The ratios 125 to  and  to 125 are equvalent, so

By the cross-product property,

Without any futher information, however, it cannot be determined which of  and  is the greater. For example,  and  fits the condition, as does the reverse case.

The ratios 20 to  and  to 40 are equvalent, so

By the cross-product property,

Without any futher information, however, it cannot be determined which of  and  is the greater. For example,  and  fits the condition, as does the reverse case.

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

then

Multiply by 6:

Since , it follows that , and by substitution,

.

Two ratios are equivalent if and only if their cross products are equal. Set  equal to each choice in turn and find their cross products:

(a) 

The cross products are equal, so regardless of the value of , the ratios are equivalent.

(b) 

The cross products are equal if and only if , so the ratios are not equivalent.

(c) 

The cross products are equal if and only if , so the ratios are not equivalent.

The correct response is (a) only.

We know that  of the total  pieces of fruit are apples. This means that there are:

 apples.

Thus far, we know that we must have:

 apples

and

 kiwis

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

 apples

and

 kiwis

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

 or , which can be reduced to 

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

Solve for :

Carson and Davis are  inches apart on the map;  

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

Solve for :

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 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 . Then, we could cross multiply to get 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:

 cups of butter for every  cup of flower and  cups of sugar

Summing these up, you get:

 total cups.

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

Multiply both sides by :

Simplifying, this gives you:

 or  lizard tongues.