In a dataset of N distinct test scores, score S_1 is at the 75th percentile and score S_2 … — GMAT

In a dataset of $N$ distinct test scores, score $S_1$ is at the 75th percentile and score $S_2$ is at the 40th percentile. A group of 60 new distinct scores is added to the dataset: 12 are strictly less than $S_2$ , 18 are strictly between $S_2$ and $S_1$ , and 30 are strictly greater than $S_1$ . If score $S_1$ is at the 60th percentile of the combined dataset, what is the percentile rank of score $S_2$ in the combined dataset?

A
16th percentile
B
25th percentile
28th percentileCevap
D
40th percentile
E
46th percentile

Cevap

28th percentile

In the original dataset of

N

scores,

0.75N

scores are below

S_1

and

0.40N

scores are below

S_2

. Adding 60 scores (12 below

S_2

, 18 between

S_2

and

S_1

, and 30 above

S_1

) increases the total dataset size to

N + 60

. The number of scores below

S_1

becomes

0.75N + 12 + 18 = 0.75N + 30

. Given that

S_1

is at the 60th percentile of the new dataset,

0.75N + 30 = 0.60(N + 60)

, which simplifies to

0.15N = 6

, giving

N = 40

. The original number of scores below

S_2

0.40 \times 40 = 16

. Adding the 12 new scores that are below

S_2

gives

16 + 12 = 28

scores below

S_2

in the combined dataset. Out of 100 total scores in the combined dataset, the percentile rank of

S_2

\frac{28}{100} \times 100\% = 28\%

, which corresponds to the 28th percentile.

Adım Adım Çözüm

Set up expressions for the number of scores below

S_1

and

S_2

in the original dataset.

In the original dataset of

N

scores,

0.75N

scores are strictly less than

S_1

, and

0.40N

scores are strictly less than

S_2

By definition of percentile rank,

k\%

percentile means

k\%

of the dataset scores fall strictly below that value.

Determine the number of scores strictly below

S_1

in the combined dataset.

The total number of new scores added strictly below

S_1

12 + 18 = 30

. Thus, the total number of scores below

S_1

in the combined dataset is

0.75N + 30

, while the new total dataset size is

N + 60

Scores added below

S_2

and scores added between

S_2

and

S_1

are all strictly less than

S_1

Solve for

N

using the 60th percentile rank condition for

S_1

in the combined dataset.

0.75N + 30 = 0.60(N + 60) \implies 0.75N + 30 = 0.60N + 36 \implies 0.15N = 6 \implies N = 40

Setting the count of scores below

S_1

equal to

60\%

of the new total dataset size

N + 60

forms a single-variable linear equation.

Calculate the percentile rank of score

S_2

in the combined dataset.

Original scores below

S_2 = 0.40 \times 40 = 16

. Combined scores below

S_2 = 16 + 12 = 28

. Combined total dataset size =

40 + 60 = 100

. Percentile rank of

S_2 = \frac{28}{100} \times 100\% = 28\%

Dividing the total count of scores strictly below

S_2

in the combined set by the total combined dataset size yields the updated percentile rank.

Anahtar Kavram

Percentile Rank and Combined Sets

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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