There are 900 bottles to be filled. Jim and Molly working independently but at the same time take 30 minutes to fill the bottles. How long should it take Molly working by herself to fill the bottles?
Statement 1 - Molly fills half as many bottles as Jim.
Statement 2 - Jim would take 45 minutes by himself.
Which of the statements above make it possible to answer the question.
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Statement 1 alone is sufficient, but statement 2 alone is not sufficient.
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Statement 2 alone is sufficient, but statement 1 alone is not sufficient.
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Both statements together are sufficient, but neither statement alone is sufficient.
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Each statement alone is sufficient
To solve this question, you need to know how to set up a system of equations using the given information and how to solve for the unknown variable. Here is one possible explanation:
Let $x$ be the number of bottles that Jim fills in one minute and $y$ be the number of bottles that Molly fills in one minute. Then, we have the following equations:
- $x + y = 30$ (the total number of bottles filled by Jim and Molly in one minute $\frac{900}{30}$)
Statement 1 tells us that $y = x/2$, which means that Molly fills half as many bottles as Jim in one minute. We can substitute this into the first equation and get:
- $x + x/2 = 30$
- $3x/2 = 30$
- $x = 20$
This means that Jim fills 20 bottles in one minute and Molly fills 10 bottles in one minute. We can use this to find how long it takes Molly to fill the bottles by herself:
- $30y = 900$
- $30(10) = 900$
- $y = 30$
So, Molly takes 30 minutes by herself to fill the bottles. Therefore, statement 1 alone is sufficient to answer the question.
Statement 2 tells us that $45x = 900$, which means that Jim would take 45 minutes by himself to fill the bottles. We can solve for x and get:
- $x = 20$
This is the same as what we found from statement 1, so we can use it to find how long it takes Molly to fill the bottles by herself:
- $y = x/2$
- $y = 20/2$
- $y = 10$
So, Molly fills 10 bottles in one minute and takes 30 minutes by herself to fill the bottles. Therefore, statement 2 alone is also sufficient to answer the question.
Since both statements alone are sufficient, the correct answer is D. Each statement alone is sufficient.
Correct answer: Each statement alone is sufficient.
Together, Jim and Molly fill 900 bottles in 30 minutes, so their combined rate is 30 bottles/min.
Statement 1: If Molly fills half as many bottles as Jim in that same 30-minute window, she fills 300 and Jim fills 600. Molly's rate = 300/30 = 10 bottles/min, so alone she'd take 900/10 = 90 minutes. Sufficient.
Statement 2: If Jim alone takes 45 minutes, his rate is 900/45 = 20 bottles/min. Since the combined rate is 30/min, Molly's rate = 30 - 20 = 10 bottles/min, giving the same 90 minutes. Sufficient.
Both statements independently pin down Molly's rate (and both agree on 90 minutes), so each alone answers the question — making the other three options, which claim one or both statements are insufficient, wrong.