frac{468 - 3A}{3} \]

### Step 6: Multiply through by 3 to eliminate the fraction
Multiplying both sides by 3 gives:
\[ 3A = 468 - 3A \]

### Step 7: Solve for \( A \)
Adding \( 3A \) to both sides:
\[ 3A + 3A = 468 \]
\[ 6A = 468 \]
Dividing both sides by 6:
\[ A = \frac{468}{6} = 78 \]

### Conclusion
The average of the given numbers is:
\[ \boxed{78} \]