📐 Concept diagram

00-02 — Fractions

Phase: 0 — Arithmetic & Number Foundations Subject: 00-02 Prerequisites: 00-01 — Whole Number Arithmetic Next subject: 00-03 — Decimals

Learning Objectives

By the end of this subject, you will be able to:

1. Identify proper fractions, improper fractions, and mixed numbers, and convert fluently between all three forms
2. Generate equivalent fractions by multiplying or dividing both numerator and denominator by the same non-zero number, and simplify fractions to lowest terms using the GCF
3. Add and subtract fractions with both like and unlike denominators, including mixed numbers
4. Multiply and divide fractions confidently, including reciprocals for division and cross-cancellation for efficiency
5. Convert between fractions and decimals, and apply fraction operations to real-world problems

Core Content

1. What Is a Fraction?

A fraction represents a part of a whole. It is written as:

$ numerator
-----------
denominator
$

• The numerator (top) tells you how many parts you have.
• The denominator (bottom) tells you how many equal parts the whole is divided into.

Concrete analogy: If you cut a pizza into 8 equal slices and eat 3 slices, you've eaten 3/8 (three-eighths) of the pizza. The denominator (8) is the total number of slices; the numerator (3) is how many you took.

Formally, a fraction a/b means: a parts of size 1/b each. The quantity 1/b is the unit fraction — it's what you get when you split 1 into b equal pieces.

Vocabulary:

A proper fraction is always less than 1. An improper fraction is ≥ 1. A mixed number is another way to write an improper fraction greater than 1.

2. Equivalent Fractions

Two fractions are equivalent if they represent the same amount. The golden rule:

Multiplying or dividing both the numerator and denominator by the SAME non-zero number produces an equivalent fraction.

$a       a × k
—  =   —————       (k ≠ 0)
b       b × k
$

Why this works: Multiplying by k/k is multiplying by 1 (since any number divided by itself is 1, as long as k ≠ 0). Multiplying by 1 doesn't change the value.

Examples of generating equivalent fractions:

$1     1 × 2    2      1    1 × 5    5
—  =  ————— = —      —  =  ————— = ——
2     2 × 2    4      2    2 × 5    10

3     3 × 4    12
—  =  ————— = ——
5     5 × 4    20
$

Visual check: 1/2 of a chocolate bar is the same amount as 2/4 of it (you just cut it into smaller pieces).

3. Simplifying (Reducing) Fractions

Simplifying means finding the equivalent fraction with the smallest possible numerator and denominator. This is the fraction in lowest terms (or simplest form).

Method: 1. Find the GCF of the numerator and denominator (see 00-01 for GCF review). 2. Divide both the numerator and denominator by the GCF.

Example: Simplify 24/36

$Step 1: GCF(24, 36)
  24 = 2³ × 3
  36 = 2² × 3²
  GCF = 2² × 3 = 4 × 3 = 12

Step 2: Divide both by 12
  24 ÷ 12    2
  ———————  = —
  36 ÷ 12    3
$

Check: 24/36 = 24 ÷ 12 / 36 ÷ 12 = 2/3. And 2 ÷ 3 = 0.666..., 24 ÷ 36 = 0.666... ✓

Shortcut method: If you can't find the GCF immediately, simplify in steps:

$24    12     6     2     2
—— = —— = —— = —— = ——
36    18     9     3     3
$

(Divide by 2, then by 2 again, then by 3 — same result, just slower.)

Common pitfall: Some people cancel individual digits instead of dividing both numbers by the same factor. For example, "simplifying" 16/64 to 1/4 by crossing out the 6s — this happens to give the right answer accidentally, but it's the WRONG method. Using the correct method: GCF(16, 64) = 16, so 16/64 = 1/4. ✓ The method works, but "digit cancellation" is not mathematics — it's coincidence.

4. Converting Between Improper Fractions and Mixed Numbers

Improper → Mixed: 1. Divide the numerator by the denominator (integer division with remainder). 2. The quotient is the whole number part. 3. The remainder is the numerator of the fractional part. 4. The denominator stays the same.

Formula: If a ÷ b = q remainder r, then:

$a                   r
—  =  q +  —        where 0 ≤ r < b
b          b
$

Example: Convert 17/5 to a mixed number.

$17 ÷ 5 = 3 remainder 2
So 17/5 = 3 2/5
$

Check: 3 2/5 = 3 + 2/5 = 15/5 + 2/5 = 17/5 ✓

Mixed → Improper: 1. Multiply the whole number by the denominator. 2. Add the numerator. 3. Put the result over the original denominator.

Formula:

$   c         a × c + b
a  —    =    —————————
   b             c
$

Derivation: a b/c means a + b/c. Convert the whole number to a fraction with denominator c: a = a × c / c. Then add: (a × c)/c + b/c = (a × c + b)/c.

Example: Convert 4 2/3 to an improper fraction.

$4 × 3 = 12
12 + 2 = 14
4 2/3 = 14/3
$

⚠️ THIS IS CRITICAL — Converting mixed numbers to improper fractions is an essential skill you'll use constantly in algebra when working with rational expressions, equations with fractions, and any problem where you need common denominators.

5. Adding and Subtracting Fractions

Case 1: Same Denominator (Like Fractions)

When denominators are the same, add or subtract only the numerators and keep the denominator unchanged.

$a     c     a + c
—  +  —  =  —————
b     b       b

a     c     a − c
—  −  —  =  —————
b     b       b
$

Why this works: If you have 2/8 of a pizza and add 3/8, you're adding like units. It's no different from 2 apples + 3 apples = 5 apples — the "eighths" are the unit.

Example: 2/9 + 5/9 = (2 + 5)/9 = 7/9

Case 2: Different Denominators (Unlike Fractions)

When denominators differ, you must first convert both fractions to equivalent fractions with a common denominator. The most efficient common denominator is the LCM of the two denominators.

Method: 1. Find the LCM of the denominators. 2. Convert each fraction to an equivalent fraction with the LCM as its denominator. 3. Add or subtract the numerators; keep the common denominator. 4. Simplify the result if possible.

Example: Calculate 2/3 + 1/4

$Step 1: LCM(3, 4) = 12

Step 2: Convert both fractions
  2     2 × 4     8
  —  =  —————  = ——
  3     3 × 4    12

  1     1 × 3     3
  —  =  —————  = ——
  4     4 × 3    12

Step 3: Add numerators
   8      3     11
  ——  +  ——  =  ——
  12     12     12

Step 4: Simplify? 11/12 is already in lowest terms (GCF(11, 12) = 1).
$

Answer: 11/12

The LCM shortcut: For any two numbers a and b:

LCM(a,b) = (a × b) / GCF(a,b)

Common pitfall — the "butterfly trap": Some students learn to add fractions by drawing a butterfly — cross-multiplying and multiplying denominators. For 2/3 + 1/4 this gives (2×4 + 1×3)/(3×4) = (8+3)/12 = 11/12. This works because the product of denominators IS a common denominator, so it produces the right answer. BUT it's NOT the LCM, so for 1/6 + 1/4 it gives (1×4 + 1×6)/(6×4) = 10/24 = 5/12. The LCM approach: LCM(6,4) = 12, so 2/12 + 3/12 = 5/12 — same answer but smaller numbers to work with. Both are correct; the LCM approach is cleaner and avoids extra simplification steps.

Case 3: Adding/Subtracting Mixed Numbers

Method 1 (recommended): Convert to improper fractions, then add/subtract, then convert back.

Example: Calculate 2 1/3 + 1 3/4

$Step 1: Convert to improper fractions
  2 1/3 = (2×3 + 1)/3 = 7/3
  1 3/4 = (1×4 + 3)/4 = 7/4

Step 2: Find LCM(3, 4) = 12
  7/3 = 28/12
  7/4 = 21/12

Step 3: Add
  28/12 + 21/12 = 49/12

Step 4: Convert back to mixed number
  49 ÷ 12 = 4 remainder 1
  49/12 = 4 1/12
$

Answer: 4 1/12

6. Multiplying Fractions

Multiplying fractions is simpler than addition — you don't need a common denominator.

$a     c     a × c
—  ×  —  =  —————
b     d     b × d
$

Rule: Multiply numerators together and denominators together.

Why this works: 2/3 × 4/5 means "two-thirds of four-fifths." Imagine a rectangle divided into 3 rows and 5 columns (15 equal parts). Shade 4 columns (4/5) in one direction, then 2 rows (2/3) in the other. The overlap is 2 × 4 = 8 pieces out of 3 × 5 = 15 total. So 2/3 × 4/5 = 8/15.

Example: Calculate 3/7 × 2/5

$3 × 2     6
—————  =  ——
7 × 5    35
$

Cross-cancellation (efficiency tip):

Before multiplying, cancel any common factors between any numerator and any denominator.

$ 4     15     4 × 15
——  ×  ——  =  ——————     ← Don't multiply yet
 9     8      9 × 8

Cancel: 4 and 8 share factor 4 → 1 and 2
Cancel: 15 and 9 share factor 3 → 5 and 3

  4¹   15⁵    1 × 5     5
 ——  × ——  = —————  = ——
  9₃    8²    3 × 2     6
$

Without cancellation: (4 × 15)/(9 × 8) = 60/72 = (÷12) = 5/6. Same answer, but with cross-cancellation you avoid multiplying large numbers and then simplifying.

Multiplying with Mixed Numbers

Convert mixed numbers to improper fractions first, then multiply.

Example: 2 1/2 × 1 1/3

$2 1/2 = 5/2
1 1/3 = 4/3

5/2 × 4/3 = (5×4)/(2×3) = 20/6 = 10/3 = 3 1/3
$

Multiplying a Fraction by a Whole Number

A whole number n can be written as n/1:

$3 × 2/5 = 3/1 × 2/5 = (3×2)/(1×5) = 6/5 = 1 1/5
$

7. Dividing Fractions

Division is multiplication by the reciprocal. The reciprocal of a/b is b/a (flip the fraction).

$a     c     a     d     a × d
—  ÷  —  =  —  ×  —  =  —————
b     d     b     c     b × c
$

Derivation — why keep-change-flip works:

Division is the inverse of multiplication. a/b ÷ c/d = ? means: "what number × c/d = a/b?"

Let the answer be x. Then x × (c/d) = a/b. Solve: x = a/b × d/c. This is exactly "multiply by the reciprocal."

The common mnemonic is KCF: Keep the first fraction, Change ÷ to ×, Flip the second fraction.

Example: Calculate 3/4 ÷ 2/5

$3/4 ÷ 2/5 = 3/4 × 5/2 = (3×5)/(4×2) = 15/8 = 1 7/8
$

Visual understanding: 3/4 ÷ 2/5 means "how many 2/5 portions fit into 3/4?" Since 2/5 = 0.4 and 3/4 = 0.75, we're asking how many 0.4s fit into 0.75. The answer 15/8 = 1.875 means 1.875 portions of size 0.4 fit into 0.75. Check: 1.875 × 0.4 = 0.75 ✓.

Example dividing by a whole number: 5/6 ÷ 2

$5/6 ÷ 2 = 5/6 ÷ 2/1 = 5/6 × 1/2 = 5/12
$

Check: If you split 5/6 into 2 equal parts, each part is 5/12. And 5/12 + 5/12 = 10/12 = 5/6 ✓.

Common pitfall: Dividing by 1/2 is NOT the same as dividing by 2. - 6 ÷ 2 = 3 (splitting into 2 equal groups) - 6 ÷ 1/2 = 6 × 2/1 = 12 (asking how many halves fit into 6 — there are 12 halves in 6 wholes)

⚠️ THIS IS CRITICAL — Division by a fraction appears everywhere in algebra: rational expressions, rates, proportions, and solving fractional equations. Mastering "multiply by the reciprocal" now saves enormous confusion later.

8. Fraction-Decimal Conversion

Any fraction a/b can be converted to a decimal by performing the division a ÷ b.

Fraction → Decimal: Divide numerator by denominator.

$3/8 = 3 ÷ 8 = 0.375
1/3 = 1 ÷ 3 = 0.333... (recurring)
$

Decimal → Fraction: 1. Write the decimal as a fraction with denominator 10, 100, 1000, etc. (based on the number of decimal places). 2. Simplify.

$0.75 = 75/100 = (÷ 25) = 3/4
0.625 = 625/1000 = (÷ 125) = 5/8
0.2 = 2/10 = 1/5
$

For recurring decimals: This is an advanced topic covered in 00-03.

Key Terms

• Division
• Equivalent fraction
• Equivalent fractions
• Fraction-decimal conversion
• Improper fraction
• Mixed number
• Multiplication
• Proper fraction
• Unit fraction

Worked Examples

Example 1: Simplify 84/126

Problem: Reduce 84/126 to its lowest terms.

Solution:

$Step 1: Find GCF(84, 126)
  84 = 2² × 3 × 7
  126 = 2 × 3² × 7

  Common primes: 2¹ (min exponent 1), 3¹ (min exponent 1), 7¹ (min exponent 1)
  GCF = 2 × 3 × 7 = 42

Step 2: Divide both numerator and denominator by 42
  84 ÷ 42 = 2
  126 ÷ 42 = 3

84/126 = 2/3
$

Check: 84 ÷ 126 = 0.666..., 2 ÷ 3 = 0.666... ✓

Answer: 2/3

Example 2: Add 3/5 + 7/10 + 1/2

Problem: Add three fractions with different denominators.

Solution:

$Step 1: Find the LCM of all three denominators
  Denominators: 5, 10, 2
  LCM(5, 10, 2)
  5 = 5, 10 = 2 × 5, 2 = 2
  LCM = 2 × 5 = 10

Step 2: Convert each fraction to denominator 10
  3/5 = (3×2)/(5×2) = 6/10
  7/10 already has denominator 10
  1/2 = (1×5)/(2×5) = 5/10

Step 3: Add numerators
  6/10 + 7/10 + 5/10 = (6+7+5)/10 = 18/10

Step 4: Simplify
  18/10 = 9/5 = 1 4/5
$

Answer: 9/5 or 1 4/5

Example 3: Compute 2 2/3 ÷ 1 1/6

Problem: Divide two mixed numbers.

Solution:

$Step 1: Convert both to improper fractions
  2 2/3 = (2×3 + 2)/3 = 8/3
  1 1/6 = (1×6 + 1)/6 = 7/6

Step 2: Apply KCF (Keep-Change-Flip)
  8/3 ÷ 7/6 = 8/3 × 6/7

Step 3: Cross-cancel before multiplying
  8 and 7 share no common factor
  6 and 3 share factor 3: 6→2, 3→1

  8¹/1 × 2/7 = (8×2)/(1×7) = 16/7

Step 4: Convert to mixed number
  16 ÷ 7 = 2 remainder 2
  16/7 = 2 2/7
$

Check: 2 2/7 × 1 1/6 = 16/7 × 7/6 = (16×7)/(7×6) = 16/6 = 8/3 = 2 2/3 ✓

Answer: 16/7 or 2 2/7

Example 4: Solve a Real-World Fraction Problem

Problem: A recipe requires 3/4 cup of sugar for one batch of cookies. If you want to make 2 1/2 batches, how much sugar do you need?

Solution:

$This is multiplication: (3/4) × (2 1/2)

Step 1: Convert the mixed number
  2 1/2 = 5/2

Step 2: Multiply
  3/4 × 5/2 = (3×5)/(4×2) = 15/8

Step 3: Convert to mixed number
  15 ÷ 8 = 1 remainder 7
  15/8 = 1 7/8
$

Answer: 1 7/8 cups of sugar

Practice Problems

(Answers are below. Try each problem before checking.)

Problem 1: Simplify 48/72 to lowest terms.

Problem 2: Convert 29/6 to a mixed number.

Problem 3: Calculate: 5/8 + 2/3

Problem 4: Calculate: 3 1/4 − 1 5/6

Problem 5: Multiply: 6/7 × 14/15 (use cross-cancellation)

Problem 6: Divide: 5/9 ÷ 2/3

Problem 7: Mary has 2 3/4 metres of ribbon. She cuts it into pieces each 5/8 metre long. How many pieces does she get?

Summary

1. A fraction a/b represents a parts out of b equal parts; proper fractions (a < b) are less than 1, improper fractions (a ≥ b) are ≥ 1, and mixed numbers combine a whole number with a proper fraction
2. Equivalent fractions are generated by multiplying or dividing both numerator and denominator by the same non-zero number; we simplify by dividing both by their GCF
3. To add or subtract fractions, find a common denominator (the LCM of the denominators), convert each fraction, then add/subtract numerators and simplify
4. Multiplication is straightforward (numerator × numerator, denominator × denominator); use cross-cancellation to cancel common factors before multiplying
5. Division uses KCF — Keep the first fraction, Change ÷ to ×, Flip the second fraction (multiply by the reciprocal). Always convert mixed numbers to improper fractions first
6. Fraction-decimal conversion: a/b = a ÷ b; decimals convert to fractions by writing over a power of 10 and simplifying

Pitfalls

• Adding denominators as well as numerators. 1/2 + 1/3 is NOT 2/5. You must find a common denominator first; only the numerators are added. The denominator stays the same once it's common.
• Forgetting to flip the second fraction when dividing. Division is multiplication by the reciprocal: a/b ÷ c/d = a/b × d/c. Skipping the flip (or flipping the wrong fraction) is one of the most persistent errors.
• Cancelling digits instead of dividing by common factors. \"Simplifying\" 16/64 to 1/4 by crossing out the 6s happens to work by coincidence but is not a valid method. Always divide numerator and denominator by their GCF.
• Confusing a fraction with a ratio. 3/4 of a pizza means 3 parts out of 4 total; a ratio 3:4 compares two separate quantities (e.g., 3 boys to 4 girls). The denominator in a fraction represents the whole, not just the other part.
• Neglecting to simplify the final answer. Many students do all the hard work of finding LCMs and converting fractions, then leave 18/24 as the answer instead of simplifying to 3/4.

Quiz

Answer each question, then read the explanation for your choice.

Q1: Which of the following is an improper fraction?

A) 3/4 B) 5/6 C) 8/8 D) 1/2

Q2: Simplify 36/60 to lowest terms.

A) 6/10 B) 9/15 C) 3/5 D) 2/3

Q3: Calculate: 2/5 + 1/3

A) 3/8 B) 3/15 C) 11/15 D) 1/5

Q4: Convert 5 3/4 to an improper fraction.

A) 15/4 B) 23/4 C) 20/4 D) 17/4

Q5: Multiply: 3/8 × 4/9

A) 12/72 B) 1/6 C) 7/17 D) 3/2

Q6: Divide: 7/10 ÷ 3/5

A) 21/50 B) 7/6 C) 10/21 D) 1 1/6

Q7: Which pair of fractions are equivalent?

A) 2/5 and 6/15 B) 3/4 and 9/16 C) 1/3 and 2/9 D) 5/6 and 10/13

Q8: A water tank contains 3/4 of its capacity. After using 2/5 of a tank, how much water remains as a fraction of the tank's capacity?

A) 1/20 B) 7/20 C) 3/20 D) 5/9

Next Steps

Move on to 00-03 — Decimals to learn about place value in decimals, terminating and recurring decimals, rounding, significant figures, decimal operations, and scientific notation.

Q5: Simplify 48/72 to lowest terms.

A) 2/3 B) 4/6 C) 8/12 D) 6/9

Answer: A) 2/3

GCF(48, 72) = 24. 48 ÷ 24 = 2, 72 ÷ 24 = 3. Result: 2/3.