5 Fractions, Decimals, and Percentages

Fractions, Decimals, and Percentages is the study of the three interchangeable ways rational quantities are written and manipulated in elementary algebra — as ratios of integers, as place-value decimal expansions, and as parts-per-hundred comparisons — together with the rules for converting between them, performing arithmetic within each form, and choosing the most convenient form for a given calculation.

Fraction Meaning and Structure

A fraction represents a part of a whole, written as a/b, where the numerator a counts how many parts are taken and the denominator b indicates how many equal parts the whole is divided into, with b ≠ 0. A fraction is described as proper when its numerator is smaller than its denominator (value less than one), improper when the numerator is greater than or equal to the denominator (value one or greater), and a mixed number expresses an improper fraction as a whole number combined with a proper fraction, such as 2 3/4.

Equivalent Fractions and Simplest Form

Equivalent fractions represent the same value while using different numerators and denominators, produced by multiplying or dividing both parts of a fraction by the same nonzero number, since this operation is equivalent to multiplying by 1:

A fraction is in simplest form (or lowest terms) when its numerator and denominator share no common factor other than 1, found by dividing both by their greatest common factor.

Fraction Addition and Subtraction

Adding or subtracting fractions requires a common denominator, since only parts of equally sized wholes can be combined directly. Fractions with the same denominator are combined by adding or subtracting numerators and keeping the denominator unchanged. Fractions with different denominators must first be rewritten as equivalent fractions sharing a common denominator, typically the least common multiple of the original denominators:

Fraction Multiplication and Division

Multiplying fractions requires no common denominator: numerators are multiplied together, and denominators are multiplied together.

Dividing by a fraction is defined as multiplying by its reciprocal, the fraction obtained by swapping numerator and denominator:

Decimal Structure and Representation

A decimal represents a rational or irrational number using place value, where digits to the right of the decimal point represent tenths, hundredths, thousandths, and so on, in successively smaller powers of ten. A decimal is terminating if it has a finite number of nonzero digits, repeating if a digit or block of digits recurs infinitely, and neither if the number is irrational. Every terminating or repeating decimal corresponds to a rational number and can be converted to a fraction.

Decimal Operations

Adding and subtracting decimals requires aligning digits by place value, effectively aligning decimal points before combining. Multiplying decimals is performed as if the decimal points were absent, with the decimal point placed in the result according to the total number of decimal places in the original factors. Dividing decimals is performed by shifting the decimal point in both the divisor and dividend by the same number of places until the divisor is a whole number, then dividing normally.

Fraction and Decimal Conversion

Converting a fraction to a decimal is performed by dividing the numerator by the denominator. Converting a terminating decimal to a fraction is performed by writing the decimal digits over the appropriate power of ten and simplifying, as in 0.75 = 75/100 = 3/4. Converting a repeating decimal to a fraction uses an algebraic technique involving multiplying by a power of ten to shift the repeating block and subtracting to eliminate the infinite repetition, producing an exact fractional value.

Percentage Representation and Conversion

A percentage expresses a quantity as a number of parts per one hundred, using the % symbol. Converting a percentage to a decimal divides by 100 (equivalently, moves the decimal point two places left); converting a decimal to a percentage multiplies by 100 (moves the decimal point two places right). Converting a fraction to a percentage first converts it to a decimal, then to a percentage; converting a percentage to a fraction writes it over a denominator of 100 and simplifies.

Fundamental Percentage Calculations

The core percentage relationship connects a part, a whole, and a percent:

This single relationship, rearranged algebraically, solves the three fundamental percentage problem types: finding the part when the percent and whole are known (multiply the whole by the percent as a decimal), finding the percent when the part and whole are known (divide part by whole), and finding the whole when the part and percent are known (divide part by the percent as a decimal). Percentage increase and decrease are computed by comparing the change in a quantity to its original value.

Representation Comparison and Error Analysis

Because the same quantity can be written as a fraction, a decimal, or a percentage, this area also addresses how to choose the most efficient representation for a given task and how to catch conversion errors: verifying a fraction-to-decimal conversion by long division, verifying a decimal-to-percentage conversion by checking the decimal point shifted exactly two places, and verifying that a simplified fraction is truly in lowest terms. Common errors include misplacing the decimal point during percentage conversion, adding fractions without first finding a common denominator, and forgetting to invert the second fraction when dividing.