Question

The batting average of a cricket player for 40 innings is 50 runs. His highest score in an innings exceeded his lowest score by 172 runs. If these two innings are excluded, then the average of the remaining 38 innings is 48 runs. Determine his highest score, scored in one innings.
(a) 175
(b) 180
(c) 174
(d) 185

Answer 1

Hint: We start solving the problem by assigning the variables for highest score, lowest score and the sum of the runs in remaining 38 innings. We then recall the definition of average and use it for the 40 innings first and then to the remaining 38 innings to get the sum of the highest and lowest score by the batsmen. We then find another relation using the condition that the highest score in an innings exceeds his lowest score by 172 runs. We use these conditions to find the highest score of batsman.

Complete step by step answer:
According to the problem, we are given that the batting average of a cricket player for 40 innings is 50 runs. If the two innings which contain his lowest and highest score are excluded, then his average in the remaining 38 innings is 48 runs. We need to find the highest score if his highest score in an innings exceeds his lowest score by 172 runs.
Let us assume the highest and lowest scores are ‘x’, ‘y’ runs and the sum of the runs in the remaining 38 innings is S.
According to the problem, we are given that the batting average of a cricket player for 40 innings is 50 runs.
We know that the mathematical average is defined as the ratio of the sum of the terms given to the total number of terms.
So, we get $\dfrac{S+x+y}{40}=50$.
$\Rightarrow S+x+y=2000$.
$\Rightarrow S=2000-x-y$ ---(1).
According to the problem, we are given that the average in the remaining 38 innings is 48 runs if the innings with highest and lowest scores are subtracted from the 40 innings.
So, we get $\dfrac{S}{38}=48$.
$\Rightarrow S=1824$.
From equation (1), we get $2000-x-y=1824$.
$\Rightarrow x+y=176$ ---(2).
According to the problem, we are given that the highest score in an innings exceed his lowest score by 172 runs.
So, we get $x=y+172$ ---(3).
Let us substitute equation (3) in equation (2).
So, we get $y+172+y=176$.
$\Rightarrow 2y=4$.
$\Rightarrow y=2$ ---(4).
Let us substitute equation (4) in equation (3).
So, we get $x=2+172=174$.
So, we have found the highest score of the batsman as 174 runs.

The correct option for the given problem is (c).

Note:
We can see that the problem contains a huge amount of calculations, so we need to do each step carefully while solving this problem. We can also assume the variables to represent runs in the remaining 38 innings but it will lead us to confusion and making mistakes. We should not just assume the lowest score as 0 while solving this problem. Similarly, we can expect problems to find the average if the sum of the runs in next runs is given.