Katy has  $\displaystyle \small 250$ friends that are girls and $\displaystyle \small 100$ friends from Michigan, so the ratio is $\displaystyle \small 250:100$.  When we divide both numbers by fifty to simplify, the ratio becomes $\displaystyle \small 5:2$, since

$\displaystyle \small \frac{250}{50}=5$ and $\displaystyle \small \frac{100}{50}=2$.

We can compare miles per hour.

(a) 50 miles over a one-hour period is 50 miles per hour.

(b) 70 miles over a one-and-one-fourth-hour period is

 $\displaystyle 70 \div 1\frac{1}{4} = \frac{70}{1} \div \frac{5}{4} = \frac{70}{1} \times \frac{4}{5} = \frac{14}{1} \times \frac{4}{1} = 56$

Mike drove an average of 56 miles per hour over the last 70 miles, making (b) greater.

Travis drove 40 miles in three-fourths of an hour, so we can divide:

$\displaystyle 40 \div \frac{3}{4} = \frac{40}{1} \times \frac{4} {3} = \frac{160} {3} = 53 \frac{1} {3}$ miles per hour,

which is less than 55.

The closest that Coleman and Garrett can be to each other is 200 miles (if Garrett is between Wilsonville and Coleman); the farthest is 440 miles (if Wilsonville is between Coleman and Garrett). 

Call the map distance between Coleman and Garrett $\displaystyle N$.

The two extremes of $\displaystyle N$ can be calculated using proportion statements.

The minimum:

$\displaystyle \frac{N}{200} = \frac{6}{320}$

$\displaystyle \frac{N}{200} \cdot 200 = \frac{6}{320} \cdot 200$

$\displaystyle N = \frac{1,200}{320} = \frac{1,200\div 80}{320\div 80} = \frac{15}{4} = 3 \frac{3}{4}$

The maximum:

$\displaystyle \frac{N}{440} = \frac{6}{320}$

$\displaystyle \frac{N}{440} \cdot 440= \frac{6}{320} \cdot 440$

$\displaystyle N = \frac{2,640}{320} = \frac{2,640\div 80}{320\div 80} = \frac{33}{4} = 8 \frac{1}{4}$

It is therefore unclear whether the map distance is greater than or less than 5 inches.

After the new girls enter the room, you have $\displaystyle 15$ boys and $\displaystyle 12 + 5 = 17$ girls. This means that there is a total of $\displaystyle 15 + 17 = 32$ people in the room.  The ratio of girls to boys would be $\displaystyle 17 : 32$.

We know that $\displaystyle \frac{2}{3}$ of the total $\displaystyle 45$ pieces of fruit are apples. This means that there are:

$\displaystyle \frac{2}{3} * 45 = 30$ apples.

Thus far, we know that we must have:

$\displaystyle 30$ apples

and

$\displaystyle 15$ kiwis

Now, if we double the apples, we will have:

$\displaystyle 60$ apples

and

$\displaystyle 15$ kiwis

This means that the proportion of kiwis to total fruit will be:

$\displaystyle 15:(60+15)$ or $\displaystyle 15:75$, which can be reduced to $\displaystyle 1:5$

You just need to work this through step-by-step.

We know that half of the vehicles are cars; therefore, $\displaystyle 100$ of them are cars. To find the number of SUVs, multiply $\displaystyle 200$ by $\displaystyle 0.25$ (a quarter) and get $\displaystyle 50$ SUVs. To find the number of motorcycles, multiply $\displaystyle 200$ by $\displaystyle 0.05$ to get $\displaystyle 10$.  Finally, there is $\displaystyle 20$% remaining for trucks; therefore, multiply $\displaystyle 200$ by $\displaystyle 0.2$ to get $\displaystyle 40$.  

Now, if this is doubled, we have $\displaystyle 80$ trucks.  This means that the total number of vehicles is:

$\displaystyle 100+50+10+80 = 240$ vehicles

Therefore, the ratio of motorcycles to total vehicles will be:

$\displaystyle 10 : 240$

Reducing this, you get:

$\displaystyle 1:24$

The ratios 125 to $\displaystyle M$ and $\displaystyle N$ to 125 are equvalent, so

$\displaystyle \frac{125}{M} = \frac{N }{125}$

By the cross-product property,

$\displaystyle MN = 125 \cdot 125 = 15,625$

Without any futher information, however, it cannot be determined which of $\displaystyle M$ and $\displaystyle N$ is the greater. For example, $\displaystyle M = 25$ and $\displaystyle N = 625$ fits the condition, as does the reverse case.

The ratios 20 to $\displaystyle M$ and $\displaystyle N$ to 40 are equvalent, so

$\displaystyle \frac{20}{M} = \frac{N }{40}$

By the cross-product property,

$\displaystyle MN = 20 \cdot 40 = 800$

Without any futher information, however, it cannot be determined which of $\displaystyle M$ and $\displaystyle N$ is the greater. For example, $\displaystyle M = 10$ and $\displaystyle N = 80$ fits the condition, as does the reverse case.

The cross products of two equivalent fractions are themselves equivalent, so if

$\displaystyle \frac{M}{6} = \frac{N}{7}$

then

$\displaystyle 6N= 7 M$

Multiply by 6:

$\displaystyle 6 \cdot 6N=6 \cdot 7 M$

$\displaystyle 36N = 42 M$

Since $\displaystyle 42 < 49$, it follows that $\displaystyle 42 M < 49 M$, and by substitution,

$\displaystyle 36N < 49 M$.