Question

Jon, John and Jan worked together to type a manuscript in 3 hours. If Jon and Jan are to work together, they can finish typing the same manuscript in 4 hours. On the other hand, if john worked alone for 7 hours and then is joined by Jan, they can finish typing it in two more hours. How many hours will each need to type the manuscript alone?

Answer 1

So basically, you have to create 3 separate equations in order to solve

Answer 2

I've done that, but I'm not sure how to solve them.

Answer 3

The third one is a trick

Answer 4

You probably don't have the same equations as me. What did you use as variables?

Answer 5

Let me see your equations

Answer 6

@satellite73

Answer 7

For some reason, I was getting these three equations using your method:

Answer 8

i have a headache already

Answer 9

a = Jon
b = John
c = Jan

\[\large \frac{abc}{ab + bc + ac} = 3\]

\[\large \frac{ac}{a + c} = 4\]

\[\large \frac{bc}{b+ c} = 2\]

Answer 10

\[r_1+r_2+r_3=\frac{1}{3}\]
\[r_1+r_3=\frac{1}{4}\]
\[r_2=\frac{1}{7}\]

Answer 11

But apparently there's something wrong with my last equation, but I don't know why

Answer 12

since \(r_2=\frac{1}{7}\) we get the first equation is ... no no that was wrong

Answer 13

I posted equations of my own satellite. Look at those

Answer 14

\(r_2\) is not \(\frac{1}{7}\)

Answer 15

Do you see the ones I posted above?

Answer 16

yeah, i think the last one is wrong
last statement is tricky

Answer 17

this one is tricky
if john worked alone for 7 hours and then is joined by Jan, they can finish typing it in two more hours
i screwed up with mine

Answer 18

Mine is even more messed up

Answer 19

I have no clue how to write it

Answer 20

i think your first two are right.
last one is wrong

Answer 21

That's why I brought you here. To help me figure out the last one

Answer 22

ok hold on, i think we can get second rate

Answer 23

It uses your methods afterall

Answer 24

well this one is a bit different because of the mix at the end

Answer 25

lets see if slow is good
first statement
Jon, John and Jan worked together to type a manuscript in 3 hours
so \(a+b+c=\frac{1}{3}\)

Answer 26

If Jon and Jan are to work together, they can finish typing the same manuscript in 4 hours
tells us \(a+c=\frac{1}{4}\) and so we can solve for \(b\)

Answer 27

this tells us \(b=\frac{1}{12}\)

Answer 28

we get John needs 12 hours to do the job by himself,
now lets solve for Jan

Answer 29

How come you're not using the

\(\frac{MN}{M + N}\)

Answer 30

because that works when question says "how long will it take them both" given the rate of each
i think your first and second equations are correct, but the third one is wrong
and besides, we need the rate of each, so it makes more sense to solve for them individually

Answer 31

at any rate, (no pun intended) we got John's rate at \(\frac{1}{12}\) and using that we can get Jan's rate

Answer 32

john works for 7 hours, completes \(\frac{7}{12}\) of the job, leaving \(\frac{5}{12}\) to complete,
then it takes two hours with the additional rate of jan, so
\[(\frac{1}{12}+c)\times 2=\frac{5}{12}\]

Answer 33

\[\frac{2}{12}+2c=\frac{5}{12}\]
\[2c=\frac{3}{12}\]
\[c=\frac{1}{8}\] by this time my arithmetic and algebra are probably suspect

Answer 34

and finally
\[a+\frac{1}{12}+\frac{1}{8}=\frac{1}{3}\] and i guess i should actually solve because now i am curious

Answer 35

I already knew what the answers were.

Answer 36

Just couldn't figure out the equation for the last one

Answer 37

you knew? lol ok did you get 8 12 and 8?

Answer 38

Yeah

Answer 39

the problem with that slick method i like so much is that it doesn't work in exactly these types of problems, where one person works for a while and then the other joins
that is because the right hand side is no long one (one job) but something like five twelfths of a job

Answer 40

Oh, I see what you mean

Answer 41

But when you did it, you only used the first two equations as well.

Answer 42

So my way should work for that too

Answer 43

yeah maybe, but how much of the job has one completed before the other joins? you need to know how much is left
i bet you can come up with an equation in \(b\) and \(c\) that would give you that, but i don't know it

Answer 44

it would be rather complicated i think

Answer 45

If anyone can, @satellite73 can

Answer 46

thanks for the vote of confidence
now it is time for the twilight zone

Answer 47

I'm disappointed that there's no way to find b using the method I chose