In a cricket match highest score in a innings is 2 / 9 of the total score and second highest score is 2 / 9 of the rest score. If difference between highest and second highest scores is 8 runs. Then the total score is?

Options

A. 160
B. 172
C. 180
D. 162

Answer

To solve the problem step by step, let's denote the total score of the cricket match as \( T \).

### Step 1: Define the highest score
According to the problem, the highest score is \( \frac{2}{9} \) of the total score. Therefore, we can express the highest score \( H \) as:
\[ H = \frac{2}{9} T \]

### Step 2: Calculate the remaining score after the highest score
The remaining score after accounting for the highest score is:
\[ \text{Remaining Score} = T - H = T - \frac{2}{9} T = \frac{7}{9} T \]

### Step 3: Define the second highest score
The second highest score is \( \frac{2}{9} \) of the remaining score. Thus, we can express the second highest score \( S \) as:
\[ S = \frac{2}{9} \left(\frac{7}{9} T\right) = \frac{14}{81} T \]

### Step 4: Set up the equation based on the difference
We know that the difference between the highest score and the second highest score is 8 runs. Therefore, we can write the equation:
\[ H - S = 8 \]
Substituting the expressions for \( H \) and \( S \):
\[ \frac{2}{9} T - \frac{14}{81} T = 8 \]

### Step 5: Find a common denominator and simplify
The common denominator for the fractions is 81. We can rewrite \( \frac{2}{9} T \) as \( \frac{18}{81} T \):
\[ \frac{18}{81} T - \frac{14}{81} T = 8 \]
This simplifies to:
\[ \frac{4}{81} T = 8 \]

### Step 6: Solve for \( T \)
To find \( T \), multiply both sides by 81:
\[ 4T = 8 \times 81 \]
\[ 4T = 648 \]
Now divide both sides by 4:
\[ T = \frac{648}{4} = 162 \]

### Conclusion
The total score \( T \) is:
\[ \boxed{162} \]