Subtracting Fractions: A Simple Guide

Hey everyone! Today, we're diving into the world of fractions and tackling a common task: subtracting and simplifying fractions. It might sound a bit daunting, but trust me, guys, once you get the hang of it, it's a piece of cake. We're going to break down a specific problem, rac{11}{16}-rac{6}{20}, and walk through each step so you can confidently subtract any fractions that come your way. Getting a good handle on fraction subtraction is super important, whether you're acing a math test, baking a recipe that calls for specific measurements, or even just understanding how things are divided up in the real world. So, buckle up, and let's make subtracting fractions less of a headache and more of a win!

Understanding the Basics of Fraction Subtraction

Before we jump into our example, let's quickly chat about what's going on when we subtract fractions. The main rule, and it's a biggie, is that you can only subtract fractions directly if they have the same denominator. Think of the denominator as the 'size' of the pieces you're working with. If you're trying to take away small pieces from big pieces, or vice versa, it gets messy. You need the pieces to be the same size first. If they aren't, don't sweat it! That's where finding a 'common denominator' comes in. This means we'll adjust the fractions so they have the same bottom number, making the subtraction possible. It's like making sure you're comparing apples to apples, not apples to oranges. The numerator, the top number, tells you how many of those pieces you have. When you subtract fractions with the same denominator, you simply subtract the numerators and keep the denominator the same. For example, rac{5}{8} - rac{2}{8} = rac{5-2}{8} = rac{3}{8}. See? The 'eighths' stay as eighths. The trickier part, and the part we'll focus on in our problem, is when the denominators are different, like in rac{11}{16}-rac{6}{20}. Here, our 'pieces' are sixteenths and twentieths – different sizes! So, our mission is to find a way to make them the same size before we can subtract. This involves finding the least common multiple (LCM) of the denominators, which becomes our common denominator. Once we have that, we'll adjust our fractions accordingly, ensuring that whatever we do to the bottom, we also do to the top (to keep the value of the fraction the same). Let's get this done!

Solving rac{11}{16}-rac{6}{20}: Step-by-Step

Alright guys, let's tackle our specific problem: rac{11}{16}-rac{6}{20}. Remember our golden rule? Different denominators mean we need a common one. So, first things first, we need to find the least common multiple (LCM) of 16 and 20. This is the smallest number that both 16 and 20 can divide into evenly.

We can list out the multiples of each number:

• Multiples of 16: 16, 32, 48, 64, 80, 96, ...
• Multiples of 20: 20, 40, 60, 80, 100, ...

The smallest number that appears in both lists is 80. So, 80 is our least common denominator (LCD)! This means we're going to convert both rac{11}{16} and rac{6}{20} into equivalent fractions with a denominator of 80.

Let's start with rac{11}{16}. To get from 16 to 80, we need to multiply 16 by something. What is it? Well, 80 divided by 16 is 5 ($80 e 16 = 5$). So, we multiply 16 by 5 to get 80. Now, to keep the fraction equivalent, we must do the same thing to the numerator. So, we multiply 11 by 5 as well: $11 e 5 = 55$. Therefore, rac{11}{16} is equivalent to rac{55}{80}.

Next, let's work on rac{6}{20}. To get from 20 to 80, we need to multiply 20 by something. What is it? 80 divided by 20 is 4 ($80 e 20 = 4$). So, we multiply 20 by 4 to get 80. Again, we do the same to the numerator: $6 e 4 = 24$. Therefore, rac{6}{20} is equivalent to rac{24}{80}.

Now that we have our equivalent fractions with the same denominator, the subtraction becomes super easy! Our problem is now: rac{55}{80} - rac{24}{80}.

We simply subtract the numerators and keep the denominator the same: $55 - 24 = 31$. So, the result is rac{31}{80}.

See? We converted the fractions so they were 'like terms' and then the subtraction was straightforward. It's all about finding that common ground, that common denominator!

Simplifying the Result

After performing the subtraction, we got rac{31}{80}. The next crucial step, as the problem asks, is to simplify this fraction. Simplifying means reducing the fraction to its lowest terms, where the numerator and the denominator have no common factors other than 1. Think of it as finding the simplest way to represent that amount.

To simplify rac{31}{80}, we need to look for any common factors between 31 and 80. Let's think about the number 31. Is it a prime number? A prime number is a whole number greater than 1 that has only two factors: 1 and itself. Let's check: Can 31 be divided evenly by 2? No. By 3? No ($3+1=4$, not divisible by 3). By 5? No (doesn't end in 0 or 5). By 7? No ($7 e 4 = 28$, $7 e 5 = 35$). It turns out, 31 is a prime number! Its only factors are 1 and 31.

Now, we need to see if 80 has 31 as a factor. Can 80 be divided by 31? No, it cannot. $31 e 1 = 31$, $31 e 2 = 62$, $31 e 3 = 93$. Since 31 is prime and it's not a factor of 80, the only common factor between 31 and 80 is 1.

When the only common factor between the numerator and the denominator is 1, the fraction is already in its simplest form. So, rac{31}{80} cannot be simplified any further. It's already as simple as it gets!

This is why when we look at our answer choices, rac{31}{80} is the correct one. We performed the subtraction correctly and then checked for simplification, finding that our result was already in its simplest form. It's a good habit to always check for simplification after subtracting (or adding) fractions, just in case!

Comparing with Answer Choices

Now that we've meticulously worked through the problem rac{11}{16}-rac{6}{20} and arrived at our answer, rac{31}{80}, let's quickly compare this with the given options:

a. rac{22}{36}

b. rac{19}{80}

c. rac{31}{80}

d. rac{13}{40}

Looking at our calculation, our simplified result is rac{31}{80}. This directly matches option c. It's always a good idea to double-check your work and make sure you haven't made any silly mistakes, but we've gone through the steps carefully. Finding the LCD, converting the fractions, performing the subtraction, and then simplifying (or confirming it's already simplified) are the key steps. Option a, rac{22}{36}, looks like it might have come from incorrectly finding a common denominator or incorrectly simplifying. Option b, rac{19}{80}, might be a result of subtracting the numerators incorrectly ($55-24$ is not 19). Option d, rac{13}{40}, is a common denominator, but it's not the least common denominator, and using it would lead to a different result, which might then require simplification, but it's unlikely to end up as rac{13}{40} from the original problem without further errors. Our rigorous process confirms that c. rac{31}{80} is indeed the correct answer. High-fives all around!

Conclusion: Mastering Fraction Subtraction

So, there you have it, guys! We've successfully navigated the process of subtracting and simplifying fractions using the example rac{11}{16}-rac{6}{20}. The key takeaway is that when denominators differ, the first and most crucial step is finding a common denominator. We found the least common multiple (LCM) of 16 and 20, which was 80. Then, we converted our original fractions into equivalent fractions with this common denominator: rac{11}{16} became rac{55}{80}, and rac{6}{20} became rac{24}{80}. With a common denominator in place, subtracting was a breeze: rac{55}{80} - rac{24}{80} = rac{31}{80}. Finally, we checked if our result, rac{31}{80}, could be simplified. Since 31 is a prime number and not a factor of 80, the fraction was already in its simplest form. This led us directly to the correct answer, option c.

Remember, practice makes perfect! The more you work with fractions, the more intuitive these steps will become. Don't be afraid to write out the multiples, find the LCM, and perform the conversions. Every step is a building block to mastering these math skills. Whether you're dealing with recipes, measurements, or complex equations, understanding how to subtract and simplify fractions is a fundamental skill that will serve you well. Keep practicing, stay curious, and you'll be a fraction whiz in no time! You've got this!