In a dataset of N distinct values arranged in ascending order, the p\text{th} percentile i… — GMAT

In a dataset of $N$ distinct values arranged in ascending order, the $p\text{th}$ percentile is defined as the value at position $\left\lceil \frac{p}{100} \times N \right\rceil$ .

Initially, a dataset contains $100$ distinct test scores. The $40\text{th}$ percentile score is $S_1$ and the $70\text{th}$ percentile score is $S_2$ . A researcher adds $x$ new scores that are strictly less than $S_1$ , and $y$ new scores that are strictly between $S_1$ and $S_2$ , where $x$ and $y$ are positive integers.

If the $40\text{th}$ percentile score of the expanded dataset remains $S_1$ and the $70\text{th}$ percentile score of the expanded dataset remains $S_2$ , what is the maximum possible value of $x + y$ ?

A
$2$
$3$ Cevap
C
$4$
D
$5$
E
$6$

Cevap

The maximum possible value of

x + y

3

The correct choice is

3

. By setting up the percentile rank equations using the ceiling function, the position of

S_1

in the expanded dataset requires

\lceil 0.4(x + y) \rceil = x

, while the position of

S_2

requires

\lceil 0.7(x + y) \rceil = x + y

. Letting

k = x + y

, the ceiling condition

\lceil 0.7k \rceil = k

implies

k - 1 < 0.7k \le k

, which simplifies to

0.3k < 1

k < 3.33

. Since

x

and

y

are positive integers,

k

can take integer values of

2

3

. Thus, the maximum possible value of

x + y

3

Adım Adım Çözüm

Determine the initial rank positions of

S_1

and

S_2

in the 100-element dataset.

Position of

S_1 = \lceil 0.40 \times 100 \rceil = 40

; Position of

S_2 = \lceil 0.70 \times 100 \rceil = 70

Applying the given percentile formula to

N = 100

gives the exact index of each score in ascending order.

Express the new rank positions of

S_1

and

S_2

after inserting

x

scores below

S_1

and

y

scores between

S_1

and

S_2

In the new dataset of size

100 + x + y

S_1

is at position

40 + x

and

S_2

is at position

70 + x + y

Adding

x

scores less than

S_1

shifts

S_1

's rank by

x

. Adding

y

scores between

S_1

and

S_2

further shifts

S_2

's rank by

x + y

relative to its original position.

Formulate the percentile conditions for the expanded dataset of size

M = 100 + x + y

For

S_1

\lceil 0.40(100 + x + y) \rceil = 40 + x \implies \lceil 0.4(x + y) \rceil = x

.
For

S_2

\lceil 0.70(100 + x + y) \rceil = 70 + x + y \implies \lceil 0.7(x + y) \rceil = x + y

The percentile rank of

S_1

must equal its new position

40 + x

, and the percentile rank of

S_2

must equal its new position

70 + x + y

Solve for the total added elements

k = x + y

From

\lceil 0.7k \rceil = k

, we get

k - 1 < 0.7k \le k

, which simplifies to

0.3k < 1 \implies k < \frac{10}{3} \approx 3.33

By definition of the ceiling function,

\lceil z \rceil = k \iff k - 1 < z \le k

for integer

k

Evaluate integer solutions for

k = x + y

given

x, y \ge 1

For

k = 2

x = \lceil 0.4(2) \rceil = 1

y = 1

(valid).
For

k = 3

x = \lceil 0.4(3) \rceil = 2

y = 1

(valid).
The maximum integer value for

k = x + y

3

Since

x

and

y

are positive integers,

k \ge 2

. The maximum integer

k < 3.33

3

Anahtar Kavram

Percentile Rank and Indexing with Ceiling Functions

Tahmini Süre:2m 30s

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