Lesson rate of work

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This Lesson (rate of work ) was created by by richwmiller(17219)

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There are two basic types of "Rate of work" problems.
The more common FIRST type A is when each person works at a different rate and we want to find out how fast they work either alone or with others.
An example of type "A"
If Sally can paint a house in 4 hours and Jim can paint the same house in 6 hours, then how long will it take for both of them to paint that house together?

The other type B is where we assume that each individual in a group works at the same rate.
 An example of type "B" 
"If 2 persons can paint 2 house in 2 hours then, then how many person can paint 18 house in 6 hours?"

I'll start with type A.
The more common type A is when each person works at a different rate and we want to find out how fast they work either alone or with others.

The equation has two basic forms
1) a/b+a/c=1 or 
2) 1/b+1/c=1/a
The lone "1" in the first equation represents the one task at hand.
We derive the second equation from the first by dividing both sides by "a".

Each person does a fraction of the job which all adds up to one job.
The letter "a"  represents the time working together.
The letters "b" and "c" represent the time needed by person a and person b to complete the job alone.
If Sally can paint a house in 4 hours and Jim can paint the same house in 6 hours, then how long will it take for both of them to paint that house together?
Using our example of Sally and Jim we know b and c and want to find "a". The second equation is easiest to use for finding "a"

1/b+1/c=1/a
fill in b and c
1/4+1/6=1/a
find common denominator
3/12+2/12=1/a
5/12=1/a
a=12/5
a=2 2/5 hours or 2.4 hours or 2 hours 24 minutes

The SECOND type of Type A asks how fast one of the workers can work alone.
We will need to know how fast they work together. 
Either equation /formula can be used.
1) a/b+a/c=1 or 
2) 1/b+1/c=1/a
"Marie and Isabel can paint an apartment in 5 hours working together. If Marie can do the same job alone in 7 hours, how long will it take Isabel to finish it alone?"
remember a is time working together.

1) a/b+a/c=1 or 
2) 1/b+1/c=1/a
1) 5/7+5/c=1 or 
2) 1/7+1/c=1/5

In either case we will solve for c
5/7+5/c=1
subtract 5/7 from both sides
5/c=2/7
cross multiply
35=2c
17.5=c
Does this make sense? Yes. Why? They only save 2 hours when both work together. If they worked at the same rate then they would only need 3.5 hours. (half of 7 hours) So we know that Isabella is much slower.

Let's do it with equation 2.
1/7+1/c=1/5
get common denominator
5/35+1/c=7/35
1/c=2/35
c=35/2
c=17.5

The THIRD type of type A is when we have a relationship between workers. One works twice as fast or some such.
"It takes John three times as long to clean an office building as Lisa. If they work together, it takes 6 hours to clean. How long does it take Lisa by herself? "
Lisa (b) will be 3x and John (c) will be x  
We could also have John as x/3 and Lisa as x

1) a/b+a/c=1 or 
2) 1/b+1/c=1/a
a=6
b=3x
c=x
1) 6/(3x)+6/x=1 or 
2) 1/(3x)+1/x=1/6

6/(3x)+6/x=1
6/3x+18/3x=1

24/3x=1
b=3x=24
c=x=8

1/(3x)+1/x=1/6

1/3x+3/3x=1/6
4/(3x)=1/6
24/3x=1
b=3x=24
c=x=8

A FOURTH type A problem is when someone leaves the job or there are more than two people involved

Question 839596: A, B, & C can do a piece of work in 10, 9, 15 days respectively. They started the work together but C left after 2 days. In how many days will the remaining work be completed by A & B?

2/10+2/9+2/15+x/10+x/9=1
(19x)/90+5/9 = 1
(19 x)/90=4/9 
(19 x)/10=4 
19x=40
x = 40/19
x = 2.1053 days more to finish
BTW They would have finished in 3.6 days total with all three.

A FIFTH  type A when the work isn't completed
Question 843775: Five sixths of a room is now painted.Carlos did two fifths of the painting.How much of the room did he paint?

2/5*5/6=1/3 
Carlos did 1/3 or 2/6 which means that someone else did 3/6 or 1/2

A SIXTH type A is when we have two groups of people working at different rates
If 6 men and 5 women complete a task in 6 days while the same task complete by 3 men and 4 women in 10 days. How long will take for 9 men and 15 women to complete this task?

6*6=36 md
5*6=30 wd 
3*10=30 md
4*10=40 wd 
36md+30wd=30md+40wd
6md=10wd
.6m=1w 
36mh+30*wd*.6=(36+18) md =54 md 
30md+.6*40wd=(30+24)md=54 md 
9d+.6*15d=54
9d+9d=54
18x=54
d=3 days for 9 men and 15 women 
another way
6/x+5/y=1/6
3/x+4/y=1/10 
6/x+5/y=1/6
6/x+8/y=2/10 
6/x+8/y=6/30
6/x+5/y=5/30
subtract
3/y=1/30

x = 54, y = 90 
9/54+15/90=1/d
1/6+1/6=1/d
2/6=1/d
d=3

Next we will discuss Type B rate of work problems

An example of type "B" 
"If 2 persons can paint 2 houses in 2 hours then, then how many person can paint 18 houses in 6 hours?"
First calculate how many man hours are used.
They use 2*2=4 person/man hours to paint 2 houses.
now find how many man hours are used per unit/house
4/2 =2 person/man hours per house.

Now for the question.
Find how many man hours are needed for the job
18 houses at 2 hours per house=36 person/man hours.
Divide the  total man hours need by the time allowed to find number of people needed.
36/6 =6 persons 
6 persons will need 6 hours to paint 18 houses.

The variations that you might see are 
1) if some people leave after a few hours/days and 
2) more people are added
3) if you want to find how many hours a certain number of men will need.

1) if some people leave after a few hours/days

A certain number of men can finish a piece of work in 10 days.
If however there were 10 men less, it will take 10 days more for the work to be finished.
How many men were there originally?

let m = the original number of men
then
10m = the number of man-days required to complete the job
:
"If there were 10 men less, it will take 10 days more for the work to be finished"
That means that it would take 20 days
20(m-10) = 10m
20m - 200 = 10m
20m - 10m = 200
10m = 200
m = 200/10
m = 20 men originally
:
:
Check this; find the number of man-hrs to do the job.
20*10 = 200 man-hrs (20 men working 10 days)
or
10*20 = 200 man-hrs (10 men working 20 days)

2) more people are added
12 clerks are assigned to enter certain data on index cards. The number of clerks could perform the task in 18 days. After these clerks have worked on this assignment for 6 days, 4 more clerks are added to the staff to do this work. Assuming that all the clerks work at the same rate of speed, how many days will it take to complete the entire task?

12*18=216 clerk days
216=12*6+16x
144=16x
x=9 more days

3) if you want to find the time a certain number of men will need.

It took 24 workers working 10 hours per day for 24 days to complete a task. At this rate, how many days will it takes 60 workers to complete the same task working 8 hours per day?

24 * 10 * 24 = 5760 man-hours

Let d = no. of days required to complete the job
60 * 8 * d = 5760
480d = 5760
d = 5760/480
d = 12 days

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