Working Together - Math Story Problems

Question

Tasha can do a job in 3 hours, and Ben can do the same job in 12 hours. How long will it take the two of them to do the job working together?

Answer 1

Tasha can do a job in 3 hours, and Ben can do the same job in 12 hours. How long will it take the two of them to do the job working together? 

Interesting problem! Because they don’t tell us any specifics about what the job is, I would create an imaginary job that fits the parameters of the problem.

I’m going to imagine that the job is filling boxes with merchandise. Next I’ll choose the number of boxes they need to fill to complete the job. I’ll choose 24, because it’s a multiple of both 3 and 12. So to finish the job, we'll say that the worker has to fill 24 boxes.

The problem tells us that Tasha can do it in 3 hours, and Ben can do it in 12 hours. 

Translated into an equation, where T is the number of boxes Tasha can fill in an hour and B is the number that Ben can fill in an hour, it looks like this:

3T=24   3 hours times the number of boxes Tasha can fill in an hour equals 24 boxes filled

12B=24  12 hours times the number of boxes Ben can fill in an hour equals 24 boxes filled

Next we solve both equations to get the values of T and B.

3T=24

3T/3= 24/3   Divide both sides by 3

T=8          Tasha can fill 8 boxes in an hour

12B=24

12B/12=24/12  Divide both sides by 12

B=2          Ben can fill 2 boxes in an hour

Now we envision what will happen if Tasha and Ben work together. 

8+2=10      In 1 hour, Tasha can fill 8 boxes, and Ben can fill 2, for a total of 10.

Next we’ll create a table and plug in how much work gets done each hour.

In each of the first 2 hours, Tasha and Ben filled 10 boxes, for a total of 20. But they still had 4 more boxes to fill. Since we know they can fill 10 boxes per hour, we can divide 4 by 10 to find out how long it will take them to fill the last 4 boxes. 

4÷10 = 0.4

	Hours	Boxes Filled
	1	10
	1	10
	0.4	4
TOTAL	2.4	24

Solution: If Tasha and Ben work together, it will take them 2.4 hours to finish the job.

You can try it with different numbers for the total imaginary job, and you should always get the same result.

I was a decent math student in school, but just like the solution above, I often had to come at problems from an unconventional angle so they made sense to me. I like to draw diagrams, tables, etc. to find visual solutions. If your math brain works this way, let's talk! I'd love to help you prep for the GED math test or other high school level mathematical challenges.

Answer 2

Sometimes it is important to have the physical understanding of a problem so that we know where things come from and not just simply blindly apply a formula.

.

Let us call the J the "total job" that needs to be completed. This can be counting beans, for example.

Speed at which Tasha works is J divided by 3 hours or J/3

Speed at which Ben works is J divided by 12 hours or J/12

The combined speed that Tasha and Ben work together is (J/3 +J/12)

But the combined speed of the two together is also equal to J/t, where t is the time that the two take together

Thus:

J/3 + J/12 = J/t

J cancels out and we are left with:

1/3 + 1/12 = 1/t

Therefore, adding the fractions:

5/12 =1/t

or t -12/5

or t= 2 2/5 hours

Once you get the picture of where things come from, then it is appropriate to use shortcuts.

Answer 3

Tasha can do a job in 3 hours, and Ben can do the same job in 12 hours. How long will it take the two of them to do the job working together?

First of all take the time to review the word problem and establish a method of solving the math problem.

Step 1: Word Problem Analysis:

Let T be the time it takes for Tasha to do the job alone.

Let B the time it takes for Ben to do the job alone.

How long take to do the job alone?

We are given that Tasha (T) = 3 hours and Ben (B) = 12 hours

Step 2:

How long will it take the two of them to do the job working together?

We first need to find a common denominator for the fractions.

1/3 + 1/12 the common denominator is 12

1/3 x 12/1 = 12/3 =4 =4/12

1/12 x 12/1=12/12 =1/12

4/12 +1/12 = 5/12

Their combined work rate is 5/12 of the job per hour.

Step 3:

Time is the inverse of the work rate: 1/5/12

Simplify 1/5/12 = 12/5

Convert to mixed number: 12/5 = 2 2/5

Convert the fraction of hour to minutes: 2/5 times 60 = 24 minutes

The time it takes them to complete the job together is 2 hours and 24 minutes.

Mathematical Solution:

It will take Tasha and Ben 2 hours and 24 minutes to complete the job together.

I hope the mathematical calculations (step by step approach) was helpful. Please let me know if you require any more assistance. If anyone in my neighborhood is interested in setting up an in-person math tutoring session. I look forward to hearing from them. Have an amazing day. Doris H.

Answer 4

Working together problems can be tricky.

But let x = the number of hours it will take for Tasha and Ben to do the job working together.

Then (1/3) + (1/12) = (1/x).

So, [(1/3) (4/4)] + (1/12) = (1/x) and (4/12) + (1/12) = (1/x), then (5/12) = (1/x).

(5/12) (12/5) (x) = (1/x) (12/5) (x), gives x = 12/5, which simplifies to 2 2/5 hours.