SSAT Upper Level Quantitative - How to find a ratio

In a high school, the ratio of freshmen to seniors is . If there are seniors, how many freshmen are there?

Explanation

Set up the following proportion, with being the number of freshmen.

$\frac{4}{5}=\frac{x}{150}$

Now, cross-multiply and solve for .

Scrabble

A popular word game uses one hundred tiles, each of which is marked with a letter or a blank. The distribution of the tiles is shown above, with each letter paired with the number of tiles marked with that letter. Notice that there are two blank tiles.

If the tiles marked with an "E" are removed, and the rest are placed in a box, then what is the ratio of consonant tiles to vowel tiles in the box?

Note: for purposes of this problem, "Y" is considered a consonant.

Explanation

Out of the 100 tiles, there are nine "A" tiles, twelve "E" tiles, nine "I" tiles, eight "O" tiles, and four "U" tiles.

If the "E" tiles are removed, there will be

vowel tiles.

The number of consonant tiles can most easily be found by adding the number of vowel tiles and blanks:

The rest of the tiles are consonant tiles; subtract from 100 to get

of them.

Therefore, the ratio of consonant tiles to vowel tiles in the box after removing the "E's" is

$\frac{56}{30} = \frac{56 \div 2 }{30 \div 2 } = \frac{28}{15 }$ - that is, a 28 to 15 ratio.

There are boys and girls at a playground. What is the ratio of boys to girls?

Explanation

Write the numbers of boys and girls as a fraction, then simplify.

$\frac{45}{36}=\frac{5}{4}$

$\frac{5}{4}$ can also be written as

A barn has geese, ducks, and chickens. What is the ratio of geese to chickens?

Explanation

First, add up the total number of ducks, geese, and chickens.

Now, write the fraction of these animals that are geese and the fraction of these animals that are chickens.

$\text{Geese}=\frac{15}{47}$

$\text{Chickens}=\frac{20}{47}$

Now, since we want the ratio of geese to chickens, we write the fractions as thus:

$\frac{\text{Geese}}{\text{Chickens}}=\frac{\frac{15}{47}}{\frac{20}{47}}$

Divide and simplify the resulting fraction to find the ratio.

$\frac{\frac{15}{47}}{\frac{20}{47}}=\frac{15}{47}\div\frac{20}{47}=\frac{15}{47}\times\frac{47}{20}=\frac{15}{1}\times\frac{1}{20}=\frac{15}{20}=\frac{3}{4}=3:4$

If pounds of chicken cost , how much does pounds of chicken cost?

Explanation

There is more than one way to solve this problem. You can either figure out how much the chicken costs per pound and multiply that cost by three pounds, or you can set up a proportion and solve for the cost of three pounds of chicken that way.

Solving the Problem Using Cost per Pound

First, find how much the chicken costs per pound.

$\frac{33.75}{15}=2.25$

Since chicken costs per pound, multiply this by the number of pounds we need to get the cost.

$$2.25\times3=$6.75$

pounds of chicken cost .

Solving the Problem Using a Proportion

You can set up a proportion to figure out how much pounds of chicken costs:

$\frac{3:lbs}{15:lbs}=\frac{x}{$33.75}$

Cross multiply:

$3\cdot 33.75=15x$

Solve for , the cost of pounds of chicken:

$\frac{101.25}{15}=x$

The ratio of offensive players to defensive players on a football team is $2\text{ to }5$ . If there are players on the team, how many offensive players are there?

Explanation

Let be the number of offensive players and be the number of defensive players.

Since there is a total of players on the team,

We need to find the number of offensive players, so we will need to find the value of .

$2x=6\times2=12$

At a high school, there are freshmen, sophomores, juniors, and seniors. What is the ratio of seniors to freshmen?

Explanation

Write the number of seniors and numbers of freshmen as a fraction:

$\frac{30}{36}=\frac{5}{6}$

That fraction is equivalent to $5\text{ to }6$ .

In a factory, there are glass bottles. If the ratio of red bottles to blue bottles is , how many more blue bottles than red bottles are there?

Explanation

First, find the number of red bottles and blue bottles.

Let be the number of red bottles and be the number of blue bottles. Since there is a total of bottles at the factory,

There are red bottles. Find the value of to find the number of blue bottles.

$5x=550\times5=2750$

Now, because the question wants to find how many more blue bottles than red bottles there are, subtract the number of red bottles from the number of blue bottles.