Fundamentals of Fractions

Fractions represent parts of a whole. They consist of a numerator (top number) and a denominator (bottom number). Below, you’ll find everything you need to know about fractions.

What Are Fractions and Their Uses

A fraction is made up of two parts, separated by a fraction bar:

  • Numerator (top number) - indicates how many parts are taken
  • Denominator (bottom number) - shows how many equal parts the whole is divided into

For example, the fraction \( \large \frac{3}{4} \) means the whole is divided into 4 equal parts, and we’re taking 3 of them.

Fractions - Pie

Fractions in Everyday Life:

In the Kitchen

\( \large \frac{1}{2} \) cup of flour, \( \large \frac{3}{4} \) tsp of salt

In Construction

A board \( \large 2\frac{1}{4} \) meters long

In Finance

\( \large \frac{1}{4} \) percentage point, \( \large \frac{1}{3} \) of the budget

Types of Fractions

Proper Fractions

\( \large \frac{3}{4} \quad \frac{2}{5} \quad \frac{1}{8} \)

The numerator is smaller than the denominator.
The fraction’s value is less than 1.

Improper Fractions

\( \large \frac{5}{3} \quad \frac{7}{4} \quad \frac{11}{5} \)

Mixed Numbers

\( \large 1\frac{2}{3} \quad 2\frac{3}{4} \quad 5\frac{1}{2} \)

Combines a whole number and a proper fraction.
The value is greater than 1.

Converting an Improper Fraction to a Mixed Number:

\( \large \frac{17}{5} = 3\frac{2}{5} \)
  1. Divide the numerator by the denominator: \( 17 \div 5 = 3 \) (remainder \( 2 \))
  2. The quotient is the whole number: \( 3 \)
  3. The remainder is the new numerator: \( 2 \)
  4. The denominator stays the same: \( 5 \)

Simplifying and Expanding Fractions

Simplifying Fractions

\( \large \frac{8}{12} = \frac{2}{3} \)

Simplifying involves dividing the numerator and denominator by their greatest common divisor (GCD).

  1. Find the GCD of the numerator and denominator
  2. Divide both by the GCD

For \( \large \frac{8}{12} \): GCD(8, 12) = 4, so \( \large \frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3} \)

Expanding Fractions

\( \large \frac{2}{3} = \frac{8}{12} \)

Expanding involves multiplying the numerator and denominator by the same number.

  1. Choose a number to multiply by
  2. Multiply both the numerator and denominator

For \( \large \frac{2}{3} \): Multiply by 4 to get \( \large \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \)

Note: Simplifying or expanding does NOT change the value of a fraction!

Comparing Fractions

To compare fractions:

  1. Convert the fractions to a common denominator
  2. Compare the numerators - a larger numerator means a larger fraction
\( \large \frac{3}{4} \text{ and } \frac{2}{3} \)

Solution:

  1. Find the least common multiple (LCM) of 4 and 3 = 12
  2. Convert: \( \large \frac{3}{4} = \frac{9}{12} \) and \( \large \frac{2}{3} = \frac{8}{12} \)
  3. Compare numerators: 9 > 8
  4. Conclusion: \( \large \frac{3}{4} > \frac{2}{3} \)

Operations with Fractions

Adding Fractions

\( \large \frac{a}{b} + \frac{c}{b} = \frac{a + c}{b} \)

1. Same denominators - add the numerators

\( \large \frac{2}{5} + \frac{1}{5} = \frac{3}{5} \)

2. Different denominators - find a common denominator

\( \large \frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12} \)

Subtracting Fractions

\( \large \frac{a}{b} - \frac{c}{b} = \frac{a - c}{b} \)

1. Same denominators - subtract the numerators

\( \large \frac{4}{7} - \frac{2}{7} = \frac{2}{7} \)

2. Different denominators - find a common denominator

\( \large \frac{2}{3} - \frac{1}{4} = \frac{8}{12} - \frac{3}{12} = \frac{5}{12} \)

Multiplying Fractions

\( \large \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \)

Multiply numerator by numerator and denominator by denominator

\( \large \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \)

Dividing Fractions

\( \large \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \)

Multiply the first fraction by the reciprocal of the second

\( \large \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8} \)

Quick Tips

  • Addition and subtraction require a common denominator
  • Multiplication and division don’t need a common denominator
  • For division, flip the second fraction and multiply
  • Always simplify the final result to its lowest terms

Fractions in Everyday Life

In the Kitchen

  • \( \large \frac{2}{3} \) cup of flour
  • \( \large \frac{1}{4} \) tsp of salt
  • \( \large 1\frac{1}{2} \) cups of milk

In Construction

  • Board \( \large 2\frac{1}{4} \) meters long
  • Nail \( \large \frac{3}{4} \) inch
  • Wall \( \large 3\frac{1}{2} \) meters high

In Finance

  • \( \large \frac{1}{4} \) of income for savings
  • \( \large \frac{1}{3} \) of company shares
  • \( \large 3\frac{1}{2} \)% interest rate

Understanding fractions is essential for education and everyday tasks alike.

Dive deeper into specific fraction operations by exploring our other pages!

Adding Fractions Subtracting Fractions Multiplying Fractions Dividing Fractions