# Thread: Time and Work Problem

1. ## Time and Work Problem

Hello,

I feel like this should really be simple, but I'm still having trouble in finding the right equation for this word problem:

"John can finish a job in 9 hours. After working for 5 hours, he decided to take a rest. Randy helped out John and finished the job in 2 hours and 20 minutes. How long would it take for Randy to do the job alone?"

The answer to this is 5 hours and 15 min. However, I don't quite understand how it got there.

For, the first few problems that I did with this type of word problem, I would either use,

this kind of formula: t/a + t/b=1, where t is the time spent working together, and a and b, their individual times.

or this formula: 1/a+1/b= 1, where I take into account the rate of work of each individual.

Both types of formula has helped me get to the right answer, till I got to this problem.

What I did was to add the 5 hours from John and 2 hours and 20 minutes from Randy, in order to get the total time spent working, but I don't think that's right, since it's not yielding the right answer.

I hope someone can help, preferably in layman's terms. Thanks.

2. ## Re: Time and Work Problem

I would think of the question in terms of rates, which would derive a formula similar to 1/a + 1/b = 1

I would suggest not relying on formulas, but rather focus on the reasoning behind why the formulas work. Above is an equation of two combined rates to complete a job.

First, consider the amount of the job has been completed by John in the 5 hours he worked, which means the remaining work for the job was done by Randy in 2 hours 20 minutes.
John's work rate is 1 job per 9 hours, which is equal to the rate of 1/9 job per hour.
That means John completes 5/9 of the job in 5 hours
That means Randy completed 1 - 5/9 = 4/9 of the job in 2 hours and 20 minutes = 140 minutes.

Hence Randy's rate is 4/9 job per 140 minutes
Dividing the top and bottom of the ratio by 4/9 to get an equivalent ratio yields the rate of 1 job per 140 * 9 / 4 minutes = 315 minutes = 5h 15min

EDIT: Different from romsek's answer because I assume that Randy completed the remaining part of the job alone, rather than assume both John and Randy completed the job together.
His answer is much cleaner, and can be adapted for the above assumption easily. (instead of r_j + r_R, it would be just r_R)

3. ## Re: Time and Work Problem

John can finish a job in 9 hrs.

John's rate of working is thus $r_J = \dfrac{1~job}{9~hr}$

Randy's rate of working is unknown call it $r_R$

$1~job = 5~hr \cdot r_J \dfrac {job}{hr} + \dfrac 7 3 ~hr \cdot (r_J+r_R) ~\dfrac {job}{hr}$

dropping units for a moment

$1 =5 \cdot \dfrac 1 9 + \dfrac 7 3 \left(\dfrac 1 9 + r_R\right)$

$1 = \dfrac 5 9 + \dfrac{7}{27} + \dfrac 7 3 r_R$

$1 - \dfrac{22}{27} = \dfrac 7 3 r_R$

$\dfrac{5}{27} = \dfrac 7 3 r_R$

$r_R = \dfrac 3 7 \cdot \dfrac{5}{27} = \dfrac{5}{63}~\dfrac{job}{hr}$

and thus Randy takes

$T_R = \dfrac{1}{r_R} = \dfrac{63}{5}~hr = 12~hr,~36~min$

4. ## Re: Time and Work Problem

Melikes to transform these into ye olde "speed = distance/time":

JOHN:@10................................90........ ..........................>9 hr (or anything similar!)

JOHN:@10........50...........>5 hr;RANDY:@r..........40..........>2 1/3 = 7/3 hr

r = 40 / (7/3) = 120/7

90 / r = 90 / (120/7) = 5.25 hr = 5 hr 15 min

5. ## Re: Time and Work Problem

John completed $\dfrac{5}{9}$ of the job. That leaves $1-\dfrac{5}{9} = \dfrac{4}{9}$ of the job. It takes Randy $\dfrac{7}{3}$ hours to complete $\dfrac{4}{9}$ of the job. To complete the entire job himself, Randy would need:

$$\dfrac{\dfrac{7}{3}\text{ hr}}{\dfrac{4}{9}\text{ job}} = \dfrac{21}{4}\text{hr / job} = 5\text{ hr }15\text{ min}$$ to complete one job.