Explanations:The problem involves calculating the average score of a batsman over a series of matches. The average score is defined as the total score divided by the number of matches played. In this case, the batsman's average score increases by 3 after scoring 87 runs in the 17th match. We need to find the average score at the start of the 17th match.
Let's denote the average score at the start of the 17th match as \( A \). The total score after 16 matches would be \( 16A \). After scoring 87 runs in the 17th match, the new total score becomes \( 16A + 87 \). The new average score after 17 matches is given by the total score divided by 17, which is \( \frac{16A + 87}{17} \).
According to the problem, this new average is 3 more than the initial average \( A \). Therefore, we can set up the following equation:
\[
\frac{16A + 87}{17} = A + 3
\]
To solve for \( A \), we first eliminate the fraction by multiplying both sides of the equation by 17:
\[
16A + 87 = 17(A + 3)
\]
Expanding the right side of the equation, we get:
\[
16A + 87 = 17A + 51
\]
Next, we isolate \( A \) by subtracting \( 16A \) and 51 from both sides:
\[
87 - 51 = 17A - 16A
\]
Simplifying, we find:
\[
36 = A
\]
Thus, the average score at the start of the 17th match is 36.
The key to solving this problem is understanding the relationship between the total score, the number of matches, and the average. By setting up the equation correctly and solving for the initial average, we can determine the correct answer. This type of problem is common in competitive exams and requires careful application of the average formula and algebraic manipulation.