The average score of 20 contestants in a quiz competition increased by 3.5 points when one participant withdrew. However, when a new participant entered the competition, the average score of the group decreased by 2.5 points. What is the absolute difference between the score of the participant who withdrew and the score of the newly entered participant?

The correct answer is 14.

Let us analyze the problem step-by-step by considering the two scenarios relative to the initial group of contestants.

Let the initial number of 20 contestants have an average score represented by:

$A$

The initial total score of all 20 contestants is calculated as:

$\text{Initial Total Score}=20A$

Step 1: Calculate the score of the participant who withdrew

When 1 participant withdraws from the original group of 20 contestants:

The number of remaining contestants becomes:

$20-1=19$

The average score of the group increases by 3.5 points, giving a new average of:

$A+3.5$

The total score of the remaining 19 contestants is:

$\text{Total Score (19 contestants)}=19(A+3.5)=19A+66.5$

Let the score of the participant who withdrew be represented by:

$x$

The score of the withdrawn participant is the difference between the initial total score and the new total score:

$x=20A-(19A+66.5)$

$x=A-66.5$

Step 2: Calculate the score of the newly entered participant

When a new participant enters the original group of 20 contestants:

The total number of contestants becomes:

$20+1=21$

The average score of the group decreases by 2.5 points, giving a new average of:

$A-2.5$

The total score of all 21 contestants is:

$\text{Total Score (21 contestants)}=21(A-2.5)=21A-52.5$

Let the score of the newly entered participant be represented by:

$y$

The score of the newly entered participant is the difference between the new total score and the initial total score:

$y=(21A-52.5)-20A$

$y=A-52.5$

Step 3: Calculate the absolute difference between the two scores

Now, we find the absolute difference between the score of the withdrawn participant and the score of the newly entered participant:

$|x-y|=\left|\right(A-66.5)-(A-52.5\left)\right|$

$|x-y|=|-66.5+52.5|$

$|x-y|=|-14|=14$

Therefore, the absolute difference between the score of the participant who withdrew and the score of the newly entered participant is 14.