I raked leaves for 6 hours, John did it in 7 hours, How long would it take if we worked together?

Question

I raked leaves for six hours, John raked leaves in seven hours.  How long would it take if we worked together?

Answer 1

 Let's set up an equation for this  problem.

If one person can rake leaves in 6 hours, that means they can do 1/6 of the  job in 1 hour.

If one person can rake leaves in 7 hours, that means they can do 1/7 of the  job in 1 hour.

We want to determine the time it would take them to do it together, which is our unknown. We will call our unknown t. (t = time)

Therefore our equation will be  1/6t + 1/7t =  1/t.

We will find the least common denominator which will be 42t.

We will multiply both sides of the equation by the least common denominator  42t.

42t (1/6t) + 42t (1/7t) =  42t (1/t)  

= 7t + 6t = 42 

=13t = 42

We will divide both sides by 13 (Division is the inverse operation of multiplication)

13t/13 =  42/13

t=  3.230769      which we will round to 3.23 

It will take them both 3 23/100  hours to do it together.

To convert 23/100 into minutes we will use  a proportion.

We know there are 60 minutes in an hour so we want to know what is  23% of 60

The proportion would be   23/100 = x/60
We will cross multiply     23(60) = 100 (x)

1380 = 100x

We will divide both sides by 100.

1380/100 = 100x/100

13.8 =  x  (we will round to 14)

Therefore if it take one person 6 hours to rake leaves and the other 7 hours to rake leaves.

It would take them 3 hours and 14 minutes to rake the leaves together.

The  solution is  3 hours  and 14 minutes.

 

 

 

Answer 2

Hi Judy!

Start with a guess: if both are one-job-in-6-hour workers, teaming up finishes the job in half the time - 3 hrs. With both raking one job / 7 hours, a joint effort takes half the time: 3.5 hours. Thus, the actual time would be in the middle, say 3h 15m:

1/6 + 1/7 = 1/t ... 13/42 = 1/t ... t = 42/13 ... t = 3h and (3/13)60min/hr ==> 3h 14m ... Regards, ma'am :)