A batsman scores certain runs in 72 innings. In next two innings he scores 54 and 36 respectively. Therefore, decreases his average by 3 runs. Find his average after 74th innings.

Options

A. 151
B. 153
C. 152
D. 155

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Define the Variables
Let the total runs scored in the first 72 innings be \( X \).

### Step 2: Calculate Total Runs After 74 Innings
In the next two innings, the batsman scores 54 and 36 runs. Therefore, the total runs after 74 innings can be calculated as:
\[ \text{Total runs after 74 innings} = X + 54 + 36 = X + 90 \]

### Step 3: Calculate the Average Runs
The average runs after 72 innings is given by:
\[ \text{Average after 72 innings} = \frac{X}{72} \]

The average runs after 74 innings is:
\[ \text{Average after 74 innings} = \frac{X + 90}{74} \]

### Step 4: Set Up the Equation for Average Decrease
According to the problem, the average decreases by 3 runs after the 74th innings. Therefore, we can set up the equation:
\[ \frac{X + 90}{74} = \frac{X}{72} - 3 \]

### Step 5: Cross-Multiply to Eliminate Fractions
Cross-multiplying gives:
\[ 72(X + 90) = 74X - 3 \times 74 \times 72 \]
\[ 72X + 6480 = 74X - 1596 \]

### Step 6: Rearrange the Equation
Rearranging the equation to isolate \( X \):
\[ 6480 + 1596 = 74X - 72X \]
\[ 8076 = 2X \]
\[ X = \frac{8076}{2} = 4038 \]

### Step 7: Calculate the Average After 74 Innings
Now that we have \( X \), we can find the average after 74 innings:
\[ \text{Average after 74 innings} = \frac{X + 90}{74} = \frac{4038 + 90}{74} = \frac{4128}{74} \]

### Step 8: Simplify the Average
Calculating the division:
\[ \frac{4128}{74} = 56 \]

### Step 9: Final Calculation for the New Average
Now, we need to find the average that is decreased by 3 from the previous average:
\[ \text{Previous Average} = \frac{4038}{72} = 56.0833 \text{ (approximately)} \]
So, the new average after 74 innings is:
\[ 56.0833 - 3 = 53.0833 \]

### Final Answer
Thus, the average after the 74th innings is approximately \( 53.08 \).