Question

The batting average of a batsman for 20 innings is 35 and the difference between the runs of best inning and worst inning is 50. If these two innings are not included the average becomes 32 for 18 innings. The best score of the batsman is?

Answer 1

Hint: Assume the runs of the batsman in 18 innings as \[{{x}_{1}},{{x}_{2}},{{x}_{3}},......,{{x}_{18}}\] and assume \[{{x}_{L}}\] and \[{{x}_{B}}\] as the lowest and best score of the batsman respectively. Form a linear equation by taking the difference of \[{{x}_{B}}\] and \[{{x}_{L}}\] and equating it with 50. Now, find the total runs scored by the batsman in 20 innings and 18 innings by using the formula: - total runs scored = mean \[\times \] number of innings. Form another linear equation in \[{{x}_{B}}\] and \[{{x}_{L}}\]. Solve the two equations to get the value of \[{{x}_{B}}\].

Complete step by step solution:
Here, we have been provided with the mean score of a batsman in 20 innings and 18 innings respectively and we are asked to determine the best score of the batsman if the difference between the best inning and worst inning is given.
Now, let us assume \[{{x}_{L}}\] and \[{{x}_{B}}\] as the lowest and best innings of the batsman respectively and we are considering \[{{x}_{1}},{{x}_{2}},{{x}_{3}},......,{{x}_{18}}\] as the runs scored by the batsman in the remaining 18 innings.
We have been given that the difference between the runs in best inning and worst inning is 50, so we have,
\[\Rightarrow {{x}_{B}}-{{x}_{L}}=50\] -------- (1)
Now, average of the batsman for 20 innings is 35, so using the formula: - Mean = (total runs scored / number of innings), we get,
\[\Rightarrow \] Total runs scored = mean \[\times \] number of innings
Here, the total runs scored will be the sun of runs scored in each individual inning. So, we have,
\[\Rightarrow \] Total runs scored in 20 innings = 35 \[\times \] 20
\[\Rightarrow \] Total runs scored in 20 innings = 700
\[\Rightarrow \left( {{x}_{1}}+{{x}_{2}}+....+{{x}_{18}} \right)+\left( {{x}_{L}}+{{x}_{B}} \right)=700\] ------- (2)
Now, it is said to us that is we do not include the best and worst inning then the average runs scored in 18 innings becomes 32, so we have,
\[\Rightarrow \] Total runs scored in 18 innings = 32 \[\times \] 18
\[\Rightarrow \left( {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+....+{{x}_{18}} \right)=576\] ------- (3)
Subtracting equation (3) from (2), we get,
\[\Rightarrow {{x}_{L}}+{{x}_{B}}=700-576\]
\[\Rightarrow {{x}_{L}}+{{x}_{B}}=124\] ------- (4)
So, we have two equations for two variables \[{{x}_{L}}\] and \[{{x}_{B}}\], we need to find the value of \[{{x}_{B}}\], so solving equations (1) and (4) by adding the two equations, we get,
\[\begin{align}
  & \Rightarrow 2{{x}_{B}}=124+50 \\
 & \Rightarrow 2{{x}_{B}}=174 \\
 & \Rightarrow {{x}_{B}}=87 \\
\end{align}\]
Hence, the best score of the batsman is 87.

Note: One may note that here we cannot find the runs of the batsman in each inning because there is not enough information regarding all the 20 innings. For finding the scores of 20 innings we would have required 20 equations. Now, you must remember the formula of the arithmetic mean to solve the above question. Note that you can also determine the value of \[{{x}_{L}}\] by substituting the obtained value of \[{{x}_{B}}\] in equation (1) or (4).