Question

A cricket player after playing 15 tests scored 124 runs in the 16th test. As a result, the average of his runs is increased by 4. The present average of runs is

• A. 55
• B. 64
• C. 60
• D. 68

Hint:

When an average increases after adding a new term, equate the new average formula with the given condition. Always express in terms of \(n\) and \(n+1\) items for clarity.

Answer 1

The Correct Option is B

Step 1: Assume the initial average.
Let the average runs for the first 15 tests be \(x\).
So, the total runs in 15 tests = \(15x\).
Step 2: Add the 16th test runs.
In the 16th test, he scores 124 runs.
So, the total runs after 16 tests = \(15x + 124\).
Step 3: Write the condition for increased average.
The new average = \(x+4\).
Also, new average = \(\dfrac{15x + 124}{16}\).
\[ \dfrac{15x + 124}{16} = x+4 \]
Step 4: Solve the equation.
\(15x + 124 = 16x + 64\)
\(\Rightarrow 124 - 64 = 16x - 15x\)
\(\Rightarrow x = 60\).
Step 5: Find the new average.
New average = \(x+4 = 60+4 = 64\).
\[\boxed{64}\]

Answer 2

There is a quicker way to see this without setting up an equation for the old average: the extra runs scored in the 16th test must cover both the player's own excess above the new average and the boost needed to lift each of the previous 15 innings by 4 runs.

1. 55: Checking \(15\times4+55=60+55=115\), which does not equal the 124 runs actually scored in the 16th test, so this cannot be the present average.
2. 64: Checking \(15\times4+64=60+64=124\), which exactly matches the 124 runs scored in the 16th test. This is consistent with everything given.
3. 60: Checking \(15\times4+60=60+60=120\), which does not equal 124, so this fails the check. (This value is actually the player's average before the 16th test, not the "present" average asked for.)
4. 68: Checking \(15\times4+68=60+68=128\), which does not equal 124, so this also fails.

The shortcut \(\text{runs in last innings} = \text{new average} + (\text{number of previous innings})\times(\text{increase in average})\) confirms that the new, present average is \(64\), since \(124 = 64 + 15\times4\).

So the correct answer is 64.