Simplifying Fractions: CorbettMaths Textbook Answers Guide

Complete textbook answers and detailed explanations for simplifying fractions: methods, definitions, and visual representations for 4th grade math students.

Complete Summary: Simplifying Fractions
Detailed Learning Summary

This comprehensive guide covers everything you need to know about simplifying fractions, including definitions, rules, methods, and practice problems with detailed solutions.

Definitions

Numerator: The top number in a fraction representing the number of parts taken.

Denominator: The bottom number in a fraction representing the total number of equal parts.

Simplifying Fractions: Reducing a fraction to its lowest terms by dividing both numerator and denominator by their greatest common divisor.

Greatest Common Divisor (GCD): The largest number that divides both numerator and denominator evenly.

Core Rules

• Basic Rule: \(\frac{a}{b} = \frac{a ÷ d}{b ÷ d}\) where \(d = \gcd(a,b)\)

• Lowest Terms: A fraction is in lowest terms when \(\gcd(numerator, denominator) = 1\)

• Equivalence: Simplified fractions have the same value as the original fraction

• Verification: Use cross multiplication to verify equivalence: \(\frac{a}{b} = \frac{c}{d}\) means \(a×d = b×c\)

Methods

• Factor Listing: List all factors and find the greatest common one

• Prime Factorization: Break numbers into prime factors and cancel common ones

• Euclidean Algorithm: Efficient method for large numbers

• Quick Division: Divide by common factors (2, 3, 5) repeatedly

Tips & Tricks

• Start with small primes (2, 3, 5) for quick simplification

• Use divisibility rules: numbers ending in 0,2,4,6,8 are divisible by 2

• If sum of digits is divisible by 3, the number is divisible by 3

• Always verify your answer by cross-multiplying

Textbook Answers: Problems 1 to 3
1 Basic Fraction Reduction
Problem 1
Simplify: \(\frac{15}{25}\)
Definition:

Simplifying fractions: Reducing a fraction to its lowest terms by dividing both numerator and denominator by their Greatest Common Divisor (GCD)

Solution method:
  1. Find factors of numerator (15): 1, 3, 5, 15
  2. Find factors of denominator (25): 1, 5, 25
  3. Identify the Greatest Common Divisor: 5
  4. Divide both numerator and denominator by GCD
Original Fraction
\(\frac{15}{25}\)
GCD
GCD(15,25)=5
Simplified
\(\frac{3}{5}\)
Step 1: Find factors of 15

15 = 1 × 15, 3 × 5

Factors of 15: 1, 3, 5, 15

Step 2: Find factors of 25

25 = 1 × 25, 5 × 5

Factors of 25: 1, 5, 25

Step 3: Identify GCD

Common factors: 1, 5

Greatest Common Divisor: 5

Step 4: Divide by GCD

\(\frac{15}{25} = \frac{15 ÷ 5}{25 ÷ 5} = \frac{3}{5}\)

\(\frac{15}{25} = \frac{3}{5}\)
Textbook Answer:

\(\frac{15}{25} = \frac{3}{5}\)

Applied rules:

• GCD Method: Divide both numerator and denominator by their greatest common divisor

• Equivalence: The value of the fraction remains unchanged

• Lowest Terms: A fraction is in lowest terms when GCD of numerator and denominator is 1

2 Medium Difficulty
Problem 2
Simplify: \(\frac{28}{42}\)
Definition:

Prime factorization: Breaking down numbers into their prime factors to identify common factors

Original Fraction
\(\frac{28}{42}\)
Prime Factors
\(\frac{2^2 × 7}{2 × 3 × 7}\)
Simplified
\(\frac{2}{3}\)
Step 1: Find prime factors of 28

28 = 2 × 14 = 2 × 2 × 7 = \(2^2 × 7\)

Step 2: Find prime factors of 42

42 = 2 × 21 = 2 × 3 × 7

Step 3: Cancel common factors

\(\frac{2^2 × 7}{2 × 3 × 7} = \frac{2^{2-1} × 7^{1-1}}{3} = \frac{2}{3}\)

Step 4: Verify result

\(\frac{2}{3}\) cannot be simplified further since GCD(2,3) = 1

\(\frac{28}{42} = \frac{2}{3}\)
Textbook Answer:

\(\frac{28}{42} = \frac{2}{3}\)

Applied rules:

• Prime Factorization: Express numbers as products of prime factors

• Canceling Common Factors: Remove identical factors from numerator and denominator

• Lowest Terms: Result is in lowest terms when no common factors remain

3 Advanced Simplification
Problem 3
Simplify: \(\frac{45}{75}\)
Definition:

Euclidean Algorithm: An efficient method to find GCD of large numbers by repeated division

Original Fraction
\(\frac{45}{75}\)
GCD Process
GCD(45,75)=15
Result
\(\frac{3}{5}\)
Step 1: Apply Euclidean Algorithm

\(75 ÷ 45 = 1\) remainder \(30\)

\(45 ÷ 30 = 1\) remainder \(15\)

\(30 ÷ 15 = 2\) remainder \(0\)

So GCD(45,75) = 15

Step 2: Divide by GCD

\(\frac{45}{75} = \frac{45 ÷ 15}{75 ÷ 15} = \frac{3}{5}\)

Step 3: Verify equivalence

\(3 × 75 = 225\) and \(5 × 45 = 225\), so cross products are equal

Step 4: Confirm lowest terms

GCD(3,5) = 1, so \(\frac{3}{5}\) is in lowest terms

\(\frac{45}{75} = \frac{3}{5}\)
Textbook Answer:

\(\frac{45}{75} = \frac{3}{5}\)

Applied rules:

• Euclidean Algorithm: Efficient GCD calculation for large numbers

• Cross Multiplication: Verification method for equivalent fractions

• Lowest Terms Verification: GCD of numerator and denominator equals 1

Core Concepts, Methods, and Rules
\(\frac{a}{b} = \frac{a ÷ d}{b ÷ d}\) where \(d = \gcd(a,b)\)
Fundamental Simplification Rule
Definition
Simplified Fraction
A fraction is simplified when numerator and denominator have no common factors other than 1
Method
GCD Approach
Divide both parts by greatest common divisor
Technique
Prime Factorization
Cancel common prime factors in numerator and denominator
Key definitions:

Numerator: Top number in a fraction (the dividend)

Denominator: Bottom number in a fraction (the divisor)

Greatest Common Divisor (GCD): Largest number that divides both numerator and denominator evenly

Equivalent Fractions: Fractions that represent the same value but have different numerators and denominators

Lowest Terms: A fraction is in lowest terms when the only common factor of numerator and denominator is 1

Complete methodology:
  1. Identify the fraction: Recognize the numerator and denominator
  2. Find GCD: Use factor listing, prime factorization, or Euclidean algorithm
  3. Divide both parts: Divide numerator and denominator by GCD
  4. Verify result: Check that GCD of new numerator and denominator is 1
  5. Check equivalence: Use cross multiplication to confirm
Tip 1: Start by checking if both numbers are even (divide by 2).
Tip 2: If both numbers end in 0 or 5, divide by 5.
Tip 3: Sum of digits rule: If sum is divisible by 3, the number is divisible by 3.
Tip 4: Always verify by cross-multiplying: \(\frac{a}{b} = \frac{c}{d}\) means \(a×d = b×c\).
Tip 5: A fraction is fully simplified when numerator and denominator are both prime numbers or coprime.
Common errors: Forgetting to divide both numerator and denominator, not checking for lowest terms, stopping too early before full simplification, arithmetic errors in division.
Key notes: A fraction is fully simplified when the only common factor of numerator and denominator is 1. Always double-check your work!
Essential formulas and rules:

• Basic Simplification: \(\frac{a}{b} = \frac{a ÷ \gcd(a,b)}{b ÷ \gcd(a,b)}\)

• Verification: Two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) are equivalent if \(a×d = b×c\)

• Lowest Terms: A fraction \(\frac{a}{b}\) is in lowest terms if \(\gcd(a,b) = 1\)

• Prime Factorization: Cancel identical prime factors from numerator and denominator

• Euclidean Algorithm: For finding GCD of large numbers efficiently

Fraction Simplification Infographic

📊
Visual Fraction Representation
15
25

Represents 15 parts out of 25 total parts

Factor Tree Example
15
3
5
25
5
5
Lowest Terms Check

GCD of numerator and denominator = 1

\(\gcd(3,5) = 1\)

Therefore \(\frac{3}{5}\) is in lowest terms

Simplification Workflow
1
Identify
2
Find GCD
3
Divide
4
Verify
5
Confirm
Common Mistakes
  • Only dividing the numerator
  • Not checking if result is in lowest terms
  • Missing common factors
  • Arithmetic errors in division
  • Stopping before full simplification
📚 Master the GCD method for efficient fraction simplification!
Textbook Answers: Problems 4 to 5
4 Complex Simplification
Problem 4
Simplify: \(\frac{56}{84}\)
Definition:

Multiple simplification steps: Some fractions require multiple rounds of division before reaching lowest terms

Original
\(\frac{56}{84}\)
First Step
\(\frac{28}{42}\)
Second Step
\(\frac{2}{3}\)
Step 1: Initial simplification

Both 56 and 84 are divisible by 2

\(\frac{56}{84} = \frac{56 ÷ 2}{84 ÷ 2} = \frac{28}{42}\)

Step 2: Continue simplification

Both 28 and 42 are divisible by 14

\(\frac{28}{42} = \frac{28 ÷ 14}{42 ÷ 14} = \frac{2}{3}\)

Step 3: Verify lowest terms

Factors of 2: 1, 2

Factors of 3: 1, 3

Common factor: 1, so \(\frac{2}{3}\) is in lowest terms

Step 4: Alternative direct approach

GCD(56,84) = 28

\(\frac{56}{84} = \frac{56 ÷ 28}{84 ÷ 28} = \frac{2}{3}\)

\(\frac{56}{84} = \frac{2}{3}\)
Textbook Answer:

\(\frac{56}{84} = \frac{2}{3}\)

Applied rules:

• Progressive Simplification: Multiple division steps if needed

• Direct GCD Method: Single-step approach using greatest common divisor

• Verification: Confirm final fraction has GCD of 1

5 Real-World Application
Problem 5
A pizza was cut into 36 slices. If 24 slices were eaten, what fraction of the pizza was eaten? Simplify your answer.
Definition:

Real-world fractions: Represent ratios of quantities in practical scenarios

Fraction Setup
\(\frac{24}{36}\)
GCD Calculation
GCD(24,36)=12
Simplified
\(\frac{2}{3}\)
Step 1: Set up the fraction

Slices eaten = 24

Total slices = 36

Fraction eaten = \(\frac{24}{36}\)

Step 2: Find GCD of 24 and 36

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

GCD = 12

Step 3: Simplify the fraction

\(\frac{24}{36} = \frac{24 ÷ 12}{36 ÷ 12} = \frac{2}{3}\)

Step 4: Interpret the result

\(\frac{2}{3}\) of the pizza was eaten

This means 2 out of every 3 slices were eaten

\(\frac{24}{36} = \frac{2}{3}\) of the pizza was eaten
Textbook Answer:

\(\frac{2}{3}\) of the pizza was eaten

Applied rules:

• Fraction Setup: Numerator = part, Denominator = whole

• Real-world Context: Interpret mathematical results in practical terms

• Simplification: Always reduce to lowest terms for clarity

Questions & Answers

Question: Why do we need to simplify fractions? Why can't we just leave them as they are?

Answer: Great question! There are several important reasons to simplify fractions:

  • Standard Form: Simplified fractions are considered the standard way to write fractions. Just like we write 2 instead of 4/2, we prefer \(\frac{2}{3}\) over \(\frac{4}{6}\).
  • Easier Comparison: It's much easier to compare \(\frac{2}{3}\) and \(\frac{3}{4}\) than \(\frac{4}{6}\) and \(\frac{6}{8}\).
  • Clearer Understanding: A simplified fraction gives a clearer picture of the relationship between the parts and the whole.
  • Future Calculations: When adding, subtracting, multiplying, or dividing fractions, working with simplified forms often makes calculations easier.

Think of it like this: both \(\frac{4}{6}\) and \(\frac{2}{3}\) mean the same thing, but \(\frac{2}{3}\) is cleaner and more efficient to work with!

Question: How do I know when a fraction is fully simplified? How do I know I'm done?

Answer: A fraction is fully simplified when the numerator and denominator have no common factors other than 1. Here are ways to check:

  • GCD Method: Find the greatest common divisor of the numerator and denominator. If GCD = 1, then the fraction is in lowest terms.
  • Prime Check: If both numerator and denominator are prime numbers, the fraction is simplified.
  • Factor Check: List the factors of both numbers. If the only common factor is 1, you're done!

Example: For \(\frac{5}{7}\), factors of 5 are {1, 5} and factors of 7 are {1, 7}. The only common factor is 1, so \(\frac{5}{7}\) is fully simplified.

If you can still divide both numerator and denominator by the same number greater than 1, continue simplifying!

Question: What's the fastest way to find the GCD? I sometimes take too long finding all the factors.

Answer: Here are the most efficient methods for finding GCD:

  1. Quick Division Method: Look for obvious divisors first (2, 3, 5) and divide both numbers repeatedly until no more common factors exist.
  2. Prime Factorization: Break both numbers into prime factors and multiply the common factors together.
  3. Euclidean Algorithm: For larger numbers, repeatedly divide and take remainders until you reach a remainder of 0.

For smaller numbers (like in 4th grade), start with the Quick Division Method:

  • If both numbers are even, divide by 2
  • If both end in 0 or 5, divide by 5
  • If the sum of digits of both numbers is divisible by 3, divide by 3

Example: For \(\frac{24}{36}\), both are even (÷2), both divisible by 3 (÷3), giving us \(\frac{2}{3}\).