Australia scored a total of x runs in 50 overs India tied the score in `20%` less over. If India's average run rate had been `y%` higher, then score would have been tied 10 overs earlier. Find the value of `y%` ?

Options

A. `33(1)/(3)%`
B. `23(1)/(3)%`
C. `38%`
D. `40%`

Answer

To solve the problem, let's break it down step by step.

### Step 1: Understand the problem
Australia scored a total of \( x \) runs in 50 overs. India tied the score in 20% less overs, which means they took 40 overs to score \( x \) runs.

### Step 2: Calculate India's run rate
India's run rate can be calculated as:
\[ \text{Run Rate} = \frac{\text{Total Runs}}{\text{Total Overs}} = \frac{x}{40} \]

### Step 3: Determine the increased run rate
If India's average run rate had been \( y\% \) higher, we can express this as:
\[ \text{New Run Rate} = \left(1 + \frac{y}{100}\right) \cdot \frac{x}{40} \]

### Step 4: Calculate the time to tie the score with the new run rate
According to the problem, if the run rate was \( y\% \) higher, India would have tied the score 10 overs earlier, which means they would have tied the score in \( 30 \) overs.

### Step 5: Set up the equation for the new run rate
Using the new run rate, the total runs scored in 30 overs would be:
\[ \text{Total Runs} = \text{New Run Rate} \times \text{Total Overs} = \left(1 + \frac{y}{100}\right) \cdot \frac{x}{40} \cdot 30 \]

### Step 6: Set the equation equal to \( x \)
Since India tied the score, this total must equal \( x \):
\[ \left(1 + \frac{y}{100}\right) \cdot \frac{x}{40} \cdot 30 = x \]

### Step 7: Simplify the equation
We can simplify this equation:
\[ \left(1 + \frac{y}{100}\right) \cdot \frac{30}{40} = 1 \]
\[ \left(1 + \frac{y}{100}\right) \cdot \frac{3}{4} = 1 \]

### Step 8: Solve for \( y \)
Now, multiply both sides by \( \frac{4}{3} \):
\[ 1 + \frac{y}{100} = \frac{4}{3} \]
Subtract 1 from both sides:
\[ \frac{y}{100} = \frac{4}{3} - 1 = \frac{4}{3} - \frac{3}{3} = \frac{1}{3} \]
Multiply both sides by 100:
\[ y = \frac{100}{3} \approx 33.33 \]

### Conclusion
Thus, the value of \( y\% \) is approximately \( 33.33\% \).

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