Solving 7 15/20 - 2 1/10 - 1 2/5 A Step By Step Guide

Hey guys! Today, we're going to break down the solution to the math problem: $7 \frac{15}{20} - 2 \frac{1}{10} - 1 \frac{2}{5}$. It might look a bit intimidating at first, but don't worry, we'll take it step by step and make sure it's super clear. We'll start by finding a common denominator, then convert the fractions, and finally, do the subtraction. So, let's dive in and get this sorted!

Step 1: Finding a Common Denominator

Okay, so the first thing we need to do is find a common denominator for our fractions. This is super important because we can't directly add or subtract fractions unless they have the same denominator. Our fractions have denominators of 20, 10, and 5. To find the common denominator, we need to list the multiples of each number and see what's the smallest multiple they all share. Think of it like finding the smallest meeting point for these numbers. It's like saying, "Hey, what's the smallest number that all these guys can divide into evenly?"

Let's list the multiples:

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

Now, let's look for the smallest number that appears in all three lists. Can you spot it? Yep, it's 20! So, the common denominator we're going to use is 20. This means we'll convert all our fractions to have a denominator of 20. This makes the subtraction process much smoother and easier. Finding the common denominator is like setting the stage for the rest of the problem – it's a crucial first step that makes everything else fall into place. Without it, we'd be trying to compare apples and oranges, which, as you know, doesn't really work in math!

What is the common denominator? 20

Step 2: Creating Equivalent Fractions

Now that we've found our common denominator, which is 20, the next step is to create equivalent fractions. This means we need to convert each fraction in our original problem so that they all have the same denominator of 20. Remember, an equivalent fraction is just a different way of writing the same fraction – it has the same value, but the numbers look different. Think of it like exchanging money; you can have different combinations of coins that still add up to the same amount. We are essentially doing the same thing with fractions here.

Let's break down each fraction:

• For $7 \frac{15}{20}$, the fraction part is already $\frac{15}{20}$, so we don't need to change it. This one is ready to go! It's like showing up to the party already dressed in the right outfit.
• For $2 \frac{1}{10}$, we need to convert $\frac{1}{10}$ to have a denominator of 20. To do this, we ask ourselves, "What do we multiply 10 by to get 20?" The answer is 2. So, we multiply both the numerator (1) and the denominator (10) by 2: $\frac{1 imes 2}{10 imes 2} = \frac{2}{20}$. So, $2 \frac{1}{10}$ becomes $2 \frac{2}{20}$.
• For $1 \frac{2}{5}$, we need to convert $\frac{2}{5}$ to have a denominator of 20. Again, we ask, "What do we multiply 5 by to get 20?" The answer is 4. So, we multiply both the numerator (2) and the denominator (5) by 4: $\frac{2 imes 4}{5 imes 4} = \frac{8}{20}$. Thus, $1 \frac{2}{5}$ becomes $1 \frac{8}{20}$.

Now, we have all our fractions with a common denominator of 20. Our problem now looks like this: $7 \frac{15}{20} - 2 \frac{2}{20} - 1 \frac{8}{20}$. See how much cleaner that looks? By converting to equivalent fractions, we've made the problem much easier to handle. It's like organizing your tools before starting a project – having everything in the right place makes the job so much smoother.

What are the equivalent fractions?

• $2 \frac{1}{10} = 2 \frac{2}{20}$
• $1 \frac{2}{5} = 1 \frac{8}{20}$

Step 3: Subtracting the Fractions

Alright, guys, now for the fun part – subtracting the fractions! We've done the prep work of finding a common denominator and creating equivalent fractions, so now we can actually perform the subtraction. Remember, when we subtract mixed numbers (whole numbers with fractions), we can subtract the whole numbers and the fractions separately, as long as the first fraction is bigger than the fractions we are subtracting. It’s like having separate piggy banks for dollars and cents – you can subtract from each one individually.

Let's rewrite our problem with the equivalent fractions we found:

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

First, let's subtract the whole numbers:

$7 - 2 - 1 = 4$

So, we have 4 as our whole number part of the answer. Now, let's subtract the fractions. We'll do it step by step:

$\frac{15}{20} - \frac{2}{20} = \frac{15 - 2}{20} = \frac{13}{20}$

Now, we subtract the next fraction:

$\frac{13}{20} - \frac{8}{20} = \frac{13 - 8}{20} = \frac{5}{20}$

So, the fraction part of our answer is $\frac{5}{20}$. Now, let's put the whole number and the fraction together:

$4 \frac{5}{20}$

But wait, we're not quite done yet! We can simplify the fraction $\frac{5}{20}$. Both 5 and 20 can be divided by 5:

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

So, our final answer is $4 \frac{1}{4}$. Woohoo! We did it! Subtracting fractions can seem tricky, but by breaking it down into smaller steps – finding a common denominator, creating equivalent fractions, and then subtracting – it becomes much more manageable. It's like climbing a staircase; each step gets you closer to the top.

What is the result of the subtraction? $4 \frac{5}{20}$

Step 4: Simplify the Fraction (If Possible)

Okay, we've got our answer, but the job's not quite done until we simplify the fraction, if we can. Simplifying a fraction means reducing it to its lowest terms. It's like making sure your answer is in its most elegant form. We want to make sure that the numerator (the top number) and the denominator (the bottom number) don't have any common factors other than 1. Think of it like making sure your pizza is cut into the fewest possible slices while still representing the same amount of pizza.

In our case, we have the fraction $\frac{5}{20}$. To simplify it, we need to find the greatest common factor (GCF) of 5 and 20. The GCF is the largest number that divides evenly into both numbers. Let's list the factors of 5 and 20:

• Factors of 5: 1, 5
• Factors of 20: 1, 2, 4, 5, 10, 20

The greatest common factor of 5 and 20 is 5. So, we can divide both the numerator and the denominator by 5:

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

So, our simplified fraction is $\frac{1}{4}$. This means that $\frac{5}{20}$ and $\frac{1}{4}$ are equivalent fractions, but $\frac{1}{4}$ is in its simplest form. Now, let's put this simplified fraction back into our mixed number. Our original answer was $4 \frac{5}{20}$, so now it becomes $4 \frac{1}{4}$.

Simplifying fractions is a crucial step because it ensures that our answer is in the most concise and clear form. It's like proofreading your work before you submit it – you want to make sure everything is just right. Plus, simplified fractions are easier to work with in future calculations. So, always remember to simplify your fractions whenever possible!

Can the fraction be simplified? If so, what is the simplified fraction? Yes, $\frac{1}{4}$

Final Answer

Alright, guys, we've reached the end of our journey! We've taken a mixed number subtraction problem, broken it down into manageable steps, and conquered it. We started by finding a common denominator, then creating equivalent fractions, subtracting the fractions, and finally, simplifying our answer. It's been quite the adventure, but we've emerged victorious!

So, let's recap our steps and state our final answer:

1. Original Problem: $7 \frac{15}{20} - 2 \frac{1}{10} - 1 \frac{2}{5}$
2. Common Denominator: 20
3. Equivalent Fractions: $7 \frac{15}{20} - 2 \frac{2}{20} - 1 \frac{8}{20}$
4. Subtraction: $4 \frac{5}{20}$
5. Simplified Fraction: $4 \frac{1}{4}$

Therefore, our final answer is $4 \frac{1}{4}$.

Great job, everyone! You've tackled a multi-step fraction problem like pros. Remember, the key to mastering math is to break down complex problems into smaller, more manageable steps. And always, always double-check your work and simplify when you can. Keep practicing, and you'll become fraction ninjas in no time!

Final Answer: $4 \frac{1}{4}$