Subtracting Mixed Fractions A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of subtracting mixed fractions. Specifically, we're going to tackle the problem: $7 \frac{15}{20} - 2 \frac{1}{10} - 1 \frac{2}{5} = ?$ Don't worry, it might look a little intimidating at first, but we'll break it down into easy-to-follow steps. By the end of this guide, you'll be a pro at solving these types of problems! We'll explore not just the solution, but the why behind each step, making sure you truly grasp the concepts involved. Think of this as not just learning to solve one problem, but gaining a superpower to tackle countless others!

Step 1: Finding the Common Denominator – The Key to Fraction Harmony

The very first thing we need to do when subtracting fractions is to make sure they have the same denominator. Why? Because we can only directly add or subtract fractions that represent parts of the same whole. Imagine trying to compare slices of a pizza cut into 20 pieces with slices cut into 10 pieces – it's much easier if all the slices are the same size! Finding the common denominator is like finding a universal language for our fractions, allowing us to accurately compare and combine them. This first step is crucial because without a common denominator, we're essentially trying to subtract apples from oranges – it just doesn't work! So, let's roll up our sleeves and find that common ground for our fractions.

In our problem, we have the fractions $\frac{15}{20}$, $\frac{1}{10}$, and $\frac{2}{5}$. To find a common denominator, we need to identify a number that is a multiple of all three denominators: 20, 10, and 5. One way to do this is to list out the multiples of each number. Remember, multiples are just what you get when you multiply a number by 1, 2, 3, and so on. This process might seem a little tedious at first, but it's a foolproof way to find that magic number that will allow us to combine our fractions seamlessly. Think of it as detective work – we're searching for the common thread that connects these fractions.

Let's list them out:

Multiples of 20: 20, 40, 60, 80, ... Multiples of 10: 10, 20, 30, 40, ... Multiples of 5: 5, 10, 15, 20, 25, ...

Now, we look for the smallest number that appears in all three lists. This is called the least common multiple (LCM), and it will be our common denominator. By using the LCM, we ensure that we're working with the smallest possible numbers, making our calculations easier down the line. It's like choosing the right tool for the job – the LCM is the most efficient tool for combining these fractions. Can you spot the LCM in our lists? It's the number 20! So, our common denominator is 20.

What is the common denominator? 20

Step 2: Creating Equivalent Fractions – Speaking the Same Language

Now that we've found our common denominator, 20, we need to rewrite each fraction as an equivalent fraction with a denominator of 20. Equivalent fractions are fractions that represent the same amount, even though they have different numerators and denominators. Think of it like this: $\frac{1}{2}$ is the same as $\frac{2}{4}$ – you're still talking about half of something, just using different numbers to express it. To create equivalent fractions, we multiply both the numerator and the denominator of the original fraction by the same number. This is crucial because it maintains the fraction's value – we're just changing how it looks, not what it represents. It's like translating a sentence into another language – the meaning stays the same, but the words are different.

For the first fraction, $\frac{15}{20}$, we already have a denominator of 20, so we don't need to do anything! It's already in the form we need. This is a nice little shortcut that saves us some time and effort. Always be on the lookout for these opportunities to simplify the process!

For the second fraction, $\frac{1}{10}$, we need to figure out what to multiply 10 by to get 20. The answer is 2! So, we multiply both the numerator and the denominator of $\frac{1}{10}$ by 2: $\frac{1}{10} \times \frac{2}{2} = \frac{2}{20}$. Remember, multiplying by $\frac{2}{2}$ is the same as multiplying by 1, so we're not changing the value of the fraction, just its appearance. This step is all about finding the right multiplier to transform our fractions into their equivalent forms.

For the third fraction, $\frac{2}{5}$, we need to figure out what to multiply 5 by to get 20. The answer is 4! So, we multiply both the numerator and the denominator of $\frac{2}{5}$ by 4: $\frac{2}{5} \times \frac{4}{4} = \frac{8}{20}$. Again, we're using the principle of equivalent fractions to rewrite $\frac{2}{5}$ in a way that allows us to combine it with the other fractions. By the end of this step, all our fractions will be speaking the same language, ready for subtraction.

What are the equivalent fractions? $\frac{15}{20}= \frac{15}{20}$ $\frac{1}{10}=$ $\frac{2}{5}=$

Step 3: Rewriting the Original Problem – Setting the Stage for Subtraction

Now that we have our equivalent fractions, it's time to rewrite the original problem using these new fractions. This step is all about clarity and organization. By rewriting the problem, we make it visually clear what we're doing and minimize the risk of making mistakes. It's like tidying up your workspace before starting a project – a clean setup leads to a smoother process. So, let's replace our original fractions with their equivalent counterparts, setting the stage for the actual subtraction.

Our original problem was: $7 \frac{15}{20} - 2 \frac{1}{10} - 1 \frac{2}{5} = ?$

We found that $\frac{1}{10}$ is equivalent to $\frac{2}{20}$ and $\frac{2}{5}$ is equivalent to $\frac{8}{20}$. So, we can rewrite the problem as:

$7 \frac{15}{20} - 2 \frac{2}{20} - 1 \frac{8}{20} = ?$

See how much clearer the problem looks now? All the fractions have the same denominator, which means we're ready to perform the subtraction. This rewriting step is a crucial bridge between finding the equivalent fractions and actually solving the problem. It's like setting up the dominoes before you start the chain reaction – each step prepares the way for the next.

What does the problem look like now? 7 rac{15}{20}-2 rac{?}{20}-1 rac{?}{20}=

Step 4: Subtracting the Mixed Numbers – Whole Numbers and Fractions Unite!

Now comes the fun part – actually subtracting the mixed numbers! There are a couple of ways we can approach this. One method is to subtract the whole numbers and the fractions separately. This is often the easiest approach when the fraction we're subtracting is smaller than the fraction we're subtracting from. It's like breaking the problem down into smaller, more manageable pieces. The other method is to convert the mixed numbers into improper fractions and then subtract. This method is particularly useful when we need to borrow from the whole number, as we'll see in a moment. Think of these two methods as different tools in your mathematical toolbox – you can choose the one that best suits the problem at hand.

Let's start by trying the first method: subtracting the whole numbers and fractions separately. We have:

$7 \frac{15}{20} - 2 \frac{2}{20} - 1 \frac{8}{20}$

First, we subtract the whole numbers: 7 - 2 - 1 = 4. That's the whole number part of our answer. This is a straightforward step that simplifies the problem significantly. It's like taking care of the big pieces of the puzzle first.

Next, we subtract the fractions: $\frac{15}{20} - \frac{2}{20} - \frac{8}{20}$. Since the fractions have a common denominator, we can simply subtract the numerators: 15 - 2 - 8 = 5. So, we have $\frac{5}{20}$. This part of the process is where our hard work in finding the common denominator pays off. It allows us to combine the fractional parts of our mixed numbers effortlessly.

Combining the whole number and the fraction, we get $4 \frac{5}{20}$. But wait, we're not quite done yet!

What is the result? Whole Numbers: $ewline$ Fractions: $ewline$

Step 5: Simplifying the Fraction – Putting the Final Touches

The final step is to simplify the fraction. Simplifying a fraction means reducing it to its lowest terms. We do this by dividing both the numerator and the denominator by their greatest common factor (GCF). The GCF is the largest number that divides evenly into both the numerator and the denominator. Simplifying fractions is like polishing a gemstone – it brings out its true beauty and makes it easier to understand. A simplified fraction is easier to work with and gives a clearer picture of the quantity it represents.

In our case, we have the fraction $\frac{5}{20}$. What's the greatest common factor of 5 and 20? It's 5! So, we divide both the numerator and the denominator by 5:

$\frac{5}{20} \div \frac{5}{5} = \frac{1}{4}$

Now our fraction is in its simplest form. This step is crucial for presenting our answer in the most elegant and understandable way. It's like writing the final draft of an essay – we're making sure our answer is clear, concise, and correct.

So, our final answer is $4 \frac{1}{4}$. We've successfully navigated the world of mixed fraction subtraction! You've seen how breaking down a complex problem into smaller, manageable steps can make it much easier to solve. And you've learned the importance of finding common denominators, creating equivalent fractions, and simplifying your answers. These are valuable skills that will serve you well in all your mathematical adventures.

What is the simplified fraction?

Final Answer

Therefore, $7 \frac{15}{20} - 2 \frac{1}{10} - 1 \frac{2}{5} = 4 \frac{1}{4}$.

Congratulations! You've successfully subtracted mixed fractions. Keep practicing, and you'll become a master in no time! Remember, the key is to break down the problem into smaller steps, understand the why behind each step, and don't be afraid to ask for help when you need it. Happy calculating!