# Thread: A basic arithmatic problem

1. ## A basic arithmatic problem

Can someone explain why do we need to determine the amount of work done in 1 hour (as indicated by the underlined part). Please explain the logic behind taking the ratio of the number of hours. Also, does "1" in the numerators show a job or "1 hour".

CB

2. We want to find the combined rate. So in the example, if machines X and Y work together we know it will take less than 4 hours intuitively. The question is how long? So after 4 hours machine X completes 100% of the task. After 5 hours, machine Y completes 100% of the task. So to compare apples to apples, how much will machines X and Y complete in 1 hour? It is $\frac{1}{4}$ and $\frac{1}{5}$ of the task respectively. To find how much of the task they complete together in 1 hour, we add the two and get $\frac{9}{20}$ of the task completed. But this is only the amount they have finished in one hour. Then we set up a ratio and proportion to find how long it will take both of them to complete 100% of the task. You don't have to use 1 hour. We could have done the following:

How much will machines X and Y complete in 2 hours? It is $\frac{1}{2}$ and $\frac{2}{5}$ respectively. Then together in 2 hours they can complete $\frac{9}{10}$ of the task. Then we set up a ratio and proportion to find out how long it would take both of them to complete 100% of the task. The key idea is that we want to compare apples to apples (i.e. keep times constant to calculate rates).

3. Hello, CB!

I use a different approach to "Work" problems.
. . See if you like it . . .

$A$ can do a job in 4 hours.
$B$ can do the same job in 5 hours.
How long will it take them to complete job if they work together?
$A$ can do the job in 4 hours.
. . In one hour, he can do $\frac{1}{4}$ of the job.
. . In $x$ hours, he can do $\frac{x}{4}$ of the job.

B can do the job in 5 hours.
. . In one hour, he can do $\frac{1}{5}$ of the job.
. . In $x$ hours, he can do $\frac{x}{5}$ of the job.

In $x$ hours, working together, they can do . $\frac{x}{4} + \frac{x}{5}$ .of the job.

But in $x$ hours, we expect them to compete the job (one job).

. . There is our equation! . . . . . $\frac{x}{4} + \frac{x}{5}\;=\;1$

Multiply by 20: . $5x + 4x \:=\:20\quad\Rightarrow\quad 9x \:=\:20\quad\Rightarrow\quad x \,=\,\frac{20}{9}$

. . Therefore, working together, it will take them $2\frac{2}{9}$ hours.