Two automated data processing algorithms, Algorithm X and Algorithm Y, operate at their re… — GRE General Test

Two automated data processing algorithms, Algorithm X and Algorithm Y, operate at their respective constant rates. Algorithm X takes $x$ hours to process a full dataset when working alone. Algorithm Y operates at a constant rate that is $50\%$ faster than Algorithm X.

In Scenario 1, Algorithm X processes the dataset alone for $2$ hours, after which Algorithm Y joins, and both algorithms work together for an additional $t$ hours to complete the dataset.

In Scenario 2, Algorithm Y processes the dataset alone for $3$ hours, after which Algorithm X joins, and both algorithms complete the remaining work together in $\frac{4}{5}t$ hours.

Which of the following statements must be true? Indicate all such statements.

Algorithm X operating alone requires 14 hours and 30 minutes to process one full dataset.Answer
B
Algorithm Y operating alone requires 10 hours and 15 minutes to process one full dataset.
If both algorithms operate together from the beginning, they will complete one full dataset in 5 hours and 48 minutes.Answer
D
The total time spent by Algorithm X working across Scenario 1 is 5 hours.
E
Algorithm Y processes exactly 60 percent of the dataset in Scenario 2.

Answer

The true statements are that Algorithm X operating alone requires 14 hours and 30 minutes to process one full dataset, and that operating together from the start, both algorithms complete one full dataset in 5 hours and 48 minutes.

The statement specifying that Algorithm X takes 14 hours and 30 minutes alone is correct because solving the system yields

x = 14.5

hours. The statement regarding the combined time of 5 hours and 48 minutes is correct because

\frac{1}{R_X + R_Y} = 5.8

hours, which equals 5 hours and 48 minutes.

Step-by-Step Solution

Define individual and combined work rates in terms of

x

Rate of Algorithm X is

R_X = \frac{1}{x}

datasets/hr. Since Algorithm Y is

50\%

faster, its rate is

R_Y = 1.5 R_X = \frac{3}{2x}

datasets/hr. Their combined rate is

R_{X+Y} = \frac{1}{x} + \frac{3}{2x} = \frac{5}{2x}

datasets/hr.

Establishing accurate rate expressions is essential for modeling combined work scenarios.

Formulate equations for both scenarios and solve for

t

and

x

From Scenario 1:

2 \cdot \frac{1}{x} + t \cdot \frac{5}{2x} = 1 \implies 2 + 2.5t = x

.
From Scenario 2:

3 \cdot \frac{3}{2x} + \frac{4}{5}t \cdot \frac{5}{2x} = 1 \implies 4.5 + 2t = x

.
Equating the two expressions:

2 + 2.5t = 4.5 + 2t \implies 0.5t = 2.5 \implies t = 5

hours.
Substituting

t = 5

yields

x = 2 + 2.5(5) = 14.5

hours (14 hours 30 minutes).

Setting work completed equal to 1 dataset in both scenarios yields a solvable system of equations.

Evaluate the given statement choices using the derived parameters.

1. Algorithm X alone time:

x = 14.5

hours = 14 hours 30 minutes (True).
2. Algorithm Y alone time:

\frac{1}{R_Y} = \frac{2x}{3} = \frac{29}{3} = 9

hours 40 minutes (False).
3. Combined time from start:

\frac{1}{R_{X+Y}} = \frac{2x}{5} = \frac{29}{5} = 5.8

hours = 5 hours 48 minutes (True).
4. Total time Algorithm X works in Scenario 1:

2 + t = 2 + 5 = 7

hours (False).
5. Work fraction by Algorithm Y in Scenario 2:

(3 + \frac{4}{5}(5)) \cdot \frac{3}{2x} = 7 \cdot \frac{3}{29} = \frac{21}{29} \approx 72.4\%

(False).

Direct numerical verification reveals which statements hold true.

Key Concept

Combined work rate equations with variable initial delays and relative rate multipliers.

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