A batsman has scored an average of 46 runs for a certain number of innings played in England. When he came back to India, he played another two test matches of two innings each and scored at an average of 55 runs. For the innings in England and in India taken together, he has improved his average by 2 runs over the matches played in England. The number of innings played in England was

Answer

To solve the problem step by step, we will define the variables and set up the equations based on the information provided in the question.

### Step 1: Define Variables
Let \( X \) be the number of innings played by the batsman in England.

### Step 2: Calculate Total Runs Scored in England
The batsman has an average of 46 runs in England. Therefore, the total runs scored in England can be expressed as:
\[ \text{Total Runs in England} = 46 \times X \]

### Step 3: Calculate Total Runs Scored in India
In India, the batsman played 2 test matches with 2 innings each, which totals to 4 innings. He scored at an average of 55 runs in these innings. Thus, the total runs scored in India can be calculated as:
\[ \text{Total Runs in India} = 55 \times 4 = 220 \]

### Step 4: Calculate Total Runs Scored Overall
The total runs scored by the batsman after playing in both England and India will be:
\[ \text{Total Runs Overall} = \text{Total Runs in England} + \text{Total Runs in India} = 46X + 220 \]

### Step 5: Calculate Total Innings Played Overall
The total number of innings played by the batsman overall is:
\[ \text{Total Innings Overall} = X + 4 \]

### Step 6: Set Up the Average Equation
According to the problem, the batsman’s average has improved by 2 runs after playing in India. Therefore, his new average is:
\[ \text{New Average} = 46 + 2 = 48 \]
We can set up the equation for the average:
\[ \frac{46X + 220}{X + 4} = 48 \]

### Step 7: Cross Multiply to Solve for X
Cross-multiplying gives us:
\[ 46X + 220 = 48(X + 4) \]
Expanding the right side:
\[ 46X + 220 = 48X + 192 \]

### Step 8: Rearranging the Equation
Now, we rearrange the equation to isolate \( X \):
\[ 46X + 220 - 192 = 48X \]
\[ 46X + 28 = 48X \]
\[ 28 = 48X - 46X \]
\[ 28 = 2X \]

### Step 9: Solve for X
Dividing both sides by 2 gives:
\[ X = 14 \]

### Conclusion
The number of innings played in England is \( \boxed{14} \).