Solving Simplifying Fractions: CorbettMaths Practice Questions

Master simplifying fractions: Greatest Common Divisor (GCD), factor trees, step-by-step methods, and visual representations through 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Fraction Reduction
Exercise 1
Simplify the fraction:
\(\frac{12}{18}\)
Definition:

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

Simplification method:
  1. Find the factors of the numerator
  2. Find the factors of the denominator
  3. Identify the Greatest Common Divisor (GCD)
  4. Divide both numerator and denominator by the GCD
Original Fraction
\(\frac{12}{18}\)
Factors
GCD(12,18)=6
Simplified
\(\frac{2}{3}\)
Step 1: Find factors of 12

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

Step 2: Find factors of 18

Factors of 18: 1, 2, 3, 6, 9, 18

Step 3: Identify GCD

Common factors: 1, 2, 3, 6
Greatest Common Divisor: 6

Step 4: Divide by GCD

\(\frac{12}{18} = \frac{12 ÷ 6}{18 ÷ 6} = \frac{2}{3}\)

\(\frac{12}{18} = \frac{2}{3}\)
Final answer:

\(\frac{12}{18} = \frac{2}{3}\)

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 Prime Factorization Method
Exercise 2
Simplify the fraction:
\(\frac{24}{36}\)
Definition:

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

Original Fraction
\(\frac{24}{36}\)
Prime Factors
\(\frac{2^3 × 3}{2^2 × 3^2}\)
Simplified
\(\frac{2}{3}\)
Step 1: Find prime factors of 24

\(24 = 2 × 2 × 2 × 3 = 2^3 × 3\)

Step 2: Find prime factors of 36

\(36 = 2 × 2 × 3 × 3 = 2^2 × 3^2\)

Step 3: Cancel common factors

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

Step 4: Verify result

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

\(\frac{24}{36} = \frac{2}{3}\)
Final answer:

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

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 Large Numbers Simplification
Exercise 3
Simplify the fraction:
\(\frac{48}{72}\)
Definition:

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

Original Fraction
\(\frac{48}{72}\)
GCD Process
GCD(48,72)=24
Result
\(\frac{2}{3}\)
Step 1: Apply Euclidean Algorithm

\(72 ÷ 48 = 1\) remainder \(24\)
\(48 ÷ 24 = 2\) remainder \(0\)
So GCD(48,72) = 24

Step 2: Divide by GCD

\(\frac{48}{72} = \frac{48 ÷ 24}{72 ÷ 24} = \frac{2}{3}\)

Step 3: Verify equivalence

\(2 × 36 = 72\) and \(3 × 24 = 72\), so cross products are equal

Step 4: Confirm lowest terms

GCD(2,3) = 1, so \(\frac{2}{3}\) is in lowest terms

\(\frac{48}{72} = \frac{2}{3}\)
Final answer:

\(\frac{48}{72} = \frac{2}{3}\)

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

Rules, Methods, Laws, and Definitions
\(\frac{a}{b} = \frac{a ÷ d}{b ÷ d}\) where \(d = \gcd(a,b)\)
Fraction Simplification Formula
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

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
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 \(a×d = b×c\).
Common errors: Forgetting to divide both numerator and denominator, not checking for lowest terms, stopping too early before full simplification.
Key notes: A fraction is fully simplified when the only common factor of numerator and denominator is 1.
Formulas to know by heart:

• 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

Fraction Simplification Infographic

📊
Visual Fraction Representation
12
18

Represents 12 parts out of 18 total parts

Factor Tree Example
12
2
6
2
3
Lowest Terms Check

GCD of numerator and denominator = 1

\(\gcd(2,3) = 1\)

Therefore \(\frac{2}{3}\) is in lowest terms

Simplification Workflow
1
Identify
2
Find GCD
3
Divide
4
Verify
Common Mistakes
  • Only dividing the numerator
  • Not checking if result is in lowest terms
  • Missing common factors
  • Arithmetic errors in division
📚 Master the GCD method for efficient fraction simplification!
Solution: Exercises 4 to 5
4 Complex Simplification
Exercise 4
Simplify the fraction:
\(\frac{60}{84}\)
Definition:

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

Original
\(\frac{60}{84}\)
First Step
\(\frac{30}{42}\)
Second Step
\(\frac{5}{7}\)
Step 1: Initial simplification

Both 60 and 84 are divisible by 2
\(\frac{60}{84} = \frac{60 ÷ 2}{84 ÷ 2} = \frac{30}{42}\)

Step 2: Continue simplification

Both 30 and 42 are divisible by 6
\(\frac{30}{42} = \frac{30 ÷ 6}{42 ÷ 6} = \frac{5}{7}\)

Step 3: Verify lowest terms

Factors of 5: 1, 5
Factors of 7: 1, 7
Common factor: 1, so \(\frac{5}{7}\) is in lowest terms

Step 4: Alternative direct approach

GCD(60,84) = 12
\(\frac{60}{84} = \frac{60 ÷ 12}{84 ÷ 12} = \frac{5}{7}\)

\(\frac{60}{84} = \frac{5}{7}\)
Final answer:

\(\frac{60}{84} = \frac{5}{7}\)

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
Exercise 5
In a class of 45 students, 27 passed the math test. What fraction of the class passed? Simplify the fraction.
Definition:

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

Fraction Setup
\(\frac{27}{45}\)
GCD Calculation
GCD(27,45)=9
Simplified
\(\frac{3}{5}\)
Step 1: Set up the fraction

Students who passed = 27
Total students = 45
Fraction = \(\frac{27}{45}\)

Step 2: Find GCD of 27 and 45

Factors of 27: 1, 3, 9, 27
Factors of 45: 1, 3, 5, 9, 15, 45
GCD = 9

Step 3: Simplify the fraction

\(\frac{27}{45} = \frac{27 ÷ 9}{45 ÷ 9} = \frac{3}{5}\)

Step 4: Interpret the result

\(\frac{3}{5}\) of the class passed the math test
This means 3 out of every 5 students passed

\(\frac{27}{45} = \frac{3}{5}\) of the class passed
Final answer:

\(\frac{3}{5}\) of the class passed the math test

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

Comprehensive Guide: Laws, Methods, Rules, Definitions
\(\frac{a}{b} = \frac{a ÷ d}{b ÷ d}\)
Fundamental Simplification Rule
Key definitions:

Equivalent Fractions: Different fractions that represent the same value (e.g., \(\frac{1}{2} = \frac{2}{4} = \frac{3}{6}\))

Lowest Terms: A fraction where the numerator and denominator share no common factors other than 1

Greatest Common Divisor (GCD): The largest positive integer that divides both numbers without remainder

Prime Number: A number greater than 1 that has exactly two factors: 1 and itself

Complete methodology:
  1. Identify Components: Determine numerator and denominator
  2. Calculate GCD: Use one of the available methods
  3. Perform Division: Divide both numerator and denominator by GCD
  4. Verify Result: Ensure new fraction is in lowest terms
  5. Check Work: Cross-multiply to verify equivalence
Tip 1: For quick simplification, try dividing by 2, 3, or 5 first if possible.
Tip 2: A fraction is completely simplified when numerator and denominator are both prime numbers or coprime.
Tip 3: Use divisibility rules: numbers ending in 0, 2, 4, 6, 8 are divisible by 2.
Tip 4: The product of the original numerator and simplified denominator equals the product of the original denominator and simplified numerator.
Common errors: Forgetting to divide both parts, not continuing to lowest terms, miscalculating GCD, misidentifying factors.
Exam preparation: Memorize divisibility rules, practice GCD calculations, master prime factorization, understand equivalence verification.
Essential formulas and rules:

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

• Equivalence Check: \(\frac{a}{b} = \frac{c}{d}\) if and only if \(a×d = b×c\)

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

• Prime Factorization Method: Cancel identical prime factors from top and bottom

• Euclidean Algorithm: For finding GCD of large numbers efficiently

Questions & Answers

Question: I don't understand why 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 \(\frac{2}{3}\).