Question

A CAT aspirant appears for a certain number of tests. His average score increases by 1 if the first 10 tests are not considered, and decreases by 1 if the last 10 tests are not considered. If his average scores for the first 10 and the last 10 tests are 20 and 30, respectively, then the total number of tests taken by him is

Answer 1

Correct Answer: 60

Let the total number of tests be \( n \) and the overall average score be \( A \). Let's break down the problem:

1. When the first 10 tests (average = 20) are not considered, the average rises by 1. 

2. When the last 10 tests (average = 30) are omitted, the average falls by 1.

These conditions can be expressed with equations:

1. \((A + 1)(n - 10) = An - 200\)

2. \((A - 1)(n - 10) = An - 300\)

Let's solve for \( A \) and \( n \):

Expanding the first equation gives:

\( An - 10A + n - 10 = An - 200 \)

\( n - 10A - 10 = -200 \)

\( n - 10A = -190 \)

From the second equation:

\( An - 10A - n + 10 = An - 300 \)

\(-n - 10A + 10 = -300 \)

\( -n - 10A = -310 \)

Now we have the equations:

\( n - 10A = -190 \)

\(-n - 10A = -310 \)

Add these to eliminate \( n \):

\( (n - 10A) + (-n - 10A) = -190 + (-310) \)

\(-20A = -500\)

\( A = 25 \)

Substitute \( A = 25 \) back into \( n - 10A = -190 \):

\( n - 10(25) = -190 \)

\( n - 250 = -190 \)

\( n = 60 \)

The total number of tests taken is \( 60 \). This value falls within the expected range of 60, confirming accuracy.