What a Fraction Is
A fraction represents part of a whole. The numerator on top counts how many parts you have; the denominator underneath says how many equal parts make up the whole. In 3/4, the whole is divided into four parts and you have three of them.
A fraction is also a division problem waiting to be done: 3/4 is exactly 3 ÷ 4 = 0.75. This equivalence is why every fraction has a decimal form and why the two are interchangeable.
Types of Fractions
| Type | Definition | Example |
|---|---|---|
| Proper | Numerator smaller than denominator | 3/4 |
| Improper | Numerator equal to or larger than denominator | 7/3 |
| Mixed number | A whole number with a proper fraction | 2 1/3 |
| Equivalent | Different numbers, same value | 1/2 = 2/4 = 50/100 |
| Unit | Numerator of 1 | 1/8 |
Improper fractions and mixed numbers describe the same quantity: 7/3 = 2 1/3. Improper form is easier to calculate with; mixed form is easier to read. Convert between them by dividing: 7 ÷ 3 = 2 remainder 1, giving 2 1/3.
Adding and Subtracting
Fractions can only be added when the parts are the same size, which means finding a common denominator first.
a/b + c/d = (a×d + c×b) ÷ (b×d)
1/2 + 3/4 becomes (1×4 + 3×2) ÷ (2×4) = 10/8, which simplifies to 5/4, or 1 1/4. Multiplying the denominators always works, though using the lowest common denominator keeps the numbers smaller: for 2 and 4 the LCD is 4, giving 2/4 + 3/4 = 5/4 directly.
Multiplying and Dividing
Multiplication is the easy one — no common denominator is needed:
a/b × c/d = (a×c) ÷ (b×d)
a/b ÷ c/d = a/b × d/c = (a×d) ÷ (b×c)
Division is multiplication by the reciprocal — flip the second fraction and multiply. 1/2 ÷ 3/4 = 1/2 × 4/3 = 4/6 = 2/3. This is why dividing by a fraction less than 1 makes the answer larger, which is counter-intuitive until you read it as "how many three-quarters fit into a half?".
Simplifying
A fraction is in lowest terms when the numerator and denominator share no common factor above 1. Divide both by their greatest common divisor:
84/126: the GCD is 42, so 84 ÷ 42 = 2 and 126 ÷ 42 = 3, giving 2/3.
The Euclidean algorithm finds the GCD quickly: repeatedly replace the larger number with the remainder of dividing it by the smaller, until the remainder is zero. For 84 and 126: 126 mod 84 = 42, then 84 mod 42 = 0, so the GCD is 42.
Common Fraction, Decimal and Percentage Equivalents
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.33% |
| 2/3 | 0.666… | 66.67% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/6 | 0.1666… | 16.67% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
| 1/10 | 0.1 | 10% |
| 1/16 | 0.0625 | 6.25% |
| 1/32 | 0.03125 | 3.125% |
Terminating and Repeating Decimals
Whether a fraction produces a terminating decimal is decided entirely by the prime factors of its reduced denominator. If the only prime factors are 2 and 5 — the factors of ten — the decimal terminates. Any other prime factor produces a repeating decimal.
| Fraction | Denominator factors | Decimal |
|---|---|---|
| 3/8 | 2 × 2 × 2 | 0.375 — terminates |
| 7/20 | 2 × 2 × 5 | 0.35 — terminates |
| 1/3 | 3 | 0.333… — repeats |
| 5/6 | 2 × 3 | 0.8333… — repeats |
| 1/7 | 7 | 0.142857142857… — repeats with period 6 |
Converting a Decimal to a Fraction
For a terminating decimal, write the digits over the appropriate power of ten and simplify. 0.375 = 375/1000, and dividing both by their GCD of 125 gives 3/8.
Repeating decimals need algebra. For 0.333…, let x = 0.333…; then 10x = 3.333…; subtracting gives 9x = 3, so x = 1/3. The same trick handles longer periods by multiplying by 10 raised to the length of the repeat.
Where Fractions Beat Decimals
- Exactness. 1/3 is exact; 0.333 is not, and rounding errors accumulate through a calculation.
- Imperial measurement. Construction, plumbing and machining in the U.S. work in halves, quarters, eighths, sixteenths and thirty-seconds.
- Cooking. Recipes scale naturally in fractions — half of 3/4 cup is 3/8 cup.
- Music. Note durations and time signatures are fractions by construction.
- Probability and algebra. Symbolic work is almost always cleaner in fractional form.
Frequently Asked Questions
Why can't I just add the tops and the bottoms?
Because the parts are different sizes. 1/2 + 1/3 is not 2/5 — that would be less than 1/2, which is impossible when adding a positive quantity. Converting to a common denominator makes the parts comparable: 3/6 + 2/6 = 5/6.
What is the lowest common denominator?
The smallest number both denominators divide into — their least common multiple. For 4 and 6 it is 12, not 24. Using the product of the denominators always works but leaves larger numbers to simplify afterwards.
Can a denominator be zero?
No. Division by zero is undefined, so a fraction with a zero denominator has no value.
How do I compare two fractions?
Cross-multiply: 3/7 versus 4/9 becomes 3×9 = 27 against 4×7 = 28, so 4/9 is larger. Converting both to decimals also works.