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Chapter 2: Fractions & Decimals

Pre-Algebra  •  Chapter 2 of 10  •  Updated March 2026

Fractions and decimals are two ways of expressing numbers between integers. This chapter builds each skill one technique at a time — starting with the vocabulary of fractions, then the tools (prime factorization, GCF, LCM) needed to work with them, and finally all four arithmetic operations.

📋 Chapter Contents

  1. Types of Fractions — proper, improper, mixed numbers
  2. Prime Factorization — factor trees, writing prime factorization
  3. Greatest Common Factor (GCF) — using prime factorization
  4. Least Common Multiple (LCM) — using prime factorization
  5. Equivalent Fractions & Simplifying
  6. Adding & Subtracting Fractions — same and different denominators, mixed numbers
  7. Multiplying Fractions — cross-canceling, mixed numbers
  8. Dividing Fractions — reciprocal, KCF method
  9. Decimals & Conversions — place value, fraction ↔ decimal, repeating decimals

2.1 Types of Fractions

Definition: Fraction

A fraction $\dfrac{a}{b}$ represents $a$ parts of a whole divided into $b$ equal parts, where $b \neq 0$.

Proper Fraction

Numerator < Denominator
Value between 0 and 1

$\dfrac{3}{5}, \quad \dfrac{7}{10}, \quad \dfrac{1}{4}$

Improper Fraction

Numerator ≥ Denominator
Value ≥ 1

$\dfrac{9}{4}, \quad \dfrac{7}{3}, \quad \dfrac{5}{5}$

Mixed Number

Integer + proper fraction
Value ≥ 1

$2\dfrac{1}{4}, \quad 5\dfrac{3}{7}, \quad 1\dfrac{1}{2}$

Converting: Improper Fraction → Mixed Number

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

Example 1 — Improper to Mixed (easy)

Convert $\dfrac{17}{5}$ to a mixed number.

Step 1: Divide 17 ÷ 5.   $17 = 5 \times 3 + 2$. Quotient = 3, remainder = 2.
Step 2: Write: integer part = 3, numerator = 2, denominator stays 5.
Answer: $\dfrac{17}{5} = 3\dfrac{2}{5}$

Example 2 — Improper to Mixed (larger numbers)

Convert $\dfrac{47}{6}$ to a mixed number.

Step 1: 47 ÷ 6 = 7 remainder 5   (since $6 \times 7 = 42$, $47 - 42 = 5$).
Answer: $\dfrac{47}{6} = 7\dfrac{5}{6}$

Converting: Mixed Number → Improper Fraction

Multiply the integer by the denominator, then add the numerator. The result becomes the new numerator.

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

Example 3 — Mixed to Improper

Convert $4\dfrac{3}{7}$ to an improper fraction.

Step 1: Multiply integer × denominator: $4 \times 7 = 28$.
Step 2: Add the numerator: $28 + 3 = 31$.
Step 3: Place over the original denominator.
Answer: $4\dfrac{3}{7} = \dfrac{31}{7}$

Example 4 — Mixed to Improper (practice)

Convert $6\dfrac{5}{8}$ to an improper fraction.

$6 \times 8 + 5 = 48 + 5 = 53$
Answer: $6\dfrac{5}{8} = \dfrac{53}{8}$

✏ Practice 2.1 — Types of Fractions

Convert each improper fraction to a mixed number:

  1. $\dfrac{22}{7}$    Hint: 22 ÷ 7 = ?
  2. $\dfrac{35}{4}$
  3. $\dfrac{100}{9}$

Convert each mixed number to an improper fraction:

  1. $3\dfrac{1}{5}$
  2. $8\dfrac{2}{3}$
  3. $12\dfrac{7}{10}$
Show Answers

1) $3\frac{1}{7}$   2) $8\frac{3}{4}$   3) $11\frac{1}{9}$   4) $\frac{16}{5}$   5) $\frac{26}{3}$   6) $\frac{127}{10}$

2.2 Prime Factorization

Key Vocabulary

💡 Primes to memorize (up to 50)

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Divisibility shortcuts: divisible by 2 if last digit is even; by 3 if digit sum is divisible by 3; by 5 if last digit is 0 or 5.

The Factor Tree Method

A factor tree splits a number into two factors, then keeps splitting until every branch ends in a prime number (circled). The prime factorization is the product of all the primes at the ends of the branches.

Example 5 — Factor Tree for 36

Find the prime factorization of 36.

36
/    \
4     9
/ \   / \
2  2  3  3
Step 1: Split 36 = 4 × 9.
Step 2: Split 4 = 2 × 2 (both prime — stop). Split 9 = 3 × 3 (both prime — stop).
Step 3: Collect all primes at the ends: 2, 2, 3, 3.
Answer: $36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$

💡 The answer is always the same — no matter how you split!

Even if you start with 36 = 6 × 6, or 36 = 12 × 3, you always end up with $2^2 \times 3^2$. This is the Fundamental Theorem of Arithmetic: every integer > 1 has exactly one prime factorization.

Example 6 — Factor Tree for 60

Find the prime factorization of 60.

60
/    \
6     10
/ \   / \
2  3  2  5
Primes collected: 2, 3, 2, 5. Sort them: 2, 2, 3, 5.
Answer: $60 = 2^2 \times 3 \times 5$

Example 7 — Factor Tree for 180 (multi-level)

Find the prime factorization of 180.

180
/    \
4     45
/ \    / \
2  2  9  5
         / \
        3  3
Primes: 2, 2, 3, 3, 5.
Answer: $180 = 2^2 \times 3^2 \times 5$

Ladder (Repeated Division) Method

Alternatively, divide by the smallest prime that fits, and keep dividing until you reach 1.

Example 8 — Ladder Method for 84

Find the prime factorization of 84 using repeated division.

2 | 84
2 | 42
3 | 21
  |  7
Answer: $84 = 2 \times 2 \times 3 \times 7 = 2^2 \times 3 \times 7$

✏ Practice 2.2 — Prime Factorization

Write the prime factorization of each number. Draw a factor tree or use the ladder method.

  1. 24
  2. 45
  3. 56
  4. 72
  1. 90
  2. 120
  3. 150
  4. 210
Show Answers

1) $2^3 \times 3$   2) $3^2 \times 5$   3) $2^3 \times 7$   4) $2^3 \times 3^2$   5) $2 \times 3^2 \times 5$   6) $2^3 \times 3 \times 5$   7) $2 \times 3 \times 5^2$   8) $2 \times 3 \times 5 \times 7$

2.3 Greatest Common Factor (GCF)

Definition: GCF

The Greatest Common Factor of two (or more) integers is the largest integer that divides all of them evenly.

Used for: simplifying fractions.

GCF Using Prime Factorization — Method

  1. Write the prime factorization of each number.
  2. Identify the common prime factors (primes that appear in both).
  3. For each common prime, take the lowest (smaller) exponent.
  4. Multiply those together — that's the GCF.

Example 9 — GCF(12, 18)

Step 1: Factor: $12 = 2^2 \times 3$    $18 = 2 \times 3^2$
Step 2: Common primes: 2 and 3.
Step 3: Lowest exponents: $2^{\min(2,1)} = 2^1$ and $3^{\min(1,2)} = 3^1$.
Step 4: GCF $= 2^1 \times 3^1 = 6$.
Check: Factors of 12: {1,2,3,4,6,12}. Factors of 18: {1,2,3,6,9,18}. Largest common = 6 ✓

Example 10 — GCF(24, 60)

Step 1: $24 = 2^3 \times 3$    $60 = 2^2 \times 3 \times 5$
Step 2: Common primes: 2 and 3 (5 is only in 60).
Step 3: $2^{\min(3,2)} = 2^2 = 4$ and $3^{\min(1,1)} = 3^1 = 3$.
Answer: GCF(24, 60) $= 4 \times 3 = \mathbf{12}$

Example 11 — GCF(36, 48, 60) — three numbers

$36 = 2^2 \times 3^2$    $48 = 2^4 \times 3$    $60 = 2^2 \times 3 \times 5$
Common to all three: 2 and 3 (5 only in 60, extra 2's not shared with 36).
$2^{\min(2,4,2)} = 2^2 = 4$    $3^{\min(2,1,1)} = 3^1 = 3$
Answer: GCF $= 4 \times 3 = \mathbf{12}$

✏ Practice 2.3 — GCF

Find the GCF using prime factorization:

  1. GCF(16, 24)
  2. GCF(30, 45)
  3. GCF(42, 70)
  4. GCF(56, 84)
  5. GCF(18, 27, 45)
  6. GCF(100, 150, 250)
Show Answers

1) 8   2) 15   3) 14   4) 28   5) 9   6) 50

2.4 Least Common Multiple (LCM)

Definition: LCM

The Least Common Multiple of two (or more) integers is the smallest positive integer that is a multiple of all of them.

Used for: finding common denominators when adding or subtracting fractions.

LCM Using Prime Factorization — Method

  1. Write the prime factorization of each number.
  2. List all prime factors that appear in any of the numbers.
  3. For each prime, take the highest exponent that appears.
  4. Multiply those together — that's the LCM.

Example 12 — LCM(12, 18)

$12 = 2^2 \times 3$    $18 = 2 \times 3^2$
All prime factors: 2 and 3.
Highest exponents: $2^{\max(2,1)} = 2^2 = 4$   and   $3^{\max(1,2)} = 3^2 = 9$.
Answer: LCM $= 4 \times 9 = \mathbf{36}$
Check: Multiples of 12: 12, 24, 36, …   Multiples of 18: 18, 36, …   ✓

Example 13 — LCM(8, 15)

$8 = 2^3$    $15 = 3 \times 5$
No common prime factors — they share nothing!
All primes: $2^3$, $3^1$, $5^1$.
Answer: LCM $= 8 \times 3 \times 5 = \mathbf{120}$

When two numbers share no common factors, LCM = their product.

Example 14 — LCM(12, 20, 30)

$12 = 2^2 \times 3$    $20 = 2^2 \times 5$    $30 = 2 \times 3 \times 5$
All primes: 2, 3, 5. Highest: $2^2$, $3^1$, $5^1$.
Answer: LCM $= 4 \times 3 \times 5 = \mathbf{60}$

💡 GCF × LCM = Product of the two numbers

$$\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b$$

Check: GCF(12,18) × LCM(12,18) = 6 × 36 = 216 = 12 × 18 ✓   This is a great way to verify your answers.

✏ Practice 2.4 — LCM

Find the LCM using prime factorization:

  1. LCM(4, 6)
  2. LCM(9, 15)
  3. LCM(8, 14)
  4. LCM(6, 10, 15)
  5. LCM(4, 9, 12)
  6. Verify using GCF × LCM = a × b for problem 3.
Show Answers

1) 12   2) 45   3) 56   4) 30   5) 36   6) GCF(8,14)=2, 2×56=112=8×14 ✓

2.5 Equivalent Fractions & Simplifying

Fundamental Property of Fractions

Multiplying or dividing both numerator and denominator by the same nonzero number gives an equivalent fraction — same value, different appearance:

$$\frac{a}{b} = \frac{a \times k}{b \times k} \qquad \text{and} \qquad \frac{a}{b} = \frac{a \div k}{b \div k}$$

Building Equivalent Fractions (scaling up)

Example 15 — Find an equivalent fraction

Write $\dfrac{3}{5}$ as a fraction with denominator 20.

Step 1: What do we multiply 5 by to get 20?   $5 \times 4 = 20$.
Step 2: Multiply both numerator and denominator by 4.
$$\frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20}$$

Simplifying Fractions (reducing to lowest terms)

A fraction is in simplest form when GCF(numerator, denominator) = 1. To simplify: divide both by their GCF.

Example 16 — Simplify $\dfrac{36}{48}$

Step 1: Find GCF(36, 48).   $36 = 2^2 \times 3^2$   $48 = 2^4 \times 3$.   GCF $= 2^2 \times 3 = 12$.
Step 2: Divide both by 12.
$$\frac{36}{48} = \frac{36 \div 12}{48 \div 12} = \frac{3}{4}$$
Check: GCF(3, 4) = 1 — no more simplifying possible ✓

Example 17 — Simplify $\dfrac{45}{75}$

$45 = 3^2 \times 5$    $75 = 3 \times 5^2$
GCF $= 3^1 \times 5^1 = 15$
$$\frac{45}{75} = \frac{45 \div 15}{75 \div 15} = \frac{3}{5}$$

💡 Simplifying in steps — OK to do it gradually

If you don't immediately spot the GCF, simplify in multiple steps by dividing by any common factor each time.

$\dfrac{36}{48} \xrightarrow{\div 2} \dfrac{18}{24} \xrightarrow{\div 2} \dfrac{9}{12} \xrightarrow{\div 3} \dfrac{3}{4}$   (same result, just three smaller steps)

✏ Practice 2.5 — Simplifying

  1. Simplify $\dfrac{20}{30}$
  2. Simplify $\dfrac{42}{56}$
  3. Simplify $\dfrac{72}{96}$
  4. Write $\dfrac{5}{8}$ with denominator 40.
  5. Write $\dfrac{7}{12}$ with denominator 60.
Show Answers

1) $\frac{2}{3}$   2) $\frac{3}{4}$   3) $\frac{3}{4}$   4) $\frac{25}{40}$   5) $\frac{35}{60}$

2.6 Adding & Subtracting Fractions

The Golden Rule

You can only add or subtract fractions that have the same denominator.

If denominators differ → find the LCD (Least Common Denominator = LCM of the denominators), rewrite each fraction, then add/subtract the numerators.

Case 1: Same Denominator

Example 18 — Same denominator

Compute $\dfrac{5}{9} + \dfrac{7}{9}$ and $\dfrac{11}{12} - \dfrac{5}{12}$.

$\dfrac{5}{9} + \dfrac{7}{9} = \dfrac{5+7}{9} = \dfrac{12}{9} = \dfrac{4}{3} = 1\dfrac{1}{3}$   (simplify after adding)
$\dfrac{11}{12} - \dfrac{5}{12} = \dfrac{11-5}{12} = \dfrac{6}{12} = \dfrac{1}{2}$   (GCF(6,12)=6)

Case 2: Different Denominators

Example 19 — Different denominators

Compute $\dfrac{5}{6} + \dfrac{3}{8}$.

Step 1: LCD(6, 8).   $6 = 2 \times 3$,   $8 = 2^3$.   LCD $= 2^3 \times 3 = 24$.
Step 2: Convert each fraction. $$\frac{5}{6} = \frac{5 \times 4}{24} = \frac{20}{24} \qquad \frac{3}{8} = \frac{3 \times 3}{24} = \frac{9}{24}$$
Step 3: Add. $\dfrac{20}{24} + \dfrac{9}{24} = \dfrac{29}{24} = 1\dfrac{5}{24}$

Example 20 — Subtraction with different denominators

Compute $\dfrac{7}{10} - \dfrac{2}{15}$.

LCD(10, 15):   $10 = 2 \times 5$,   $15 = 3 \times 5$.   LCD $= 2 \times 3 \times 5 = 30$.
$$\frac{7}{10} = \frac{21}{30} \qquad \frac{2}{15} = \frac{4}{30}$$
$$\frac{21}{30} - \frac{4}{30} = \frac{17}{30}$$

Case 3: Adding Mixed Numbers

Example 21 — Adding mixed numbers

Compute $2\dfrac{3}{4} + 1\dfrac{2}{3}$.

Method A — Add parts separately:
LCD(4, 3) = 12.   $\dfrac{3}{4} = \dfrac{9}{12}$,   $\dfrac{2}{3} = \dfrac{8}{12}$.
Integer parts: $2 + 1 = 3$.   Fraction parts: $\dfrac{9}{12} + \dfrac{8}{12} = \dfrac{17}{12} = 1\dfrac{5}{12}$.
Total: $3 + 1\dfrac{5}{12} = \mathbf{4\dfrac{5}{12}}$

Case 4: Subtracting Mixed Numbers (with borrowing)

Example 22 — Subtracting mixed numbers

Compute $5\dfrac{1}{3} - 2\dfrac{3}{4}$.

Method — convert to improper fractions:
$5\dfrac{1}{3} = \dfrac{16}{3}$    $2\dfrac{3}{4} = \dfrac{11}{4}$
LCD(3, 4) = 12.   $\dfrac{16}{3} = \dfrac{64}{12}$   $\dfrac{11}{4} = \dfrac{33}{12}$
$$\frac{64}{12} - \frac{33}{12} = \frac{31}{12} = 2\frac{7}{12}$$

⚠ Most common mistake

$\dfrac{1}{3} + \dfrac{1}{4} \neq \dfrac{2}{7}$ — never add the denominators! Denominators tell you the size of each piece. You must find equivalent fractions with the same-size pieces first.

✏ Practice 2.6 — Adding & Subtracting

  1. $\dfrac{3}{8} + \dfrac{5}{8}$
  2. $\dfrac{7}{10} + \dfrac{3}{4}$
  3. $\dfrac{5}{6} - \dfrac{1}{4}$
  4. $3\dfrac{5}{6} - 1\dfrac{7}{9}$
  5. $4\dfrac{2}{5} + 2\dfrac{3}{4}$
  6. A board is $9\dfrac{1}{2}$ feet long. After cutting off $3\dfrac{3}{8}$ feet, how much remains?
Show Answers

1) 1   2) $\frac{29}{20}=1\frac{9}{20}$   3) $\frac{7}{12}$   4) $2\frac{1}{18}$   5) $7\frac{3}{20}$   6) $6\frac{1}{8}$ ft

2.7 Multiplying Fractions

Multiplication Rule

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$

Multiply numerators together and denominators together. No common denominator needed.

Then simplify the result — or better, simplify before multiplying using cross-canceling.

Basic Multiplication

Example 23 — Basic multiplication

Compute $\dfrac{2}{5} \times \dfrac{3}{7}$.

$\dfrac{2}{5} \times \dfrac{3}{7} = \dfrac{2 \times 3}{5 \times 7} = \dfrac{6}{35}$   (already in lowest terms)

Cross-Canceling (Simplify Before Multiplying)

Look for common factors between any numerator and any denominator before you multiply. Cancel them to keep numbers small.

Example 24 — Cross-canceling

Compute $\dfrac{9}{14} \times \dfrac{7}{12}$.

Spot: 9 and 12 share factor 3  |  7 and 14 share factor 7.
Cancel: $9 \div 3 = 3$, $12 \div 3 = 4$.   $7 \div 7 = 1$, $14 \div 7 = 2$.
$$\frac{\overset{3}{\cancel{9}}}{\underset{2}{\cancel{14}}} \times \frac{\overset{1}{\cancel{7}}}{\underset{4}{\cancel{12}}} = \frac{3 \times 1}{2 \times 4} = \frac{3}{8}$$
Without cross-canceling: $\dfrac{63}{168}$ → GCF = 21 → $\dfrac{3}{8}$. Same answer, more work.

Example 25 — Multiplying three fractions

Compute $\dfrac{4}{9} \times \dfrac{3}{8} \times \dfrac{6}{5}$.

Cross-cancel: 4 and 8 (÷4 → 1 and 2); 3 and 9 (÷3 → 1 and 3); 6 and 2 (÷2 → 3 and 1).
$$\frac{1}{3} \times \frac{1}{1} \times \frac{3}{5} = \frac{1 \times 1 \times 3}{3 \times 1 \times 5} = \frac{3}{15} = \frac{1}{5}$$

Multiplying Mixed Numbers

Always convert to improper fractions first. Never multiply mixed numbers directly.

Example 26 — Mixed numbers

Compute $3\dfrac{1}{2} \times 1\dfrac{3}{5}$.

Convert: $3\dfrac{1}{2} = \dfrac{7}{2}$    $1\dfrac{3}{5} = \dfrac{8}{5}$
Cross-cancel: 8 and 2 share factor 2 → $\dfrac{7}{1} \times \dfrac{4}{5}$
$= \dfrac{28}{5} = 5\dfrac{3}{5}$

✏ Practice 2.7 — Multiplying

  1. $\dfrac{3}{5} \times \dfrac{5}{9}$
  2. $\dfrac{4}{9} \times \dfrac{3}{8}$   (use cross-canceling)
  3. $\dfrac{7}{10} \times \dfrac{15}{21}$
  4. $2\dfrac{2}{3} \times 1\dfrac{1}{8}$
  5. A recipe needs $\dfrac{3}{4}$ cup of sugar. If you make $2\dfrac{1}{2}$ batches, how much sugar is needed?
Show Answers

1) $\frac{1}{3}$   2) $\frac{1}{6}$   3) $\frac{1}{2}$   4) $3$   5) $1\frac{7}{8}$ cups

2.8 Dividing Fractions

The Reciprocal

The reciprocal of $\dfrac{a}{b}$ is $\dfrac{b}{a}$ — just flip numerator and denominator.

The product of a number and its reciprocal is always 1: $\dfrac{a}{b} \times \dfrac{b}{a} = 1$.

Examples: reciprocal of $\dfrac{3}{5}$ is $\dfrac{5}{3}$; reciprocal of $4 = \dfrac{4}{1}$ is $\dfrac{1}{4}$.

Division Rule — KCF: Keep, Change, Flip

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

Keep the first fraction unchanged. Change ÷ to ×. Flip the second fraction (take its reciprocal).

Why? Dividing by $\dfrac{c}{d}$ is the same as asking "how many groups of $\dfrac{c}{d}$ fit in $\dfrac{a}{b}$?" Multiplying by the reciprocal gives the same count.

Dividing Two Fractions

Example 27 — Basic division

Compute $\dfrac{5}{6} \div \dfrac{2}{3}$.

Keep $\dfrac{5}{6}$, Change to ×, Flip $\dfrac{2}{3}$ to $\dfrac{3}{2}$.
$$\frac{5}{6} \times \frac{3}{2}$$
Cross-cancel: 3 and 6 share factor 3 → $\dfrac{5}{2} \times \dfrac{1}{2} = \dfrac{5}{4} = 1\dfrac{1}{4}$

Example 28 — Dividing a whole number by a fraction

Compute $6 \div \dfrac{3}{4}$.

Write $6 = \dfrac{6}{1}$. Then KCF: $\dfrac{6}{1} \times \dfrac{4}{3}$.
Cross-cancel: 6 and 3 → $\dfrac{2}{1} \times \dfrac{4}{1} = 8$.
Answer: 8   (there are 8 groups of $\frac{3}{4}$ in 6)

Dividing Mixed Numbers

Example 29 — Mixed numbers

Compute $4\dfrac{1}{2} \div 1\dfrac{1}{3}$.

Convert: $\dfrac{9}{2} \div \dfrac{4}{3}$
KCF: $\dfrac{9}{2} \times \dfrac{3}{4} = \dfrac{27}{8} = 3\dfrac{3}{8}$

Example 30 — Real-world division

A board $7\dfrac{1}{2}$ feet long is cut into pieces each $\dfrac{3}{4}$ foot. How many pieces?

$7\dfrac{1}{2} \div \dfrac{3}{4} = \dfrac{15}{2} \div \dfrac{3}{4} = \dfrac{15}{2} \times \dfrac{4}{3}$
Cross-cancel: $15$ and $3$ (÷3) → 5, 1.   $4$ and $2$ (÷2) → 2, 1.
$= \dfrac{5 \times 2}{1 \times 1} = 10$   Answer: 10 pieces

⚠ Flip the RIGHT fraction

In $\dfrac{a}{b} \div \dfrac{c}{d}$, you flip the divisor (the fraction after ÷). Never flip the first fraction. $\dfrac{5}{6} \div \dfrac{2}{3}$ → flip $\dfrac{2}{3}$, not $\dfrac{5}{6}$.

✏ Practice 2.8 — Dividing

  1. $\dfrac{3}{4} \div \dfrac{1}{2}$
  2. $\dfrac{5}{6} \div \dfrac{10}{3}$
  3. $8 \div \dfrac{2}{5}$
  4. $4\dfrac{1}{5} \div 1\dfrac{2}{3}$
  5. How many $\dfrac{1}{3}$-cup servings are in $5$ cups of cereal?
  6. A $6\dfrac{3}{4}$-mile trail is divided into equal sections of $\dfrac{3}{8}$ mile. How many sections?
Show Answers

1) $\frac{3}{2}=1\frac{1}{2}$   2) $\frac{1}{4}$   3) 20   4) $\frac{63}{25}=2\frac{13}{25}$   5) 15 servings   6) 18 sections

2.9 Decimals & Conversions

Place Value

PositionNameValueExample (in 5.3472)
1st after decimalTenths$\frac{1}{10}$3
2nd after decimalHundredths$\frac{1}{100}$4
3rd after decimalThousandths$\frac{1}{1000}$7
4th after decimalTen-thousandths$\frac{1}{10000}$2

Fraction → Decimal

Divide numerator by denominator (long division or calculator).

Example 31 — Terminating decimal

Convert $\dfrac{3}{8}$ to a decimal.

$3 \div 8 = 0.375$   (terminating — the decimal ends)
When does a fraction give a terminating decimal? When the denominator (in lowest terms) has only factors of 2 and 5.   $8 = 2^3$ ✓

Example 32 — Repeating decimal

Convert $\dfrac{5}{11}$ to a decimal.

$5 \div 11 = 0.454545\ldots = 0.\overline{45}$
The denominator 11 has the prime factor 11 (not just 2 or 5) → repeating decimal.

Decimal → Fraction

Example 33 — Terminating decimal to fraction

Convert $0.064$ to a fraction in lowest terms.

Step 1: Read the place value: last digit (4) is in the thousandths place → denominator is 1000.
$0.064 = \dfrac{64}{1000}$
Step 2: Simplify. GCF(64, 1000) = 8.   $\dfrac{64 \div 8}{1000 \div 8} = \dfrac{8}{125}$

Repeating Decimal → Fraction

Example 34 — One repeating digit $0.\overline{3}$

Convert $0.\overline{3}$ to a fraction.

Step 1: Let $x = 0.\overline{3} = 0.3333\ldots$
Step 2: Multiply by $10$ (one repeating digit): $10x = 3.3333\ldots$
Step 3: Subtract: $10x - x = 3.333\ldots - 0.333\ldots$ → $9x = 3$ → $x = \dfrac{3}{9} = \dfrac{1}{3}$

Example 35 — Two repeating digits $0.\overline{27}$

Convert $0.\overline{27}$ to a fraction.

Step 1: $x = 0.272727\ldots$
Step 2: Two-digit repeat → multiply by $100$: $100x = 27.272727\ldots$
Step 3: Subtract: $99x = 27$ → $x = \dfrac{27}{99} = \dfrac{3}{11}$

Example 36 — Mixed repeat $0.1\overline{6}$

Convert $0.1\overline{6} = 0.1666\ldots$ to a fraction.

Step 1: $x = 0.1\overline{6}$. Multiply by 10: $10x = 1.\overline{6} = 1.666\ldots$
Step 2: Multiply by 100: $100x = 16.\overline{6} = 16.666\ldots$
Step 3: Subtract: $100x - 10x = 16.666\ldots - 1.666\ldots$ → $90x = 15$ → $x = \dfrac{15}{90} = \dfrac{1}{6}$

Quick Reference — Common Fraction–Decimal Conversions

FractionDecimal
$\frac{1}{2}$0.5
$\frac{1}{4}$0.25
$\frac{3}{4}$0.75
$\frac{1}{5}$0.2
$\frac{1}{8}$0.125
FractionDecimal
$\frac{1}{3}$$0.\overline{3}$
$\frac{2}{3}$$0.\overline{6}$
$\frac{1}{6}$$0.1\overline{6}$
$\frac{1}{9}$$0.\overline{1}$
$\frac{5}{9}$$0.\overline{5}$

✏ Practice 2.9 — Decimals & Conversions

  1. Convert $\dfrac{7}{16}$ to a decimal.
  2. Convert $\dfrac{5}{12}$ to a decimal (identify as terminating or repeating).
  3. Convert $0.035$ to a fraction in lowest terms.
  4. Convert $0.\overline{4}$ to a fraction.
  5. Convert $0.\overline{18}$ to a fraction.
  6. Convert $0.2\overline{4}$ to a fraction.
Show Answers

1) 0.4375   2) $0.41\overline{6}$ (repeating)   3) $\frac{7}{200}$   4) $\frac{4}{9}$   5) $\frac{2}{11}$   6) $90x-9x=22-2$, $81x=22$, $x=\frac{22}{81}$

Chapter Summary

TopicKey Idea
Types of fractionsProper (num < den), Improper (num ≥ den), Mixed (integer + proper)
Prime factorizationWrite as product of primes. Use factor trees or ladder method.
GCFCommon primes, lowest exponents. Used to simplify fractions.
LCMAll primes, highest exponents. Used for common denominators.
SimplifyingDivide num and den by GCF.
Adding/SubtractingRequires same denominator (find LCD = LCM).
MultiplyingMultiply straight across. Cross-cancel before multiplying.
DividingKCF — Keep, Change, Flip. Always convert mixed numbers first.
Terminating decimalDenominator (lowest terms) has only factors of 2 and 5.
Repeating → fractionLet x = decimal, multiply by 10^n (n = length of repeating block), subtract.
← Chapter 1: Integers & Number Sense Chapter 3: Ratios, Rates & Proportions →

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