Let  and  be the sets of employees that speak Spanish and French, respectively. Then we know that 

 and 

Since 517 employees speak neither,  employees speak either Spanish, French, or both.

We are looking for , which can be found by using the formula:

Think of this as a rate problem, with rate being measured in "tanks per minute".

The pipes can fill the tank up at rates of , , and  tanks per minute, respectively. 

In the time  it takes to fill the tank, the three pipes fill up  of the tank,  of the tank, and  of the tank, respectively. Add the amounts of work done by the three tanks to get the total amount of work - one job.

By definition, a prime number is any number that is greater than  and is only divisible by  and itself. Therefore, by definition  is not a prime number.

Look at the work rates as "tanks per minute", not "minutes per tank".

The two pipes can fill the tank up at  tanks per minute and   tanks per minute. 

Let  be the time it took, in minutes, to fill the tank up. Then this is the amount of time that the first pipe had to let in water; the amount of time that the second pipe had, in minutes, is  .

Since rate multiplied by time is equal to work, then the two pipes fill up  and  tanks; together, they filled up  tank - one tank. This sets up the equation to be solved:

This rounds to 26 minutes after the first pipe is opened, or 8:26 AM.

This can be solved by looking at their work rates in terms of "greenhouses per hour".

Philip can paint one greenhouse in four hours, or  greenhouse per hour.

Sharon can paint one greenhouse in four and a half hours, or  greenhouses per hour. Grag can paint one greenhouse in three and a half hours, or  greenhouses per hour. If  is the number of hours that it takes for the three to paint the greenhouse, then Philip, Sharon, and Grag will paint  , , and  of the greenhouse, respectively; these three shares add up to one greenhouse, so we can set up and solve this equation:

Let's convert this to minutes by multiplying by 60: 

This rounds to one hour and 19 minutes, so the three finish at . 

To find the number of widgets created by each machine separately, divide 88 by 2:

This is the number of widgets created by 1 machine in 4 minutes.

To find the number of widgets in 1 minute, divide 44 by 4:

Use this rate to find the answer:

.

Let  be the number of minutes that it takes to drain the tank.

Think of emptying the tank as one job. Then the drain can do one job in 25 minutes, or  jobs per minute.

Now, think of the inlet pipe as doing a "negative" job - it is doing the opposite of emptying the tank, working against the drain. It is doing "negative one" job in 45 minutes, or  jobs per minute.

Now, think of this as a rate problem. In  minutes, the drain does 

 jobs

and the pipe does

 jobs.

Together they do 1 job, the draining of the whole tank. 

Set up an equation to solve for x:

, which rounds to 56 minutes.

The first seven prime numbers are:

To find the sum, all numbers must be added:

Then, the new volume is  times the old volume.

If in an hour  produces  packages and  produces  packages respectively, then both machines produce  packages in an hour altogether.

Since the order requires  packages, the factory will take:

Therefore, it will take 5 hours for the factory to complete the packages necessary for the shipment.