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Percentages, Fractions, and Ratios Explained
How percentages, fractions, and ratios relate to each other, with conversion methods and worked examples for each. A reference guide for everyday math.
What a percentage is
A percentage is a number expressed as parts per hundred. The word comes from the Latin “per centum,” meaning “by the hundred.” When you say 40%, you mean 40 out of 100.
To convert a decimal to a percentage, multiply by 100. To convert a percentage to a decimal, divide by 100.
- 0.75 = 75%
- 0.03 = 3%
- 1.5 = 150%
To convert a fraction to a percentage, divide the numerator by the denominator, then multiply by 100. For example, 3/8 = 0.375 = 37.5%.
The three basic percentage problems
Almost every percentage question falls into one of three types.
Type 1: Find X% of Y. What is 15% of 200? 0.15 x 200 = 30.
Type 2: What percent is X of Y? What percent of 80 is 12? 12 / 80 = 0.15 = 15%.
Type 3: Percentage change. A price went from $50 to $65. What is the percentage change? (65 - 50) / 50 = 15 / 50 = 0.30 = 30% increase.
The percentage change formula is:
Percentage change = (New value - Old value) / Old value x 100
A positive result means an increase. A negative result means a decrease. Always divide by the original (old) value, not the new one.
Try it yourself
Calculate percentages: find X% of Y, what percent X is of Y, percentage increase/decrease, and more.
Open Percentage CalculatorWhat a fraction is
A fraction represents a part of a whole. It has a numerator (top number) and a denominator (bottom number). In the fraction 3/4, the numerator is 3 and the denominator is 4. It means 3 parts out of 4 equal parts.
Simplifying fractions. Divide both numerator and denominator by their greatest common divisor (GCD). For 12/18, the GCD is 6. So 12/18 = 2/3.
Converting fractions to decimals. Divide the numerator by the denominator. 5/8 = 0.625.
Converting decimals to fractions. Write the decimal over the appropriate power of 10 and simplify. 0.45 = 45/100 = 9/20.
Adding fractions. Find a common denominator first. To add 1/3 + 1/4: the common denominator is 12. So 4/12 + 3/12 = 7/12.
Multiplying fractions. Multiply numerators and denominators straight across. 2/3 x 4/5 = 8/15.
Dividing fractions. Multiply by the reciprocal. 2/3 divided by 4/5 = 2/3 x 5/4 = 10/12 = 5/6.
Try it yourself
Add, subtract, multiply, and divide fractions. Simplify fractions and convert between fractions and decimals.
Open Fraction CalculatorWhat a ratio is
A ratio compares two or more quantities. If a class has 12 boys and 18 girls, the ratio of boys to girls is 12:18, which simplifies to 2:3.
A ratio is similar to a fraction but differs in meaning. The fraction 12/30 means “12 out of 30 total students.” The ratio 12:18 means “for every 12 boys, there are 18 girls.” Ratios compare parts to parts. Fractions compare a part to the whole.
Simplifying ratios. Divide all parts by their GCD. For 12:18, the GCD is 6. So 12:18 = 2:3.
Scaling ratios. A recipe calls for flour and sugar in a 3:1 ratio. If you use 9 cups of flour, you need 3 cups of sugar (multiply both parts by the same factor).
Ratios with more than two parts. A concrete mix uses cement, sand, and gravel in a 1:2:3 ratio. For every 1 part cement, use 2 parts sand and 3 parts gravel. The total is 6 parts. Cement is 1/6 of the mix, sand is 2/6 (1/3), and gravel is 3/6 (1/2).
Converting between all three
Any value can be expressed as a fraction, a decimal, a percentage, or a ratio. Here is a reference table for common conversions:
| Fraction | Decimal | Percentage | Ratio (part:whole) |
|---|---|---|---|
| 1/2 | 0.5 | 50% | 1:2 |
| 1/3 | 0.333… | 33.3% | 1:3 |
| 2/3 | 0.667… | 66.7% | 2:3 |
| 1/4 | 0.25 | 25% | 1:4 |
| 3/4 | 0.75 | 75% | 3:4 |
| 1/5 | 0.2 | 20% | 1:5 |
| 1/8 | 0.125 | 12.5% | 1:8 |
| 3/8 | 0.375 | 37.5% | 3:8 |
| 1/10 | 0.1 | 10% | 1:10 |
| 1/100 | 0.01 | 1% | 1:100 |
To go from fraction to percentage: divide, then multiply by 100. To go from percentage to fraction: put the percentage over 100 and simplify. To go from ratio to fraction: the first number becomes the numerator; the sum of all parts becomes the denominator. A 2:3 ratio means 2/(2+3) = 2/5 for the first quantity.
Real-world applications
Shopping. A store offers 30% off a $120 jacket. The discount is 0.30 x 120 = $36. The sale price is $84.
Cooking. A recipe for 4 servings uses a 2:1 ratio of broth to rice (2 cups broth, 1 cup rice). For 8 servings, double both: 4 cups broth, 2 cups rice.
Tipping. 20% of a $55 restaurant bill: 0.20 x 55 = $11 tip.
Test scores. You got 42 out of 50 questions right. Your score: 42/50 = 0.84 = 84%.
Mixing. A cleaning solution calls for a 1:9 ratio of concentrate to water. For 5 liters total, use 0.5 liters concentrate and 4.5 liters water (1/10 of the total is concentrate).
Comparing prices. Store A sells 12 oz for $3.60 (30 cents/oz). Store B sells 16 oz for $4.48 (28 cents/oz). Store B is cheaper per ounce.
Key takeaways
- A percentage is parts per hundred; multiply a decimal by 100 to get the percentage
- The three percentage problems are: find X% of Y, find what percent X is of Y, and find percentage change
- Always divide by the original value when calculating percentage change
- Fractions compare a part to the whole; ratios compare parts to other parts
- Simplify fractions and ratios by dividing by the greatest common divisor
- To convert a ratio like 2:3 to a fraction, the first part is 2/(2+3) = 2/5
- The common conversions table (1/4 = 25%, 1/3 = 33.3%, etc.) is worth memorizing for quick mental math
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Percentage Calculator
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Fraction Calculator
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Ratio Calculator
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