A fraction is one number divided by another: 3/4 means 3 ÷ 4, which is 0.75. The top number, the numerator, counts parts, and the bottom number, the denominator, names their size. Arithmetic on fractions comes down to two questions: do the denominators match, and which operation are you doing. Multiplication is straight across, division flips the second fraction, and only adding or subtracting needs a common denominator.

Key Takeaways

3/4 is the same number as 0.75: a fraction is just a division written sideways.

To add or subtract fractions, use a common denominator: 1/3 + 1/6 = 0.5.

Multiplication is straight across: (2/3) × (3/4) = 0.5.

To divide, multiply by the reciprocal: (1/2) ÷ (1/4) = 2.

What is a fraction?

A fraction is a division written as two stacked numbers instead of a division sign. In 3/4, the 3 is the numerator and the 4 is the denominator, and the whole symbol reads as “three divided by four”. The fraction line is doing the work of the ÷ sign. Type 3/4 on any calculator and you get 0.75, the decimal version of the same number. The fraction calculator handles these operations directly, and this post explains the hand method behind them.

Numerators and denominators

The numerator counts how many parts you have. The denominator says how many equal parts make up one whole, which fixes the size of each part. In 3/4 you have three parts and each part is a quarter. In 1/2 you have one part and each part is a half. The two roles are not interchangeable: 3/4 and 4/3 are different numbers, because swapping the count and the size changes the answer completely.

Equivalent fractions

Two fractions are equivalent when they name the same amount. 3/4, 6/8, and 9/12 all equal 0.75. You build an equivalent fraction by multiplying or dividing the numerator and denominator by the same number, which relabels the parts without changing the ratio. Multiplying 3/4 by 2/2 gives 6/8, and both are 0.75.

The reverse step, dividing both numbers down to the smallest working pair, is called simplifying. 6/12 becomes 1/2, and both equal 0.5. Equivalent fractions matter mainly because they let you find a common denominator, which is the whole job of the next section.

Adding and subtracting fractions

Addition and subtraction need matching denominators. When they already match, add the numerators and keep the denominator: 1/4 + 2/4 = 3/4, and 3/4 = 0.75. When the denominators differ, you rewrite one or both fractions first so they agree.

Finding a common denominator

For 1/3 + 1/6, notice that 6 is a multiple of 3. Rewrite 1/3 as 2/6, then add: 2/6 + 1/6 = 3/6 = 0.5. The whole thing works as one calculator expression: 1/3+1/6 → = 0.5. When neither denominator divides the other, multiply them to get a common denominator. For 1/2 + 1/3 the common denominator is 6, giving 3/6 + 2/6 = 5/6, which the display shows as 0.833333333333.

Multiplying and dividing fractions

Multiplication is the easiest case. Multiply the two numerators and the two denominators: (2/3) × (3/4) = 6/12 = 0.5. Type (2/3)*(3/4) to confirm. You can simplify instead of ending in six twelfths, but the decimal answers out the same.

Division uses the reciprocal. To divide (1/2) ÷ (1/4), flip the second fraction over and multiply: (1/2) × (4/1) = 4/2 = 2. Type (1/2)/(1/4). The answer makes sense: a quarter fits into a half exactly two times. When the numbers do not divide so neatly, (3/4) ÷ (1/2) = (3/4) × (2/1) = 6/4 = 1.5, and (3/4)/(1/2) returns 1.5.

Mixed numbers and improper fractions

An improper fraction has a numerator at least as large as its denominator, like 5/4. A mixed number writes the whole part separately, like 1 1/4. The two name the same amount: 5/4 = 1.25, and one and a quarter is 1.25.

To convert a mixed number to an improper fraction, multiply the whole part by the denominator and add the numerator. 2 1/3 becomes (2 × 3 + 1) ÷ 3 = 7/3, and typing 7/3 → = 2.33333333333. Going the other way, divide the numerator by the denominator and keep the remainder as the fraction part.

From fraction to decimal to percent

A fraction becomes a decimal by doing the division, and a percent is the decimal times 100. So 3/4 = 0.75 = 75%, and 1/8 = 0.125 = 12.5%. Type 3/4*100 and the calculator reports 75, the percent step on its own. The percentages guide covers the decimal to percent conversion in detail, and the percentage calculator applies it to real amounts. When numbers get very large or very small, results usually land in scientific notation, which uses the same digit counting that percents do.

Frequently asked questions

Do I need a common denominator to multiply or divide fractions?

No. Only addition and subtraction need matching denominators. Multiplication is straight across, and division is multiplication by the flipped fraction.

What is the difference between an improper fraction and a mixed number?

Both can name the same amount. 5/4 is improper because its numerator is larger than its denominator. 1 1/4 is mixed because it shows the whole part and the fraction part separately. The two are equal.

Why does dividing `1/2` by `1/4` give 2?

Because the question is how many quarters fit into a half, and the answer is two. The reciprocal rule is just the mechanical way to compute that.

How do I type fractions on the calculator?

Use the slash key. A bare fraction like 3/4 is fine, but wrap a numerator or denominator in parentheses when it has more than one term, as in (2/3)*(3/4). Without the parentheses, the order of operations can split a compound fraction into steps you did not intend.