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,

$\displaystyle 36N < 49 M$.

Two ratios are equivalent if and only if their cross products are equal. Set $\displaystyle \frac{M}{7}$ equal to each choice in turn and find their cross products:

(a) $\displaystyle \frac{M}{7} = \frac{3M}{21}$

$\displaystyle M \cdot 21 = 3M \cdot 7$

$\displaystyle 21M = 21M$

The cross products are equal, so regardless of the value of $\displaystyle M$, the ratios are equivalent.

(b) $\displaystyle \frac{M}{7} = \frac{M+ 3}{10}$

$\displaystyle M \cdot 10 =( M+ 3 )\cdot 7$

$\displaystyle 10M = 7 M + 21$

$\displaystyle 10M - 7M = 7 M + 21 - 7M$

$\displaystyle 3M = 21$

$\displaystyle M = 7$

The cross products are equal if and only if $\displaystyle M = 7$, so the ratios are not equivalent.

(c) $\displaystyle \frac{M}{7} = \frac{M- 3}{4}$

$\displaystyle M \cdot 4 =( M- 3 )\cdot 7$

$\displaystyle 4M = 7 M - 21$

$\displaystyle 4M- 7M = 7 M - 21 - 7M$

$\displaystyle -3M =- 21$

$\displaystyle M = 7$

The cross products are equal if and only if $\displaystyle M = 7$, so the ratios are not equivalent.

The correct response is (a) only.

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 $\displaystyle 1:5$

Let $\displaystyle N$ be the map distance between Carson and Davis. A proportion statement can be set up relating map inches to real miles:

$\displaystyle \frac{N}{104} = \frac{4}{260}$

Solve for $\displaystyle N$:

$\displaystyle \frac{N}{104} \cdot 104 = \frac{4}{260} \cdot 104$

$\displaystyle N = \frac{416}{260} = \frac{416\div 52 }{260\div 52} =\frac{8}{5} = 1\frac{3}{5}$

Carson and Davis are $\displaystyle 1\frac{3}{5}$ inches apart on the map; $\displaystyle 1\frac{3}{5} > 1 \frac{1}{2}$ 

Let $\displaystyle N$ be the real distance between Vandalia and Ferrell. A proportion statement can be set up relating real miles to map inches:

$\displaystyle \frac{N}{3 \frac{3}{5}} = \frac{250}{6}$

Solve for $\displaystyle N$:

$\displaystyle \frac{N}{3 \frac{3}{5}} \cdot 3 \frac{3}{5} = \frac{250}{6}\cdot 3 \frac{3}{5}$

$\displaystyle N = \frac{250}{6}\cdot \frac{18}{5} = \frac{50}{1}\cdot \frac{3}{1} = 150$

The actual distance between Vandalia and Ferrell is 150 miles.

There are a couple different ways to solve this problem. One way is to set up an equation from the given equation. Essentially, you have to find the total number of books before you can find how many paperbacks. An equation for that could be $\displaystyle 12=.6x.$ In other works, 12 is 60% of what total amount? (Remember, in equations, we convert percentages to decimals.) Then, you would solve for x to get 20 total books. Once you know the total, you can subtract the number of hardbacks from that to get 8 paperbacks. Another way to solve this equation is to set up a proportion. That would be $\displaystyle \frac{60}{100}=\frac{12}{x}$. Then, we could cross multiply to get $\displaystyle 60x=1200.$ Solving for x would again give you 20 and you would repeat the steps from above to get 8.

This is an easy case of proportions. To triple the recipe, you merely need to triple each of its component parts; therefore, you will have:

$\displaystyle 6$ cups of butter for every $\displaystyle 3$ cup of flower and $\displaystyle 51$ cups of sugar

Summing these up, you get:

$\displaystyle 6+3+51 = 60$ total cups.

To solve this, you need to set up a proportion:

$\displaystyle \frac{18}{4} = \frac{x}{3}$

Multiply both sides by $\displaystyle 3$:

$\displaystyle x=\frac{54}{4}$

Simplifying, this gives you:

$\displaystyle \frac{27}{2}$ or $\displaystyle 13.5$ lizard tongues.