How to subtract fractions with different denominators

Subtracting fractions is a foundational math skill, but how to subtract fractions with different denominators is often the trickiest part for learners of all ages. Unlike subtracting fractions with the same (like) denominators (where you simply subtract numerators), fractions with different denominators require an extra step: finding a common denominator to make the fractions comparable. This guide breaks down the process in simple, actionable steps, with examples for proper fractions, mixed numbers, and real-world scenarios – plus tips to avoid the most common errors that trip up students.

Key Terms to Know Before Subtracting Fractions with Different Denominators

Before diving into the steps, clarify these core terms to build a solid understanding:

  • Fraction: A number representing part of a whole (e.g., ⅓, 5/8), with a numerator (top number = parts you have) and denominator (bottom number = total equal parts).
  • Like Denominators: Fractions with the same bottom number (e.g., 3/7 and 2/7) – easy to subtract directly.
  • Unlike Denominators: Fractions with different bottom numbers (e.g., ½ and ⅓) – require a common denominator to subtract.
  • Least Common Denominator (LCD): The smallest number that both denominators divide into evenly (also called the Least Common Multiple, LCM, of the denominators). For example, the LCD of 2 and 3 is 6; the LCD of 4 and 6 is 12.
  • Equivalent Fractions: Fractions that have the same value but different numerators/denominators (e.g., ½ = 3/6 = 4/8) – created by multiplying numerator and denominator by the same number.
  • Mixed Number: A whole number plus a proper fraction (e.g., 2 ¼ = 2 + ¼) – requires extra steps for subtraction with different denominators if borrowing is needed.
  • Proper Fraction: A fraction where the numerator is smaller than the denominator (e.g., 3/5) – no whole number component.
  • Improper Fraction: A fraction where the numerator is larger than the denominator (e.g., 7/4) – can be converted to a mixed number for easier subtraction.

Step-by-Step: How to Subtract Fractions with Different Denominators (Proper Fractions)

Follow these 5 simple steps to subtract any two proper fractions with unlike denominators – no guesswork required:

Step 1: Find the Least Common Denominator (LCD)

The LCD is the smallest number that both denominators divide into without a remainder. To find it:

  • List multiples of each denominator until you find the smallest common one (e.g., denominators 4 and 6: multiples of 4 = 4, 8, 12; multiples of 6 = 6, 12 → LCD = 12).
  • For larger numbers, factor each denominator into primes and multiply the unique primes (highest exponents): e.g., denominators 8 (2³) and 12 (2²×3) → LCD = 2³×3 = 24.

Step 2: Rewrite Both Fractions as Equivalent Fractions with the LCD

Multiply the numerator and denominator of each fraction by the number needed to turn the original denominator into the LCD. This keeps the fraction’s value the same but makes the denominators equal.

Step 3: Check if Borrowing Is Needed (Only for Mixed Numbers)

If subtracting mixed numbers and the top fraction’s numerator is smaller than the bottom one (after rewriting with LCD), borrow 1 from the whole number (convert 1 to a fraction with the LCD: e.g., 1 = 12/12 for LCD 12). Skip this step for proper fractions.

Step 4: Subtract the Numerators (Keep the LCD)

Now that the denominators are the same, subtract the second numerator from the first – the denominator stays the LCD (do not subtract denominators!).

Step 5: Simplify the Result (If Needed)

Reduce the fraction to its lowest terms by dividing numerator and denominator by their greatest common factor (GCF). If the result is an improper fraction, convert it back to a mixed number.

Example 1: Basic Subtraction of Proper Fractions (Different Denominators)

Problem: ¾ – ⅙Solution:

  1. Find LCD: Denominators 4 and 6 → LCD = 12.
  2. Rewrite fractions:¾ = (3×3)/(4×3) = 9/12⅙ = (1×2)/(6×2) = 2/12
  3. No borrowing needed (proper fractions).
  4. Subtract numerators: 9/12 – 2/12 = 7/12.
  5. Simplify: 7/12 (already in lowest terms – GCF of 7 and 12 is 1).

Example 2: Subtraction with Larger Denominators

Problem: 5/8 – 3/10Solution:

  1. Find LCD: Denominators 8 and 10 → multiples of 8 = 8,16,24,32,40; multiples of 10 =10,20,30,40 → LCD=40.
  2. Rewrite fractions:5/8 = (5×5)/(8×5) =25/403/10 = (3×4)/(10×4)=12/40
  3. No borrowing needed.
  4. Subtract numerators:25/40 -12/40=13/40.
  5. Simplify:13/40 (GCF 1 – simplest form).

Step-by-Step: How to Subtract Mixed Fractions with Different Denominators

Mixed fractions (whole number + fraction) add a borrowing step if the fractional part of the top number is smaller than the bottom one after rewriting with LCD.

Example 1: Mixed Fractions (No Borrowing Needed)

Problem: 3 ⅖ – 1 ⅓Solution:

  1. Find LCD: Denominators 5 and 3 → LCD=15.
  2. Rewrite fractional parts:⅖ = 6/15, ⅓=5/15 → mixed numbers become 3 6/15 -1 5/15.
  3. No borrowing needed (6/15 >5/15).
  4. Subtract:Whole numbers: 3-1=2Fractions:6/15 -5/15=1/15
  5. Combine & simplify: 2 1/15.

Example 2: Mixed Fractions (Borrowing Required)

Problem: 4 ¼ – 2 ⅗Solution:

  1. Find LCD: Denominators 4 and5 → LCD=20.
  2. Rewrite fractional parts:¼=5/20, ⅗=12/20 → mixed numbers become 4 5/20 -2 12/20.
  3. Borrowing needed (5/20 <12/20):Convert 4 5/20 to 3 + 20/20 +5/20 =3 25/20.
  4. Subtract:Whole numbers:3-2=1Fractions:25/20 -12/20=13/20
  5. Combine & simplify:1 13/20.

Example 3: Subtracting a Mixed Fraction from a Whole Number

Problem: 6 – 3 ⅘Solution:

  1. Convert whole number to mixed fraction with LCD (denominator 5):6 =5 5/5.
  2. Rewrite problem:5 5/5 -3 ⅘.
  3. No extra borrowing needed (5/5 >4/5).
  4. Subtract:Whole numbers:5-3=2Fractions:5/5 -4/5=1/5.
  5. Simplify:2 1/5.

Real-World Examples of Subtracting Fractions with Different Denominators

Subtracting fractions with unlike denominators isn’t just for math class – it’s used daily in cooking, measuring, crafting, and budgeting:

Example 1: Cooking/Baking

Problem: A recipe calls for 2 ¾ cups of flour. You only have 1 ⅓ cups. How much more flour do you need?Solution:

  1. LCD of 4 and3=12.
  2. Rewrite:2 9/12 -1 4/12.
  3. Subtract:1 5/12 cups of flour needed.

Example 2: Measuring Length (DIY/Woodworking)

Problem: A board is 5 ⅝ feet long. You need to cut off 2 ¼ feet. How long will the remaining board be?Solution:

  1. LCD of8 and4=8.
  2. Rewrite:5 ⅝ -2 2/8.
  3. Subtract:3 3/8 feet.

Example 3: Time Management

Problem: You plan to study for 3 ½ hours. You’ve already studied 1 ⅔ hours. How much more time do you need to study?Solution:

  1. LCD of2 and3=6.
  2. Rewrite:3 3/6 -1 4/6.
  3. Borrow:2 9/6 -1 4/6=1 5/6 hours.

Example 4: Baking (Ingredient Reduction)

Problem: A cake recipe uses ⅚ cup of sugar. You want to reduce the sugar by ⅓ cup. How much sugar will you use?Solution:

  1. LCD of6 and3=6.
  2. Rewrite:⅚ – 2/6=3/6=½ cup of sugar.

Common Mistakes to Avoid When Subtracting Fractions with Different Denominators

These errors are the most common – fix them to get accurate results every time:

  1. Subtracting Denominators Directly: The biggest mistake (e.g., ¾ -⅙ = (3-1)/(4-6)=2/-2=-1) – denominators are never subtracted; only numerators (after LCD).
  2. Using the Wrong LCD: Choosing a common denominator (e.g., 24 for 4 and6) instead of the least common (12) makes calculations harder but not wrong – just inefficient.
  3. Forgetting to Multiply the Numerator: Only changing the denominator (e.g., ¾ →9/12 but ⅙→1/12 instead of2/12) changes the fraction’s value.
  4. Borrowing Incorrectly: When borrowing 1 from a whole number, convert it to a fraction with the LCD (e.g.,1=12/12 for LCD12, not1/12).
  5. Skipping Simplification: Leaving answers like 4/8 instead of ½ makes results incomplete and harder to use in real life.

Frequently Asked Questions (FAQs) About Subtracting Fractions with Different Denominators

Q1: Can I use any common denominator instead of the LCD?

A1: Yes – any common denominator works (e.g., 24 instead of12 for 4 and6), but the LCD makes calculations simpler and reduces simplification steps.

Q2: What if the result is a negative fraction (e.g., ⅓ – ½)?

A2: Subtract the smaller fraction from the larger and add a negative sign: ⅓ -½ = -(½ -⅓)= -(3/6 -2/6)= -1/6.

Q3: How do I check if my answer is correct?

A3: Reverse the operation – add the result to the smaller fraction to see if you get the larger one (e.g., ¾ -⅙=7/12 →7/12 +⅙=7/12+2/12=9/12=¾).

Q4: Do I have to convert mixed numbers to improper fractions?

A4: No – you can subtract whole numbers and fractions separately (as shown in examples) or convert to improper fractions (e.g.,4 ¼=17/4, 2 ⅗=13/5; LCD=20 →85/20 -52/20=33/20=1 13/20) – choose what’s easier for you.

Q5: How do I subtract fractions with different denominators and negative numbers?

A5: Follow the same steps, then apply negative sign rules (e.g., -¾ -⅙= -9/12 -2/12= -11/12; ⅓ -(-½)=⅓+½=5/6).

Conclusion

Learning how to subtract fractions with different denominators boils down to one key rule: find the LCD, rewrite the fractions to match the LCD, then subtract numerators. Whether you’re working with proper fractions or mixed numbers (with borrowing), the process is consistent – and practice with real-world examples makes it second nature. Avoid common mistakes like subtracting denominators or forgetting to multiply the numerator, and you’ll master this skill in no time.

If you have questions about subtracting specific fractions (e.g., with large denominators or negative numbers), leave a comment below!