Question

The average score of Dhoni after 48 innings is 48 and in the \[{{49}^{th}}\] innings, Dhoni scores 97 runs. In the \[{{50}^{th}}\] innings the M minimum number of runs required to increase his average score by 2 than it was before the \[{{50}^{th}}\] innings:
(a) 99
(b) 149
(c) 151
(d) Can’t be determined

Answer 1

Hint: To solve the given question, we will first find out what an average is. Then we will assume that Dhoni’s score in the first match is \[{{x}_{1}},\] the score in the second match is \[{{x}_{2}}\] and the score in the \[{{n}^{th}}\] match is \[{{x}_{n}}.\] Then, we will apply the formula of average again for the first 48 matches. From there, we will find the sum of the first 48 scores. Then, we will apply the formula of average again for the first 50 matches and equate it to the new average after \[{{49}^{th}}\] innings. From here, we will find the score in the \[{{50}^{th}}\] match.

Complete step-by-step answer:
To start with, we will first find out what an average is and what the formula to calculate is. The average is defined as the central value of a set of entities. In other words, the average is calculated by adding all the numbers in a set together and then dividing them by the quantity of numbers in the set.
Now, we will assume that Dhoni’s score in the first match is \[{{x}_{1}},\]the score in the second match is \[{{x}_{2}},\] the score in the third match is \[{{x}_{3}}\] and score in the \[{{n}^{th}}\] match is \[{{x}_{n}}.\] Now, in the question it is given that Dhoni’s average score after 48 matches is 48. Now, the average of a set of r numbers is calculated by the formula shown below.
\[\text{Average}=\dfrac{{{\alpha }_{1}}+{{\alpha }_{2}}+{{\alpha }_{3}}+.....+{{\alpha }_{r}}}{r}\]
Thus, in our case, r = 48 and average = 48. So, we will get,
\[\Rightarrow 48=\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+.....{{x}_{48}}}{48}\]
\[\Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+.....{{x}_{48}}=48\times 48\]
\[\Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+.....{{x}_{48}}=2304.....\left( i \right)\]
Now, we will calculate the average of Dhoni in the first 49 matches. Now, in the question, it is given that \[{{x}_{49}}=97.\] So, we have,
\[\text{Average in the first 49 matches}=\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+....{{x}_{48}}+{{x}_{49}}}{49}\]
Now, we will substitute the value of \[{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+....{{x}_{48}}\] from (i) to the above equation. Thus, we will get,
\[\Rightarrow \text{Average in the first 49 matches}=\dfrac{2304+97}{49}\]
\[\Rightarrow \text{Average in the first 49 matches}=\dfrac{2401}{49}\]
\[\Rightarrow \text{Average in the first 49 matches}=49\]
Thus the average of Dhoni after 50 matches = 49 + 2 = 51. Thus, we can say that,
\[51=\dfrac{{{x}_{1}}+{{x}_{2}}+.....{{x}_{48}}+{{x}_{49}}+{{x}_{50}}}{50}\]
\[\Rightarrow 51=\dfrac{2304+97+{{x}_{50}}}{50}\]
\[\Rightarrow 51\times 50=2401+{{x}_{50}}\]
\[\Rightarrow 2550=2401+{{x}_{50}}\]
\[\Rightarrow {{x}_{50}}=149\]
Hence, option (b) is the right answer.

Note: Generally, if we want to calculate the average of n numbers, we must know each number but in our case, all the scores of Dhoni are not given. So, if we want to calculate the average score of Dhoni’s first 50 matches, we will just use the sum of the scores in Dhoni’s first 50 matches. Here, we do not need to find each match score of Dhoni. We just need to assume the scores such that their average is 51.