GMAT含相对工作效率的工作速率问题通用代数解法问询

Hey John, let's clear up this subtle mix-up you're hitting—you're already using the right core formula, the issue just comes down to how you translate the "twice as fast" relationship into equations. Let's break this step by step.

First, let's make sure we're on the same page with the standard work rate formula:

$\frac{1}{A} + \frac{1}{B} = \frac{1}{t}$
Here, A and B represent the time each machine takes to complete the task alone, and t is the time they take together. The fractions $\frac{1}{A}$ and $\frac{1}{B}$ are the work rates (tasks per hour) of each machine.

Your mistake came when translating "Machine A is twice as fast as Machine B":

• If A is twice as fast, its work rate is double B's rate. So mathematically: $\text{Rate}_A = 2 \times \text{Rate}_B$
• Since rate is the inverse of time, that means $\frac{1}{A} = 2 \times \frac{1}{B}$
• Rearranging that gives $B = 2A$ (this makes sense: if A is faster, it takes half the time B does, so B's time is double A's)

Now let's plug this corrected relationship into the formula using your example:

1. Substitute $B = 2A$ into $\frac{1}{A} + \frac{1}{B} = \frac{1}{10}$:
   $\frac{1}{A} + \frac{1}{2A} = \frac{1}{10}$
2. Combine the fractions on the left:
   $\frac{2 + 1}{2A} = \frac{3}{2A} = \frac{1}{10}$
3. Solve for A:
   $2A = 30 \implies A = 15$ hours
4. Then find B using $B = 2A$:
   $B = 2 \times 15 = 30$ hours

That's the correct result you were expecting!

General Step-by-Step Approach for Relative Rate Work Problems #

• Define your variables clearly: Stick to using variables for time to complete the task alone (since GMAT questions almost always ask for time, this keeps things intuitive).
• Translate relative speed to a time relationship:
  • If X is n times as fast as Y, then $\text{Rate}_X = n \times \text{Rate}_Y$
  • Convert that to time: $\frac{1}{X_{time}} = n \times \frac{1}{Y_{time}}$, so $Y_{time} = n \times X_{time}$ (faster machines have shorter times)
• Plug into the standard work formula: $\frac{1}{X} + \frac{1}{Y} = \frac{1}{\text{combined time}}$
• Solve and verify: Check that the resulting rates add up to the combined rate to confirm you didn't mix up relationships.

Let's test this with a quick sanity check:

• A's rate is $\frac{1}{15}$ tasks per hour, B's is $\frac{1}{30}$ tasks per hour
• Combined rate: $\frac{1}{15} + \frac{1}{30} = \frac{2 + 1}{30} = \frac{1}{10}$ tasks per hour, which matches the given combined time of 10 hours. Perfect!

备注：内容来源于stack exchange，提问作者John