Ratios, proportions, and percents are essential tools in algebra and quantitative reasoning. They help you compare quantities, scale values, describe relationships, and interpret real-world situations. On the Praxis Core Math exam, you’ll often translate between ratios, proportional relationships, and percent statements. Fluency with these ideas helps you reason clearly and avoid common comparison errors.

Ratios

A ratio compares two or more quantities by describing how one amount relates to another. Ratios can compare parts within the same group (apples to oranges in a basket) or compare quantities across groups. You’ll see ratios in geometry, scale drawings, recipes, and probability.

A ratio can be written in several equivalent forms. Each form communicates the same comparison: how much of one quantity corresponds to a given amount of another. Like fractions, ratios can often be simplified by dividing each term by the greatest common divisor.

Definitions

Ratio

A ratio compares two quantities. It can be written in three forms:

1. Fraction form: $\frac{a}{b}$
2. Colon form: $a:b$
3. Word form: “$a$ to $b$”

For example, the ratio of $5$ apples to $3$ oranges can be written as:

$$\frac{5}{3}\text{or}5:3\text{or}5\text{ to }3$$

Ratios may also be simplified. For example:

$$\frac{10}{15}=\frac{2}{3}$$

because both numerator and denominator were divided by $5$.

Example: Triangle angle ratio A triangle has angles in the ratio $2:3:5$. Find the measure of each angle.

• Let the angles be $2x$, $3x$, and $5x$
• Sum of angles: $2x+3x+5x=180^{\circ }$
• Combine like terms: $10x=180^{\circ }$, so $x=18^{\circ }$
• Angles are $2x=36^{\circ }$, $3x=54^{\circ }$, and $5x=90^{\circ }$

Answer: $36^{\circ },54^{\circ },90^{\circ }$

Example: Fruit basket ratio In a basket, the ratio of oranges to apples to watermelons is $4:3:2$. The total number of pieces of fruit is $81$. If $3$ watermelons are removed, what fraction of the remaining fruit are watermelons?

(spoiler)

• Let the multiplier be $x$
• Oranges $=4x$, apples $=3x$, watermelons $=2x$
• Total fruit: $4x+3x+2x=9x=81$, so $x=9$
• Oranges $=36$, apples $=27$, watermelons $=18$
• After removing $3$ watermelons: $18-3=15$ watermelons
• Total fruit left: $81-3=78$
• Fraction that are watermelons: $\frac{15}{78}=\frac{5}{26}$

Answer: $\frac{5}{26}$

Proportions

A proportion is an equation that states two ratios are equal. Proportional reasoning shows up in scaling, unit conversions, recipes, maps, and rate problems. When two ratios form a proportion, the multiplicative relationship between corresponding quantities stays consistent.

If a proportion is written as $\frac{a}{b}=\frac{c}{d}$, then the cross-products are equal. This gives a dependable way to solve for an unknown when the other three values are known.

Definitions

Proportion

A proportion is an equation stating that two ratios are equal.

A general proportion looks like:

$$\frac{a}{b}=\frac{c}{d}$$

and satisfies

$$a\cdot d=b\cdot c.$$

Example: Recipe proportion A recipe uses $2$ cups of flour for every $3$ cups of sugar. How much flour is needed for $9$ cups of sugar?

• Set up the proportion: $\frac{2}{3}=\frac{x}{9}$
• Cross-multiply: $2\cdot 9=18=3x$
• Solve for $x$: $x=\frac{18}{3}=6$

Answer: $6$ cups of flour

Example: Test pass proportion If $4$ out of every $5$ students pass an exam, how many students out of $100$ would be expected to pass?

(spoiler)

• Set up the proportion: $\frac{4}{5}=\frac{x}{100}$
• Cross-multiply: $4\cdot 100=400=5x$
• Solve for $x$: $x=\frac{400}{5}=80$

Answer: $80$ students

Example: Map distance proportion On a map, the actual distance between two cities is $92.5$ miles, and the distance between them on the map is $3.7$ inches. If the distance between a town and a lake on the same map is $1.8$ inches, what is the actual distance between the town and the lake?

(spoiler)

• Set up the ratio of actual to map distance: $\frac{\text{actual}}{\text{map}}=\frac{92.5}{3.7}$
• Write the proportion: $\frac{x}{1.8}=\frac{92.5}{3.7}$
• Cross-multiply: $3.7x=92.5\cdot 1.8=166.5$
• Solve for $x$: $x=\frac{166.5}{3.7}=45.0$

Answer: $45.0$ miles

Percents

A percent expresses a quantity as a portion of $100$. Percents are common in finance, statistics, discounts, taxes, and growth problems. Since the same value can be written as a fraction, decimal, or percent, you’ll often switch forms depending on what makes the problem easiest.

Many percent problems ask you to find one of three things:

• the percent
• the part
• the whole

Once you identify what’s missing, you can use the appropriate equation.

$$\text{Percent}=\frac{\text{Part}}{\text{Whole}}\times 100$$

To solve for the Part:

$$\text{Part}=\text{Whole}\times \left(\frac{\text{Percent}}{100}\right)$$

To solve for the Whole:

$$\text{Whole}=100\times \left(\frac{\text{Part}}{\text{Percent}}\right)$$

Example: Finding the percent A student scored $80$ out of $100$ on a test. What percent is this?

$$\text{Percent}=\frac{80}{100}\times 100=80\%$$

Answer: $80\%$

Example: Finding the part What is $25\%$ of $200$?

• Method 1: use the formula

$$\text{Part}=200\times \left(\frac{25}{100}\right)=200\times 0.25=50$$

• Method 2: use a fraction equivalent

$$25\%=\frac{25}{100}=\frac{1}{4},200\times \frac{1}{4}=\frac{200}{4}=50$$

Answer: $50$

Example: Finding the whole If $30\%$ of a number is $45$, what is the number?

Percent change

Percent change measures how much a value increases or decreases relative to the original value. If the result is positive, it’s a percent increase. If the result is negative, it’s a percent decrease.

A consistent method is:

• subtract the original value from the new value
• divide by the original value
• convert to a percent

$$\text{Percent change}=\frac{\text{new value}-\text{original value}}{\text{original value}}\times 100\%$$

Example: Percent increase A shirt increases in price from $\$20$ to $\$25$.

$$>\frac{25-20}{20}\times 100\%=\frac{5}{20}\times 100\%=25\%$$

Answer: $25\%$ increase

Example: Percent decrease A phone’s price drops from $\$500$ to $\$400$.

Percent of a percent

Sometimes you need to find a percent of another percent (for example, layered discounts or tax applied to a discounted price). Converting each percent to a decimal makes the multiplication straightforward.

To find a percent of a percent:

• convert each percent to a decimal
• multiply
• convert the result back to a percent

Example: Percent of a percent What is $20\%$ of $50\%$?

• Convert to decimals: $0.20$ and $0.50$
• Multiply: $0.20\times 0.50=0.10$
• Convert back to a percent: $0.10=10\%$

Answer: $10\%$

Example: Percent of a percent What is $5\%$ of $80\%$?

Simplifying using zero-canceling

When both the numerator and denominator of a fraction end in zeros, you can often simplify by canceling matching zeros. Each canceled zero corresponds to dividing both numbers by $10$. This shortcut is especially useful in proportion and percent problems with large numbers.

This only works with trailing zeros in both numbers. If one number has fewer trailing zeros than the other, you can only cancel as many zeros as they have in common.

::: sidenote Simplifying zero example:

$$\frac{1200}{300}\to \frac{120}{30}\to \frac{12}{3}=4$$

If the numerator has only one trailing zero:

$$\frac{1050}{300}\to \frac{105}{30}$$

No further zeros can be canceled because the numerator has no more trailing zeros. :::

Dividing by powers of $10$ doesn’t change the overall ratio:

$$\frac{4500}{30}=\frac{450}{3}$$

The ratio stays the same, but the numbers are easier to work with.