A university research institute evaluated N technology projects completed last year. Each … — GMAT

A university research institute evaluated $N$ technology projects completed last year. Each project received grant funding from at least one of two foundations: Foundation $X$ or Foundation $Y$ . Exactly $30\%$ of the $N$ projects received funding from both foundations. If the average (arithmetic mean) grant amount per project among projects funded by Foundation $X$ was $\$50,000$ and the average grant amount per project among projects funded by Foundation $Y$ was $\$60,000$ , what was the average total grant funding per project across all $N$ projects?

(1) The number of projects that received funding from Foundation $X$ only was equal to the number of projects that received funding from Foundation $Y$ only.
(2) The total dollar amount disbursed by Foundation $Y$ exceeded the total dollar amount disbursed by Foundation $X$ by $\$1,800,000$ .

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Cevap
B
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not sufficient.

Statement (1) alone is sufficient because equating the counts of projects funded exclusively by Foundation X and exclusively by Foundation Y implies that the total number of projects funded by Foundation X equals the total number funded by Foundation Y. Combined with the principle of overlapping sets where thirty percent of projects received funding from both, this uniquely determines that sixty-five percent of projects were funded by Foundation X and sixty-five percent were funded by Foundation Y, yielding a unique overall average grant of $71,500. Statement (2) alone is insufficient because the equation relating total disbursements depends on the unknown total number of projects N.

Adım Adım Çözüm

Express the total funding and set proportions in terms of the total number of projects

N

Let

n_X

be the number of projects funded by Foundation

X

and

n_Y

be the number of projects funded by Foundation

Y

. The number of projects funded by both is

0.30N

. Since every project is funded by at least one foundation,

N = n_X + n_Y - 0.30N \implies n_X + n_Y = 1.30N

. Dividing by

N

gives the ratio sum

\frac{n_X}{N} + \frac{n_Y}{N} = 1.30

Establishing the relationship between the subset counts and total projects simplifies the target average formula.

Formulate the expression for the average total grant funding per project.

Total Funding

= 50,000 n_X + 60,000 n_Y

. Therefore, the average funding per project is

\text{Average} = \frac{50,000 n_X + 60,000 n_Y}{N} = 50,000\left(\frac{n_X}{N}\right) + 60,000\left(\frac{n_Y}{N}\right)

Since total grant funding across all projects is the sum of all money disbursed by Foundation

X

and Foundation

Y

, the overall average depends strictly on the ratios

\frac{n_X}{N}

and

\frac{n_Y}{N}

Evaluate Statement (1) independently.

Statement (1) states that the number of projects funded by

X

only equals the number funded by

Y

only:

n_X - 0.30N = n_Y - 0.30N \implies n_X = n_Y

. Since

\frac{n_X}{N} + \frac{n_Y}{N} = 1.30

and

n_X = n_Y

, we get

2\left(\frac{n_X}{N}\right) = 1.30 \implies \frac{n_X}{N} = 0.65

and

\frac{n_Y}{N} = 0.65

. Substituting these into the average formula yields

\text{Average} = 50,000(0.65) + 60,000(0.65) = 71,500

. Statement (1) ALONE is sufficient.

Knowing that the two single-foundation set sizes are equal determines the exact proportions of

N

funded by each foundation.

Evaluate Statement (2) independently.

Statement (2) states that

60,000 n_Y - 50,000 n_X = 1,800,000

. Dividing by

N

gives

60,000\left(\frac{n_Y}{N}\right) - 50,000\left(\frac{n_X}{N}\right) = \frac{1,800,000}{N}

. Because

N

is unknown, the right-hand side is not fixed, so

\frac{n_X}{N}

and

\frac{n_Y}{N}

cannot be uniquely determined. Statement (2) ALONE is not sufficient.

An absolute dollar equation introduces a dependency on the total count

N

, preventing a unique calculation of the relative proportions.

Anahtar Kavram

Overlapping Sets and Weighted Averages in Data Sufficiency

Tahmini Süre:2m 0s

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