Multiple choice

Three machines, D, E, and F, operating at constant rates, can together produce a batch of components in 8 hours. How long would machines E and F, working together, take to produce the same batch? (1) Machine D produces twice as many components per hour as machine E, and machine E produces twice as many components per hour as machine F. (2) The time machine D takes to produce 12 components is the same as the time machines E and F together take to produce 9 components.

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

  4. EACH statement ALONE is sufficient.

  5. Statements (1) and (2) TOGETHER are NOT sufficient.
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D Correct answer
Explanation

Let rates be D, E, F. D+E+F = 1/8. (1) D=2E, E=2F => D=4F, E=2F. 4F+2F+F = 1/8 => 7F = 1/8 => F=1/56. E=2/56. E+F = 3/56. Time = 56/3 hours. (2) D/12 = (E+F)/9 => 9D = 12(E+F) => 3D = 4(E+F). Since D+E+F=1/8, E+F = 1/8 - D. 3D = 4(1/8 - D) => 7D = 0.5 => D=1/14. E+F = 1/8 - 1/14 = 3/56. Both are sufficient.

AI explanation

The combined rate of machines D, E, and F is 1/8 of the batch per hour. Statement (1) gives the rates as D = 2E and E = 2F; substituting these into the combined rate equation 2F + 2F + F = 1/8 yields 5F = 1/8, so the combined rate of machines E and F is 2/8 + 1/8, or 3/8 of the batch per hour. Statement (2) provides a direct ratio of the rates, stating the rate of D is to the combined rate of E and F as 12 is to 9, or 4 to 3; using the formula for combined rates, if D + (E + F) = 1/8 and D / (E + F) = 4/3, you can solve for the combined rate of E and F. Each statement independently provides enough data to find the time taken by machines E and F, so both statements independently are sufficient.

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