The rate of draining is \(\displaystyle 40\textup{ gallons per hour}\) once the new pipe is added. Recall that:

\(\displaystyle W=RT\), where \(\displaystyle W\) is the total work output. For our data, this means the total amount of water. Now, we are measuring our rate in hours, so we should translate the three days' time into hours. This is easily done:

\(\displaystyle 3*24=72\)

Now, based on this, we can set up the equation:

\(\displaystyle W=40 * 72=2880\)

Now, this means that there will be \(\displaystyle 200000-2880\) or \(\displaystyle 197120\) gallons in the reservoir after three days.

This problem is a variation on the standard equation \(\displaystyle \small W=RT\).  The \(\displaystyle \small R\) variable contains all twenty bakers, however, instead of just one.  Still, let's start by substituting in our data:

\(\displaystyle \small 720 = R * 8\)

Solving for \(\displaystyle \small R\), we get \(\displaystyle \small 90\).

Now, this represents how many dozen cookies the whole group of \(\displaystyle \small 20\) make per hour.  We can find the individual rate by dividing \(\displaystyle \small 90\) by \(\displaystyle \small 20\), which gives us \(\displaystyle \small 4.5\).  Notice, however, that the question asks for the number of cookies—not the number of dozens.  Therefore, you need to multiply \(\displaystyle \small 4.5\) by \(\displaystyle \small 12\), which gives you \(\displaystyle \small 54\).

This problem is a variation on the standard equation \(\displaystyle \small W=RT\).  The \(\displaystyle \small R\) variable contains all the workers.  Therefore, we could rewrite this as \(\displaystyle \small W=NIT\), where \(\displaystyle \small N\) is the number of workers and \(\displaystyle \small I\) is the individual rate of work.  Thus, for our first bit of data, we know:

\(\displaystyle \small 90 = 30 * 6 * I = 180 I\)

Solving for \(\displaystyle \small I\), you get \(\displaystyle \small I = 0.5\)

Now, for the actual question, we can fill out the complete equation based on this data:

\(\displaystyle \small 60=40 * 0.5 * T\)

\(\displaystyle \small 60=20 * T\)

Solving for \(\displaystyle \small T\), you get \(\displaystyle \small 3\).

Recall that in general \(\displaystyle \small R=\frac{D}{T}\)

Now, let's gather our three rates:

Rate 1: \(\displaystyle \small 5\)

Rate 2: \(\displaystyle \small 5*1.5 = 7.5\)

Rate 3: \(\displaystyle \small 7.5 * 1.25=9.375\)

Now, we know that the time is a total of \(\displaystyle \small 2\) hours.  Based on our data, we can write:

\(\displaystyle \small R=\frac{5*1 + 7.5 * 0.5 + 9.375 * 0.5}{2}=\frac{13.4375}{2}\)

This is \(\displaystyle \small 6.71875\) miles per hour, which rounds to \(\displaystyle \small 6.72\).

Begin by setting up the standard equation \(\displaystyle \small D=RT\)

However, for our data, we know the distance and the rate only.  We do not know the time that it took for the person's return.  It is \(\displaystyle \small 30+x\), where \(\displaystyle \small x\) is the return time.  Thus, we can write:

\(\displaystyle \small 2500 = 52(30+x)\)

Solving for \(\displaystyle \small x\), we get:

\(\displaystyle \small \small 2500 = 1560 + 52x\)

\(\displaystyle \small \small 2500 = 1560 + 52x\)

\(\displaystyle \small \small 940 = 52x\)

\(\displaystyle \small \small x=18.07692307692308\), which rounds to \(\displaystyle \small 18.08\) minutes.

Here, we need to do some unit conversions, knowing that there are \(\displaystyle 60\;minutes\) in an \(\displaystyle hour\).  We have two different rates, which result in two different equations, which we need to add to get a total time.  

\(\displaystyle 80 miles * \frac{hour}{70 miles} * \frac{60 minutes}{1 hour} = 68.57 \;minutes\)

\(\displaystyle 20 miles * \frac{hour}{20 miles} * \frac{60 minutes}{1 hour} = 60 \;minutes\)

\(\displaystyle 60\;minutes \;+68.57\;minutes=128.57\;minutes\).

We know that it takes Max an hour to drive 25 miles. We also know that there are 60 minutes in an hour. Using this information we can create the following ratio:

\(\displaystyle \frac{25\textup{ miles}}{60\textup{ minutes}}\)

We are trying to calculate the the amount of time it will take to drive 15 miles. Let's create a proportion and use a variable for the unknown time.

\(\displaystyle \frac{25\textup{ miles}}{60\textup{ minutes}}=\frac{15\textup{ miles}}{x\textup{ minutes}}\)

Cross-multiply and solve for the time.

\(\displaystyle 25x=15\times 60\)

\(\displaystyle 25x=900\)

\(\displaystyle x=36 \textup{ minutes}\) 

A pie is made up of  \(\displaystyle \frac{1}{9}\) crust, \(\displaystyle \frac{1}{3}\) apples, \(\displaystyle \frac{1}{4}\) sugar, and the rest is jelly. What is the ratio of crust to jelly?

To compute this ratio, you must first ascertain how much of the pie is jelly. This is:

\(\displaystyle 1-\frac{1}{4}-\frac{1}{3}-\frac{1}{9}\)

Begin by using the common denominator \(\displaystyle 36\):

\(\displaystyle 1-\frac{1}{4}-\frac{1}{3}-\frac{1}{9}=\frac{36}{36}-\frac{9}{36}-\frac{12}{36}-\frac{4}{36}\)

\(\displaystyle \frac{36}{36}-\frac{9}{36}-\frac{12}{36}-\frac{4}{36}=\frac{11}{36}\)

So, the ratio of crust to jelly is:

\(\displaystyle \frac{1}{9}:\frac{11}{36}\)

This can be written as the fraction:

\(\displaystyle \frac{\frac{1}{9}}{\frac{11}{36}}=\frac{1}{9}*\frac{36}{11}=\frac{4}{11}\), or \(\displaystyle 4:11\)

This problem is really an easy fraction division. You should first divide the lemon juice amount by the water amount:

\(\displaystyle \frac{lemon\:juice}{water}=\frac{\frac{7}{15}}{\frac{1}{3}}\)

Remember, to divide fractions, you multiply by the reciprocal:

\(\displaystyle \frac{\frac{7}{15}}{\frac{1}{3}} = \frac{7}{15}*3=\frac{7}{5}\)

This is the same as saying: 

\(\displaystyle 7:5\)

To find a ratio like this, you simply need to make the fraction that represents the division of the two values by each other. Therefore, we have:

\(\displaystyle x:y=\frac{\frac{11}{100}}{\frac{15}{8}}\)

Recall that division of fractions requires you to multiply by the reciprocal:

\(\displaystyle \frac{\frac{11}{100}}{\frac{15}{8}} = \frac{11}{100}*\frac{8}{15}=\frac{11}{25}*\frac{2}{15}\), 

which is the same as:

\(\displaystyle \frac{22}{375}\)

This is the same as the ratio:

\(\displaystyle 22:375\)