Unlike Fractions

What Are Unlike Fractions?

Unlike fractions are fractions that have different denominators. For example, $\frac{1}{3}$ and $\frac{1}{4}$ are unlike fractions because their denominators, $3$ and $4$, are not the same. In contrast, fractions with the same denominator, such as $\frac{1}{5}$ and $\frac{3}{5}$, are called like fractions.

Think of a pizza. If you have $\frac{1}{8}$ of a pizza and your friend has $\frac{2}{8}$ of the same pizza, you can easily combine your shares. You have $1+2=3$ slices of the same size, so together you have $\frac{3}{8}$ of the pizza. This is simple because the slices (the denominators) are the same size.

But what if you have $\frac{1}{2}$ of a pizza and your friend has $\frac{1}{3}$ of another pizza? The slices are different sizes. You can't just add the numerators and say you have $2$ slices. Two of what? The core challenge with unlike fractions is that they represent parts of a whole that have been divided differently. Before we can add or subtract them, we must find a way to express them using a common unit of measurement. This is where the concept of the Least Common Denominator comes in.

How Do You Find the Least Common Denominator (LCD)?

The key to working with unlike fractions is to make them 'like' each other. We do this by finding a Least Common Denominator (LCD). The LCD is the smallest number that is a multiple of all the denominators in the problem. In other words, the LCD is simply the Least Common Multiple (LCM) of the denominators.

There are two primary methods for finding the LCD:

Method 1: Listing Multiples

This method is effective for smaller, simpler denominators. You simply list the multiples of each denominator until you find the first one they have in common.

Let's find the LCD of $\frac{2}{5}$ and $\frac{1}{4}$. The denominators are $5$ and $4$.

• Multiples of 5: 5, 10, 15, 20, 25, 30, ...
• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, ...

The first multiple they share is $20$. Therefore, the LCD is $20$.

Method 2: Prime Factorization

This method is more powerful and reliable for larger or more complex denominators. It involves breaking each denominator down into its prime factors.

Let's find the LCD for $\frac{5}{12}$ and $\frac{7}{18}$. The denominators are $12$ and $18$.

1. Find the prime factorization of each denominator.
   $12=2\times 6=2\times 2\times 3=2^2\times 3^1$
   $18=2\times 9=2\times 3\times 3=2^1\times 3^2$
2. Identify all the unique prime factors from both numbers.
   The unique prime factors are $2$ and $3$.
3. Take the highest power of each unique prime factor.
   The highest power of $2$ is $2^2$ (from 12).
   The highest power of $3$ is $3^2$ (from 18).
4. Multiply these highest powers together to get the LCD.
   LCD = $2^2\times 3^2=4\times 9=36$

So, the LCD of $12$ and $18$ is $36$. This method guarantees you find the smallest possible common denominator, which makes the subsequent calculations much easier.

How Do You Add Unlike Fractions?

Once you know how to find the LCD, adding unlike fractions becomes a systematic, four-step process. The goal is to convert the unlike fractions into equivalent like fractions, which you can then add easily.

Let's walk through an example.

How Do You Subtract Unlike Fractions?

The great news is that subtracting unlike fractions follows the exact same process as adding them. The only difference is the final operation: you subtract the numerators instead of adding them. The first two steps—finding the LCD and converting the fractions—are identical.

Which is Bigger? How to Compare Unlike Fractions

Sometimes you don't need to add or subtract fractions, but simply determine which one is larger or smaller. There are two excellent methods for comparing unlike fractions.

Method 1: The LCD Method

This is the most intuitive method because it builds on what we've already learned. Just like with addition and subtraction, you convert both fractions to have a common denominator (the LCD). Once they have the same denominator, you can simply compare their numerators. The fraction with the larger numerator is the larger fraction.

Method 2: The Cross-Multiplication Method

This is a fantastic shortcut for comparing just two fractions. To compare $\frac{a}{b}$ and $\frac{c}{d}$, you multiply the numerator of each fraction by the denominator of the other. You then compare the resulting products.

What About Mixed Numbers with Unlike Fractions?

When you encounter mixed numbers (like $2\frac{1}{4}$) in an addition or subtraction problem with unlike fractions, the most reliable strategy is to first convert them into improper fractions. Trying to work with the whole numbers and fractions separately can sometimes lead to borrowing and carrying, which can be confusing. Converting to improper fractions simplifies the problem into the steps we've already mastered.

Let's consider the problem $3\frac{1}{2}+1\frac{2}{5}$.

1. Convert mixed numbers to improper fractions.
   To convert a mixed number, multiply the whole number by the denominator and add the numerator. This result becomes the new numerator.
   $$3\frac{1}{2}=\frac{(3\times 2)+1}{2}=\frac{6+1}{2}=\frac{7}{2}$$
   $$1\frac{2}{5}=\frac{(1\times 5)+2}{5}=\frac{5+2}{5}=\frac{7}{5}$$
   Our problem is now $\frac{7}{2}+\frac{7}{5}$.
2. Find the LCD.
   The LCD of $2$ and $5$ is $10$.
3. Convert to equivalent fractions.
   $$\frac{7}{2}=\frac{7\times 5}{2\times 5}=\frac{35}{10}$$
   $$\frac{7}{5}=\frac{7\times 2}{5\times 2}=\frac{14}{10}$$
4. Add the numerators.
   $$\frac{35}{10}+\frac{14}{10}=\frac{35+14}{10}=\frac{49}{10}$$
5. Simplify and convert back to a mixed number (optional, but good practice).
   To convert $\frac{49}{10}$ back to a mixed number, we perform the division: $49÷10=4$ with a remainder of $9$.
   So, $\frac{49}{10}=4\frac{9}{10}$.

What Are Common Mistakes When Working with Unlike Fractions?

Working with unlike fractions is straightforward once you learn the rules, but there are a few common pitfalls students fall into. Being aware of these can help you avoid them in your own work.

• Adding or Subtracting Denominators: This is the most frequent error. Never add or subtract the denominators. For example, $\frac{1}{2}+\frac{1}{3}$ is NOT $\frac{2}{5}$. The denominator represents the size of the slices; you can't add them. You must find a common denominator first.
• Forgetting to Change the Numerator: When you find the LCD and change the denominator of a fraction, you must also change the numerator by multiplying it by the same factor. Forgetting this step means you are changing the value of the fraction. For example, $\frac{2}{3}$ is not equal to $\frac{2}{12}$; it is equal to $\frac{8}{12}$.
• Using a Common Multiple instead of the LCD: You can use any common multiple as a common denominator, but using the least common denominator (LCD) keeps the numbers smaller and easier to work with. Using a larger multiple will still get you the correct answer, but it will require more difficult simplification at the end.
• Errors in Simplification: Always check if your final answer can be simplified. A fraction is simplified when its numerator and denominator have no common factors other than $1$. Leaving an answer as $\frac{6}{8}$ instead of $\frac{3}{4}$ might lose you points.
• Cross-Multiplying to Add: The cross-multiplication trick is a shortcut for comparing fractions, not for adding or subtracting them. Applying it to addition will give you the wrong answer.

Quick Reference: The 4-Step Process

Feeling overwhelmed? Just remember this core process for adding or subtracting any pair of unlike fractions. Follow these steps every time, and you'll get the right answer.

The 4 Steps for Adding/Subtracting Unlike Fractions

1. Find LCD: Determine the Least Common Denominator of the fractions. Use the listing multiples method for small numbers or prime factorization for larger ones.
2. Convert: Rewrite each fraction as an equivalent fraction that uses the LCD as its new denominator. Remember to multiply the numerator by the same number you multiplied the denominator by.
3. Operate: Perform the operation. Add or subtract the numerators of your new, 'like' fractions. The denominator stays the same.
4. Simplify: Reduce the resulting fraction to its simplest form by dividing the numerator and denominator by their greatest common factor. If the result is an improper fraction, convert it to a mixed number if required.

Quick Tip for Comparing Fractions

To quickly compare $\frac{a}{b}$ and $\frac{c}{d}$, use cross-multiplication. Compare the products $a\times d$ and $b\times c$. The larger product corresponds to the larger fraction.