Please help me with this math problem. What formula do I use?

Question

TIm usually mows his lawn in 4hrs. John can mow the same lawn in 7hrs. How much time would it take for them to mow the lawn together?

Answer 1

To do this you must come up with a lawnmowing rate for both.  In other words, Tim mows at a rate of 1/4 lawn-per-hour (lph).  That's 1 lawn divided by 4 hours.  John's rate is 1/7 lph.

Now the time they mow is the unknown variable, t.

Tim's rate multiplied by time t + John's rate multiplied by time t must equal 1 lawn.  

The best way to understand this is to look at the units.  Rate is in lawn/hour while time is in hour.  Multiplied, they equal lawn.

(1/4)*t + (1/7)*t = 1

t = 28/11 or 2.54 hours.

Answer 2

If they divided the lawn up into equal halves, Tim would finish in 2 hrs. and John in 3.5, so the lawn would be finished in 3.5 hrs.  But that leaves 1.5 hrs for Tim to just stand around when he could help finish the lawn.  So the trick is to find how they divide up the lawn so that the two guys finish their respective parts at the same time.

T = Tim's mowing rate = 1 lawn / 4 hrs = 1/4 lawn/hr

J = John's mowing rate = 1 lawn / 7 hrs = 1/7 lawn/hr

t = the time it takes each guy to mow his part

A = total area of lawn = 1 lawn

Rate x time = Area

Tt + Jt = A

(T+J)t = 1

(1/4 + 1/7)t = 1

(11/28)t = 1

t = 28/11 ≈ 2.55 hrs.

Answer 3

Take product over sum, getting:

4*7/(4+7) = 28/11 = 2 6/11 hours

Answer 4

Tim = 4

John = 7

4+7 divided by 2 = x

Average time = 5.5 hrs

Answer 5

(4+7)/2

11/2

5.5

Answer 6

1/(1/4 + 1/7)

=1/(11/28)

=28/11

= 2  6/11 hr

Answer 7

Convert their times into rates by inverting:

In one hour, Tim mows 1/4 of the lawn.

In one hour, John mows 1/7 of the lawn.

In one hour, Tim and John would mow 1/4 + 1/7 of the lawn.

1/4 + 1/7

= 7/28 + 4/28

= 11/28

In one hour they would cut 11/28 of the lawn.  Now invert again to get your answer for cutting the whole lawn:

28/11 = 2 6/11

A short cut formula is:

AB / (A + B)

= 4*7 / (4 + 7)

= 28/11

= 2 6/11

You might want to multiply by 60 to figure the number of minutes.

2 --> 2 hours

6/11 x 60 = 360/11 --> 32 8/11 minutes

Answer:

Approx. 2 hours 33 minutes

P.S.  Don't make the mistake of just averaging their times.  It obviously doesn't make sense that Tim can mow the lawn in 4 hours, but it would take him longer (5.5 hours) if John were helping.  Working together it should take *less* time.