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Zorluk: KolayRates, Ratios, Percentages, and Applied Word Problems

A water reservoir is filled by two pumps, Pump X and Pump Y, each operating continuously at its own constant rate. How many hours does it take for Pump X and Pump Y working together to fill the empty reservoir?

(1) Working alone at its constant rate, Pump X fills the reservoir in 6 hours.
(2) Working alone at its constant rate, Pump Y fills the reservoir in 12 hours.

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

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
The correct answer is that both statements together are sufficient, but neither statement alone is sufficient. Statement (1) gives only the rate of Pump X, and Statement (2) gives only the rate of Pump Y. Neither statement alone allows calculation of the combined rate. When used together, the combined rate is 16+112=14\frac{1}{6} + \frac{1}{12} = \frac{1}{4} reservoirs per hour, which yields a unique solution of 4 hours.

Adım Adım Çözüm

1
Rephrase the question stem mathematically
Let rXr_X be the hourly rate of Pump X and rYr_Y be the hourly rate of Pump Y. The target combined time TT is given by T=1rX+rYT = \frac{1}{r_X + r_Y}. We need the value of rX+rYr_X + r_Y.
Simplifying the target variable helps determine what information is necessary for sufficiency.
2
Evaluate Statement (1) independently
Statement (1) states rX=16r_X = \frac{1}{6} reservoir per hour, but gives no information about rYr_Y. Therefore, rX+rYr_X + r_Y cannot be calculated.
One variable in a two-variable sum remains unknown.
3
Evaluate Statement (2) independently
Statement (2) states rY=112r_Y = \frac{1}{12} reservoir per hour, but gives no information about rXr_X. Therefore, rX+rYr_X + r_Y cannot be calculated.
One variable in a two-variable sum remains unknown.
4
Evaluate Statement (1) and Statement (2) together
Combining both statements: rX+rY=16+112=312=14r_X + r_Y = \frac{1}{6} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}. Thus, T=11/4=4T = \frac{1}{1/4} = 4 hours. A single, definitive numerical answer is obtained.
Both individual rates are known, allowing exact computation of the combined rate and total time.

Anahtar Kavram

Combined Work Rates in Data Sufficiency
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Rates, Ratios, Percentages, and Applied Word Problems

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