Training Fractions Fraction Basics and Equivalent Fractions
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Fraction Basics and Equivalent Fractions

18 min Fractions

Fraction Basics

A fraction represents a part of a whole. When you slice a pizza into eight pieces and eat three, you have eaten three-eighths of the pizza. Fractions appear everywhere in daily life — in recipes, measurements, statistics, and financial calculations.

This lesson introduces the numerator and denominator, proper and improper fractions, mixed numbers, and the crucial concept of equivalent fractions. You will learn that the same quantity can be expressed in infinitely many ways — one-half, two-fourths, three-sixths — and that simplifying fractions means finding the cleanest representation.

Equivalent fractions are the key to adding, subtracting, and comparing fractions in the lessons that follow, so take the time to master the cross-multiplication and GCF techniques introduced here.

A fraction represents a part of a whole or a quotient of two integers.

Definition

A fraction $\dfrac{a}{b}$ has two parts:

  • Numerator ($a$): the number of parts you have
  • Denominator ($b$): the total number of equal parts ($b \neq 0$)

Types of Fractions

TypeDefinitionExample
ProperNumerator $<$ Denominator$\dfrac{3}{4}$
ImproperNumerator $\ge$ Denominator$\dfrac{7}{4}$
Mixed NumberWhole number + proper fraction$1\dfrac{3}{4}$

Converting Between Mixed Numbers and Improper Fractions

Rule — Mixed to Improper

$$a\frac{b}{c} = \frac{a \cdot c + b}{c}$$

Example 1

Convert $2\dfrac{3}{5}$ to an improper fraction.

$$2\frac{3}{5} = \frac{2 \times 5 + 3}{5} = \frac{13}{5}$$

Example 2

Convert $5\dfrac{2}{7}$ to an improper fraction.

$$5\frac{2}{7} = \frac{5 \times 7 + 2}{7} = \frac{37}{7}$$

Rule — Improper to Mixed

Divide the numerator by the denominator. The quotient is the whole part; the remainder is the new numerator.

Example 3

Convert $\dfrac{29}{6}$ to a mixed number.

$29 \div 6 = 4$ remainder $5$.

$$\frac{29}{6} = 4\frac{5}{6}$$

Equivalent Fractions

Multiplying or dividing both numerator and denominator by the same nonzero number gives an equivalent fraction:

$$\frac{a}{b} = \frac{a \times n}{b \times n}$$

Example 4

Write three fractions equivalent to $\dfrac{2}{3}$.

$$\frac{2}{3} = \frac{4}{6} = \frac{6}{9} = \frac{8}{12}$$

Simplifying Fractions

Procedure

Divide both numerator and denominator by their greatest common factor (GCF).

Example 5

Simplify $\dfrac{18}{24}$.

GCF of 18 and 24 is 6.

$$\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$$

Example 6

Simplify $\dfrac{45}{60}$.

GCF = 15. $\dfrac{45}{60} = \dfrac{3}{4}$

Practice Problems

1. Convert $3\dfrac{4}{5}$ to an improper fraction.
2. Convert $\dfrac{23}{7}$ to a mixed number.
3. Simplify $\dfrac{12}{18}$.
4. Convert $7\dfrac{1}{3}$ to an improper fraction.
5. Simplify $\dfrac{24}{36}$.
6. Convert $\dfrac{41}{8}$ to a mixed number.
7. Write two fractions equivalent to $\dfrac{5}{8}$.
8. Simplify $\dfrac{56}{72}$.
9. Convert $10\dfrac{3}{4}$ to an improper fraction.
10. Simplify $\dfrac{90}{120}$.
11. Convert $\dfrac{100}{9}$ to a mixed number.
12. Is $\dfrac{3}{5}$ equivalent to $\dfrac{9}{15}$?
13. Simplify $\dfrac{42}{56}$.
14. Convert $6\dfrac{5}{9}$ to an improper fraction.
15. What is the GCF of 28 and 42?
Show Answer Key

1. $\dfrac{19}{5}$

2. $3\dfrac{2}{7}$

3. $\dfrac{2}{3}$

4. $\dfrac{22}{3}$

5. $\dfrac{2}{3}$

6. $5\dfrac{1}{8}$

7. $\dfrac{10}{16}$ and $\dfrac{15}{24}$

8. $\dfrac{7}{9}$

9. $\dfrac{43}{4}$

10. $\dfrac{3}{4}$

11. $11\dfrac{1}{9}$

12. Yes — $\dfrac{3 \times 3}{5 \times 3} = \dfrac{9}{15}$

13. $\dfrac{3}{4}$

14. $\dfrac{59}{9}$

15. $14$

Adding and Subtracting Fractions