A bowler whose bowling average is 24.85 runs per wicket, takes 5 wickets for 52 runs in next inning and thereby decreases his average by 0.85. Find the number of wickets before last match.

Answer

To solve the problem step by step, we will use the concept of averages and the relationship between runs, wickets, and averages.

### Step 1: Understand the initial conditions
The bowler's initial average is given as 24.85 runs per wicket. This means that for every wicket he takes, he has given away an average of 24.85 runs.

### Step 2: Calculate the new average
After taking 5 wickets for 52 runs in the next inning, his average decreases by 0.85. Therefore, the new average can be calculated as:
\[ \text{New Average} = \text{Old Average} - 0.85 = 24.85 - 0.85 = 24.00 \]

### Step 3: Set up the equation for the total runs and wickets
Let \( n \) be the number of wickets taken before the last match. The total runs given by the bowler before the last match can be expressed as:
\[ \text{Total Runs} = \text{Old Average} \times n = 24.85n \]

After taking 5 more wickets, the total number of wickets becomes \( n + 5 \), and the total runs given becomes:
\[ \text{Total Runs After Last Match} = 24.85n + 52 \]

### Step 4: Set up the equation for the new average
The new average after taking 5 wickets can also be expressed as:
\[ \text{New Average} = \frac{\text{Total Runs After Last Match}}{\text{Total Wickets After Last Match}} \]
Substituting the values, we have:
\[ 24.00 = \frac{24.85n + 52}{n + 5} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 24.00(n + 5) = 24.85n + 52 \]
Expanding both sides results in:
\[ 24n + 120 = 24.85n + 52 \]

### Step 6: Rearranging the equation
Rearranging the equation to isolate \( n \):
\[ 120 - 52 = 24.85n - 24n \]
This simplifies to:
\[ 68 = 0.85n \]

### Step 7: Solve for \( n \)
Now, divide both sides by 0.85 to find \( n \):
\[ n = \frac{68}{0.85} = 80 \]

### Conclusion
Thus, the number of wickets taken before the last match is:
\[ \boxed{80} \]