Quick reference
What ratios and proportions are
A ratio compares two quantities by division. The ratio of 3 to 4 can be written as 3:4, 3/4, or 3 to 4. All three representations are equivalent. The ratio 3:4 does not specify the actual quantities — it only states their relationship. A class with 3:4 boys to girls could have 3 boys and 4 girls, or 6 boys and 8 girls, or 15 boys and 20 girls. All satisfy the ratio 3:4.
A proportion is an equation stating that two ratios are equal: a/b = c/d. If we know three of the four values, we can solve for the fourth. This is the core operation in almost all scaling problems.
Ratios can compare more than two quantities. A concrete mix ratio of 1:2:3 (cement:sand:gravel) means for every 1 part cement, use 2 parts sand and 3 parts gravel, regardless of the actual amounts.
Solving a proportion by cross-multiplication
Worked examples
Set up the proportion: 5/2,50 = 8/x. Cross-multiply: 5x = 8 x 2,50 = 20. Solve: x = 20/5 = 4,00. Alternatively, use the unit rate: cost per apple = 2,50/5 = 0,50. Cost of 8 apples = 8 x 0,50 = 4,00. Both methods give the same answer.
Scale factor = 10/4 = 2,5. Multiply every ingredient by 2,5. Flour: 300 x 2,5 = 750g. Eggs: 2 x 2,5 = 5 eggs. Milk: 150 x 2,5 = 375ml. The ratio between all ingredients remains the same — only the total quantities change.
Scale 1:50.000 means 1 cm on the map = 50.000 cm in reality = 500 m = 0,5 km. Real distance = 6,4 cm x 50.000 = 320.000 cm = 3.200 m = 3,2 km. Map scales are ratios. The proportion is: 1/50.000 = 6,4/x, so x = 6,4 x 50.000 = 320.000 cm.
Find the Greatest Common Divisor (GCD) of 36 and 48. GCD = 12. Divide both terms by 12: 36/12 = 3, 48/12 = 4. Simplified ratio: 3:4. A ratio in simplest form has no common factors other than 1. This is equivalent to reducing a fraction to lowest terms.
Calculate and simplify ratios
Enter any two quantities to see their ratio in simplest form, or solve a proportion with one unknown value.
Scaling quantities accurately
Scaling means multiplying all quantities in a set by the same factor (the scale factor) while preserving the ratios between them. The scale factor is found by dividing the new value by the original value for any one of the quantities.
Scale factor = New quantity / Original quantity
All other quantities are then multiplied by this same scale factor. This ensures the ratios between all quantities are preserved exactly.
In engineering and architecture, scale models use a fixed scale ratio such as 1:100. Every dimension on the model is 100 times smaller than the real object. A wall that is 5 m tall in reality is 5 cm tall on a 1:100 scale model.
In photography and design, scaling maintains the aspect ratio of an image. If an image is 1.200 pixels wide and 800 pixels tall (ratio 3:2), scaling the width to 600 pixels requires scaling the height to 400 pixels to maintain the same 3:2 aspect ratio. Scaling one dimension without the other distorts the image.
Common ratio applications
| Application | Ratio type | Example |
|---|---|---|
| Recipe scaling | Part to part | Flour:sugar:butter = 3:2:1 |
| Map reading | Scale ratio | 1:25.000 (1 cm = 250 m) |
| Architectural drawing | Scale ratio | 1:100 (1 mm = 10 cm) |
| Financial analysis | Part to whole | Debt-to-equity = 0,4 (40% debt) |
| Photography | Aspect ratio | 16:9 widescreen format |
| Concrete mix | Part to part | Cement:sand:gravel = 1:2:3 |
| Currency exchange | Exchange rate | 1 EUR = 1,08 USD |
Common mistakes
Methodology
Ratio simplification uses the Greatest Common Divisor (GCD) to reduce both terms. Proportion solving uses cross-multiplication: a/b = c/d implies a x d = b x c. Scaling uses a constant scale factor applied to all quantities uniformly.
These methods apply to direct proportions where quantities increase together. Inverse proportions (where one quantity increases as another decreases) require the formula a x b = c x d instead of a/b = c/d.
Calculate and simplify any ratio
Enter two quantities to get the simplified ratio, or solve a proportion with one unknown value.
Frequently asked questions
Formula based on standard mathematical and financial methods. Results are for informational purposes. Last reviewed May 2026. Version 1.