Jonathan can type a 20 page document in 40 minutes, Susan can type it in 30 minutes, and Jack can type it in 24 minutes. Working together, how much time will it take them to type the same document?
Here is a similar problem that will guide you in the solution of your version.
It takes Al 5 hours to paint a shed, Ben 10 hours and Charlie 15 hours. How long would it take all three to paint the shed working together?
1--A can paint the shed in 5 hours.
2--B can paint the shed in 10 hours.
3--C can paint the shed in 15 hours.
4--A's rate of painting is 1 shed per A hours (5 hours) or 1/A (1/5) shed/hour.
5--B's rate of painting is 1 shed per B hours (10 hours) or 1/B (1/10) shed/hour.
6--C's rate of painting is 1 shed per C hours (15 hours) or 1/C (1/15 shed/hour.
7--Their combined rate of painting is therefore 1/A + 1/B + 1/C = (AC + BC + AB)/ABC = (1/5 + 1/10 + 1/15) = (11/30 sheds /hour.
8--Therefore, the time required for all of them to paint the 1 shed working together is 1 shed/(AC+BC+AB)/ABC sheds/hour = ABC/(AC+BC+AB) = 5(10)15/[5(15)+10(15)+5(10) = 30/11 hours = 2.7272 hours = 2hr-43min-38.18sec.
Note - The time required to complete a single "specific task" by three individuals working together, who can complete the task individually in A, B, and C units of time is ABC/(AC + BC + AB).
To find out how much time it will take them to type the document together, we need to calculate their combined typing speed.
Let's first calculate the typing speed of each person:
Jonathan's typing speed: 20 pages / 40 minutes = 1/2 page per minute
Susan's typing speed: 20 pages / 30 minutes = 2/3 page per minute
Jack's typing speed: 20 pages / 24 minutes = 5/6 page per minute
Now, let's calculate their combined typing speed:
Combined typing speed = Jonathan's typing speed + Susan's typing speed + Jack's typing speed
= 1/2 + 2/3 + 5/6
To add fractions with different denominators, we need to find a common denominator. In this case, the least common multiple (LCM) of 2, 3, and 6 is 6. So, let's convert the fractions to have a denominator of 6:
Combined typing speed = (1/2) * (3/3) + (2/3) * (2/2) + (5/6)
= 3/6 + 4/6 + 5/6
= 12/6
= 2 pages per minute
Now, to find out how much time it will take them to type the document together, we can divide the total number of pages (20) by their combined typing speed:
Time required = Total number of pages / Combined typing speed
= 20 pages / 2 pages per minute
= 10 minutes
So, it will take Jonathan, Susan, and Jack 10 minutes to type the same document together.