arithmetic

Fractions and Decimals

Pre-Algebra25 minDifficulty

You should know: variables and expressions

Overview

Fractions represent parts of a whole: a/b means a parts out of b equal parts (b not zero). Decimals are an alternate notation using powers of ten. Every fraction can be written as a decimal (terminating or repeating) and vice versa. Operations on fractions (add, subtract, multiply, divide) follow specific rules.

Intuition

A fraction a/b asks: if you cut a pizza into b equal slices, how many slices is a? To add fractions, you need slices of the same size (common denominator). To multiply fractions, multiply tops and multiply bottoms. To divide by a fraction, flip it and multiply. Decimals are just fractions with denominators that are powers of 10: 0.3 = 3/10, 0.25 = 25/100 = 1/4.

Formal Definition

Definition

A fraction is a/b with a, b integers, b not 0. Equivalent fractions: a/b = (ka)/(kb) for k not 0. Operations: (a/b) + (c/d) = (ad+bc)/(bd), (a/b)*(c/d) = ac/(bd), (a/b)/(c/d) = ad/(bc). A fraction is in lowest terms when gcd(a,b)=1. A decimal terminates iff its denominator in lowest terms has no prime factors other than 2 and 5.

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

Addition of fractions

\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

Multiplication

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}

Division

Notation

Notation	Meaning
$\frac{a}{b}$	Fraction: a numerator, b denominator
$0.\overline{d}$	Repeating decimal

Theorems

Theorem 1: Decimal Representation

A fraction a/b in lowest terms has a terminating decimal expansion if and only if b has no prime factors other than 2 and 5. Otherwise the decimal repeats.

Worked Examples

1
Common denominator is 12.
$\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12}$
2
Add numerators.
$= \frac{11}{12}$

✓ Answer

11/12

Practice Problems

Easyfill in blank

Compute 3/5 + 1/3.

Easyfill in blank

Convert 0.625 to a fraction in lowest terms.

Common Mistakes

Common Mistake

Adding fractions by adding numerators and adding denominators: 1/2 + 1/3 = 2/5.

Find a common denominator. 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

Common Mistake

Dividing fractions by dividing numerator by numerator and denominator by denominator.

Divide by a fraction by multiplying by its reciprocal: (a/b)/(c/d) = (a/b)*(d/c).

Quiz

Which fraction equals 0.75?

2/33/44/57/10

Historical Background

Fractions appear in the oldest mathematical records. The Rhind Papyrus (c. 1650 BCE) shows Egyptians using unit fractions (1/n). Babylonians used sexagesimal (base-60) fractions. The modern fraction notation a/b developed in medieval Islamic and European mathematics. Decimal notation was popularized by Simon Stevin in 1585 with his work 'De Thiende' (The Tenth).

1650 BCE
Egyptian unit fractions in Rhind Papyrus
Ahmes
1585
Simon Stevin popularizes decimal notation
Simon Stevin

Summary

Fraction a/b: a parts of b equal total parts.
To add/subtract fractions: find a common denominator, then combine numerators.
To multiply: multiply tops, multiply bottoms. To divide: flip the divisor and multiply.
Terminating decimal iff denominator (in lowest terms) has only factors 2 and 5.

References

BookLarson, R. Pre-Algebra. Holt McDougal, 2010.