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

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

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.

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 :)