Candy and Tim share a paper route. It takes Candy 70min to deliver all the papers, and it takes Tim 80min. How long does it take the two when they work together?
The answer is 37min and 20s. Here's what I have. 7/6t + 8/6t = 15/6
I have t = 5/2. I don't think that's 37min and 20s. Is it?
how did you get 3(route)/112 ?
You may want to consider a slightly different approach. It makes it easier for me to understand.
Assume the paper route is 100 papers.
Then Candy can deliver 100 papers/70 min = 1.42857 papers/min.
Tim can deliver 100 papers/80 min = 1.25 papers/min.
Together they can deliver 1.42857/min + 1.25/min = 2.67857 papers/min.
So if there are 100 papers, it will take 100 papers x (1 min/2.67857) = 37.3333 minutes. This method has the disadvantage that the decimal equivalent of 1/3 is a repeating fraction; however, the answer is 37 min and 19.999999 seconds which for all practical purposes is 37 min and 20 seconds.
Sean, if in DrBob's way you replace "100 papers" with my "route" the solutions are the same, except Bob changed his fractions to decimals.
in my case, "route" canceled, in Bob's case "100 papers" canceled.
In these "working together" kind of problems, if one person can do the job in a hours, another in b hours, c hours for the next and so on,
then when all are working together, the time taken will be
1/(1/a + 1/b + 1/c + ...)
To find the time it takes for Candy and Tim to complete the paper route together, we need to use the concept of rates.
Let's say that Candy's rate of delivering papers is r1 (papers per minute) and Tim's rate is r2 (papers per minute). The total number of papers to deliver is the same regardless of who delivers them, so we can set up an equation using their rates:
Candy's rate * time taken by Candy + Tim's rate * time taken by Tim = Total number of papers
In this case, it would be:
r1 * 70 + r2 * 80 = Total number of papers
Since the total number of papers is the same, we can simplify this equation to:
7r1 + 8r2 = Total number of papers
Now, we need to find the combined rate of Candy and Tim when they work together. Let's call it r_combined. The equation will be:
r_combined * time taken when they work together = Total number of papers
Since they are working together, their rates add up:
r_combined = r1 + r2
Now we can substitute this equation into the previous one:
7(r1 + r2) + 8r2 = Total number of papers
Expanding the equation:
7r1 + 7r2 + 8r2 = Total number of papers
Combining like terms:
7r1 + 15r2 = Total number of papers
Now, we know that it takes Candy 70 minutes to deliver all the papers and Tim 80 minutes, so their rates are:
r1 = 1/70 (papers per minute)
r2 = 1/80 (papers per minute)
Replacing the rates in our equation:
7(1/70) + 15(1/80) = Total number of papers
Simplifying the equation:
1/10 + 3/16 = Total number of papers
Finding a common denominator:
8/80 + 30/80 = Total number of papers
(8 + 30)/80 = Total number of papers
38/80 = Total number of papers
Simplifying:
19/40 = Total number of papers
So, the total number of papers is 19/40.
Now, let's find the combined rate (r_combined):
r_combined = r1 + r2
r_combined = 1/70 + 1/80
To add these fractions, we need a common denominator:
r_combined = (8/560) + (7/560)
r_combined = 15/560
Now we can find the time taken when they work together by using the equation:
r_combined * time taken when they work together = Total number of papers
(15/560) * t = 19/40
To solve for t, we can multiply both sides by the reciprocal of (15/560):
t = (19/40) * (560/15)
Simplifying the equation:
t = 37/2
t = 18.5 minutes
So, it takes Candy and Tim 18.5 minutes (or 18 minutes and 30 seconds) to complete the paper route when they work together, not 37 minutes and 20 seconds as mentioned in your answer.
Candy's rate = route/70
Tim's rate = route/80
combined rate = route/80 + route/70
= 3(route)/112
so time to do route with combined rate
= route/[3route/112]
= 112/3
= 37 1/3 minutes
and of course 1/3 minute is 20 seconds,
so 37 minutes, 20 seconds