12 Days of Christmas GMAT Competition 2025 - Day 7: A librarys late : Two-part Analysis (TPA)

A library’s late-return fine increases each day by a fixed amount. The fine on the first overdue day is $1, and each subsequent day the fine increases by $0.5 more than the previous day’s fine.

Select for Number of overdue days and for Total fine in dollars numbers that could be the number of days a book is overdue and the total fine charged that would be jointly consistent with the given information. Make only two selections, one in each column.

lets start wuth 4 days -
1 + 1.5 + 2 + 2.5 = 7

now 5 days
1 + 1.5 + 2 + 2.5 + 3 = 10

10 is here on table

so ans is 5,10

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23 Dec 2025, 10:26

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Fine on day 1 = $1 and it increases by 0.5$ everyday
Day 1= $1
Day 2= 1.5
Day 3= 2
Day 4=2.5 ....Sum = (4/2)(2(1)+(4-1)0.5)=7
Day 5= 3...... sum = (5/2)(2(1)+(5-1)0.5)=10....both choices are in options

Number of overdue days = 5
Total fine = 10

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23 Dec 2025, 10:34

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D1 = 1
D2 = 1 + 0.5 = 1.5
D3 = 1.5 + 0.5 = 2

This is AP with a = 1, d= 0.5
an = a + (n-1)d = 1+(n-1)*0.5 = 1 + 0.5n - 0.5 = 0.5(n+1)
S = n*(a+an)/2 = [1+0.5(n+1)]/2 = n*(3+n)/4

All values are integers,
Checking options n = 5,8 or 12 would work. But if n = 12, then we do not have a bigger value for S, so only 5 or 8 are possible values
S = 5*(3+5)/4 = 10
This works 5,10

Bunuel

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23 Dec 2025, 10:44

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Bunuel

This is an arithmetic sequence with
First term a = 1
common difference d = 0.5

Sum formula:
Sn = n/2[2a + (n-1)d]
Sn = n/2[2(1) + (n-1)(0.5)]

Lets look at the options:
n=4, then Sn = 7 NOT LISTED
n=5, Sn = 10 LISTED
n=6, Sn=13.5 NOT LISTED
n=8, Sn=22 NOT LISTED
n=10, Sn=32.5 NOT LISTED
n=12, Sn = 45 NOT LISTED

Final answer:
no of overdue days = 5
total fine = 10

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23 Dec 2025, 10:44

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Fine on day 1 = $1
Increase of fixed amount of $0.5 each day, i.e. day wise fine would be

1, 1.5, 2, 2.5, 3, 3.5, 4

Total fine for each day would be
1, 2.5, 4.5, 7, 10, 13.5, 17.5 and so on

As we can see, on Day 5 the total fine is $10 which is consistent with given information. Hence the answer.

Alternatively we can solve using Sum of AP if number of days were higher
Sn = n/2 (2a + (n-1) d)
Where n = number of days
a = fine on day one, $1
d = increase in fine, $0.5
Putting values of n in the above equation will give satisfactory answer for 5 days fine as $10.

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23 Dec 2025, 10:49

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Late return fine on the first overdue day = $1
Late return fine on the second overdue day = $1 + $0.5 = $1.5; Total fine = $2.5
Late return fine on the third overdue day = $1.5 + $0.5 = $2; Total fine = $4.5
Late return fine on the fourth overdue day = $2.5; Total fine = $7
Late return fine on the fifth overdue day = $3; Total fine = $10

Number of overdue days	Total fine
5	10

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23 Dec 2025, 11:08

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fine each day is 1$ + 0.5$ * (n - 1), where n is the number of overdue days.
day 1 -> 1
day 2 -> 1.50
day 3 -> 2
day 4 -> 2.50
day 5 -> 3
...

for the total fine you have to add up each previous fine up to the day considered
total fine
day 1 -> 1
day 2 -> 2.50
day 3 -> 4.50
day 4 -> 7
day 5 -> 10

the only options consistent with the data are:
days overdue: 5
total fine = 1 + 1.5 + 2 + 2.5 + 3 = 10

Answers: overdue days = 5 and total fine = 10

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23 Dec 2025, 11:12

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so the fine amount will be 1,1.5,2 , and so on

if the number of days overdue is 5
fine amount is 1+1.5+2+2.5+3=10

answer=5,10

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23 Dec 2025, 11:31

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Number of days	Fine for that day	Total fine overdue
1	$1.00	$1.00
2	$1.00+0.50 = $1.50	$2.50 (1+1.5)
3	$1.50+0.50 = $2.00	$4.5 (2.5+2)
4	$2.00+0.50 = $2.50	$7.00 (4.5+2.5)
5	$2.50+0.50 = $3.00	$10.00 (7+3)
6	$3.00+0.50 = $3.50	$13.5 (10+3.5)

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23 Dec 2025, 11:34

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Fine in 4 days: 1+1.5+2+2.5 = 7
so on 5th day, 7+3 =10

Ans 5 & 10

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23 Dec 2025, 12:00

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We are given that the fine amount is 1$ with 0.5 increase every day

If the number of overdue days were to be 5, the fine amounts would be 1,1.5,2,2.5, and 3 summing up to 10

Since both the values are present as part of answer options, this is the combination that we are looking at.

Therefore, Number of overdue days =5 and Total fine = 10

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23 Dec 2025, 12:34

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given first overdue day fine is $1 which increases by $0.5 each next day.
need to find total no of overdue days and total value of fine up to that day??

overdue day	1	2	3	4	5
fine in $	1	1.5	2	2.5	3
total value of fine up to in $	1	2.5	4.5	7	10

above are the correct choices

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23 Dec 2025, 12:51

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Bunuel

Overdue day = 1 2 3 4 5
Fine got on day = 1 1.5 2 2.5 3
Fine accumulated = 1 2.5 4.5 7 10

Thus from option 5 for overdue days and 10 for total fine.

Alternatively, if n is the overdue day, Fine on that day will be F= 1+(n-1)0.5 : since its an arithmetic sequence and total sum will be then S= (n/2)(1+F)
or S = (n/2)(2+(n-1)0.5)
By putting in values for n from options can get sum.

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23 Dec 2025, 14:43

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Bunuel

this a question of arithmetic progression
first term = 1
no of overdue days =n
sum of all the fine of each day till day n= n/2(2x1+ (n-1) 0.5)------formula of sum of n - term of A,.P
using options one by one we can see that when n= 5 sum = 10 and thus the answer
Ans- 5,10
please note- the question seems vague in its wording....if the fine is increasing each day then why would we need to pay the sum of fines of all the overdue days. This very fact whether the sum of all the fines or the fine on the last day is to be calculated is not clearly specified in the question

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23 Dec 2025, 14:57

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for first overdue day fine is 1$. after that each day it increases by 0.5 more than previous day.

for 4 days= 1+1.5+2+2.5=7$ fine. not in choice
for 5 days= 1+1.5+2+2.5+3= 10$ fine

ans is days= 5, fine= 10$

Bunuel