Solve for unknown values in any proportion using cross-multiplication. Understand direct and inverse proportions, convert units, scale recipes, and solve rate problems with ease.
Key concepts and formulas for proportion calculations
Enter three known values and our calculator finds the missing fourth value using cross-multiplication.
Open Percentage Calculator →A proportion is a statement that two ratios are equal: a/b = c/d. This is read "a is to b as c is to d." Proportions express a consistent relationship between four quantities where the ratio between the first pair equals the ratio between the second pair.
Proportions are foundational in mathematics, science, cooking, construction, photography, finance, and medicine. Any time you need to scale a relationship while keeping the ratio constant, you're working with proportions.
Cross-multiplication is the universal method for solving proportions with one unknown. Given a/b = c/d, cross-multiplying gives a×d = b×c.
To solve for any unknown variable:
Find x: a/b = c/x → x = (b × c) ÷ a
Find a: a/b = c/d → a = (b × c) ÷ d
Find b: a/b = c/d → b = (a × d) ÷ c
Example: If 8 gallons of paint covers 400 sq ft, how much paint is needed for 650 sq ft?
8/400 = x/650 → x = (8 × 650) ÷ 400 = 5,200 ÷ 400 = 13 gallons
Direct proportion: As one quantity increases, the other increases proportionally. The ratio y/x remains constant (y = kx where k is the constant of proportionality). Examples: distance traveled at constant speed (more time = more distance), cost of items (more units = higher total cost), earnings at hourly rate.
Inverse proportion: As one quantity increases, the other decreases proportionally. The product x×y remains constant (y = k/x). Examples: speed and travel time for a fixed distance (faster = less time), workers and job completion time (more workers = fewer days), gear ratios (larger gear = slower rotation).
The key question to determine which type applies: "If I increase one quantity, does the other increase (direct) or decrease (inverse)?"
A recipe for 4 servings calls for 3 cups of broth. How much broth for 10 servings?
4/3 = 10/x → x = (3 × 10) ÷ 4 = 7.5 cups of broth
A medication dosage is 5mg per 10kg of body weight. For a 68kg patient:
10/5 = 68/x → x = (5 × 68) ÷ 10 = 34mg
A blueprint uses a scale of 1:25. A wall measures 8 cm on the blueprint. Actual length?
1/25 = 8/x → x = 25 × 8 = 200 cm = 2 meters
3 workers complete a project in 12 days. How long for 5 workers? (Inverse proportion)
3 × 12 = 5 × x → x = 36 ÷ 5 = 7.2 days
Unit conversion is one of the most practical applications of proportions. The method: set up a proportion using a known conversion factor, then solve for the unknown.
Miles to kilometers: 1 mile = 1.609 km. Convert 26.2 miles (marathon):
1/1.609 = 26.2/x → x = 26.2 × 1.609 = 42.16 km
Pounds to kilograms: 1 lb = 0.4536 kg. Convert 185 lbs:
1/0.4536 = 185/x → x = 185 × 0.4536 = 83.9 kg
Gallons to liters: 1 gallon = 3.785 L. Convert 20 gallons:
1/3.785 = 20/x → x = 20 × 3.785 = 75.7 liters
Use cross-multiplication. For a/b = c/x: x = (b×c)÷a. Example: 3/5 = 12/x → x = (5×12)÷3 = 20. Always verify by checking both ratios equal the same decimal.
Direct: both increase together (y=kx). More hours = more pay. Inverse: one increases as other decreases (y=k/x). More workers = fewer days to finish. Key question: does increasing one raise or lower the other?
Set up using a known conversion. To convert 45 miles to km (1 mile = 1.609 km): 1/1.609 = 45/x → x = 45 × 1.609 = 72.4 km. Works for any unit conversion.
Direct rate: 3 workers lay 120 bricks/day, 7 workers lay 7×40=280 bricks/day. Inverse rate: 3 workers finish in 8 days, 5 workers: 3×8=5×x → x=4.8 days. Identify direct vs inverse first.