Enter three of the four values in a proportion (a/b = c/d) and this calculator will solve for the missing one. Leave the unknown value as 0.
A proportion is a statement that two ratios are equal. Written a/b = c/d (or a:b = c:d). Proportions are powerful tools for solving problems involving scaling, conversions, similarity, and rates. Whenever you've scaled a recipe, converted units, or solved a word problem like "if 3 apples cost $2, how much do 12 apples cost?", you've worked with proportions.
The standard solving method is **cross multiplication**: if a/b = c/d, then a × d = b × c. From this, you can solve for any unknown. If you know three of four values, the fourth follows.
Example: 3/4 = x/20. Cross multiply: 3 × 20 = 4 × x. 60 = 4x. x = 15.
Verify: 3/4 = 15/20 (both equal 0.75) ✓.
Proportions underlie many practical calculations: - **Recipe scaling**: ingredients scale proportionally. - **Unit conversion**: miles to km uses a fixed ratio. - **Maps and scale models**: distance on map = distance in reality / scale. - **Currency exchange**: amounts scale with exchange rate. - **Similar triangles**: corresponding sides are proportional. - **Speed/time/distance**: any one is proportional to the others (holding the third fixed).
Common applications: recipe scaling, unit conversions, scale models, financial calculations, similar shapes (geometry), rate problems, dilution calculations (chemistry), and any problem involving "X is to Y as Z is to W".
**Scenario:** Recipe for 4 people uses 200 g flour. Scale for 7 people. **Calculation:** 200/4 = x/7. Cross multiply: 200 × 7 = 4 × x. 1400 = 4x. x = 350 g. **Result:** Use 350 g flour for 7 people. Direct proportion — more people, more flour at same rate per person.
**Scenario:** Map scale 1 inch = 50 miles. Distance on map: 3.5 inches. Actual distance? **Calculation:** 1/50 = 3.5/x. Cross multiply: x = 50 × 3.5 = 175 miles. **Result:** Actual distance: 175 miles. Useful for trip planning when GPS unavailable. Scale-based reading is fundamental skill in cartography.
**Scenario:** Two similar right triangles. Small: legs 3 and 4. Large: leg 12 corresponding to 3. Find other leg. **Calculation:** 3/4 = 12/x. Cross multiply: 3x = 48. x = 16. **Result:** Other leg = 16. Triangles are 4× scale ratio. Verify: hypotenuse small = 5, hypotenuse large = 20 (4×).
**Use proportions for:**
- **Recipe scaling**: adjusting ingredients. - **Unit conversions**: with known ratio. - **Map reading**: distance scaling. - **Scale models**: trains, dollhouses, architectural models. - **Similar geometric figures**: scaling shapes. - **Rate problems**: speed, work, flow. - **Percentage calculations**: alternative method. - **Dilution problems**: chemistry concentration. - **Financial allocation**: dividing money by shares. - **Sports statistics**: projecting from partial seasons.
**Direct vs inverse:**
- **Direct**: as one increases, other increases proportionally. y = kx. - **Inverse**: as one increases, other decreases proportionally. y = k/x.
Examples: - Direct: cost of apples vs number of apples. - Inverse: speed vs time for fixed distance. - Direct: weight vs height (roughly for adults). - Inverse: pressure vs volume for fixed amount of gas.
**Cross multiplication trick:**
For a/b = c/d: a × d = b × c.
Solve for x in any position: - x/b = c/d → x = bc/d - a/x = c/d → x = ad/c - a/b = x/d → x = ad/b - a/b = c/x → x = bc/a
**Common applications:**
- **Cooking**: scaling recipes up or down. - **Construction**: scaling architectural drawings. - **Sewing**: pattern adjustments. - **Chemistry**: dilutions, concentrations. - **Photography**: aspect ratios. - **Investments**: distributing returns. - **Sports statistics**: pace and projection. - **Pharmaceutical**: dosing by weight.
**Real-world examples:**
- **Tip**: 18% of meal cost = tip amount. - **Sale price**: 25% off original = sale price. - **Speed**: 60 mph for 2 hours = 120 miles. - **Gas mileage**: 30 mpg for 100 miles = 3.33 gallons. - **Currency**: $100 at 1.1 EUR/USD = 110 EUR.
**Educational uses:**
Proportions appear in: - 6th grade math: ratio and proportion problems. - 7th-8th grade: percentages, similar figures. - High school algebra: linear equations. - High school geometry: similar triangles. - Test prep (SAT, ACT, GRE): heavy emphasis on word problems.
**Software:**
- **Excel**: simple formula. - **Programming**: trivial arithmetic. - **Mental math**: cross multiply, then divide. - **Online calculators**: instant for any 3 values.
**Pitfalls:**
- **Unit confusion**: numerator and denominator must align. - **Direct vs inverse**: not all problems are direct proportion. - **Cross-multiplying wrong values**: keep diagonal pairs. - **Negative values**: usually proportions work with positives. - **Word problems**: parse carefully — "x for every y" vs "x out of y".
**Setting up correctly:**
For "x is to y as a is to b": x/y = a/b.
For "if A produces B in C time, how much in D time": B/C = x/D (where x is what you want for time D).
Identify what stays constant (the rate) and what scales.
**Pitfalls (continued):**
- **Implicit unit assumption**: sometimes proportions hide unit dependencies. - **Reversed direction**: ensure variables on same side of equation. - **Multiplying instead of dividing**: cross multiply, then divide both sides. - **Mixing up scale and unit conversion**: 1 inch = 1 ft is unit, not scale.
Simplify ratios and scale them to a desired total.
Calculate percentages: find X% of Y, what percent X is of Y, or X is Y% of what.
Calculate golden ratio relationships using phi (1.618...).
Calculate factorial (n!), double factorial (n!!), and subfactorial (!n).
Calculate triangle area, perimeter, and angles from three side lengths using Heron's formula and the law of cosines.
Calculate the area and perimeter of a trapezoid from its bases, legs, and height.
Set the unknown to 0
Solved for C
15
Equation
3/4 = 15.0000/20
Cross Product (a x d)
60
Cross Product (b x c)
60
Proportional?
Yes