Fraction Formulas and Explanations

This guide provides the mathematical formulas and steps used in the calculator to solve problems involving fractions.

Basic Fraction Operations

Addition and Subtraction ➕➖

To add or subtract fractions, you must first find a common denominator. The most efficient way is to find the Least Common Multiple (LCM) of the denominators.

Formula:

Alternative (Cross-Multiplication):

Explanation:

• Find the LCM of the denominators ($b$ and $d$).
• Convert both fractions to equivalent fractions with the LCM as their new denominator.
• Add or subtract the new numerators.
• Keep the common denominator.
• Simplify the final fraction by dividing both the numerator and denominator by their Greatest Common Divisor (GCD).

Multiplication ✖️

Multiplying fractions is straightforward. You multiply the numerators together and the denominators together.

Formula:

Explanation:

• Multiply the numerators ($a$ and $c$) to get the new numerator.
• Multiply the denominators ($b$ and $d$) to get the new denominator.
• Simplify the resulting fraction.

Division ➗

To divide fractions, you use the “Keep, Change, Flip” (KCF) method. You multiply the first fraction by the reciprocal of the second fraction.

Formula:

Explanation:

• Keep the first fraction ($a/b$).
• Change the division sign (÷) to a multiplication sign (×).
• Flip the second fraction ($c/d$) to its reciprocal ($d/c$).
• Follow the rules for fraction multiplication.
• Simplify the resulting fraction.

Mixed Numbers

A mixed number combines a whole number with a fraction (e.g., $3 \frac{1}{2}$). To perform calculations, it’s often easiest to convert them into improper fractions first.

Mixed Number to Improper Fraction

Formula:

Explanation:

• Multiply the whole number by the denominator.
• Add the numerator to that result.
• Place this new value over the original denominator.

Operations with Mixed Numbers

Method:

• Convert all mixed numbers to improper fractions.
• Perform the operation (addition, subtraction, multiplication, or division) using the formulas above.
• Simplify the final improper fraction.
• If needed, convert the result back to a mixed number.

Simplifying Fractions

Simplifying a fraction means reducing it to its lowest terms.

Formula:

Explanation:

• Find the Greatest Common Divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both values without a remainder.
• Divide both the numerator and the denominator by the GCD.
• The result is the simplified fraction.

Conversions

Decimal to Fraction

To convert a decimal to a fraction, you use the place value of the last digit.

Method:

Count the number of digits after the decimal point. Let this number be N.

Write the decimal as a fraction with the decimal number (without the point) as the numerator and $10^N$ as the denominator.

Simplify the resulting fraction.

Example:

To convert 0.75 to a fraction:

• There are two decimal places, so the denominator is $10^2 = 100$.
• The numerator is 75.
• The fraction is
  75100
  .
• The GCD of 75 and 100 is 25.
• Simplified, the fraction is
  75 ÷ 25100 ÷ 25
  =
  34
  .

Fraction to Decimal

Converting a fraction to a decimal is a simple division problem.

Formula:

Explanation:

• Divide the numerator by the denominator.
• The result is the decimal equivalent.