Math Question- HELP!

YourDentition said:

I can't get this answer. It's from Top Score.

If Sarah can do work in 8 days and Julie can do the same work in 12 days, which of the combinations can complete the work in the least amount of time?
The answer is 40% work done by Sarah and the rest with Julie.
But I figured that since Sarah did the work the fastest, she should do it alone.

HELP!

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If Sarah works alone she'll do it in 8 days but if they work together it will take less - up to a point (apparently 40%). After that it will shift towards what Julie can do (12 days) until she does 100% of the work and it takes her 12 days.

Streetwolf said:

If Sarah works alone she'll do it in 8 days but if they work together it will take less - up to a point (apparently 40%). After that it will shift towards what Julie can do (12 days) until she does 100% of the work and it takes her 12 days.

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haha love when you make cameo appearances for math questions...nice work man

how do we set up an equation for this type of question?

I don't know about a specific equation for this type of problem, but my line of reasoning is: Sarah's rate of work is 1job/8days or 3jobs/24 days.Julie's rate of work is 1job/12days or 2jobs/24days. Together they work at a rate of 5 jobs/24days, or 4.8days/job. In that 4.8 days, Sarah does 4.8d/(8d/job) = .60 of a job, or 60% of one job. Julie does the other 40%.

jlt0530 said:

how do we set up an equation for this type of question?

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The least amount of time on the job is when they work together, at the maximum amount per day per person. If the amount that one person works from there goes down then the other person will have to work more and so it will only take longer. Thus we do what we normally do in these problems:

Sarah does 1/8 work per day.
Julie does 1/12 work per day.

Together they do 5/24 work per day.
Thus it takes 24/5 days.

So Sarah does 1/8 * 24/5 = 3/5 of the job = 60%.
And Julie does 1/12 * 24/5 = 2/5 of the job = 40%.

I think you got your percentages mixed up.

The book's answer can't be right. Here's why...

Start with the answer we have. It takes 4.8 days to do the work with the percentages I got.

What would happen if one of the two girls did LESS work? Then it would take them LESS time to do it. That's good.

But how about the other girl? She would have to do MORE work. And as a result it would take her MORE time. And so even though the first girl would be done sooner, we still have to wait the extra time for the second girl to finish.

Thus the scenario where they finish the fastest is that in which they finish at the same time.

Streetwolf has it right.

Also, if you want a simple formula for these types of problems it is:

If it takes A days for someone to do the work and B days for another, it will take (A*B)/(A+B) days for them to complete it together.

(12*8)/(12+8) = 96/20 = 4.8 days