Four friends keep a record of their average rank in exams. After the first four exams, the average ranks are as given in the table. But after the fifth exam, the average rank changed as shown. If so, who got the 1st rank in the fifth exam?

Friends Anna Mark Jake Layla

Avg. rank after the first 4 exams 1 2 3 4

Avg. rank after the 5th exam 1.6 2.2 2.6 3.6

Mark

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Layla

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Anna

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Jake

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Solution

The correct option is D Jake

Friends Anna Mark Jake Layla

Avg. rank after the first 4 exams 1 2 3 4

Avg. rank after the 5th exam 1.6 2.2 2.6 3.6

$A v e r a g e = \frac{S u m o f t h e r a n k s}{N u m b e r o f e x a m s} ⟹ S u m o f t h e r a n k s = A v e r a g e \times N u m b e r o f e x a m s$

Calculating the sum of the ranks of each friend for the first 4 exams, we get:

$A n n a^{'} s s u m o f r a n k s = A v e r a g e \times N u m b e r o f e x a m s = 1 \times 4 = 4$

$M a r k^{'} s s u m o f r a n k s = A v e r a g e \times N u m b e r o f e x a m s = 2 \times 4 = 8$

$J a k e^{'} s s u m o f r a n k s = A v e r a g e \times N u m b e r o f e x a m s = 3 \times 4 = 12$

$L a y l a^{'} s s u m o f r a n k s = A v e r a g e \times N u m b e r o f e x a m s = 4 \times 4 = 16$

Now, calculating the sum of ranks of each friend for the first 5 exams, we get:

$A n n a^{'} s s u m o f r a n k s = A v e r a g e \times N u m b e r o f e x a m s = 1.6 \times 5 = 8$

$M a r k^{'} s s u m o f r a n k s = A v e r a g e \times N u m b e r o f e x a m s = 2.2 \times 5 = 11$

$J a k e^{'} s s u m o f r a n k s = A v e r a g e \times N u m b e r o f e x a m s = 2.6 \times 5 = 13$

$L a y l a^{'} s s u m o f r a n k s = A v e r a g e \times N u m b e r o f e x a m s = 3.6 \times 5 = 18$

Finally, calculating the difference between the sum of the ranks after 5 exams and the sum of the ranks after 4 exams, we will get the rank secured by each person in the 5th exam.

$A n n a^{'} s r a n k i n t h e 5 t h e x a m = 8 - 4 = 4$

$M a r k^{'} s r a n k i n t h e 5 t h e x a m = 11 - 8 = 3$

$J a k e^{'} s r a n k i n t h e 5 t h e x a m = 13 - 12 = 1$

$L a y l a^{'} s r a n k i n t h e 5 t h e x a m = 18 - 16 = 2$

Hence, in the 5th exam, it is Jake who secured the 1st rank.

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