What is a proportion and how is it different from a ratio or rate?

A proportion is an equation that states two ratios or rates are equal. While a ratio compares two quantities, and a rate compares quantities with different units, a proportion specifically shows that two ratios or rates represent the same relationship. For example, the proportion 48 = 12 means that the ratio 4 to 8 is equal to the ratio 1 to 2. Proportions can be written as fractions or expressed in words like "5 is to 12 as 30 is to 72." Understanding proportions helps in comparing relationships and solving problems involving equivalent ratios.

How can you determine if a proportion is true?

To determine if a proportion is true, you can use two methods: simplifying ratios or using cross products. First, simplify each ratio to its simplest form by dividing numerator and denominator by their greatest common factors. If both simplified ratios are equal, the proportion is true. For example, 412 simplifies to 13, and 515 also simplifies to 13, so the proportion is true. Alternatively, use cross products: multiply the numerator of one ratio by the denominator of the other and compare. If the products are equal, the proportion holds. For 412 = 515, cross products are 4 × 15 = 60 and 5 × 12 = 60, confirming the proportion is true.

6. Ratio, Proportion, and Triangle Applications

Proportions

6. Ratio, Proportion, and Triangle Applications

Proportions: Videos & Practice Problems

Video Lessons

Topic summary

Proportions compare two equal ratios or rates. A proportion can be written as a fraction statement, such as $\frac{a}{b}=\frac{c}{d}$ , or read from words like “is to” and “as.” When writing a proportion, keep the same units in the same positions so the comparison represents the same relationship.

To determine whether a proportion is true, either reduce each ratio to simplest form and compare the results, or use cross products. For $\frac{a}{b}=\frac{c}{d}$, a true proportion satisfies $ad=bc$ . Ratios may include whole numbers, fractions, mixed numbers, or decimals, but they must be rewritten carefully before comparing.

You can also solve a proportion with an unknown by using cross products to form an equation, then solving for the variable and checking that the resulting ratios are equal. This helps find missing values in equivalent rates and ratio relationships.

Concept

Intro To Proportions

Video duration:

Problem

Express the sentence as a proportion.
Ten is to fifteen as four is to six.

$\frac{10}{15} = \frac{6}{4}$

$\frac{10}{15} = \frac{4}{6}$

$\frac{10}{4} = \frac{15}{6}$

$\frac{15}{10} = \frac{4}{6}$

Problem

Express the sentence as a proportion.
Two thirds is to one fifth as two fifths is to one ninth.

$\frac{\frac{2}{3}}{\frac{1}{5}} = \frac{\frac{1}{9}}{\frac{2}{5}}$

$\frac{\frac{2}{5}}{\frac{1}{9}} = \frac{\frac{2}{3}}{\frac{1}{5}}$

$\frac{2}{3} = \frac{1}{5} = \frac{2}{5} = \frac{1}{9}$

$\frac{\frac{2}{3}}{\frac{1}{5}} = \frac{\frac{2}{5}}{\frac{1}{9}}$

Problem

Express the sentence as a proportion.
A recipe uses 3 cups of rice for 5 cups of broth, and 15 cups of rice for 25 cups of broth.

$\frac{3 c . r i c e}{5 c . b r o t h} = \frac{25 c . r i c e}{15 c . b r o t h}$

$\frac{3 c . r i c e}{15 c . b r o t h} = \frac{5 c . r i c e}{25 c . b r o t h}$

$\frac{3 c . r i c e}{5 c . b r o t h} = \frac{15 c . r i c e}{25 c . b r o t h}$

$\frac{15 c . r i c e}{5 c . b r o t h} = \frac{25 c . r i c e}{3 c . b r o t h}$

Problem

Express the sentence as a proportion.
A car travels 80 miles in 2 hours and 120 miles in 3 hours.

$\frac{80 m i}{2 h r s} = \frac{120 m i}{3 h r s}$

$\frac{80 m i}{3 h r s} = \frac{120 m i}{2 h r s}$

$\frac{2 m i}{80 h r s} = \frac{3 m i}{120 h r s}$

$\frac{80 m i}{120 h r s} = \frac{2 m i}{3 h r s}$

Problem

Express the sentence as a proportion.
A paint mixture uses 2.5 liters of blue paint for every 4 liters of white paint, and another mixture uses 7.5 liters of blue paint for every 12 liters of white paint.

$\frac{2.5 L b l u e}{12 L w h i t e} = \frac{7.5 L b l u e}{4 L w h i t e}$

$\frac{2.5 L b l u e}{4 L w h i t e} = \frac{7.5 L b l u e}{12 L w h i t e}$

$\frac{4 L b l u e}{2.5 L w h i t e} = \frac{12 L b l u e}{7.5 L w h i t e}$

$\frac{2.5 L b l u e}{7.5 L w h i t e} = \frac{4 L b l u e}{12 L w h i t e}$

Problem

Express the sentence as a proportion.
A trail map shows 3.6 miles in 1.2 hours, while another section shows 7 miles in $2 and one third$ hours.

$\frac{3.6 m i}{2 \frac{1}{3} h r s} = \frac{7 m i}{1.2 h r s}$

$\frac{1.2 m i}{3.6 h r s} = \frac{2 \frac{1}{3} m i}{7 h r s}$

$\frac{3.6 m i}{7 h r s} = \frac{1.2m i}{2 \frac{1}{3} h r s}$

$\frac{3.6 m i}{1.2 h r s} = \frac{7 m i}{2 \frac{1}{3} h r s}$

Concept

Determine If Proportions Are True

Video duration:

Problem

Determine whether the proportion is true or false.
$\frac{1 \frac{2}{3}}{3} = \frac{3 \frac{1}{3}}{6}$

True

False

Cannot be determined

Problem

Determine whether the proportion is true or false.
$\frac{2.4}{6} = \frac{1.2}{4}$

True

False

Cannot be determined

Problem

Use cross products to determine whether the proportion is true or false.
$\frac{7}{4} = \frac{20}{12}$

True

False

Cannot be determined

Problem

Use cross products to determine whether the proportion is true or false.
$\frac{4}{9} = \frac{12}{27}$

True

False

Cannot be determined

Problem

Write the sentence as a proportion then determine whether the proportion is true or false.
One and eight tenths is to two as four and five tenths is to five.

$\frac{1 \frac{8}{10}}{2} = \frac{4 \frac{5}{10}}{5}$ ; False

$\frac{2}{1 \frac{8}{10}} = \frac{5}{4 \frac{5}{10}}$ ; True

$\frac{1 \frac{8}{10}}{2} = \frac{4 \frac{5}{10}}{5}$ ; True

$\frac{2}{1 \frac{8}{10}} = \frac{5}{4 \frac{5}{10}}$ ; False

Problem

Write the sentence as a proportion then determine whether the proportion is true or false.
Four bags of flour costs \$12, and eight bags of flour costs \$18.

$\frac{4}{12} = \frac{8}{18}$ ; True

$\frac{4}{12} = \frac{8}{16}$ ; True

$\frac{4}{12} = \frac{8}{18}$ , False

$\frac{4}{12} = \frac{8}{16}$ ; False

Concept

Solving Proportions

Video duration:

Problem

Solve the proportion using cross products.
$\frac{5}{8} = \frac{k}{24}$

$k equals 15$

$k equals 12$

$k equals 18$

$k = \frac{15}{8}$

Problem

Solve the proportion using cross products.
$\frac{2.5}{5} = \frac{p}{12}$

$p equals 6$

$p equals 5$

$p equals 4.8$

$p equals 12$

Problem

Solve the proportion using cross products.
$\frac{\frac{1}{3}}{\frac{3}{8}} = \frac{\frac{2}{5}}{n}$

$n = \frac{16}{5}$

$n = \frac{5}{16}$

$n = \frac{1}{20}$

$n = \frac{9}{20}$

Example

Solving Proportions Example 1

Video duration:

Here's what students ask on this topic:

A proportion is an equation that states two ratios or rates are equal. While a ratio compares two quantities, and a rate compares quantities with different units, a proportion specifically shows that two ratios or rates represent the same relationship. For example, the proportion $\frac{4}{8} = \frac{1}{2}$ means that the ratio 4 to 8 is equal to the ratio 1 to 2. Proportions can be written as fractions or expressed in words like "5 is to 12 as 30 is to 72." Understanding proportions helps in comparing relationships and solving problems involving equivalent ratios.

To determine if a proportion is true, you can use two methods: simplifying ratios or using cross products. First, simplify each ratio to its simplest form by dividing numerator and denominator by their greatest common factors. If both simplified ratios are equal, the proportion is true. For example, $\frac{4}{12}$ simplifies to $\frac{1}{3}$ , and $\frac{5}{15}$ also simplifies to $\frac{1}{3}$ , so the proportion is true. Alternatively, use cross products: multiply the numerator of one ratio by the denominator of the other and compare. If the products are equal, the proportion holds. For $\frac{4}{12} = \frac{5}{15}$ , cross products are 4 × 15 = 60 and 5 × 12 = 60, confirming the proportion is true.

To solve a proportion with a missing value, use the cross products method. Set up the proportion as $\frac{a}{b} = \frac{c}{d}$ , where one variable is unknown. Multiply across the diagonals to form an equation: $a × d = b × c$ . Then solve for the unknown variable. For example, in $\frac{4}{12} = \frac{5}{x}$ , cross multiply to get $4 × x = 5 × 12$ , which simplifies to $4x = 60$ . Dividing both sides by 4 gives $x = 15$ . Always check your answer by substituting back into the original proportion to ensure it is true.

Keeping units consistent in proportions is crucial because proportions compare two ratios or rates that represent the same relationship. If units are mixed or placed inconsistently, the comparison becomes invalid. For example, when writing a proportion comparing apples to cost, the number of apples should be in the numerator and the cost in the denominator for both ratios. This ensures the proportion accurately reflects the relationship between quantity and price. Consistent units allow meaningful comparisons and correct problem-solving when working with proportions.

Yes, proportions can involve fractions, decimals, or mixed numbers. When working with these, it is important to rewrite all ratios carefully into a consistent form before comparing. For example, convert mixed numbers to improper fractions and decimals to fractions if needed. This allows you to simplify ratios or use cross products accurately. Handling these forms correctly ensures the proportion is evaluated properly and helps avoid errors in determining if the proportion is true or in solving for unknown values.

Your Prealgebra tutors

Physics and Math Lead Instructor

Proportions: Videos & Practice Problems

Intro To Proportions

Express the sentence as a proportion.
Ten is to fifteen as four is to six.

Express the sentence as a proportion.
Two thirds is to one fifth as two fifths is to one ninth.

Express the sentence as a proportion.
A recipe uses 3 cups of rice for 5 cups of broth, and 15 cups of rice for 25 cups of broth.

Express the sentence as a proportion.
A car travels 80 miles in 2 hours and 120 miles in 3 hours.

Express the sentence as a proportion.
A paint mixture uses 2.5 liters of blue paint for every 4 liters of white paint, and another mixture uses 7.5 liters of blue paint for every 12 liters of white paint.

Express the sentence as a proportion.
A trail map shows 3.6 miles in 1.2 hours, while another section shows 7 miles in $2 and one third$ hours.

Determine If Proportions Are True

Determine whether the proportion is true or false.
$\frac{1 \frac{2}{3}}{3} = \frac{3 \frac{1}{3}}{6}$

Determine whether the proportion is true or false.
$\frac{2.4}{6} = \frac{1.2}{4}$

Use cross products to determine whether the proportion is true or false.
$\frac{7}{4} = \frac{20}{12}$

Use cross products to determine whether the proportion is true or false.
$\frac{4}{9} = \frac{12}{27}$

Write the sentence as a proportion then determine whether the proportion is true or false.
One and eight tenths is to two as four and five tenths is to five.

Write the sentence as a proportion then determine whether the proportion is true or false.
Four bags of flour costs \$12, and eight bags of flour costs \$18.

Solving Proportions

Solve the proportion using cross products.
$\frac{5}{8} = \frac{k}{24}$

Solve the proportion using cross products.
$\frac{2.5}{5} = \frac{p}{12}$

Solve the proportion using cross products.
$\frac{\frac{1}{3}}{\frac{3}{8}} = \frac{\frac{2}{5}}{n}$

Solving Proportions Example 1

Here's what students ask on this topic:

What is a proportion and how is it different from a ratio or rate?

How can you determine if a proportion is true?

How do you solve a proportion with a missing value?

What is the importance of keeping units consistent when writing proportions?

Can proportions involve fractions, decimals, or mixed numbers, and how are they handled?

Your Prealgebra tutors

Proportions: Videos & Practice Problems

Intro To Proportions

Express the sentence as a proportion.Ten is to fifteen as four is to six.

Express the sentence as a proportion.Two thirds is to one fifth as two fifths is to one ninth.

Express the sentence as a proportion.A recipe uses 3 cups of rice for 5 cups of broth, and 15 cups of rice for 25 cups of broth.

Express the sentence as a proportion.A car travels 80 miles in 2 hours and 120 miles in 3 hours.

Express the sentence as a proportion.A paint mixture uses 2.5 liters of blue paint for every 4 liters of white paint, and another mixture uses 7.5 liters of blue paint for every 12 liters of white paint.

Express the sentence as a proportion.A trail map shows 3.6 miles in 1.2 hours, while another section shows 7 miles in 2132\(\frac\)13 hours.

Determine If Proportions Are True

Determine whether the proportion is true or false. 1233=3136\(\frac{1\frac23}{3}\)=\(\frac{3\frac13}{6}\)

Determine whether the proportion is true or false.2.46=1.24\(\frac{2.4}{6}\)=\(\frac{1.2}{4}\)

Use cross products to determine whether the proportion is true or false.74=2012\(\frac\)74=\(\frac{20}{12}\)

Use cross products to determine whether the proportion is true or false.49=1227\(\frac\)49=\(\frac{12}{27}\)

Write the sentence as a proportion then determine whether the proportion is true or false.One and eight tenths is to two as four and five tenths is to five.

Write the sentence as a proportion then determine whether the proportion is true or false.Four bags of flour costs \$12, and eight bags of flour costs \$18.

Solving Proportions

Solve the proportion using cross products.58=k24\(\frac\)58=\(\frac{k}{24}\)

Solve the proportion using cross products.2.55=p12\(\frac{2.5}{5}\)=\(\frac{p}{12}\)

Solve the proportion using cross products.1338=25n\(\frac{\frac13}{\frac38}\)=\(\frac{\frac25}{n}\)

Solving Proportions Example 1

Here's what students ask on this topic:

What is a proportion and how is it different from a ratio or rate?

What is a proportion and how is it different from a ratio or rate?

How can you determine if a proportion is true?

How can you determine if a proportion is true?

How do you solve a proportion with a missing value?

How do you solve a proportion with a missing value?

What is the importance of keeping units consistent when writing proportions?

What is the importance of keeping units consistent when writing proportions?

Can proportions involve fractions, decimals, or mixed numbers, and how are they handled?

Can proportions involve fractions, decimals, or mixed numbers, and how are they handled?

Your Prealgebra tutors

Express the sentence as a proportion.
Ten is to fifteen as four is to six.

Express the sentence as a proportion.
Two thirds is to one fifth as two fifths is to one ninth.

Express the sentence as a proportion.
A recipe uses 3 cups of rice for 5 cups of broth, and 15 cups of rice for 25 cups of broth.

Express the sentence as a proportion.
A car travels 80 miles in 2 hours and 120 miles in 3 hours.

Express the sentence as a proportion.
A paint mixture uses 2.5 liters of blue paint for every 4 liters of white paint, and another mixture uses 7.5 liters of blue paint for every 12 liters of white paint.

Express the sentence as a proportion.
A trail map shows 3.6 miles in 1.2 hours, while another section shows 7 miles in $2 and one third$ hours.

Determine whether the proportion is true or false.
$\frac{1 \frac{2}{3}}{3} = \frac{3 \frac{1}{3}}{6}$

Determine whether the proportion is true or false.
$\frac{2.4}{6} = \frac{1.2}{4}$

Use cross products to determine whether the proportion is true or false.
$\frac{7}{4} = \frac{20}{12}$

Use cross products to determine whether the proportion is true or false.
$\frac{4}{9} = \frac{12}{27}$

Write the sentence as a proportion then determine whether the proportion is true or false.
One and eight tenths is to two as four and five tenths is to five.

Write the sentence as a proportion then determine whether the proportion is true or false.
Four bags of flour costs \$12, and eight bags of flour costs \$18.

Solve the proportion using cross products.
$\frac{5}{8} = \frac{k}{24}$

Solve the proportion using cross products.
$\frac{2.5}{5} = \frac{p}{12}$

Solve the proportion using cross products.
$\frac{\frac{1}{3}}{\frac{3}{8}} = \frac{\frac{2}{5}}{n}$