Ratio and Proportion Guide 2026: Simplify, Scale and Solve
Table of Contents
Simplifying Ratios
Example: 12:18 - GCD is 6: 2:3. Recipe 4 cups flour : 2 cups sugar : 1 cup butter simplifies to 4:2:1 (already simplest).
Ratios compare like units. "3 boys to 2 girls" = 3:2. As a fraction of the whole: boys = 3/5 of the class.
Decimal ratios simplify by clearing the decimals first. For 1.5 : 2.5, multiply both parts by 2 to get 3:5. For 0.75 : 1.25, multiply by 4: 3:5. Use the smallest power of 10 that clears every decimal, then divide by the GCD as usual. Exchange rates and gear ratios often arrive as decimals, so this two-step cleanup comes up constantly.
Part-to-part and part-to-whole are different questions with different answers. A class with 12 boys and 18 girls has ratio 2:3, but boys are 12/30 = 2/5 of the class - the whole is 5 parts, not 3. Mixing these up is the standard error: a 2:3 ratio does not mean 2/3 of anything. Always ask whether the problem wants a comparison of parts or a fraction of the total.
Scaling Recipes and Batches
Concrete mix 1:2:3 (cement:sand:gravel). Need 120 lb total: parts = 6, each part 20 lb.
- Cement: 1 x 20 = 20 lb
- Sand: 2 x 20 = 40 lb
- Gravel: 3 x 20 = 60 lb
Cocktail scaling: 2:1:1 (rum:lime:syrup), want 2 liters: parts = 4, each 500 ml: 1L rum, 500ml lime, 500ml syrup.
Ratios scale down just as cleanly as they scale up. A 1:2:3 concrete mix needed at only 45 lb total: 6 parts means 7.5 lb each, giving 7.5 lb cement, 15 lb sand, 22.5 lb gravel. Fractional parts are perfectly fine - the ratio constrains proportions, not roundness. Small test batches use exactly this math before committing to full-scale mixes.
Baker's percentages are ratios with flour fixed at 100%. A dough with 65% hydration and 500 g flour needs 0.65 x 500 = 325 g water. Scaling to 750 g flour: water becomes 0.65 x 750 = 487.5 g. Because every ingredient is a percentage of flour weight, scaling a recipe is one multiplication per ingredient - the professional baker's answer to ratio cooking.
Solving Proportions
Example: 3/4 = x/20. Cross: 3 x 20 = 4x, 60 = 4x, x = 15.
Map scale: 1 inch = 50 miles. 3.5 inches on map: 1/50 = 3.5/x, x = 175 miles.
Inverse proportion flips the relationship: when one quantity rises, the other falls so their product stays constant. Four workers take 6 days (24 worker-days); six workers take 24/6 = 4 days. Setting up a/b = c/d here gives the wrong answer - inverse problems need a x b = c x d. The tell is the wording: "more workers, fewer days" signals inverse; "more apples, more cost" signals direct.
Unit-price questions are proportions wearing a shopping disguise. If 3 lb of apples cost $5.40, what do 7 lb cost? Set 3/5.40 = 7/x and cross-multiply: 3x = 37.8, so x = $12.60. The proportion framework also catches bad deals: a "5 lb bag for $8" versus "$5.40 for 3 lb" compares as 5/8 vs 3/5.40 - cross-multiplying 27 > 25.6... precisely, 5 x 5.40 = 27.0 versus 8 x 3 = 24, so the bag is cheaper per pound.
Ratios in Finance and Business
- Debt-to-income: $2,000 debts / $6,000 income = 1:3 (33% DTI).
- Price ratios: Comparing 24 oz at $4.80 vs 16 oz at $3.50: unit prices $0.20/oz vs $0.219/oz - bigger wins.
- Aspect ratios: 16:9 screen, 1920x1080: 1920/1080 = 16/9. Correct.
- Gear ratios: 3:1 means the small gear spins 3x per large-gear turn.
The current ratio is a lender's favorite: current assets divided by current liabilities. A business with $180K in current assets and $90K in current liabilities sits at 2:1 - comfortably able to cover short-term bills. Below 1:1, obligations exceed liquid resources and borrowing gets expensive. Banks check this ratio before quoting rates, so it directly prices your loans.
Partnership profit splits are ratios with dollar signs. Splitting $50,000 in profit 3:2:1 means 6 total parts at $8,333 each: $25,000, $16,667, and $8,333. Partnership agreements are written as ratios precisely because the dollar amounts change yearly while the split stays fixed. The math is just division with extra steps - total the parts, divide the profit, multiply.
Frequently Asked Questions
A comparison of two or more quantities: 3:2 means 3 parts of one to 2 parts of another.
Divide all parts by their greatest common divisor. 12:18 becomes 2:3.
An equation stating two ratios are equal: a/b = c/d. Solve by cross-multiplication.
Find the total parts, divide your target total by parts to get the per-part amount, then multiply each ratio part.
A ratio 3:2 compares parts to parts; as a fraction of the whole, it's 3/5 and 2/5.