A municipal water treatment plant fills a main reservoir by pumping water from Inlet Pipe … — GMAT

A municipal water treatment plant fills a main reservoir by pumping water from Inlet Pipe $A$ and Inlet Pipe $B$ simultaneously for 6 hours. The water entering through Pipe $A$ has a mineral concentration of $a$ milligrams per liter ( $\text{mg/L}$ ), and the water entering through Pipe $B$ has a mineral concentration of $b\text{ mg/L}$ , where $a < b$ . Is the final mineral concentration of the water in the reservoir less than $\frac{a + b}{2}\text{ mg/L}$ ?

(1) The rate at which Pipe $A$ pumps water is 20 percent greater than the rate at which Pipe $B$ pumps water.
(2) The total volume of water pumped by Pipe $B$ during the 6 hours is 4,500 liters.

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, but statement (2) alone is not sufficient.

Rephrasing the question stem shows that the overall mineral concentration is less than the midpoint

\frac{a+b}{2}

if and only if a larger volume of water is drawn from the pipe with lower concentration (Pipe A), i.e.,

V_A > V_B

. Statement (1) establishes that Pipe A's pumping rate is

1.2

times Pipe B's rate, so over any identical time frame,

V_A > V_B

. This yields a definitive 'Yes' answer, making Statement (1) alone sufficient. Statement (2) provides only the total volume from Pipe B without any information about Pipe A, which is insufficient.

Adım Adım Çözüm

Simplify and rephrase the question target mathematically.

The final concentration is

C = \frac{V_A a + V_B b}{V_A + V_B}

. The question asks if

C < \frac{a + b}{2}

, which expands to

2(V_A a + V_B b) < (V_A + V_B)(a + b) \iff 2V_A a + 2V_B b < V_A a + V_A b + V_B a + V_B b \iff V_A(a - b) < V_B(a - b)

. Since

a < b

, we know that

(a - b) < 0

. Dividing both sides of the inequality by

(a - b)

reverses the inequality sign, yielding

V_A > V_B

. Thus, the question asks: 'Is the volume of water supplied by Pipe A greater than the volume supplied by Pipe B?'

Rephrasing the question stem reveals the underlying relationship required to determine sufficiency.

Evaluate Statement (1) independently.

Statement (1) states that Pipe A's pumping rate

R_A

is 20 percent greater than Pipe B's rate

R_B

, so

R_A = 1.2 R_B

. Since both pipes operate for the exact same duration of 6 hours,

V_A = 6 R_A = 7.2 R_B

and

V_B = 6 R_B

. Since

R_B > 0

V_A = 1.2 V_B > V_B

is guaranteed to be true. Statement (1) gives a definitive 'Yes' to the rephrased question.

A statement that yields a definitive 'Yes' answer to a Yes/No question is sufficient.

Evaluate Statement (2) independently.

Statement (2) gives

V_B = 4,500

liters, but provides no information regarding Pipe A's pumping rate or total volume

V_A

. If

V_A = 5,000

liters, then

V_A > V_B

(Yes); if

V_A = 3,000

liters, then

V_A < V_B

(No). Because multiple outcomes are possible, Statement (2) is not sufficient.

An inability to determine a unique 'Yes' or 'No' answer makes a statement insufficient.

Anahtar Kavram

Question Stem Simplification and Weighted Average Inequalities in Data Sufficiency

Tahmini Süre:2m 0s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru