Ratio Calculator: Simplify, Scale, and Solve Proportions

Table of Contents - Ratio

How to Use This Calculator
The Core Principle: Proportional Relationships
How to Work with Ratios Manually
Real-World Applications
Scenarios People Actually Run Into
Trade-Offs and Decisions People Underestimate
Common Mistakes and How to Recover
Related Topics
How This Calculator Works
FAQs

How to Use This Calculator - Ratio

Select your Calculation Type:

Simplify — Reduce a ratio to lowest terms
Scale — Enlarge or reduce a ratio proportionally
Solve Proportion — Find a missing value in a proportion

For simplifying, enter both terms of your ratio (e.g., 18 and 24).

For scaling, enter the original ratio and the scaling factor.

For solving proportions, enter three known values to find the fourth (e.g., 5:7 = x:49).

Click "Calculate" to see results. The output displays:

Simplified ratio in lowest terms
Scaled values maintaining proportion
Missing value in proportions
Expression showing the calculation performed

The Core Principle: Proportional Relationships

A ratio compares two or more quantities by division. A proportion states that two ratios are equal.

Basic ratio: 3:4 means "3 parts to 4 parts" or "3 for every 4."

Equivalent ratios: 3:4 = 6:8 = 9:12 = 15:20. Multiplying or dividing both terms by the same number creates equivalent ratios.

Simplifying ratios: Find the Greatest Common Divisor (GCD) of both terms and divide by it. 18:24 → GCD is 6 → 3:4.

Proportions: If a:b = c:d, then a×d = b×c (cross-multiplication). This lets you solve for unknown values.

Scaling: To scale a ratio up or down, multiply all terms by the same factor. 2:3 scaled by 4 becomes 8:12.

Ratios are fundamental to mixing, cooking, finance, construction, and countless other applications where maintaining proportions matters.

How to Work with Ratios Manually

Simplifying a ratio: Find GCD and divide both terms.

Example: Simplify 48:64 Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Factors of 64: 1, 2, 4, 8, 16, 32, 64 GCD = 16 48 ÷ 16 = 3, 64 ÷ 16 = 4 Result: 3:4

Solving a proportion: Use cross-multiplication.

Example: 5:7 = x:49 5 × 49 = 7 × x 245 = 7x x = 35 Verify: 5:7 = 35:49 (both reduce to 5:7) ✓

Scaling a ratio: Multiply all terms by the same factor.

Example: Scale 2:3:5 to make the first term 10 Factor = 10 ÷ 2 = 5 2 × 5 = 10, 3 × 5 = 15, 5 × 5 = 25 Result: 10:15:25

Converting ratio to fractions: 3:4 means 3 parts + 4 parts = 7 total parts First quantity = 3/7 of total Second quantity = 4/7 of total

Example: Split £70 in ratio 3:4 First share = £70 × 3/7 = £30 Second share = £70 × 4/7 = £40

Real-World Applications

Cooking and recipes. A recipe for 4 servings needs scaling to 10 servings. Multiply all ingredients by 10/4 = 2.5.

Mixing concrete. Cement:sand:gravel in 1:2:4 ratio. For 6 buckets of gravel: cement = 6 × (1/4) = 1.5 buckets.

Financial ratios. Debt-to-equity ratio, price-to-earnings ratio, and current ratio assess company health.

Map scales. A 1:50,000 map means 1 cm represents 50,000 cm (500 m) in reality.

Architecture and design. Golden ratio (≈1:1.618) appears in aesthetically pleasing proportions.

Paint mixing. Mix colors in specific ratios: 3 parts blue to 1 part yellow creates a specific green.

Scenarios People Actually Run Into

The concrete mixing problem. You have 6 buckets of gravel and need to find cement and sand for 1:2:4 ratio. Gravel is 4 parts, so 1 part = 6/4 = 1.5. Cement = 1.5, Sand = 3, Gravel = 6.

The recipe scaling trap. Doubling all ingredients works for most recipes, but some ingredients (like baking powder) don't scale linearly. Know which ratios hold and which don't.

The aspect ratio distortion. Resizing an image from 4:3 to 16:9 without proper cropping or letterboxing stretches the image. Maintain aspect ratio to avoid distortion.

The decimal ratio. 0.8:2 doesn't look like a clean ratio. Multiply both by 10: 8:20. Simplify by GCD 4: 2:5.

The multi-part ratio. Dividing £100 among three people in ratio 2:3:5. Total parts = 10. Shares: £20, £30, £50.

Trade-Offs and Decisions People Underestimate

Simplest form versus context. 1:2 is simpler than 50:100, but sometimes context requires specific numbers (like 50% market share).

Part-to-part versus part-to-whole. 3:4 is part-to-part. 3:7 would be part-to-whole (if total is 7). Know which your context requires.

Exact versus rounded. Scaling by 7/3 creates fractions. In practice, you might round—but this slightly changes the actual ratio.

Unit consistency. Ratios require same units. 2 kg to 500 g isn't 2:500; convert first (2000:500 = 4:1).

Ratio as fraction. 3:4 can represent "3 divided by 4" (0.75) or "3 parts to 4 parts." Context determines interpretation.

Common Mistakes and How to Recover

Confusing ratio and fraction. A ratio 3:4 compares parts (3 to 4). A fraction 3/4 compares part to whole. Ratio 3:4 means fractions 3/7 and 4/7 of the whole.

Not converting units. 2 meters to 50 cm isn't 2:50. Convert: 200 cm to 50 cm = 200:50 = 4:1.

Wrong cross-multiplication. In a:b = c:d, cross-multiply as a×d = b×c, not a×c = b×d.

Incomplete simplification. 24:36 → 4:6 is not fully simplified. Continue: 4:6 → 2:3.

Order reversal. Ratio 3:4 is different from 4:3. Maintain consistent order based on what you're comparing.

How This Calculator Works

Greatest Common Divisor (Euclidean algorithm):

gcd(a, b):
  while b ≠ 0:
    temp = b
    b = a mod b
    a = temp
  return a

Simplify ratio:

divisor = gcd(term1, term2)
simplified1 = term1 / divisor
simplified2 = term2 / divisor

Solve proportion (a:b = c:x):

x = (b × c) / a

Scale ratio:

factor = targetValue / originalValue
newTerm1 = term1 × factor
newTerm2 = term2 × factor

Multi-term ratio scaling:

for each term:
  scaledTerm = term × (knownValue / correspondingPart)

All calculations happen locally in your browser.

FAQs

How do I simplify a ratio with decimals?

Multiply both terms by 10, 100, etc., to eliminate decimals, then find GCD. Example: 0.8:2 → 8:20 → 2:5.

Can this handle three-part ratios?

Yes. Enter ratios like 1:2:4 for concrete mix or investment allocation.

What if I only know one part of a proportion?

Enter the known ratio and one value from the second ratio. The calculator solves for the missing term using cross-multiplication.

Is a ratio the same as a fraction?

Related but different. Ratio 3:4 compares parts (3 to 4). As a fraction, it can represent 3/4 = 0.75, or the parts 3/7 and 4/7 of a whole.

How do I scale a recipe for different servings?

Find the scaling factor: new servings / original servings. Multiply all ingredient quantities by this factor.

Can I use this for aspect ratios?

Yes. Enter screen dimensions to simplify (1920:1080 → 16:9) or find equivalent dimensions.

Why does my simplified ratio have decimals?

If inputs are decimal, output may be too. Multiply to get whole numbers first, then simplify.

How do I divide an amount in a given ratio?

Sum the ratio parts, divide total amount by this sum, then multiply by each part. Example: £100 in 2:3 ratio = £100/5 = £20 per part = £40 and £60.

What's a rate versus a ratio?

A rate compares quantities with different units (miles per hour, price per pound). A ratio compares quantities with the same unit (3 apples to 4 apples). Both use the colon notation.

How do I convert a ratio to a percentage?

For ratio a:b, percentage of first = a/(a+b) × 100. Ratio 3:7 means 3/10 = 30% for first part, 70% for second.

What is direct proportion?

When one quantity increases, the other increases at the same rate. If y = kx for constant k, y and x are directly proportional.

What is inverse proportion?

When one quantity increases, the other decreases proportionally. If y = k/x for constant k, y and x are inversely proportional.

How do similar triangles relate to ratios?

Similar triangles have equal ratios of corresponding sides. This principle enables surveying, shadow calculations, and geometric scaling.

What is the golden ratio?

Approximately 1:1.618 (φ). A rectangle with these proportions is aesthetically pleasing. Found in nature, art, architecture, and the Fibonacci sequence.

How do gear ratios work?

A 2:1 gear ratio means the driven gear rotates once for every two rotations of the driving gear. This trades speed for torque or vice versa.

What are financial ratios?

Metrics comparing financial statement items: debt-to-equity, current ratio, profit margin. Used to assess company health and compare across industries.

How do I compare ratios?

Convert to the same form: decimals (3:4 = 0.75, 5:6 = 0.833) or common denominators. Or cross-multiply: a:b vs c:d, compare ad to bc.

What is unit rate?

A ratio with denominator 1. "60 miles per hour" is unit rate (60:1). Useful for direct comparison and calculations.

How do aspect ratios affect displays?

Screen dimensions in ratio form: 16:9, 4:3, 21:9. Content must match aspect ratio to avoid letterboxing, pillarboxing, or stretching.

What is a mixing ratio versus a dilution ratio?

Mixing ratio: parts combined (1:4 means 1 part A + 4 parts B = 5 parts total). Dilution: final volume relative to concentrate (1:4 dilution = 1 part concentrate in 4 parts total).

How do exchange rates relate to ratios?

Currency exchange rates are ratios: $1 = €0.92 is ratio 1:0.92 or approximately 100:92. Use cross-multiplication to convert amounts.

What is odds ratio in probability?

Odds compare success to failure: 3:1 odds means 3 successes for every 1 failure, or 75% probability. Different from probability (3/4).

How do ratios apply to music and harmony?

Musical intervals are frequency ratios. Octave = 2:1, perfect fifth = 3:2. These simple ratios create harmonious sounds.

What is dimensional analysis using ratios?

Convert units by multiplying by ratios equal to 1. To convert 60 mph to ft/s: 60 miles/hour × 5280 feet/mile × 1 hour/3600 seconds = 88 ft/s.

Additional Notes

Ratios underlie countless practical applications from cooking to engineering. Understanding how to manipulate and apply ratios enables accurate scaling, fair distribution, and proper proportions in any quantitative task. Master ratios and proportions to solve problems across many domains effectively.

Ratio Calculator — Proportion & Ratio Solver

Calculation Type