SOLUTION: A farmer can plow a field in 4 days by using a tractor. A hired hand can plow the same field in 6 days by using a smaller tractor. How many days will be required for the plowing if

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Question 1000927: A farmer can plow a field in 4 days by using a tractor. A hired hand can plow the same field in 6 days by using a smaller tractor. How many days will be required for the plowing if they work together?
Found 2 solutions by josgarithmetic, szk17:
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
1%2F%281%2F4%2B1%2F6%29


The combined rate of both machines working together is the sum of their individual rates. The unit of rate will be in FIELD per DAY. 
Machine          RATE in Fields per Day
Tractor           1/4
Smaller Tractor   1/6

The rate for both working together is 1%2F4%2B1%2F6 but you want the number as DAYS per FIELD, and this would be the reciprocal of 1%2F4%2B1%2F6, so that means 1%2F%281%2F4%2B1%2F6%29 DAYS for 1 field.

This question is not about finding any equation, but only handling a rate. Maybe you wanted something like D is how many days, and 1 field would be done according to rate 1%2FD, and then 1%2F4%2B1%2F6=1%2FD?

Answer by szk17(2) About Me  (Show Source):
You can put this solution on YOUR website!
Imagine that the field has an area A given in square feet (the specific unit doesn't matter but it will help understand). Each tractor will cover the field at a different rate (because they operate at different speeds, etc) expressed in square feet per day. For example, 500 square feet/day means the tractor would only complete 500 square feet in one day. If we divide the area of the field A by the rate covered by a given tractor, we get the total time it takes to process the entire field.
For tractor 1:
A/rate1 = 4 days
For tractor 2:
A/rate2 = 6 days
Quick dimensional analysis check: square feet / (square feet / day) = days
Express the equations in terms of A, the quantity that relates to both events.
A = 4 days x rate1
A = 6 days x rate2
Now, the area A is the same so we can do:
4 days x rate1 = 6 days x rate2
rate2 = (4/6) x rate1
rate2 = (2/3) x rate1
Thus, we now know how the performances of the tractors relate to each other.
If both tractors work together, the number of days is still the area of the field divided by the rate. However, this time, you use two rates!
time for combined performance = A / (rate1 + rate2)
= A / (rate1 + (2/3) x rate1)
= A / ((5/3) x rate1)
= (3/5) x (A/rate1)
Remember that A/rate1 = 4 days, so we get:
= (3/5) x 4 days = 2.4 days
There's an equation you can use or memorize to work such problems but it's better to know what's going on.