Use a Visible Multiplier
Best when one known value is an easy multiple or fraction of its corresponding value.
4 → 20 means × 5
Free Math Guide
Solving Proportions Explained
Learn how to set up equivalent ratios, solve for a missing value with scale factors or cross multiplication, and check that your answer preserves the proportional relationship.
A proportion is an equation stating that two ratios are equivalent. Solving a proportion means finding the missing value that makes both ratios describe the same relationship. You can often see the answer by scaling one ratio up or down. When that scale factor is not obvious, cross multiplication provides a reliable equation-solving method.
A Proportion Connects Equivalent Ratios
Ratios are equivalent when one can be created by multiplying or dividing both parts of the other ratio by the same nonzero number. A proportion writes those equal ratios as an equation.
General Form
a ÷ b = c ÷ d or a/b = c/d
Example of Equivalent Ratios
The ratio 3/4 becomes 15/20 when both 3 and 4 are multiplied by 5.
3/4 = 15/20 because 3 × 5 = 15 and 4 × 5 = 20
The Relationship Stays Constant
The numbers may change, but the comparison does not. Both 3/4 and 15/20 equal 0.75, so they represent the same proportional relationship.
How to Set Up a Proportion
The most important setup rule is consistency. The first and second ratios must compare the same kinds of quantities in the same order.
Choose a ratio order
Decide what belongs on top and bottom, such as miles over hours or dollars over notebooks.
Keep the order
If miles are on top in the first ratio, miles must remain on top in the second ratio.
Place the unknown
Use x in the position that represents the quantity the problem asks you to find.
180 miles / 3 hours = x miles / 5 hours
Units Reveal Setup Errors
The incorrect setup 180 miles / 3 hours = 5 hours / x miles reverses the order. Writing the units beside the numbers makes this mismatch easier to notice.
Scale Factor or Cross Multiplication?
Both methods preserve equivalent ratios. Choose the one that makes the relationship easiest to see and calculate.
Build an Equation
Best when the scale factor is unclear, decimal, or when the unknown is in a less convenient position.
a/b = c/d means ad = bc
| Situation | Helpful Method | Why |
|---|---|---|
| 3/4 = x/20 | Scale Factor | 4 becomes 20 by multiplying by 5. |
| 7/9 = 21/x | Scale Factor | 7 becomes 21 by multiplying by 3. |
| 5/8 = x/14 | Cross Multiplication | The scale factor from 8 to 14 is not a whole number. |
| 2.4/7 = 6/x | Cross Multiplication | An equation keeps the decimal relationship organized. |
Solve a Proportion With a Scale Factor
Compare corresponding known values. Determine what multiplication or division changes one into the other, then apply that same operation to the remaining value.
Example: Solve 3/4 = x/20
- Compare the denominators: 4 × 5 = 20.
- Use the same scale factor on the numerators.
- Compute 3 × 5 = 15.
x = 15
Scaling Down Works Too
In 18/24 = x/8, divide 24 by 3 to get 8. Divide 18 by the same 3, so x = 6. A scale factor may be a multiplier, divisor, fraction, or decimal.
Solve a Proportion With Cross Multiplication
Multiply the numerator of each ratio by the denominator of the other ratio. Set the two cross products equal, then solve the resulting equation for x.
Cross-Product Property
If a/b = c/d, then a × d = b × c
Example: Solve 5/8 = x/14
- Cross multiply: 5 × 14 = 8 × x.
- Simplify the products: 70 = 8x.
- Divide both sides by 8: x = 8.75.
x = 70 ÷ 8 = 8.75
Cross Multiplication Is Not the Final Step
Cross multiplication usually produces an equation such as 8x = 70. You must still divide by the number multiplying x to isolate the unknown.
How to Solve Proportion Word Problems
A proportion can model a situation when two quantities change at a constant rate. First identify two matching pairs of quantities, then write the ratios in a consistent order.
Recipes
Scale ingredient amounts when the number of servings changes.
Maps and Scale Drawings
Connect a measurement on a drawing with the corresponding actual distance or size.
Prices and Unit Rates
Find a cost or quantity when the price per item stays constant.
Similar Figures
Use corresponding side lengths to find a missing measurement.
Ask Whether the Relationship Is Proportional
- The ratio or unit rate remains constant.
- Doubling one quantity doubles the other.
- The relationship would include the point (0, 0).
- There is no fixed starting fee or added amount changing the ratio.
Not Every Two-Quantity Problem Is Proportional
A taxi charge with a fixed starting fee is not a direct proportion because the cost per mile does not stay constant for every trip length.
Worked Proportion Examples
Missing Numerator
Solve 6/7 = x/21.
- 7 becomes 21 by multiplying by 3.
- Multiply 6 by the same scale factor.
- 6 × 3 = 18.
x = 18
Missing Denominator
Solve 7/9 = 21/x.
- Cross multiply: 7x = 9 × 21.
- Simplify: 7x = 189.
- Divide by 7: x = 27.
x = 27
Notebook Cost
Four notebooks cost $10. How much do 14 notebooks cost at the same rate?
- Keep dollars over notebooks: 10/4 = x/14.
- Cross multiply: 4x = 140.
- Divide by 4: x = 35.
14 notebooks cost $35.
Scale a Recipe
A recipe uses 3 cups of flour for 8 servings. How many cups are needed for 20 servings?
- Write cups over servings: 3/8 = x/20.
- Cross multiply: 8x = 60.
- Divide by 8: x = 7.5.
The recipe needs 7.5 cups of flour.
Map Distance
On a map, 2 inches represents 15 miles. What distance does 6 inches represent?
- Write map inches over miles: 2/15 = 6/x.
- Cross multiply: 2x = 90.
- Divide by 2: x = 45.
6 inches represents 45 miles.
Corresponding Sides
Two similar rectangles have corresponding widths of 4 and 10. A side of the smaller rectangle is 7. Find the corresponding larger side.
- Write smaller over larger: 4/10 = 7/x.
- Cross multiply: 4x = 70.
- Divide by 4: x = 17.5.
The corresponding side is 17.5 units.
How to Check a Proportion Answer
Substitute your value for x and confirm that the ratios are equivalent. You can compare decimal values, simplify both ratios, or verify that the cross products match.
Compare decimals
For 3/4 = 15/20, both ratios equal 0.75.
Simplify the ratios
The ratio 15/20 simplifies to 3/4.
Compare cross products
For 3/4 = 15/20, 3 × 20 and 4 × 15 both equal 60.
Check the context
Make sure the size, sign, unit, and meaning of the answer are reasonable.
If the two cross products are equal, the completed proportion is true.
Common Proportion Mistakes
Reversing Only One Ratio
If the first ratio is dollars per item, the second ratio must also be dollars per item. Reversing both ratios is valid; reversing only one is not.
Using a Scale Factor on One Part Only
Equivalent ratios require the same multiplication or division on both parts of a ratio.
Multiplying the Wrong Pair of Numbers
Cross products use diagonal pairs: top left with bottom right, and bottom left with top right.
Stopping Before Isolating x
An equation such as 8x = 70 is an intermediate step. Divide both sides by 8 to finish solving.
Forcing a Nonproportional Situation Into a Proportion
A proportion applies only when the relationship has a constant ratio or unit rate.
Rounding Too Early
Keep the full calculator value during intermediate work and round only the final answer as the problem directs.
Leaving Off the Unit
A correct value should be interpreted as dollars, miles, cups, items, or the quantity named in the question.
How to Solve Proportions Step by Step
1. Identify the pairs
Find the two corresponding quantities in each situation or ratio.
2. Keep a consistent order
Place matching units in matching numerator and denominator positions.
3. Write the proportion
Use x for the missing value and set the two ratios equal.
4. Choose a method
Use a visible scale factor when convenient; otherwise use cross multiplication.
5. Solve for x
Carry out the same scaling operation or isolate x in the cross-product equation.
6. Check and interpret
Verify equivalent ratios, then state the answer with its correct unit and meaning.
Practice Solving Proportions
Practice with scale factors, cross multiplication, unit rates, and real-world word problems. Generate randomized questions, request hints, use the built-in calculator, and receive immediate feedback.
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Interactive Proportions Practice
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