Enter two or three values to simplify the ratio to its lowest terms. Optionally scale the ratio to a desired total amount for distribution calculations.
A ratio expresses the relative size of two or more quantities. Written as a:b (or a/b), it tells you the relationship between parts without specifying absolute amounts. A 3:2 ratio could mean 3 cups to 2 cups, 3 ounces to 2 ounces, or 30 hours to 20 hours — the proportional relationship is the same.
Ratios are ubiquitous: aspect ratios for screens (16:9), gear ratios in vehicles, mixing ratios for paints and chemicals, financial ratios, betting odds, recipe ingredient proportions, and demographic comparisons. Wherever you compare quantities, ratios appear.
Simplifying ratios uses the same idea as simplifying fractions: divide all parts by their GCF (greatest common factor). For 12:8, GCF = 4, so 12:8 simplifies to 3:2. For 15:25:35, GCF = 5, simplifies to 3:5:7.
Scaling ratios to a desired total is a key practical operation. If you want to divide $100 in ratio 3:2: - Total parts: 3 + 2 = 5. - First share: 3/5 × 100 = $60. - Second share: 2/5 × 100 = $40.
Sum: $100 ✓. Used for splitting bills, distributing inheritances, allocating budgets, blending ingredients.
Ratios connect to other concepts: - **Fractions**: 3:2 = 3/2. - **Percentages**: 3 of 5 = 60%. - **Proportions**: ratios that are equal (a:b = c:d).
Common applications: mixing ratios (paint, concrete, fuel), financial analysis (P/E ratio, debt-to-equity), recipe ingredients, aspect ratios (screens, prints), gear ratios, demographics, betting odds, and any proportional distribution problem.
**Scenario:** Pancake recipe ratio: 2:1:0.5 (flour:milk:eggs). Total for 12 pancakes. Need for 18 pancakes. **Calculation:** Scale factor: 18/12 = 1.5. New ratio: 3:1.5:0.75. Or as proportion: increase each by 50%. **Result:** 50% more of each ingredient. Recipe scales linearly when ratios are preserved.
**Scenario:** Need 0.5 m³ concrete in ratio cement:sand:gravel = 1:2:4. How much of each? **Calculation:** Total parts: 7. Cement: 1/7 × 0.5 ≈ 0.071 m³ (~ 0.18 tonnes). Sand: 2/7 × 0.5 ≈ 0.143 m³ (~ 0.21 tonnes). Gravel: 4/7 × 0.5 ≈ 0.286 m³ (~ 0.46 tonnes). **Result:** ~180 kg cement, ~210 kg sand, ~460 kg gravel. Plus water (~0.25 × cement = ~45 kg). Standard structural concrete ratio.
**Scenario:** $450,000 inheritance divided 3:2:1 among three children. Each share? **Calculation:** Total parts: 6. Eldest: 3/6 × 450K = $225K. Middle: 2/6 × 450K = $150K. Youngest: 1/6 × 450K = $75K. **Result:** $225K, $150K, $75K. Sum: $450K ✓. Verify ratios: 225:150:75 = 3:2:1 ✓.
**Use ratios for:**
- **Recipe scaling**: ingredient proportions. - **Mixing**: concrete, paint, chemicals. - **Distribution**: dividing amounts proportionally. - **Comparison**: financial, demographic, etc. - **Engineering**: gear ratios, mechanical advantage. - **Photography/Media**: aspect ratios. - **Statistics**: standardized comparisons. - **Music**: harmonic intervals.
**Most common operations:**
1. **Simplify**: divide all parts by GCF. 2. **Scale**: distribute given total. 3. **Compare**: equivalent ratios for verification. 4. **Convert**: to percentages or fractions.
**Simplifying steps:**
1. Find GCF of all parts. 2. Divide each part by GCF. 3. Result is simplest form.
For 24:18: GCF = 6. Simplified: 4:3.
**Scaling steps:**
1. Add all parts (call sum P). 2. For each part, compute: (part / P) × total. 3. Verify sum equals total.
For 5:3:2 with total 100: Total parts: 10. Shares: 50, 30, 20. Sum 100 ✓.
**Common applications:**
- **Cooking**: scaling recipes, ingredient ratios. - **Construction**: mortar, concrete, plaster mixing. - **Engineering**: gear ratios in transmissions. - **Media**: photo/video aspect ratios. - **Photography**: f-stops as ratios (focal length / aperture diameter). - **Music**: intervals (perfect fifth = 3:2 frequency ratio). - **Finance**: stock ratios, debt ratios. - **Demographics**: population ratios. - **Sports**: betting odds.
**Music intervals (Just Intonation):**
- Unison: 1:1 - Octave: 2:1 - Perfect fifth: 3:2 - Perfect fourth: 4:3 - Major third: 5:4 - Minor third: 6:5
Small integer ratios sound "consonant" (pleasing). Why music theory uses ratios.
**Paper sizes:**
- **A series** (ISO): each size has 1:√2 ratio. Cut in half = next size. - **US Letter**: 8.5:11 (≈ 1:1.29). - **Legal**: 8.5:14 (≈ 1:1.65).
A-series brilliance: any A-paper folded in half is same shape.
**Aspect ratios history:**
- 4:3: classic TV, computer monitors before 2000s. - 16:9: HDTV, modern computer displays. - 21:9: ultrawide displays. - 2.39:1: most cinema films. - 1:1 (square): Instagram, Polaroid. - 9:16 (vertical): phone videos, TikTok.
**Software:**
- **Excel**: simple division for simplification. - **Wolfram Alpha**: ratio operations. - **Programming**: simple loops and GCF. - **Online calculators**: many free.
**Pitfalls:**
- **Order matters**: 3:2 ≠ 2:3 in most uses. - **Units must match**: 3 cm : 2 m needs conversion. - **For distribution**: total parts ÷ sum of parts × total. - **Confusing ratio and percentage**: 3:2 = 60%:40%. - **For three+ parts**: scale all proportionally. - **Zero in ratio**: usually doesn't make sense.
**Educational notes:**
Ratios introduced in elementary school: - 4th-5th grade: basic ratio concepts. - 6th-7th grade: scaling, proportions. - 8th-9th grade: similar figures. - Higher math: continued ratios, ratio tests.
Foundational concept; appears throughout life.
**Practical mixing example:**
For 1 gallon of green paint, ratio yellow:blue = 2:3: Total parts: 5. Yellow: 2/5 = 0.4 gal. Blue: 3/5 = 0.6 gal.
For 5 gallon batch: yellow 2 gal, blue 3 gal.
**Recipe ratios:**
Common ratios: - **Bread**: flour:water:yeast:salt around 100:60:1:2 (baker's percentages). - **Coffee**: 1:15 to 1:18 (coffee:water). - **Pancakes**: 2:1.5:1 (flour:milk:eggs) varies.
**Software:**
- Excel: division by GCD for simplification. - Programming: simple GCD-based simplification. - Online: many ratio simplification calculators. - CAD: aspect ratio operations.
**Pitfalls:**
- **Confusing with fractions**: 3:2 ratio = 3/5 of total (when interpreted as parts of whole). - **Mixing units**: ensure consistent units. - **Negative values**: rarely sensible in ratios. - **Decimals in ratios**: usually scale to integer. - **Order convention**: a:b means "a to b", important for direction.
Solve a/b = c/d for any missing value in a proportion.
Calculate percentages: find X% of Y, what percent X is of Y, or X is Y% of what.
Calculate golden ratio relationships using phi (1.618...).
Calculate factorial (n!), double factorial (n!!), and subfactorial (!n).
Calculate triangle area, perimeter, and angles from three side lengths using Heron's formula and the law of cosines.
Calculate the area and perimeter of a trapezoid from its bases, legs, and height.
Distribute this total according to the ratio
Simplified Ratio
3:2
Original Ratio
12:8
A:B as Decimal
1.500000
Scaled Distribution
Enter a scale total to distribute