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.

math percentages fractions ratios

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. This standard base of 100 makes percentages useful for comparing quantities of different sizes, because everything is normalized to the same scale.

To convert a decimal to a percentage, multiply by 100: 0.75 becomes 75%, 0.03 becomes 3%, and 1.5 becomes 150%. To go the other direction, divide by 100. 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%.

Percentages above 100% are valid and common. A stock that doubles in price has a 100% gain. A city whose population grows from 50,000 to 130,000 has experienced a 160% increase. The math is the same; the result just happens to exceed the original whole.

The three basic percentage problems

Almost every percentage question falls into one of three types, and the post on how to calculate percentages covers each one with multiple worked examples.

The first type is finding X% of Y. What is 15% of 200? Convert the percentage to a decimal and multiply: 0.15 x 200 = 30. This is the calculation behind tips, discounts, tax, and commission.

The second type is finding what percent X is of Y. What percent of 80 is 12? Divide the part by the whole and multiply by 100: 12 / 80 = 0.15 = 15%. This is how test scores, completion rates, and market shares are calculated. The percentage calculator handles all three types.

The third type is percentage change. A price went from $50 to $65. The change is:

\[\text{Percent change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100 = \frac{65 - 50}{50} \times 100 = 30\%\]

Always divide by the original (old) value, not the new one. This matters because the same dollar difference produces different percentages depending on which direction you are measuring. A rise from $50 to $65 is a 30% increase, but a drop from $65 to $50 is only a 23.1% decrease. The base number changes, so the percentage changes too.

Try it yourself

Calculate percentages: find X% of Y, what percent X is of Y, percentage increase/decrease, and more.

Open Percentage Calculator

Common percentage mistakes

One of the most frequent errors is confusing percentage points with percentages. If an interest rate rises from 4% to 5%, that is a 1 percentage point increase but a 25% relative increase (because 1/4 = 0.25). News headlines often mix these two measures, which can make a change sound larger or smaller than it really is.

Another common mistake is stacking percentages incorrectly. A 20% discount followed by another 10% discount is not a 30% discount. The first discount reduces $100 to $80, and the second reduces $80 to $72. The combined discount is 28%, not 30%. Each successive percentage applies to a smaller base. This same principle applies to investment losses and gains: a 50% loss followed by a 50% gain does not return you to breakeven. You end up at 75% of the original value.

Reversing percentage increases also trips people up. If a price increased by 25%, you cannot subtract 25% to get back to the original. A $100 item marked up 25% costs $125. Taking 25% off $125 gives $93.75, not $100. To reverse a 25% increase, you divide by 1.25 instead.

What a fraction is

A fraction represents a part of a whole, with a numerator (top number) and a denominator (bottom number). In 3/4, the numerator is 3 and the denominator is 4, meaning 3 parts out of 4 equal parts. Fractions are the underlying math behind percentages; 75% is just another way of writing 3/4.

To simplify a fraction, divide both numerator and denominator by their greatest common divisor (GCD). For 12/18, the GCD is 6, so 12/18 simplifies to 2/3. To convert a fraction to a decimal, divide the numerator by the denominator: 5/8 = 0.625. To convert a decimal to a fraction, write it over the appropriate power of 10 and simplify: 0.45 = 45/100 = 9/20.

Adding fractions requires a common denominator because you can only add pieces of the same size. To add 1/3 + 1/4, the common denominator is 12, giving 4/12 + 3/12 = 7/12. Multiplying fractions goes straight across:

\[\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\]

Dividing fractions means multiplying by the reciprocal: 2/3 divided by 4/5 = 2/3 x 5/4 = 10/12 = 5/6. The post on how to add, subtract, and multiply fractions explains the reasoning behind each rule. The fraction calculator handles all four operations.

Try it yourself

Add, subtract, multiply, and divide fractions. Simplify fractions and convert between fractions and decimals.

Open Fraction Calculator

Improper fractions and mixed numbers

An improper fraction has a numerator larger than its denominator, like 7/4. This means the value is greater than 1. A mixed number expresses the same value as a whole number plus a proper fraction: 7/4 = 1 3/4. To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number part and the remainder is the new numerator. For 22/5, 22 divided by 5 = 4 remainder 2, so the mixed number is 4 2/5.

Going the other direction, multiply the whole number by the denominator and add the numerator: 3 1/8 = (3 x 8 + 1) / 8 = 25/8. When performing arithmetic with mixed numbers, it is usually easier to convert to improper fractions first, do the math, and then convert back.

What 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 (divide both by the GCD of 6).

The difference between a ratio and a fraction is what they compare. The fraction 12/30 means “12 out of 30 total students,” comparing a part to the whole. The ratio 12:18 means “for every 12 boys, there are 18 girls,” comparing parts to other parts. This distinction matters when converting between the two. To go from a ratio like 2:3 to a fraction, add the parts: 2 + 3 = 5. The first quantity is 2/5 of the total, and the second is 3/5. The ratio calculator handles simplification and scaling.

Ratios with more than two parts work the same way. A concrete mix of 1:2:3 (cement:sand:gravel) means 6 total parts, with cement as 1/6, sand as 2/6 (1/3), and gravel as 3/6 (1/2).

Scaling ratios up and down

A ratio stays equivalent when you multiply or divide all parts by the same number. The ratio 3:5 is the same as 6:10 or 15:25. This is how recipes scale. If a pancake batter calls for flour and milk in a 3:2 ratio, making a single batch might use 3 cups flour and 2 cups milk. A double batch uses 6 cups flour and 4 cups milk. The proportions stay the same.

To find an unknown in a proportion, cross-multiply. If the ratio of flour to sugar is 4:1 and you have 6 cups of flour, set up 4/1 = 6/x. Cross-multiplying gives 4x = 6, so x = 1.5 cups of sugar. The proportion calculator solves these problems directly.

Converting between all three

Any value can be expressed as a fraction, decimal, percentage, or 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

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 and the sum of all parts becomes the denominator.

Real-world applications

When shopping, percentages determine sale prices. A store offers 30% off a $120 jacket. The discount is 0.30 x 120 = $36, making the sale price $84. See the discount calculator for stacked discounts.

In cooking, ratios control proportions. A recipe calls for broth and rice in a 2:1 ratio. For 4 servings, that is 2 cups broth and 1 cup rice. For 8 servings, double both: 4 cups broth, 2 cups rice.

Test scores are a percentage calculation. You got 42 out of 50 questions right. Your score: 42/50 = 0.84 = 84%.

Price comparisons use unit rates, which are ratios expressed per single unit. Store A sells 12 oz for $3.60 (30 cents per oz). Store B sells 16 oz for $4.48 (28 cents per oz). Store B is cheaper per ounce. The price per unit calculator automates this comparison.

Finance relies on all three concepts together. An interest rate of 6.5% means you pay 6.5 cents per dollar per year. A debt-to-income ratio of 1:3 means for every $1 of debt payment, you earn $3. A savings rate of 20% means you save 1/5 of your income. The underlying math is the same in each case; only the context differs.