If A farmer uses a large tractor,she can plow a filed in eight hours. Her sister, using a smaller tractor can plow the same field in 12 hrs. How many hours would it take to plow the field if they worked together ? Express your answer as a mixed number and show your work.
Here is a generic way to solve these type of problems
farmer's rate = job/8
Sister's rate = job/12
combined rate = job/8 + job/12
= 5job/24
so time for combinded rate = job/(5job/24)
= 24/5 or 4.8 hours.
notice "job" is just a phoney variable, it will cancel out anyway.
To solve this problem, we need to calculate the combined rate at which the sisters can plow the field. Let's start by finding their individual rates.
The farmer can plow the field in 8 hours, so her rate is 1 field per 8 hours.
The sister can plow the field in 12 hours, so her rate is 1 field per 12 hours.
To find their combined rate, we can add their rates together: 1/8 + 1/12.
To add these fractions, we need to find a common denominator, which is the least common multiple (LCM) of 8 and 12.
The multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, ...
The multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, ...
From the above lists, we see that 24 is the least common multiple of 8 and 12.
Now, we can rewrite the fractions with a common denominator of 24:
1/8 becomes 3/24 (multiplied by 3/3)
1/12 becomes 2/24 (multiplied by 2/2)
So, the combined rate of the two sisters working together is 3/24 + 2/24 = 5/24 fields per hour.
To find the time it would take to plow the field when they work together, we can use the formula:
Time = Total Work / Combined Rate
In this case, the total work is 1 field, and the combined rate is 5/24 fields per hour.
Time = 1 / (5/24)
Time = 1 * (24/5)
Time = 24/5
The time it would take for the sisters to plow the field when they work together is 24/5 hours.
As a mixed number, this can be written as 4 4/5 hours.
So, it would take the sisters approximately 4 hours and 48 minutes to plow the field if they worked together.