Proportions Solver

Proportions express the equality of two ratios — they are the mathematical backbone of scaling recipes, maps, unit conversions, similar triangles, and countless real-world problems. Cross-multiplication is the standard method for solving a proportion with one unknown.

Proportion notation: a : b = c : d (read: “a is to b as c is to d”)

Equivalently written as a fraction equation: a/b = c/d

Cross-multiplication rule: If a/b = c/d, then: a × d = b × c

To solve for any unknown (say x): a/b = x/d → x = (a × d) ÷ b

Percentage as a proportion: Part/Whole = Percent/100 → Part = (Percent × Whole) ÷ 100

Inverse proportion: When one quantity increases and the other decreases proportionally: a × b = c × d (constant product, not constant ratio) Example: speed × time = constant distance.

Scale factor from a proportion: Scale Factor = New Value ÷ Original Value

In similar triangles: all corresponding sides share the same scale factor.

Practical applications:

• Recipe scaling: If 3 cups flour makes 24 cookies, how much for 40 cookies? → 3/24 = x/40 → x = (3 × 40) ÷ 24 = 5 cups
• Map reading: 1 inch = 50 miles; distance on map = 3.5 inches → real distance = 3.5 × 50 = 175 miles
• Unit conversion: 1 kg = 2.205 lb; 7.5 kg = ? → 1/2.205 = 7.5/x → x = 7.5 × 2.205 = 16.54 lbs
• Shadow/height (similar triangles): 6 ft person casts 4 ft shadow; tree casts 28 ft shadow → 6/4 = x/28 → x = (6 × 28) ÷ 4 = 42 feet tall

Worked example — mixing mortar: A mortar recipe calls for 1 part cement to 3 parts sand by volume. You have 7 liters of cement.

• Ratio: cement/sand = 1/3
• Set up: 1/3 = 7/x
• Cross-multiply: x = 7 × 3 = 21 liters of sand needed
• Total mortar volume: 7 + 21 = 28 liters

Checking a proportion: Verify that a × d = b × c: 1 × 21 = 3 × 7 → 21 = 21 ✓