Proportion Calculator
Solve proportions by finding the missing value in A/B = C/D. See also Cross Multiplication Calculator and Ratio Calculator.
How to Solve Proportions
A proportion states that two ratios are equal: A/B = C/D. To find a missing value, use cross multiplication. Multiply the two diagonal values that are known, then divide by the remaining known value. This method works for any proportion where exactly one value is unknown.
Proportion Formula
A / B = C / D
Cross multiply: A × D = B × C
Solve for unknown:
A = (B × C) / D
B = (A × D) / C
C = (A × D) / B
D = (B × C) / A
Example
Solve: 3/6 = ?/12
Cross multiply: 3 × 12 = 6 × ?
36 = 6 × ?
? = 36 / 6 = 6
Check: 3/6 = 6/12 = 0.5 ✓
Frequently Asked Questions
What is a proportion?
A proportion is an equation that states two ratios are equal. For example, 1/2 = 3/6 is a proportion because both ratios simplify to the same value (0.5).
How is a proportion different from a ratio?
A ratio compares two quantities (e.g., 3:4), while a proportion states that two ratios are equal (e.g., 3/4 = 6/8). A proportion always involves four values.
Can proportions have negative numbers?
Yes, proportions can include negative numbers. The cross multiplication method works the same way. Just be careful with signs when multiplying and dividing.
What if the denominator is zero?
Division by zero is undefined, so neither B nor D can be zero in a proportion A/B = C/D. If solving for B or D, the result will be zero only if the corresponding numerator is also zero.
Solved Examples — Proportions
Example: If 3 notebooks cost $7.50, how much do 8 notebooks cost?
Solution:
Step 1: Set up proportion: 3/7.50 = 8/x
Step 2: Cross multiply: 3x = 7.50 × 8 = 60
Step 3: x = 60/3 = 20
Answer: $20.00
Example: A 6-foot person casts a 4-foot shadow. A tree casts a 20-foot shadow. How tall is the tree?
Solution:
Step 1: Height/Shadow = Height/Shadow → 6/4 = x/20
Step 2: Cross multiply: 4x = 6 × 20 = 120
Step 3: x = 120/4 = 30
Answer: The tree is 30 feet tall
Example: A car travels 180 miles on 6 gallons. How far can it go on 10 gallons?
Solution:
Step 1: Set up: 180/6 = x/10
Step 2: Cross multiply: 6x = 180 × 10 = 1800
Step 3: x = 1800/6 = 300
Answer: 300 miles
Practice Questions
Try these on your own:
- Solve: 4/9 = x/36 (Answer: x = 16)
- If 5 workers finish a job in 12 days, how long for 8 workers? (Answer: 7.5 days)
- A recipe for 4 servings needs 2.5 cups of flour. How much for 10 servings? (Answer: 6.25 cups)
- On a map, 2 cm = 15 km. What distance is 7 cm on the map? (Answer: 52.5 km)
- If 12 oranges cost $4.80, how much for 20 oranges? (Answer: $8.00)
- Solve: 7/x = 21/15 (Answer: x = 5)
Common Mistakes to Avoid
The most common mistake is setting up the proportion with mismatched units. Both ratios must compare the same types of quantities in the same order. If the first ratio is miles/gallons, the second must also be miles/gallons, not gallons/miles. Another error is using proportions for inverse relationships — if more workers means less time, that is an INVERSE proportion (use: workers₁ × time₁ = workers₂ × time₂), not a direct proportion. Students also sometimes cross-multiply incorrectly, forgetting which terms are diagonal. Always double-check by plugging your answer back into the original proportion to verify both sides are equal. Lastly, denominators can never be zero in a proportion.
Key Takeaways
- A proportion states two ratios are equal: a/b = c/d.
- Solve using cross-multiplication: a × d = b × c.
- Direct proportion: as one quantity increases, the other increases at the same rate.
- Inverse proportion: as one quantity increases, the other decreases (use product = constant).
- Always verify your answer by substituting it back and checking both sides are equal.
- Common applications: recipes, maps, scale drawings, unit pricing, similar triangles, and speed/distance problems.