Ratio and Proportion Explained: Formulas & Examples

A ratio compares two quantities of the same kind by division. The ratio of 2 pencils to 3 pens is written 2:3 and means for every 2 pencils there are 3 pens. A proportion is a statement that two ratios are equal, such as 2:3 = 4:6, which says the two comparisons describe the same relationship.

Ratios and proportions matter because so much of life is about fair sharing and correct scaling. Recipes scale ingredients in ratio, maps shrink real distances in ratio, and medicines are mixed in exact proportions. Once you can set up a proportion, you can solve any scaling problem with one equation.

Key Concepts

• Ratio a:b: equals the fraction a/b. Both terms must be in the same units (rupees to rupees, cm to cm).
• Simplest form: divide both terms by their HCF. 12:18 simplifies to 2:3 (HCF 6).
• Proportion: a:b = c:d, read “a is to b as c is to d”. The first and last terms (a, d) are the extremes; b and c are the means.
• Direct proportion: both quantities grow together (more hours worked, more pay). It follows y = kx.
• Inverse proportion: one grows as the other shrinks (more workers, fewer days to finish). It follows y = k/x.

Formulas

• If a:b = c:d, then ad = bc (cross-multiplication). So d = bc/a.
• Direct proportion: y1/x1 = y2/x2.
• Inverse proportion: x1y1 = x2y2.
• Dividing a quantity in the ratio a:b:c: the shares are a/(a+b+c), b/(a+b+c) and c/(a+b+c) of the total.

Worked Example

If 2:3 = x:18, find x.

1. Write the proportion as fractions: 2/3 = x/18.
2. Cross-multiply: 2 x 18 = 3 x x, so 36 = 3x.
3. Divide both sides by 3: x = 36 ÷ 3 = 12. Check: is 12:18 the same as 2:3? Dividing 12:18 by 6 gives 2:3. Correct.

Common Misconceptions

• “Ratios can compare any two things.” Wrong. A ratio needs the same kind of quantity; “3 apples : 2 rupees” is not a ratio but a rate, and it needs different handling.
• “In a proportion, you can add the same number to both sides.” Wrong. 2:3 is not equal to 4:5. Only multiplying or dividing both terms keeps a ratio equal.
• “Direct and inverse proportion are interchangeable.” Wrong. More machines finishing a job faster is inverse proportion; using the direct formula there gives a nonsense answer.

Real-World Applications

• Cooking: scaling a recipe for 4 people up to 10 keeps ingredient ratios fixed.
• Maps and models: a 1:50,000 map means 1 cm represents 50,000 cm (500 m) of real land.
• Finance: debt-to-equity ratios, profit sharing in partnerships, and mixing investments in a fixed ratio.

A ratio is a comparison by division, and a proportion says two comparisons match. Cross-multiplication turns any proportion into a simple equation, so the only real skill is deciding whether the relationship is direct or inverse. Get that right, and scaling problems solve themselves.