# Ratio Formula — How to Find, Simplify & Divide TL;DR The ratio formula compares two quantities of the same kind by division: a ratio a:b equals the fraction \( \frac{a}{b} \) (with \( b \neq 0 \)). This article shows how to write and simplify a ratio, find equivalent ratios, and divide a total amount in a given ratio, with six worked examples and the mistakes that quietly flip answers. ## What Is the Ratio Formula? A ratio measures how many times one quantity contains another, or how a whole is split between parts. Because \( a:b = \frac{a}{b} \), every ratio is also a fraction, and most ratio work is fraction work in disguise. \[ \boxed{a : b = \frac{a}{b}\quad(b \neq 0)} \] | Symbol | Meaning | | --- | --- | | aaa | The antecedent — the first term of the ratio | | bbb | The consequent — the second term \( (b \neq 0) \) | | a:b | "aaa is to bbb", the ratio in colon form | | \( \frac{a}{b} \) | The same ratio written as a fraction | Three things follow directly from the formula, and they cover most questions readers ask: - **Simplifying a ratio.** Divide both terms by their greatest common factor (GCF). \[ 12:18 = \frac{12}{18} = \frac{2}{3} = 2:3 \] - **Equivalent ratios.** Multiply or divide both terms by the same non-zero number. \( 2:3 = 4:6 = 6:9 \) — all the same comparison, like equivalent fractions. - **Dividing a quantity in a ratio.** To split a total in the ratio \( a:b \), add the parts \( (a+b) \), divide the total by that, then give each share its number of parts. Both quantities in a ratio must share the same unit before you compare them — you cannot put 222 metres against 333 centimetres without converting first. Ratio sits right next to [proportion](/content/math/terms/proportion/index.html), which is a statement that two ratios are equal. ## How Do You Divide an Amount in a Given Ratio? This is the question that shows up most in exams: **"Divide $200$ between two people in the ratio $3 : 5$."** The method is three clean steps. 1. **Add the parts.** \( 3 + 5 = 8 \) total parts. 2. **Find one part.** \( 200 \div 8 = 25 \) per part. 3. **Multiply out each share.** \( 3 \times 25 = 75 \) and \( 5 \times 25 = 125. \) The check is that the shares add back to the total: \( 75 + 125 = 200 \). That add-back is the safety net for the whole topic — if the shares do not return the original amount, something went wrong. ## Examples of the Ratio Formula ### Example 1 **A class has 12 boys and 18 girls. Write the ratio of boys to girls in simplest form.** \[ 12:18 = \frac{12}{18} = \frac{2}{3} = 2:3. \] **Final answer:** 2:3. ### Example 2 **Divide 200 in the ratio 3 : 5.** _Wrong attempt._ A student divides 200 by 3 and by 5 to get 66.7 and 40, then stops. Check it: 66.7+40=106.7, nowhere near 200. Dividing by each term separately ignores that the parts must share the whole. _Correct._ Add the parts first: \( 3 + 5 = 8 \). One part is \( 200 \div 8 = 25 \). The shares are \( 3 \times 25 = 75 \) and \( 5 \times 25 = 125 \). **Final answer:** 75 and 125, which add to 200. ✓ ### Example 3 **Is 4:6 equivalent to 6:9?** Simplify both: \( 4:6 = \frac{2}{3} \) and \( 6:9 = \frac{2}{3} \). Equal fractions mean equal ratios. **Final answer:** Yes, both equal 2:3. ### Example 4 **The ratio of two numbers is 5:7 and their sum is 96. Find the numbers.** Total parts = \( 5 + 7 = 12 \), so one part = \( 96 \div 12 = 8 \). The numbers are \( 5 \times 8 = 40 \) and \( 7 \times 8 = 56 \). **Final answer:** 40 and 56. ### Example 5 **A 3 : 2 ratio of flour to sugar uses 600 g of flour. How much sugar?** Set the ratios equal (a proportion): \( \frac{3}{2} = \frac{600}{x} \). Cross-multiply: \( 3x = 1200 \), so \( x = 400 \). **Final answer:** 400 g of sugar. ### Example 6 **Split a $1,500 profit between three partners in the ratio $2 : 3 : 5$.** Total parts = \( 2 + 3 + 5 = 10 \); one part = \( 1500 \div 10 = 150 \). Shares: \( 2 \times 150 = 300 \), \( 3 \times 150 = 450 \), \( 5 \times 150 = 750 \). **Final answer:** 300, 450, 750 — adding to $1,500. ✓ ## Why Ratios Matter — From Kitchens to Bank Balance Sheets Ratios were formalised because comparison by division is how scaling works, and scaling is everywhere. - **Maps and scale models.** A map scale of 1:50,000 means one unit on paper is 50,000 in the world — pure ratio, no addition. - **Cooking and chemistry.** A recipe or a chemical reaction holds because the proportion between ingredients is fixed; double both terms and it still works, add to one and it breaks. - **Finance.** Companies are read through ratios — the debt-to-equity ratio, the current ratio, the price-to-earnings ratio — each a single number standing in for a whole comparison. - **The golden ratio.** The proportion 1:1.618 recurs in art, architecture, and the spiral of a sunflower seed head — the most famous ratio in mathematics. The bigger idea waiting downstream is **rates** — speed (distance to time), density (mass to volume), exchange rates (one currency to another). A rate is just a ratio between quantities of different units, which is why ratio is the foundation under so much of later science. ## Where Ratios Go Sideways ### Mistake 1: Treating a ratio as a total **Where it slips in:** "Divide in a ratio" problems. **Don't do this:** Divide the total by each term of the ratio separately and call those the shares. **The correct way:** Add the parts to find the total number of parts, find the value of one part, then multiply. ### Mistake 2: Comparing quantities in different units **Where it slips in:** Ratios of two lengths, weights, or amounts given in mixed units. **Don't do this:** Write the ratio of 222 m to 50 cm as 2:50. **The correct way:** Convert to a common unit first. ### Mistake 3: Reversing the order of the ratio **Where it slips in:** "Ratio of A to B" versus "ratio of B to A." **Don't do this:** Write the ratio of girls to boys when the question asked for boys to girls — order matters in a ratio. ## Conclusion - The **ratio formula** writes a comparison of two like quantities as \( a:b = \frac{a}{b} \), with \( b \neq 0 \). - Simplify a ratio by dividing both terms by their GCF; build equivalent ratios by multiplying both terms by the same number. - To divide an amount in a ratio, add the parts, find one part, then multiply each share — and check the shares add back to the total. - The most common mistake is treating the ratio as a total and dividing by each term separately. - Ratio is the foundation under rates, proportion, scale, and the financial ratios used to read a balance sheet. ## Try It Yourself — Three Problems 1. Simplify the ratio 24:36. 2. Divide $450 between two people in the ratio $4 : 5. 3. The ratio of two ages is 3:7 and their sum is 50. Find both ages.