Factor Calculator

In mathematics, a fraction represents a part of a whole. It consists of a numerator (the number of equal parts being represented, placed above the line) and a denominator (the total number of parts that make up the whole, placed below the line). For example, in the fraction 3/8, the numerator is 3 and the denominator is 8. If you slice a pie into 8 equal pieces and eat 3, the remaining fraction of the pie is 5/8. Note that the denominator can never be 0, as dividing by zero is mathematically undefined.

Our free Fraction Calculator is a comprehensive math solver. It handles basic fractions, mixed numbers, simplification, conversions to and from decimals, and features a big number engine for very large integers. Select your operation and get instant, simplified step-by-step solutions.

Fraction Addition & Subtraction

Unlike integers, fractions cannot be directly added or subtracted unless they share a common denominator. If denominators differ, you must find a common multiple before performing the operation.

Method 1: Common Denominator by Multiplication

The simplest way to find a common denominator is to multiply the numerator and denominator of each fraction by the denominator of the other fraction:

(a / b) ± (c / d) = (ad ± bc) / bd

Addition Example: Solve 3/4 + 1/6

3/4 + 1/6 = (3 × 6) / (4 × 6) + (1 × 4) / (6 × 4) = 18/24 + 4/24 = 22/24

Dividing the numerator and denominator by their greatest common factor (2) yields the simplified answer: 11/12.

Subtraction Example: Solve 3/4 – 1/6

3/4 - 1/6 = (3 × 6) / (4 × 6) - (1 × 4) / (6 × 4) = 18/24 - 4/24 = 14/24

Simplifying by 2 yields: 7/12.

Method 2: Least Common Multiple (LCM)

An alternative, more efficient method uses the Least Common Multiple (LCM) of the denominators. In the problem 1/4 + 1/6 + 1/2, the denominators are 4, 6, and 2:

• Multiples of 2: 2, 4, 6, 8, 10, 12…
• Multiples of 4: 4, 8, 12…
• Multiples of 6: 6, 12…

The LCM is 12. Multiply each fraction by the factor needed to make its denominator 12:

1/4 + 1/6 + 1/2 = (1 × 3)/12 + (1 × 2)/12 + (1 × 6)/12 = 3/12 + 2/12 + 6/12 = 11/12.

Fraction Multiplication & Division

Multiplying and dividing fractions is highly straightforward and does not require finding common denominators.

1. Multiplication

Simply multiply the numerators together to form the new numerator, and multiply the denominators together to form the new denominator:

(a / b) × (c / d) = ac / bd

Example: Solve 3/4 × 1/6

3/4 × 1/6 = (3 × 1) / (4 × 6) = 3/24

Simplifying by the GCF of 3 yields: 1/8.

2. Division

To divide by a fraction, multiply the first fraction by the reciprocal of the divisor. The reciprocal is found by swapping the numerator and the denominator:

(a / b) / (c / d) = (a / b) × (d / c) = ad / bc

Example: Solve 3/4 ÷ 1/6

3/4 ÷ 1/6 = 3/4 × 6/1 = (3 × 6) / (4 × 1) = 18/4

Simplifying by 2 yields the improper fraction 9/2, which can be expressed as the mixed number 4 1/2.

Converting Between Fractions & Decimals

1. Decimals to Fractions

Each decimal place to the right of the decimal point represents a power of 10 (10^1 = 10, 10^2 = 100, 10^3 = 1000). Place the numbers to the right of the decimal point over the corresponding power of 10 and simplify.

Example: Convert 0.1234 to a fraction.

Since the number extends to the 4th decimal place, the denominator is 10^4, or 10,000:

0.1234 = 1234 / 10000 = 617 / 5000 (GCF is 2).

2. Fractions to Decimals

If the fraction’s denominator is a power of 10 or can be easily converted to one, scale the fraction. For example, 1/2 is scaled by 5 to become 5/10, which translates to 0.5. If the fraction is 5/100, it translates to 0.05. For other denominators, perform long division (numerator divided by denominator).

Solve other algebraic variables with our Slope Calculator or analyze triangle dimensions using the Triangle Calculator.

Frequently Asked Questions (FAQ)

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

An improper fraction is a fraction where the numerator is equal to or greater than the denominator (e.g., 9/2). A mixed number combines a whole number with a proper fraction (e.g., 4 1/2). Both represent the same numerical value.

Why can a fraction denominator never be zero?

Because division by zero is undefined. In fraction terms, the denominator represents the total number of equal parts that make up a whole. You cannot divide a whole into zero parts, making a fraction with a denominator of 0 mathematically impossible.

What is a Greatest Common Factor (GCF) in fraction simplification?

The GCF is the largest positive integer that divides both the numerator and the denominator without leaving a remainder. Dividing both numbers by their GCF reduces the fraction to its simplest, lowest terms (e.g., dividing 22/24 by its GCF of 2 yields 11/12).

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

No. Common denominators are only required for addition and subtraction. For multiplication, you multiply straight across. For division, you multiply by the reciprocal of the second fraction.