If the average of 64 innings of a batsman is 62. If highest and lowest innings are excluded then the average of remaining 62 innings is 60. If the difference between highest and lowest innings is 180 runs. Find his highest and lowest score.

Answer

To solve the problem step by step, we will use the information provided about the averages and the difference between the highest and lowest scores.

### Step 1: Calculate the total runs scored in 64 innings.
The average of 64 innings is given as 62. Therefore, we can calculate the total runs scored by multiplying the average by the number of innings.

\[ \text{Total runs in 64 innings} = \text{Average} \times \text{Number of innings} = 62 \times 64 = 3968 \]

### Step 2: Calculate the total runs scored in the remaining 62 innings.
When the highest and lowest scores are excluded, the average of the remaining 62 innings is 60. We can calculate the total runs scored in these 62 innings.

\[ \text{Total runs in 62 innings} = \text{Average} \times \text{Number of innings} = 60 \times 62 = 3720 \]

### Step 3: Find the total runs of the highest and lowest innings.
To find the total runs scored in the highest and lowest innings, we subtract the total runs of the remaining 62 innings from the total runs of all 64 innings.

\[ \text{Total runs of highest and lowest innings} = \text{Total runs in 64 innings} - \text{Total runs in 62 innings} = 3968 - 3720 = 248 \]

### Step 4: Set up equations for the highest and lowest scores.
Let the highest score be \( H \) and the lowest score be \( L \). According to the problem, we have two equations:

1. \( H + L = 248 \) (from Step 3)
2. \( H - L = 180 \) (the difference between the highest and lowest scores)

### Step 5: Solve the equations.
Now we can solve these two equations simultaneously.

Adding the two equations:

\[ (H + L) + (H - L) = 248 + 180 \]
\[ 2H = 428 \implies H = \frac{428}{2} = 214 \]

Now, substitute \( H \) back into the first equation to find \( L \):

\[ H + L = 248 \implies 214 + L = 248 \implies L = 248 - 214 = 34 \]

### Final Result:
The highest score \( H \) is 214 and the lowest score \( L \) is 34.

### Summary:
- Highest score: 214
- Lowest score: 34