Solving Proportions
Learn to find unknown values in proportions using cross multiplication and other strategies.
Definition
- Cross multiply:
- Simplify:
- Divide:
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Worked Examples
Solve for :
Set up cross multiplication
→ Cross products are equal
Multiply the known values
→
Divide both sides by 8
→
Verify by substituting back
✓ → Both ratios equal
Answer:
Common Mistakes
Cross multiplying incorrectly: becoming
Why it's wrong: Cross multiplication means diagonal multiplication: numerator of one fraction times denominator of the other.
Correct: Correct: , so , giving
Setting up the proportion with mismatched units
Why it's wrong: Both ratios must compare the same quantities in the same order (e.g., pencils to cost, not cost to pencils).
Correct: Keep consistent: or
Forgetting to verify the answer
Why it's wrong: Without checking, you might not catch arithmetic errors.
Correct: Always substitute your answer back into the original proportion and verify both ratios are equal.
Interactive Visual
Ratio Tape Diagram
Adjust the ratio parts using + and - buttons.
Balance Scale
Solution: x = 4
Interactive Sandbox
Expression Calculator
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History
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Practice Problems
16 problemsWhat method is used to solve proportions by multiplying diagonally?
Why It Matters
- Cooking: If a recipe for 4 people needs 2 cups of flour, how much for 10 people?
- Maps and Scale Models: A map scale of 1 cm = 50 km lets you calculate real distances
- Medicine: Doctors calculate medication dosages based on patient weight
- Photography: Resizing images while maintaining aspect ratio
- Construction: Scaling blueprints to actual building dimensions
Real World Applications
Recipe Scaling
Chefs and home cooks use proportions to adjust recipe quantities for different serving sizes.
Example:
A cookie recipe makes 24 cookies using 3 eggs. To make 40 cookies: , so eggs.
A pasta recipe for 6 people requires 450 grams of pasta. You're cooking for 10 people.
How many grams of pasta do you need?
Step 1: Write the mathematical expression
Set up the proportion:
Map Reading
Cartographers use scale ratios to represent real distances on maps.
Example:
If a map scale is 1:100,000, then 1 cm on the map equals 1 km in reality. A 5.5 cm road on the map is actually 5.5 km long.
A hiking map has a scale where 3 cm represents 2 km. You measure a trail that is 12 cm on the map.
How long is the actual trail?
Step 1: Write the mathematical expression
Set up:
Medicine Dosage
Pharmacists calculate proper medication doses based on patient weight using proportions.
Example:
A medication is prescribed at 5 mg per kg of body weight. For a 60 kg patient: , so mg.
A children's medicine is dosed at 10 mg per 5 kg of body weight. A child weighs 35 kg.
What dosage should the child receive?
Step 1: Write the mathematical expression
Set up:
Key Takeaways
- 1A proportion is an equation showing two equal ratios:
- 2Cross multiplication: if , then
- 3The scale factor method multiplies both parts of a ratio by the same number
- 4Always set up proportions with matching units in the same positions
- 5Verify your answer by substituting back into the original proportion
Frequently Asked Questions
Glossary
- Proportion
- An equation stating that two ratios are equal, such as
- Cross multiplication
- A method for solving proportions by multiplying diagonally: if , then
- Scale factor
- The number you multiply by to transform one ratio into an equivalent ratio
- Equivalent ratios
- Ratios that express the same relationship, like and