Jerry drove 60 miles per hour for 4 hours - that is, \(\displaystyle 60 \times 4 = 240\) miles. 

He drove the remainder of the distance, or \(\displaystyle 320-240 = 80\) miles over a period of \(\displaystyle 6 \frac{2}{3} - 4 = 2\frac{2}{3}\) hours, so his average speed was 

\(\displaystyle 80 \div 2 \frac{2}{3} = 80 \div \frac{8}{3} = 80 \times \frac{3} {8} = 30\) miles per hour.

If Sally drives q miles in 3 hours, her rate is 3/q miles per hour.  Plug this rate into the distance equation and solve for the time:

\(\displaystyle \dpi{100} \small Distance = rate\times time\)

\(\displaystyle \dpi{100} \small r=\frac{q}{3}\times t\)

\(\displaystyle \dpi{100} \small t=\frac{3r}{q}\)

We need to convert hours into minutes and multiply this by the 10 minute time interval:

\(\displaystyle \small \frac{12\ miles}{1\ hour}x\frac{1\ hour}{60\ min}x\frac{10\ min}{1}=\frac{120\ miles}{60}=2\ miles\)

If the driver needs to drive two laps, each one mile long, at an average rate of 60 miles per hour. To find the average speed, we need to add the speed for each lap together then divide by the number of laps. The equation would be as follows:

\(\displaystyle \frac{X+Y}{2}=60\)

In our case we know lap one was driven at \(\displaystyle 40\) miles per hour. We substitute this value in for \(\displaystyle X\) and solve for \(\displaystyle Y\).

\(\displaystyle \frac{40+Y}{2}=60\)

\(\displaystyle 40+Y=60(2)\)

\(\displaystyle 40+Y=120\)

\(\displaystyle Y=120-40\)

\(\displaystyle Y=80\)

Thus to average \(\displaystyle 60\) miles per hour for two laps with lap one being \(\displaystyle 40\) miles per hour, lap two would have to have a rate of  \(\displaystyle 80\) miles per hour.

We need to pay close attention to some details here. 

1) We are given time in minutes, but asked for an answer in hours.

2) A rate can be defined as distance over time.

Taking the first detail, we convert 600 minutes to 10 hours, since there are 60 minutes in one hour.

Taking the second detail, we divide 75 kilometers by 10 hours. This gives us an answer of 7.5 kilometers per hour.

If Ray covers \(\displaystyle 75 \:km\) in three hours, that means he covers \(\displaystyle 25 \:km\) in one hour:

\(\displaystyle \frac{1\:hr}{3\:hr}=\frac{x\:km}{75\:km}\)

\(\displaystyle 3x=1\cdot75\)

\(\displaystyle x=\frac{75}{3}=25\:km\)

Perform the following calculation to find how long it takes to cover \(\displaystyle 375 \:km\).

\(\displaystyle \frac{375\:km}{1}*\frac{1\:hr}{25\:km}=\frac{375\:km\cdot hours}{25\:km}=15 \:hours\)

In combined rate problems such as this, we must first find units of the desired answer: \(\displaystyle \frac{miles}{hour}\)and then find the totals of each piece of those units. Total miles is easy as we can just add together the two legs of the trip:

\(\displaystyle 3000 mi + 3000mi = 6000 total mi\)

To find total hours, we just have to use each leg's speed:

\(\displaystyle \frac{1hour}{600 miles} \cdot 3000 miles = 5 hours\)

\(\displaystyle \frac{1 hour}{300 miles} \cdot 3000 miles = 10 hours\)

The trip therefore took 15 total hours.

Now we simply divide the totals to find the average speed:

\(\displaystyle \frac{6000 miles}{15 hours} = 400 miles/hr\)

To solve this combined rate problem, we must use the equation: \(\displaystyle \frac{AB}{A + B}\)

Where A and B are the times it takes Frank and Francine, respectively, to eat a pie.

Therefore, it takes Frank and Francine

\(\displaystyle \frac{10\cdot 15}{10 + 15} = \frac{150}{25} = 6 minutes\)

to eat the pie.

An hour hand rotates 360 degrees for every 12 hours, so the hour hand moves \(\displaystyle \frac{360^{\circ}}{12\ hours}=30^{\circ}/hr\).

There are 4.5 hours between 3 PM and 7:30 PM, so the total degree measure is

\(\displaystyle 4.5\ hours*30^{\circ}/hr = 135^{\circ}\).

One full rotation of a circle is \(\displaystyle 360^{\circ}\), so if a sector covers \(\displaystyle 20\%\) of a circle, its angle will be \(\displaystyle 20\%\) of \(\displaystyle 360^{\circ}\). This gives us:

\(\displaystyle \angle=0.20(360^{\circ})=\frac{1}{5}(360^{\circ})=72^{\circ}\)