Ajit has a certain average for 9 innings. In the tenth innings, he scores 100 runs thereby increasing his average by 8 runs. His new average is:

Question

Ajit has a certain average for 9 innings. In the tenth innings, he scores 100 runs thereby increasing his average by 8 runs. His new average is:

A. 20
B. 21
C. 28
D. 32

Core Concept

The core concept being tested in this question is “Average” and “Increased Average due to New Data”.

Definition of Average:
The average (or mean) of a set of numbers is the sum of the numbers divided by the number of items in the set. Mathematically, it is given by:

$$\text{Average} = \frac{\text{Sum of all numbers}}{\text{Number of numbers}}$$

In this problem, we need to calculate Ajit’s new average after his 10th innings and the increase in his average due to the score of 100 in the 10th innings.

Step-by-Step Solution

Let the average score for the first 9 innings be $x$.

1. Total Runs for the First 9 Innings:

If the average for the first 9 innings is $x$, the total runs scored in those 9 innings is:

$$\text{Total runs for 9 innings} = 9 \times x$$

2. Total Runs after the 10th Innings:

In the 10th innings, Ajit scores 100 runs. So, the total runs after 10 innings will be:

$$\text{Total runs after 10 innings} = 9x + 100$$

3. New Average after 10 Innings:

The new average after the 10th inning is given as $x + 8$. Using the formula for average:

$$\frac{9x + 100}{10} = x + 8$$

4. Solve for $x$:

Now, solve the equation for $x$:

$$9x + 100 = 10(x + 8)$$

Expanding the right-hand side:

$$9x + 100 = 10x + 80$$

Simplify the equation:

$$9x – 10x = 80 – 100$$ $$-x = -20$$ $$x = 20$$

5. New Average:

Now that we know $x = 20$, the new average after the 10th innings will be:

$$x + 8 = 20 + 8 = 28$$

Correct Answer and Option

The new average after the 10th innings is 28.

Correct Answer: C

Answer Reasoning

Ajit’s average for the first 9 innings was 20, and after scoring 100 in the 10th innings, his average increased by 8 runs. Therefore, his new average is 28. This is confirmed through solving the equation step-by-step, which shows that the average increased by 8 when his total score was increased by 100.

Explanation of Incorrect Options

• Option A (20): This would be correct only if there was no increase in the average. However, the problem states that the average increased by 8 runs after the 10th innings.
• Option B (21): This is a common mistake if one incorrectly assumes that the increase in average is 1 run instead of 8 runs. The correct increase is 8 runs.
• Option D (32): This is incorrect because the new average is 28, not 32. This could result from a miscalculation in the equation or a misunderstanding of the problem.

Common Mistakes to Avoid

1. Misunderstanding the Relationship Between the Old and New Averages:
   Ensure that the increase in the average (8 runs) is correctly accounted for in the equation.
2. Incorrect Arithmetic When Solving the Equation:
   Pay careful attention to simplifying the equation and solving it step-by-step. A small mistake in handling the terms can lead to an incorrect answer.

Simplest Way to Solve (Shortcut or Insight)

The simplest method is to use the formula for average and set up the equation based on the relationship between the old and new averages:

1. Let the original average be $x$.
2. The equation is:

$$\frac{9x + 100}{10} = x + 8$$

3. Solve for $x$ and calculate the new average.

This efficient method ensures you directly find the correct average without unnecessary steps.

Visual Aid Suggestion

A simple bar chart showing the total runs before and after the 10th inning can help visualize how the score change (100 runs) impacts the average. The bars can represent the total runs, with one for 9 innings and another for 10 innings, making the difference in the averages clear.

Practice Questions

1. Question 1:
   Ajit has an average score of 18 runs over 6 innings. In the 7th innings, he scores 40 runs, and his average increases by 2 runs. What is his new average?
   A. 20
   B. 22
   C. 24
   D. 26
   (Answer: B)
2. Question 2:
   A player has an average score of 15 over 5 innings. In the 6th innings, he scores 30 runs, and his average increases by 5 runs. What is his new average?
   A. 20
   B. 21
   C. 23
   D. 25
   (Answer: A)
3. Question 3:
   A batsman scores an average of 25 runs over 4 innings. In the 5th innings, he scores 50 runs. What is the new average after the 5th innings?
   A. 26
   B. 27
   C. 28
   D. 29
   (Answer: B)
4. Question 4:
   In the first 8 innings, a player’s average score is 30. If he scores 70 runs in the 9th innings, what will be his new average?
   A. 32
   B. 33
   C. 34
   D. 35
   (Answer: B)
5. Question 5:
   The average score of a cricketer over 12 innings is 45. He scores 60 runs in the 13th innings. What is his new average?
   A. 46
   B. 47
   C. 48
   D. 49
   (Answer: A)