2 work problems?

Question

(1)Smith works twice as fast as Legatos and three times as fast as Paruolo. If paruolo can complete a job in 12 hours, what part of the job can Legatos do in 6 hours?

I don't get how to setup the equation


(2) If 3 secretaries can type six manuscripts in 12 days, how many days will it take two secretaries to type three such manuscripts?

someone please explain how to do this one

Answer 1

In general with these problems you have to work out the rate (in portion of the job per hour) that each person does, and then you are multiplying or adding rates, and then using that to compute the time required for some other task.

(1) If paruolo can do it in 12 hours,
Smith is 3x as fast and can do it in 12/3 = 4 hours.
Legatos is twice as slow as Smith and takes 2*4 = 8 hours.

In 6 hours, Legatos can do 6/8 = 3/4 of the job.

If you want to do it in equations:

p = 1/12 (paruolo does 1/12 job per hour)
s = 3*p = 3/12 = 1/4 (smith 3 times as fast)
L = s/2 = (1/4)/2 = 1/8 (legatos twice as slow)

Legatos does 1/8 of the job per hour, so in 6 hours he'd do 6*L of the work:

6*L = 6*(1/8) = 6/8 = 3/4

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(2) The first thing to do is find the rate in manuscripts per secretary per day. This is done by dividing total manuscripts by total secretaries and total days:

rate = manuscripts/(days * secretaries)
r = 6 / (3 * 12)
r = 1/6 manuscript per secretary per day

So... each secretary completes 1/6 of a manuscript per day.

Two secretaries will complete twice as much:

2r = 2*1/6
2r = 1/3 of a manuscript per day

If three manuscripts are needed:

amount / rate = time
3 manuscripts / (1/3 manuscript/day) = t days
3 / (1/3) = t
t = 9 days

Answer 2

Smith=S
Legatos=L
Paruolo=P

1S=2L (L=1/2S)
1S=3P (P=1/3S)

To makes things as easy as possible substitute everything to be in S values.
12 hours x 1P= 12x 1/3S = 4S thus Smith could do the job in 4 hours. Legatos could do it in 8 hours. If he only has 6 hours, then...
1L=8hours
1hour=1/8L
6hours=6/8L reduces to 3/4L
Legatos could do 3/4 of the job.
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Question 2 is a lot easier....
Since everything is divisible by 3 you could just say:
3s=6m per 12d
1s=6/3m per 12d or 1 Secretary = 2 manuscript per 12 days
OR 1 secretary = 1 manuscript per 6 days
- 1 sec= 1/6 manuscript per day
OR 2 secretary = 2/6 manuscript per 1 days
since we need 3 (or 18/6) then
X times 2/6 = 18/6
multiply both sides by 6 gives you:
X times 2 = 18
X = 9 days