Fraction Subtraction: $\frac{7}{15} - \frac{1}{6}$

Hey everyone! Today, we're diving into the world of fractions and tackling a subtraction problem: $\frac{7}{15} - \frac{1}{6}$. Don't worry, it's not as scary as it looks! We'll break it down step-by-step to make sure you understand how to solve it and, most importantly, why each step is necessary. This will not only help you with this specific problem but also give you the tools to conquer any fraction subtraction problem you encounter. Ready to get started? Let's go! Our goal is to find the difference between $\frac{7}{15}$ and $\frac{1}{6}$, and we'll simplify the answer to its lowest terms. So, let's look at the multiple-choice options provided: A. $\frac{9}{30}$, B. $\frac{3}{10}$, C. $\frac{27}{94}$, and D. $\frac{6}{9}$. We'll figure out which one is the correct simplified answer. Keep reading to learn how to solve it.

Understanding the Basics: Why We Can't Directly Subtract

Before we jump into the math, it's crucial to understand why we can't just subtract the numerators (the top numbers) directly. You see, when you're adding or subtracting fractions, you're essentially combining or separating parts of a whole. But, these parts only make sense when they're of the same size. Think of it like this: if you have a pizza cut into 15 slices and another pizza cut into 6 slices, you can't easily subtract a slice from one pizza from a slice of the other pizza. They're different sizes! To make the subtraction possible, we have to make sure both fractions are using the same-sized slices, or, as mathematicians call it, the same denominator. This is where finding a common denominator comes in. The common denominator is the foundation that allows us to subtract. The common denominator is the key to successfully adding or subtracting fractions. Now, let's find that common denominator! Remember, the goal is not just to get an answer, but to understand the underlying principles of fraction arithmetic, which will benefit you in more complex mathematical problems later. That is why it is very important to get the basics down.

Finding the Least Common Denominator (LCD)

Alright, guys, let's find the Least Common Denominator (LCD) for $\frac{7}{15}$ and $\frac{1}{6}$. The LCD is the smallest number that both 15 and 6 can divide into evenly. There are a couple of ways to find it, but the most straightforward methods are listing multiples or using prime factorization.

• Listing Multiples: Let's list the multiples of 15 and 6 until we find a common one:

  • Multiples of 15: 15, 30, 45, 60, ...
  • Multiples of 6: 6, 12, 18, 24, 30, 36, ... See that 30 is the smallest number that appears in both lists? So, our LCD is 30! This means we'll rewrite both fractions with a denominator of 30.

• Prime Factorization: You can also use prime factorization. Break down both denominators into their prime factors.

  • 15 = 3 x 5
  • 6 = 2 x 3 Then, take each prime factor the greatest number of times it appears in either factorization. In this case, we have 2 (from the 6), 3 (appears in both), and 5 (from the 15). Multiply these together: 2 x 3 x 5 = 30. Voila! LCD is 30!

Both methods lead us to the same conclusion: The LCD is 30. This process of finding the LCD is really important, so make sure you understand the methods. Now that we have the LCD, let's get those fractions ready for subtraction. Finding the LCD is an important step when working with fractions.

Rewriting the Fractions with the Common Denominator

Now that we know our LCD is 30, we'll rewrite both fractions with a denominator of 30. To do this, we need to figure out what to multiply the numerator and denominator of each original fraction by to get a denominator of 30. Remember, when you multiply the numerator and denominator by the same number, you're essentially multiplying by 1, which doesn't change the fraction's value, only its appearance.

• For $\frac{7}{15}$: We need to multiply both the numerator and denominator by 2, since 15 x 2 = 30. So, $\frac{7}{15}$ becomes $\frac{7*2}{15*2}$ = $\frac{14}{30}$.
• For $\frac{1}{6}$: We need to multiply both the numerator and denominator by 5, since 6 x 5 = 30. So, $\frac{1}{6}$ becomes $\frac{1*5}{6*5}$ = $\frac{5}{30}$.

So, now our problem looks like this: $\frac{14}{30} - \frac{5}{30}$. See how both fractions now have the same denominator? That’s what makes subtraction a breeze. We are getting closer to solving the problem. We changed the appearance of our fractions to make the subtraction possible. Make sure you understand the methods used.

Subtracting the Numerators

Okay, we're in the home stretch now! With both fractions having the same denominator, we can finally subtract the numerators. The denominator stays the same – it just tells us what size pieces we’re working with, which is still 30ths. You only subtract the numerators.

So, we have: $\frac{14}{30} - \frac{5}{30} = \frac{14-5}{30} = \frac{9}{30}$.

Easy, right? We've successfully subtracted the fractions! The result is $\frac{9}{30}$. Now let's simplify our answer. The hard part is over.

Simplifying the Answer

Almost done, but we're not quite finished yet. The fraction $\frac{9}{30}$ can be simplified. Simplifying a fraction means reducing it to its lowest terms. We do this by dividing both the numerator and denominator by their greatest common factor (GCF). The GCF is the largest number that divides evenly into both the numerator and denominator.

In the case of 9 and 30, the GCF is 3. So, we divide both the numerator and denominator by 3:

$\frac{9 \div 3}{30 \div 3} = \frac{3}{10}$.

Therefore, the simplified answer is $\frac{3}{10}$. Always remember to simplify your answers! It's good mathematical practice and ensures you have the fraction in its most basic form. It is also very helpful to know when taking a test. If you chose A, you would have gotten the answer before it was simplified. Always double check your work.

Choosing the Correct Answer

Let's go back to our multiple-choice options:

A. $\frac{9}{30}$ B. $\frac{3}{10}$ C. $\frac{27}{94}$ D. $\frac{6}{9}$

We found that the simplified answer is $\frac{3}{10}$. Looking at our choices, option B. $\frac{3}{10}$ is the correct answer. Congratulations, you solved the problem!

Conclusion: Fraction Mastery

So, there you have it! We've successfully subtracted $\frac{7}{15} - \frac{1}{6}$ and simplified the answer. We learned the importance of finding a common denominator, rewriting fractions, subtracting numerators, and simplifying the final result. Remember, practice makes perfect. The more you work with fractions, the more comfortable you'll become. Keep practicing, and you'll be a fraction master in no time! Remember to always simplify your answers. Keep up the great work, everyone! You can do it!