Ratios and Proportions: From Basics to Word Problems (with Examples)

What is a ratio?

A ratio is a way to compare two quantities. If a class has $12$ girls and $18$ boys, the ratio of girls to boys is

Ratios can be written three ways and they all mean the same thing:

• with a colon: $12 : 18$
• as a fraction: $\dfrac{12}{18}$
• with the word "to": $12 \text{ to } 18$

Ratios obey the same simplification rules as fractions. Divide top and bottom by their greatest common factor:

So the ratio of girls to boys is $2 : 3$, meaning that for every $2$ girls there are $3$ boys.

Equivalent ratios

Two ratios are equivalent if they simplify to the same simplest form. You can also create equivalent ratios by multiplying or dividing both parts by the same non-zero number.

Example 1. Are $4 : 6$ and $10 : 15$ equivalent?

Simplify each: $4 : 6 = 2 : 3$ and $10 : 15 = 2 : 3$. Both reduce to $2 : 3$, so yes, they are equivalent.

Example 2. Find an equivalent ratio of $3 : 5$ with a second number of $20$.

You need to multiply $5$ by $4$ to get $20$, so multiply $3$ by $4$ as well: $3 : 5 = 12 : 20$.

What is a proportion?

A proportion is an equation that says two ratios are equal:

Proportions are how you solve "if X is to Y, then ? is to Z" problems.

Cross multiplication

The fastest way to solve a proportion is the cross-multiplication rule:

You multiply the two diagonals and set them equal. Then you have a one-step linear equation to solve.

Example 3. Solve $\dfrac{x}{6} = \dfrac{4}{3}$.

Cross multiply: $x \cdot 3 = 6 \cdot 4$, so $3x = 24$, and $x = 8$.

Example 4. Solve $\dfrac{5}{x} = \dfrac{15}{21}$.

Cross multiply: $5 \cdot 21 = 15 \cdot x$, so $105 = 15x$, and $x = 7$.

Unit rates

A unit rate is a ratio in which the second quantity is exactly $1$ — answering "how much per one?" Common examples include miles per hour, dollars per pound, and pages per minute.

To convert a ratio to a unit rate, divide the first quantity by the second.

Example 5. A car travels $180$ miles in $3$ hours. What is its unit rate (miles per hour)?

Example 6. A package of $12$ apples costs \4.80$. What is the unit price?

Unit rates make comparisons fair: $4$ apples for \1.80$($$0.45$each) is more expensive per apple than$12$for$$4.80$($$0.40$ each).

Solving word problems with proportions

Most ratio word problems follow the same template: identify two equivalent ratios, set up a proportion, and cross multiply.

Example 7. A recipe calls for $2$ cups of flour for every $3$ cups of milk. If you use $9$ cups of milk, how much flour do you need?

Set up a proportion comparing flour-to-milk in both situations:

Cross multiply: $2 \cdot 9 = 3 \cdot x$, so $18 = 3x$, giving $x = 6$ cups of flour.

Example 8. A map has a scale of $1$ inch to $25$ miles. Two cities are $4.5$ inches apart on the map. How far apart are they in real life?

Cross multiply: $1 \cdot x = 25 \cdot 4.5 = 112.5$. The cities are $\mathbf{112.5}$ miles apart.

Example 9. Maria can read $30$ pages in $20$ minutes. How long will it take her to read $90$ pages at the same rate?

Cross multiply: $30x = 20 \cdot 90 = 1800$, so $x = 60$ minutes.

Three setup tips

• Keep units consistent. Both ratios in the proportion must compare the same things in the same order — for example, "flour over milk" on both sides, not "flour over milk" on one side and "milk over flour" on the other.
• Label everything. Writing "cups flour" and "cups milk" next to the numbers prevents flipping ratios accidentally.
• Convert mixed units first. If one quantity is in feet and the other in inches, convert one to match the other before setting up the proportion.

Common mistakes

• Mixing up the order. "$3 : 5$" is not the same as "$5 : 3$." Pay attention to which quantity goes first.
• Adding instead of multiplying for equivalent ratios. Equivalent ratios come from multiplying both parts by the same number, not adding the same number to both.
• Forgetting to simplify. $\dfrac{12}{18}$ and $\dfrac{2}{3}$ describe the same ratio, but answers are usually expected in simplest form.
• Cross multiplying when the ratio is not a proportion yet. You can only cross multiply once you have written an equation $\dfrac{a}{b} = \dfrac{c}{d}$.