Ratios, proportions, and percents

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 here helps you avoid two frequent traps: dividing by the wrong reference value in percent change, and misaligning units when setting up a proportion.

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 can also be simplified. For example:

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

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

When a ratio describes parts of a whole, multiply each ratio term by a common unknown $x$. This preserves the ratio while letting you solve for the actual values using the total.

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

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 reliable 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: 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

Common pitfall - proportion unit alignment: When setting up a proportion, keep like units in matching positions. For example, if the left ratio is miles/inches, the right ratio must also be miles/inches - not inches/miles. Flipping one ratio gives a wrong answer even if the cross-multiplication looks correct.

Constant rates

A rate is a ratio of two quantities with different units, such as miles per hour, gallons per mile, or cubic feet per minute. A unit rate is the amount for one unit of the second quantity: $150$ miles in $3$ hours is $150÷3=50$ miles per hour. When a rate stays constant, the amount grows in proportion to time:

$$\text{distance}=\text{rate}\times \text{time}\text{time}=\frac{\text{distance}}{\text{rate}}$$

The same pattern works for any constant rate: a pump that moves $8$ cubic feet per minute fills a $200$-cubic-foot tank in $200÷8=25$ minutes. Let the units check the setup: $\frac{\text{miles}}{\text{hour}}\times \text{hours}$ leaves miles.

Example: a trip in two parts

A traveler drives $120$ miles at $60$ mph, then bikes $15$ miles at $10$ mph. How long does the trip take?

• Driving: $120÷60=2$ hours
• Biking: $15÷10=1.5$ hours
• Total: $2+1.5=3.5$ hours

Answer: $3.5$ hours

Find the time for each part separately. Averaging the two speeds to $35$ mph gives the wrong total, because the traveler spends more time at the slower speed.

A current or wind adds to your speed when it moves with you and subtracts when it moves against you. A kayak that goes $5$ mph in still water, on a river flowing at $1$ mph, travels $6$ mph downstream and $4$ mph upstream.

Example: upstream and downstream

That kayak goes $12$ miles upstream and then $12$ miles back. How long does the round trip take?

(spoiler)

• Upstream: $12÷4=3$ hours
• Downstream: $12÷6=2$ hours

Answer: $3+2=5$ hours

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.

Example: Converting between forms

Express $\frac{3}{8}$ as a decimal and as a percent.

• Divide the numerator by the denominator: $3÷8=0.375$
• Multiply the decimal by $100$ to get the percent: $0.375\times 100=37.5\%$

Answer: $\frac{3}{8}=0.375=37.5\%$

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:

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

Example: Finding the part and the whole

(a) What is $25\%$ of $200$?

• Use the formula: $\text{Part}=200\times \left(\frac{25}{100}\right)=200\times 0.25=50$

Answer: $50$

(b) If $30\%$ of a number is $45$, what is the number?

(spoiler)

• Use the formula for the whole:

$$\text{Whole}=45÷\left(\frac{30}{100}\right)=45÷0.30=150$$

Answer: $150$

Example: Discount

A jacket originally costs $\$80$ and is on sale for $30\%$ off. What is the sale price?

(spoiler)

• Find the discount amount: $\text{Part}=80\times \left(\frac{30}{100}\right)=80\times 0.30=24$
• Subtract from the original price: $80-24=56$

Answer: $\$56$

Example: Percent increase and reversing a percent increase

(a) A town’s population is $4,000$. After an $8\%$ increase, what is the new population?

• A value growing by $8\%$ becomes $108\%$ of the original, so multiply by $1.08$.
• New population $=4000\times 1.08=4320$

Answer: $4,320$

(b) After a $20\%$ increase, the price of an item is $\$60$. What was the original price?

(spoiler)

• A $20\%$ increase means the new price is $120\%$ of the original, so multiply the original by $1.20$.
• Let $p$ be the original price: $p\times 1.20=60$
• Solve: $p=\frac{60}{1.20}=50$

Answer: $\$50$

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

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

Common pitfall - wrong denominator in percent change: Always divide by the original value, not the new value. Using the new value as the denominator is a frequent mistake that gives a different - and incorrect - result.

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 $\$400$.

(spoiler)

$$\frac{400-500}{500}\times 100\%=\frac{-100}{500}\times 100\%=-20\%$$

When stating a percent decrease in words, drop the negative sign - the word “decrease” already conveys the direction.

Answer: $20\%$ decrease

Percent of a percent

Sometimes you need to find a percent of another percent - for example, in 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: Successive discounts

A $\$100$ item is discounted by $10\%$, and then the already-reduced price is discounted by another $10\%$. What is the final price, and what is the combined percent discount?

(spoiler)

• After the first $10\%$ discount: $100\times 0.90=\$90$
• After the second $10\%$ discount: $90\times 0.90=\$81$
• Combined discount: $\frac{100-81}{100}\times 100\%=19\%$

Answer: Final price is $\$81$; combined discount is $19\%$, not $20\%$.