Subtracting Fractions: A Step-by-Step Guide

Hey guys! Let's dive into a super common and essential math skill: subtracting fractions and reducing them to their simplest form. Specifically, we're going to tackle the problem $\frac{1}{3}-\frac{1}{12}$. Don't worry; it's easier than it looks! We will break down each step, so you will understand exactly how to subtract these fractions and get the answer in its most simplified form. So grab your pencils, and let's get started!

Finding a Common Denominator

Before we can subtract fractions, they need to have the same denominator. Think of it like this: you can't directly compare or subtract apples and oranges; you need a common unit, right? The denominator is that unit for fractions. In our problem, we have $\frac{1}{3}$ and $\frac{1}{12}$. The denominators are 3 and 12.

So, how do we find a common denominator? The easiest way is to find the least common multiple (LCM) of the denominators. The LCM is the smallest number that both denominators can divide into evenly. In this case, we need to find the LCM of 3 and 12.

Multiples of 3 are: 3, 6, 9, 12, 15, and so on. Multiples of 12 are: 12, 24, 36, and so on.

The smallest number that appears in both lists is 12. Therefore, the LCM of 3 and 12 is 12. Great! This means we want to convert both fractions to have a denominator of 12. The fraction $\frac{1}{12}$ already has the denominator we want, so we don't need to change it. However, we need to convert $\frac{1}{3}$ into an equivalent fraction with a denominator of 12.

To do this, we ask ourselves: "What do we multiply 3 by to get 12?" The answer is 4. So, we multiply both the numerator (top number) and the denominator (bottom number) of $\frac{1}{3}$ by 4:

$\frac{1}{3} \times \frac{4}{4} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}$

Now we have two fractions with the same denominator: $\frac{4}{12}$ and $\frac{1}{12}$. We're ready to subtract!

Subtracting the Fractions

Now that our fractions have a common denominator, subtracting them is straightforward. We simply subtract the numerators (the top numbers) and keep the denominator the same. So, we have:

$\frac{4}{12} - \frac{1}{12} = \frac{4 - 1}{12} = \frac{3}{12}$

So, $\frac{4}{12} - \frac{1}{12}$ equals $\frac{3}{12}$. Easy peasy, right? But we're not quite done yet!

Reducing to Lowest Terms

The final step is to reduce the fraction to its lowest terms. This means we want to find the simplest form of the fraction where the numerator and denominator have no common factors other than 1. In other words, we want to divide both the numerator and the denominator by their greatest common factor (GCF).

Looking at $\frac{3}{12}$, we need to find the GCF of 3 and 12. The factors of 3 are 1 and 3. The factors of 12 are 1, 2, 3, 4, 6, and 12. The greatest common factor is 3.

So, we divide both the numerator and the denominator by 3:

$\frac{3}{12} = \frac{3 \div 3}{12 \div 3} = \frac{1}{4}$

Therefore, $\frac{3}{12}$ reduced to its lowest terms is $\frac{1}{4}$.

Putting It All Together

Let's recap the entire process:

1. Find a Common Denominator: We found the least common multiple (LCM) of 3 and 12, which was 12. We converted $\frac{1}{3}$ to $\frac{4}{12}$.
2. Subtract the Fractions: We subtracted the numerators: $\frac{4}{12} - \frac{1}{12} = \frac{3}{12}$.
3. Reduce to Lowest Terms: We found the greatest common factor (GCF) of 3 and 12, which was 3. We divided both the numerator and denominator by 3 to get $\frac{1}{4}$.

So, $\frac{1}{3} - \frac{1}{12} = \frac{1}{4}$.

And that's it! You've successfully subtracted the fractions and reduced the answer to its lowest terms. Now, let's reinforce this knowledge with a few more examples.

More Examples for Practice

Let's try another example to solidify your understanding. Suppose we want to subtract $\frac{2}{5} - \frac{1}{10}$.

Example 1: $\frac{2}{5} - \frac{1}{10}$

1. Find a Common Denominator: The denominators are 5 and 10. The LCM of 5 and 10 is 10. So, we need to convert $\frac{2}{5}$ to an equivalent fraction with a denominator of 10. To do this, we multiply both the numerator and denominator of $\frac{2}{5}$ by 2: $\frac{2}{5} \times \frac{2}{2} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}$ Now we have $\frac{4}{10}$ and $\frac{1}{10}$.
2. Subtract the Fractions: Subtract the numerators: $\frac{4}{10} - \frac{1}{10} = \frac{4 - 1}{10} = \frac{3}{10}$
3. Reduce to Lowest Terms: The fraction $\frac{3}{10}$ is already in its lowest terms because 3 and 10 have no common factors other than 1. Therefore, $\frac{2}{5} - \frac{1}{10} = \frac{3}{10}$.

Example 2: $\frac{3}{4} - \frac{1}{6}$

Let's tackle a slightly more complex example.

1. Find a Common Denominator: The denominators are 4 and 6. The LCM of 4 and 6 is 12. We need to convert both fractions to have a denominator of 12.
   • For $\frac{3}{4}$, we multiply both the numerator and denominator by 3: $\frac{3}{4} \times \frac{3}{3} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
   • For $\frac{1}{6}$, we multiply both the numerator and denominator by 2: $\frac{1}{6} \times \frac{2}{2} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}$ Now we have $\frac{9}{12}$ and $\frac{2}{12}$.
2. Subtract the Fractions: Subtract the numerators: $\frac{9}{12} - \frac{2}{12} = \frac{9 - 2}{12} = \frac{7}{12}$
3. Reduce to Lowest Terms: The fraction $\frac{7}{12}$ is already in its lowest terms because 7 and 12 have no common factors other than 1. Therefore, $\frac{3}{4} - \frac{1}{6} = \frac{7}{12}$.

Common Mistakes to Avoid

When subtracting fractions, it's easy to make a few common mistakes. Here are some things to watch out for:

• Forgetting to Find a Common Denominator: This is the most common mistake. You must have a common denominator before you can subtract the numerators.
• Subtracting the Denominators: Only subtract the numerators. The denominator stays the same once you have a common denominator.
• Not Reducing to Lowest Terms: Always check if your final answer can be simplified further. Reducing to lowest terms gives the simplest representation of the fraction.
• Incorrectly Finding the LCM or GCF: Double-check your calculations for the least common multiple (LCM) and the greatest common factor (GCF) to ensure accuracy.

Why This Matters

Understanding how to subtract fractions is crucial for many areas of mathematics and everyday life. Whether you're baking a cake, measuring ingredients for a recipe, calculating distances, or working on more advanced math problems, the ability to subtract fractions accurately is essential.

Conclusion

Alright, guys, that wraps up our guide on subtracting fractions and reducing them to their lowest terms! Remember the key steps: find a common denominator, subtract the numerators, and reduce the resulting fraction to its simplest form. Practice these steps with different examples, and you'll become a fraction subtraction pro in no time! Keep practicing, and don't hesitate to ask for help if you get stuck. Happy calculating!