rheam25 wrote:
Alan, Botham and Caddick worked on a project one after the other and never at the same time. Alan started the project and completed 20 percent of it; then Botham took over and completed another 30 percent of the project. After this Caddick started working on the project and completed it. If the time for which Alan worked on the project was 50 percent of the total time required to complete the project and Caddick worked on the project for 4 days, how much time would it have taken to complete the project if only Alan and Botham had worked together on the project from start to completion at their same respective daily rate of work?
(1) Botham worked on the project for 2 days
(2) The time for which Alan and Caddick worked on the project was 5 times the time for which Botham worked on the project.
Reading the original question, I'm thinking about using some numbers to make this simpler. In a rates problem that doesn't give you a total amount of work, it's typically pretty safe to just choose a number for the overall amount - things will tend to work out the same regardless. So let's suppose that the project is to knit 100 scarves.
So, Alan knitted 20 scarves, then Botham knitted another 30, then Caddick knitted the last 50.
It took Caddick four days to knit his 50 scarves, so he knitted at a rate of 12.5 scarves per day.
Also, Alan took half of the total time. We don't know what the total time is, though, so that doesn't help us much just yet.
That's the info - now, let's focus on the
question.
How long would it have taken A and B to knit all 100 scarves by themselves?
To answer this, we'd need to know their combined rate - we don't have a lot of info that would help with that at the moment. Let's check the statements.
(1) Botham worked on the project for 2 daysThis tells us that B's rate was 15 scarves per day. Does it let us find A's rate?
It actually does. A used half of the total time, and we know that B and C used a total of 6 days. So, A used 6 days as well, knitting at a rate of 3.33 scarves per day.
That gives us a combined rate for A and B (18.33 scarves per day), so we could figure out how long it would take them both to knit 100 scarves together.
(2) The time for which Alan and Caddick worked on the project was 5 times the time for which Botham worked on the project.Let's focus on the times:
- C took 4 days
- A took half of the total time
- A and C together took five times as long as B
Translating into math:
C = 4
A = 1/2(A + B + C)
A + C = 5B
We have three equations and three variables - we can probably figure out how much time everyone took...
A = B + C
A = B + 4
A + C = 5B
A + 4 = 5B
B + 4 + 4 = 5B
B + 8 = 5B
4B = 8
B = 2
A = B + 4 = 6
Cool - A took 6 days, B took 2 days, and C took 4 days. From there, we can answer as above.
So, both statements are sufficient on their own, and the answer is
D.