Adding and Subtracting Fractions with Unlike Denominators - A Kid-Friendly Guide

In this middle-school-friendly guide, we’ll show you how adding and subtracting fractions with unlike denominators works with easy-to-follow definitions, solved examples, and fun practice questions.  

Let's get right into it! 

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What Are Fractions with Unlike Denominators?

Fractions with unlike denominators are fractions that have different bottom numbers. 

For example, \( \Large \frac{1}{2}\) and \( \Large \frac{1}{3}\) have unlike denominators: 2 and 3.  

Why does this matter? 

Comparing fractions with unlike denominators is like trying to compare two cakes cut into different numbers and sizes of slices. It's difficult to tell how to distribute the slices to your guests fairly, right?  

You can imagine that adding and subtracting fractions with unlike denominators would be challenging as well. 

Read on to see how it works! 

How to Add Fractions with Unlike Denominators? 

The key to adding and subtracting factors with unlike denominators is to find a version of the factors that share the same denominator, i.e. turn them from fractions with unlike denominators to fractions with common denominators.  

We do this by finding equivalent fractions with common denominators. 

What finding equivalent fractions means is that we are changing the nominator and denominator of the fraction, but its value remains the same. 

For example, \( \Large \frac{2}{4}\) and \( \Large \frac{4}{8}\) are equivalent fractions because they both equal (i.e. an be reduced to) \( \Large \frac{1}{2}\).

Refresh Your Memory: How to Reduce Fractions 

To find equivalent fractions with like denominators, we need to find their least common multiple (LCM).  

A simple method to find the LCM is to list the multiples of each denominator and look for the smallest number they share. 

If you forgot what multiples are and how to find them, check out our tutorial: 

https://www.youtube.com/watch?v=KgNsaP7PVbs

Let’s say we want to find the sum of fractions \( \Large \frac{1}{4}\) and \( \Large \frac{1}{6}\).

Our denominators, the bottom numbers of our fractions, are 4 and 6. 

To find their LCM, we’ll first list the multiples for each denominator:  

• Multiples of 4 are 4, 8, 12, 16, 20, 24, 28 
• Multiples of 6 are 6, 12, 18, 24, 30 

Then, list the multiples our denominators have in common: 12 and 24 

What is the smallest, or “least,” of the two multiples they have in common? 

It is 12, correct? 

12 is the least common multiple (LCM) of 4 and 6. 

Now that we have the LCM, let's find the equivalent fractions. 

We know that our new denominator for both fractions is 12, but what are the new nominators? 

\( \Large \frac{1}{4}\)+\( \Large \frac{1}{6}\)=\( \Large \frac{x}{12}\)+\( \Large \frac{x}{12}\)

Remember, equivalent fractions means that we change the numerator and denominator, but their value remains the same. 

So, to find the nominator, we simply have to find the quotient of the new and the old denominator and multiply by it. 

The quotients are: 

12÷4=3 

12÷6=2 

Now let’s multiply our nominator and denominator by them: 

\( \Large \frac{3×1}{3×4}\)=\( \Large \frac{3}{12}\)

\( \Large \frac{2×1}{2×6}\)=\( \Large \frac{2}{12}\)

There we have it! The equivalent denominators for \( \Large \frac{1}{4}\) and \( \Large \frac{1}{6}\) are \( \Large \frac{3}{12}\) and \( \Large \frac{2}{12}\).

Now our fractions have common denominators and all that’s left to do is add them up. 

How do we add fractions with common denominators? 

We add up the numerators, while the denominator remains unchanged: 

 \( \Large \frac{3}{12}\)=\( \Large \frac{5}{12}\)

The last step is to check if this fraction can be simplified, i.e. reduced.  

In this case, \( \Large \frac{5}{12}\)is already in its simplest form, because 5 and 12 don't share any common factors other than 1.  

How to Subtract Fractions with Unlike Denominators? 

Just like with addition, to subtract fractions with unlike denominators, we first need to find their equivalents with common denominators. 

Let’s try an example: 

Say we want to subtract \( \Large \frac{2}{3}\) from \( \Large \frac{5}{6}\).

\( \Large \frac{5}{6}\)-\( \Large \frac{2}{3}\)=?

First let's find their equivalent fractions with common denominators. 

We start by finding their LCM. 

Our denominators are 6 and 3.  

To find their LCM, let’s look at the multiples of each denominator: 

• Multiples of 6 are 6, 12, 18, 24 
• Multiples of 3 are 3, 6, 12, 18 

The multiples our denominators have in common are 6, 12, and 18. 

The smallest multiple in this list is 6, which means that 6 is our LCM. 

Let’s use the LCM to find equivalent fractions with common denominators: 

\( \Large \frac{5}{6}\)-\( \Large \frac{2}{3}\)=\( \Large \frac{x}{6}\)-\( \Large \frac{x}{6}\)

What is the quotient of the new and old denominator? 

6÷6=1 

6÷3=2 

To find equivalent fractions with common denominators, we multiple by the numerator and denominator by 1 and 2: 

\( \Large \frac{1×5}{1×6}\)=\( \Large \frac{5}{6}\)

\( \Large \frac{2×2}{2×3}\)=\( \Large \frac{4}{6}\)

Now that we have our equivalent fractions with common denominators, all we have to do is subtract them: 

\( \Large \frac{5}{6}\)-\( \Large \frac{4}{6}\)

=\( \Large \frac{1}{6}\) 

Can the resulting fraction \( \Large \frac{1}{6}\) be reduced? 

No. 

Since 1 and 16 have no common factors other than 1, this fraction is already in its simplest form. 

Solved Examples

Examples for adding fraction with unlike denominators: 

Example 1: 

Step 1: Find the Least Common Multiple (LCM) 

• List the multiples of each denominator:  
  • Multiples of 2: 2, 4, 6, 8, 10, 12 
  • Multiples of 3: 3, 6, 9, 12, 18 
• The smallest number that appears on both lists is 6, so the LCM is 6. 

Step 2: Find Equivalent Fractions 

• Rewrite the fractions with the common denominator of 6:  
  • \( \Large \frac{1}{2}\)=\( \Large \frac{3}{6}\)
  • \( \Large \frac{1}{3}\)=\( \Large \frac{2}{6}\)

Step 3: Add the Numerators 

• Add the numerators: \( \Large \frac{3}{6}\)+\( \Large \frac{2}{6}\)=\( \Large \frac{5}{6}\)

Step 4: Reduce Factor 

• Factor \( \Large \frac{5}{6}\) cannot be reduced 

Example 2: 

Step 1: Find the Least Common Multiple (LCM) 

• List the multiples of each denominator:  
  • Multiples of 3: 3, 6, 9, 12, 15, 18 
  • Multiples of 4: 4, 8, 12, 16 
• The smallest number that appears on both lists is 12, so the LCM is 12. 

Step 2: Find Equivalent Fractions 

• Rewrite the fractions with the common denominator of 12:  
  • \( \Large \frac{2}{3}\)=\( \Large \frac{8}{12}\)
  • \( \Large \frac{1}{4}\)=\( \Large \frac{3}{12}\)

Step 3: Add the Numerators 

• Add the numerators: \( \Large \frac{8}{12}\)+\( \Large \frac{3}{12}\)=\( \Large \frac{11}{12}\)

Step 4: Reduce Factor 

Factor \( \Large \frac{11}{12}\)cannot be reduced 

Examples for Subtracting Fraction with Unlike Denominators: 

Example 1: 

Step 1: Find the Least Common Denominator (LCM) 

• List the multiples of each denominator:  
  • Multiples of 4: 4, 8, 12, 16, 20 
  • Multiples of 2: 2, 4, 6, 8, 10, 12 
• The smallest number that appears on both lists is 4, so the LCM is 4. 

Step 2: Find Equivalent Fractions 

• Rewrite the fractions with the common denominator of 4:  
  • \( \Large \frac{3}{4}\) stays the same 
  • \( \Large \frac{1}{2}\)=\( \Large \frac{2}{4}\)

Step 3: Subtract the Numerators 

• Subtract the numerators: \( \Large \frac{3}{4}\)-\( \Large \frac{2}{4}\)=\( \Large \frac{1}{4}\)

Step 4: Reduce Fraction 

• Fraction \( \Large \frac{1}{4}\) cannot be reduced 

 Example 2: 

Step 1: Find the Least Common Multiple (LCM) 

• List the multiples of each denominator:  
  • Multiples of 6: 6, 12, 18, 24, 30 
  • Multiples of 3: 3, 6, 9, 12, 15, 18 
• The smallest number that appears on both lists is 6, so the LCM is 6. 

Step 2: Find Equivalent Fractions 

• Rewrite the fractions with the common denominator of 6:  
  • \( \Large \frac{5}{6}\) stays the same
  • \( \Large \frac{1}{3}\)=\( \Large \frac{2}{6}\)

Step 3: Subtract the Numerators 

• Subtract the numerators: \( \Large \frac{5}{6}\)-\( \Large \frac{2}{6}\)=\( \Large \frac{3}{6}\)

Step 4: Reduce Fraction 

• Both 3 and 6 can be divided by 3, so we can simplify the fraction to \( \Large \frac{1}{3}\) 

Try it Yourself – Work Out These Fractions

Now it’s your turn!  

Solve these problems then check your answers with the answer key at the bottom of the page. 

Adding Fraction with Unlike Denominators: 

1. \( \Large \frac{1}{2}\)+\( \Large \frac{1}{4}\)
2. \( \Large \frac{2}{3}\)+\( \Large \frac{1}{6}\)
3. \( \Large \frac{1}{5}\)+\( \Large \frac{1}{10}\)
4. \( \Large \frac{1}{4}\)+\( \Large \frac{2}{3}\)
5. \( \Large \frac{3}{8}\)+\( \Large \frac{1}{2}\)

Subtracting Fraction with Unlike Denominators: 

1. \( \Large \frac{3}{4}\)-\( \Large \frac{1}{2}\)
2. \( \Large \frac{5}{6}\)-\( \Large \frac{1}{3}\)
3. \( \Large \frac{7}{8}\)-\( \Large \frac{1}{4}\)
4. \( \Large \frac{2}{3}\)-\( \Large \frac{1}{2}\)
5. \( \Large \frac{5}{7}\)-\( \Large \frac{1}{2}\)

Mistakes to Avoid When Adding or Subtracting Fractions with Unlike Denominators 

Learn from other students’ mistakes to truly master adding and subtracting fractions with unlike denominators! 

1. Forgetting to find a common denominator: This is the most common mistake. Remember, you can only add or subtract fractions when they have the same denominator, so don’t forget your first step to finding equivalent fractions is using the LCM method! 
2. Adding or subtracting both the numerator and denominator: Remember, we only add or subtract the top numbers (numerators), not the bottom numbers (denominators). 
3. Choosing the wrong common denominator: When making your lists to find the LCM, make sure you find the number that is the smallest in both lists! It is often easier to use the smallest number so that you don’t have to simplify your fractions later.  
4. Not simplifying the final answer: Always reduce your answer to its simplest form, if you need to revise how to simplify your fractions. 
5. Miscalculating the common denominator: Make sure you've accurately found the least common multiple of the denominators. Count your multiples twice, to make sure your lists are in top shape, so you get the right LCM each time! 

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PSST! Check Your Quiz Answers Here  

Here are the answers to the 10 exercise problems, let’s see how you did! 

1. \( \Large \frac{3}{4}\)
2. \( \Large \frac{5}{6}\)
3. \( \Large \frac{3}{10}\)
4. \( \Large \frac{11}{12}\)
5. \( \Large \frac{7}{8}\)
6. \( \Large \frac{1}{4}\)
7. \( \Large \frac{3}{6}\) or \( \Large \frac{1}{2}\)
8. \( \Large \frac{5}{8}\)
9. \( \Large \frac{1}{6}\)
10. \( \Large \frac{3}{14}\)