Fraction Operations Practice Problems Solutions And Explanations

Hey guys! Let's dive into some practice problems involving fraction operations. Fractions can seem tricky at first, but with a little practice, you'll become a pro in no time! We'll break down each problem step-by-step, so you understand exactly how to solve them. Whether you're a student tackling homework or just brushing up on your math skills, this guide is here to help. So grab your pencil and paper, and let's get started!

1. Multiply Fractions: 4 rac{1}{4} imes rac{2}{3}

When it comes to multiplying fractions, things can seem a bit daunting at first. But fear not! Let's break down how to tackle this problem step by step, making it super easy to understand.

First off, we've got a mixed number here: 4 rac{1}{4}. Mixed numbers are just a combo of a whole number and a fraction. To make our lives easier when multiplying, we need to turn this mixed number into an improper fraction. An improper fraction is where the top number (numerator) is bigger than the bottom number (denominator). So how do we do it?

Think of it like this: we want to know how many 'quarters' we have in total. We've got 4 whole 'ones,' and each 'one' has 4 'quarters' (because the denominator is 4). So that's 4 * 4 = 16 'quarters.' Plus, we've got that extra 1 'quarter' in the fraction part. So all together, we've got 16 + 1 = 17 'quarters.'

So our mixed number 4 rac{1}{4} turns into the improper fraction $\frac{17}{4}$. Now our problem looks like this: $\frac{17}{4} imes \frac{2}{3}$. Much better, right?

Now comes the fun part: multiplying fractions! The rule here is super simple: just multiply the top numbers together, and then multiply the bottom numbers together. That's it!

So, we multiply 17 (the numerator of the first fraction) by 2 (the numerator of the second fraction). That gives us 34. That's the new numerator for our answer.

Then, we multiply 4 (the denominator of the first fraction) by 3 (the denominator of the second fraction). That gives us 12. That's the new denominator for our answer.

So now we've got $\frac{34}{12}$. We're almost there!

This fraction looks a bit clunky, though. The numbers are kinda big, and we can definitely make it simpler. This is where simplifying fractions comes in handy. We want to find a number that divides evenly into both the top and bottom numbers. This is called finding the greatest common factor, or GCF.

Looking at 34 and 12, we can see that both of them can be divided by 2. So let's do that!

34 divided by 2 is 17, and 12 divided by 2 is 6. So our fraction simplifies to $\frac{17}{6}$. Nice!

But hold on a sec… This is still an improper fraction (the top number is bigger than the bottom number). Sometimes, it's tidier to leave our answer as a mixed number. So let's turn $\frac{17}{6}$ back into one.

To do this, we think: how many times does 6 fit into 17? It fits in 2 times (because 6 * 2 = 12). So that's our whole number part.

But we've got some leftover, right? 17 minus 12 is 5. That's how many 'sixths' we have left over. So our fraction part is $\frac{5}{6}$.

Put it all together, and we get $2 \frac{5}{6}$. Woohoo! We've solved it!

So, to recap, when multiplying fractions, remember these key steps: turn mixed numbers into improper fractions, multiply the numerators, multiply the denominators, simplify the fraction if you can, and turn improper fractions back into mixed numbers if needed. You got this!

2. Subtracting Fractions: 1 rac{7}{10} - rac{1}{4}

Let's tackle the problem of subtracting fractions, specifically 1 rac{7}{10} - rac{1}{4}. This might seem a little complex at first glance, but don't worry, we'll break it down step by step to make it super clear and manageable.

The first thing we need to address is that pesky mixed number, 1 rac{7}{10}. Just like with multiplication, mixed numbers can make subtraction a bit trickier. So, our initial move is to convert this mixed number into an improper fraction. Remember, an improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number).

So, how do we convert 1 rac{7}{10}? We think of it this way: we have one whole, and we're measuring in tenths (because the denominator is 10). That one whole is made up of 10 tenths. So, we have 10 tenths plus the additional 7 tenths in the fraction part. That gives us a total of 10 + 7 = 17 tenths. Therefore, 1 rac{7}{10} becomes $\frac{17}{10}$.

Now our problem looks like this: \frac{17}{10} - rac{1}{4}. Much cleaner already!

Here's where things get a little more interesting. To subtract fractions, they need to have the same denominator. Think of it like trying to subtract apples from oranges – it doesn't quite work! We need a common unit. This means we need to find what's called the least common denominator (LCD). The LCD is the smallest number that both denominators (in our case, 10 and 4) can divide into evenly.

So, what's the LCD of 10 and 4? If you're not sure right away, one way to find it is to list out the multiples of each number until you find one they have in common. Multiples of 10 are: 10, 20, 30, 40, and so on. Multiples of 4 are: 4, 8, 12, 16, 20, 24, and so on. Aha! We see that 20 is the smallest number that appears in both lists. So, 20 is our LCD.

Great! Now we need to convert both fractions so they have a denominator of 20. Let's start with $\frac{17}{10}$. To get the denominator from 10 to 20, we need to multiply it by 2. But remember, whatever we do to the bottom, we also have to do to the top to keep the fraction equivalent. So, we multiply both the numerator and the denominator by 2: $\frac{17 imes 2}{10 imes 2} = \frac{34}{20}$.

Next, let's convert $\frac{1}{4}$. To get the denominator from 4 to 20, we need to multiply it by 5. So, we multiply both the numerator and the denominator by 5: $\frac{1 imes 5}{4 imes 5} = \frac{5}{20}$.

Now our problem looks like this: \frac{34}{20} - rac{5}{20}. Perfect! We have a common denominator, so we can finally subtract.

Subtracting fractions with a common denominator is straightforward: just subtract the numerators and keep the denominator the same. So, 34 - 5 = 29. Our fraction is now $\frac{29}{20}$.

We've got our answer, but it's in the form of an improper fraction. It's often tidier to convert this back to a mixed number, especially for final answers. So, how many times does 20 fit into 29? It fits in once. That's our whole number part.

How much is leftover? 29 - 20 = 9. So, we have 9 leftover 'twentieths.' Our fraction part is $\frac{9}{20}$.

Putting it all together, our answer is $1 \frac{9}{20}$. Awesome!

So, when subtracting fractions, remember these steps: convert mixed numbers to improper fractions, find the least common denominator, convert the fractions to have the LCD, subtract the numerators, and simplify or convert back to a mixed number if needed. You're doing great!

3. Dividing Fractions: $1 rac{1}{8}

4$

Let's dive into dividing fractions with the problem $1 rac{1}{8}

4$. Dividing fractions might seem a little tricky at first, but trust me, once you get the hang of the key rule, it becomes surprisingly straightforward. We'll break it down step by step, just like we've been doing, so you feel confident with each part.

The very first thing we need to address, just like in our previous problems, is the mixed number: 1 rac{1}{8}. Mixed numbers aren't the most convenient form when we're dividing fractions, so our initial step is to convert this into an improper fraction. Remember, an improper fraction is where the numerator (the top number) is bigger than the denominator (the bottom number).

So, how do we turn 1 rac{1}{8} into an improper fraction? We think about how many 'eighths' we have in total. We have one whole, and since our denominator is 8, that one whole is made up of 8 'eighths.' Then we add the extra 1 'eighth' from the fraction part. So, all together, we have 8 + 1 = 9 'eighths.' This means 1 rac{1}{8} is the same as $\frac{9}{8}$.

Now, let's think about the number we're dividing by: 4. This is a whole number, but to work with it as a fraction, we can simply write it as $\frac{4}{1}$. Any whole number can be written as a fraction by putting it over 1. This doesn't change its value, but it makes it easier to apply the rules of fraction division.

So now our problem looks like this: $\frac{9}{8}

\frac{4}{1}$. We're all set to divide!

Here comes the golden rule of dividing fractions: "Keep, Change, Flip." It might sound a little quirky, but it's the key to making division a breeze. Let's break down what each part means:

• Keep: We keep the first fraction exactly as it is. So $\frac{9}{8}$ stays $\frac{9}{8}$.
• Change: We change the division sign (

) to a multiplication sign (${ imes}$). This is the core of how we handle division.

• Flip: We flip the second fraction over. This is also known as finding the reciprocal. So $\frac{4}{1}$ becomes $\frac{1}{4}$.

Following the "Keep, Change, Flip" rule, our problem transforms from $\frac{9}{8}

\frac{4}{1}$ to $\frac{9}{8} imes \frac{1}{4}$. See how we've turned a division problem into a multiplication problem? Much easier!

Now that we have a multiplication problem, we can use the rule we learned earlier: multiply the numerators together and multiply the denominators together. So, 9 (numerator of the first fraction) multiplied by 1 (numerator of the second fraction) is 9. That's our new numerator.

Then, 8 (denominator of the first fraction) multiplied by 4 (denominator of the second fraction) is 32. That's our new denominator.

So, we now have the fraction $\frac{9}{32}$.

Time to check if we can simplify our answer. Simplifying a fraction means finding a number that divides evenly into both the numerator and the denominator. Looking at 9 and 32, do they have any common factors? The factors of 9 are 1, 3, and 9. The factors of 32 are 1, 2, 4, 8, 16, and 32. The only factor they have in common is 1, which means our fraction is already in its simplest form.

Since our fraction $\frac{9}{32}$ is proper (the numerator is smaller than the denominator), we don't need to convert it to a mixed number. We're all done!

So, $1 rac{1}{8}

4 = \frac{9}{32}$.

To recap, when dividing fractions, remember these key steps: convert mixed numbers to improper fractions, rewrite whole numbers as fractions (like $\frac{4}{1}$), use the "Keep, Change, Flip" rule to turn division into multiplication, multiply the fractions, and simplify your answer if possible. You're becoming a fraction division master!

4. Adding Fractions: 3 rac{1}{6} + rac{1}{2}

Alright, let's get into adding fractions with the problem 3 rac{1}{6} + rac{1}{2}. Adding fractions might seem straightforward, but there are a couple of key steps to keep in mind, especially when dealing with mixed numbers. We'll break it down bit by bit so you can add fractions with confidence.

Just like with subtraction and division, our first step when we see a mixed number is to convert it to an improper fraction. Mixed numbers can be a little clunky to work with directly in addition, so let's make our lives easier. We're dealing with 3 rac{1}{6}, so we need to figure out how many 'sixths' we have in total.

Think of it like this: we have 3 whole 'ones,' and each 'one' has 6 'sixths' (because our denominator is 6). So that's 3 * 6 = 18 'sixths.' Plus, we have that extra 1 'sixth' from the fraction part. Add that in, and we have 18 + 1 = 19 'sixths.' So, 3 rac{1}{6} is the same as $\frac{19}{6}$.

Now our problem looks like this: \frac{19}{6} + rac{1}{2}. Much better to work with, right?

Now, here's a crucial rule for adding (and subtracting) fractions: they need to have the same denominator. It's like trying to add apples and bananas – you can't really do it directly. You need a common unit, like 'pieces of fruit.' In fractions, that common unit is the denominator.

We need to find the least common denominator (LCD) for 6 and 2. The LCD is the smallest number that both 6 and 2 divide into evenly. If you know your times tables, you might spot it right away: it's 6! 6 divides into itself (6

6 = 1), and 2 divides into 6 (6

2 = 3).

So, we want both fractions to have a denominator of 6. The first fraction, $\frac{19}{6}$, already has a denominator of 6, so we can leave it as is. That's convenient!

But we need to change $\frac{1}{2}$ so it has a denominator of 6. To get from 2 to 6, we need to multiply by 3. Remember, whatever we do to the bottom (the denominator), we also have to do to the top (the numerator) to keep the fraction equivalent. So, we multiply both the numerator and the denominator of $\frac{1}{2}$ by 3: $\frac{1 imes 3}{2 imes 3} = \frac{3}{6}$.

Now our problem looks like this: $\frac{19}{6} + \frac{3}{6}$. Perfect! We have a common denominator, so we can add the fractions.

Adding fractions with a common denominator is the easy part: just add the numerators together and keep the denominator the same. So, 19 + 3 = 22. Our fraction becomes $\frac{22}{6}$.

We have our answer, but it's an improper fraction (the numerator is bigger than the denominator). It's often a good idea to convert improper fractions back to mixed numbers, especially for final answers. So, how many times does 6 fit into 22? It fits in 3 times (because 6 * 3 = 18). That's our whole number part.

How much is leftover? 22 - 18 = 4. So, we have 4 'sixths' leftover. That's our fraction part: $\frac{4}{6}$.

Putting it together, we have $3 \frac{4}{6}$. But we're not quite done yet! We should always check if our fraction part can be simplified. Can we find a number that divides evenly into both 4 and 6? Yes! Both are divisible by 2.

So, let's divide both the numerator and the denominator of $\frac{4}{6}$ by 2: $\frac{4

2}{6

2} = \frac{2}{3}$.

Our final answer, in its simplest form, is $3 \frac{2}{3}$. Great job!

To sum it up, when adding fractions, remember these key steps: convert mixed numbers to improper fractions, find the least common denominator, convert the fractions to have the LCD, add the numerators, simplify your answer, and convert back to a mixed number if needed. Keep practicing, and you'll be a fraction-adding whiz!

5. Dividing Mixed Fractions: $1 rac{19}{48}

1 rac{1}{4}$

Let's tackle dividing mixed fractions with the problem $1 rac{19}{48}

1 rac{1}{4}$. This one looks a little complex, but don't worry, we'll break it down step by step just like before. The key here is to remember our rules for mixed numbers and division.

As you probably know by now, the first thing we need to do with mixed numbers is convert them to improper fractions. We have two mixed numbers in this problem, so we'll need to do this twice. Let's start with 1 rac{19}{48}.

To convert it, we think about how many 'forty-eighths' we have. One whole is made up of 48 'forty-eighths' (because our denominator is 48). We add the extra 19 'forty-eighths' from the fraction part: 48 + 19 = 67. So, 1 rac{19}{48} is equal to $\frac{67}{48}$.

Now let's convert the second mixed number, 1 rac{1}{4}. How many 'quarters' do we have? One whole is 4 'quarters,' and we have an extra 1 'quarter,' so that's 4 + 1 = 5 'quarters.' This means 1 rac{1}{4} is the same as $\frac{5}{4}$.

Our problem now looks like this: $\frac{67}{48}

\frac{5}{4}$. We've gotten rid of the mixed numbers, which is a great start!

Now we're ready to divide. Remember our "Keep, Change, Flip" rule? It's the key to dividing fractions.

• Keep: We keep the first fraction as it is: $\frac{67}{48}$.
• Change: We change the division sign (

) to a multiplication sign (${ imes}$).

• Flip: We flip the second fraction over (find its reciprocal). So, $\frac{5}{4}$ becomes $\frac{4}{5}$.

Applying the "Keep, Change, Flip" rule, our problem transforms to $\frac{67}{48} imes \frac{4}{5}$. Division is now multiplication – awesome!

Now we can multiply the fractions. Multiply the numerators together: 67 * 4 = 268. That's our new numerator. Multiply the denominators together: 48 * 5 = 240. That's our new denominator.

So, we have the fraction $\frac{268}{240}$.

This fraction looks pretty big, so let's see if we can simplify it. Both 268 and 240 are even numbers, which means they're both divisible by 2. Let's divide both by 2: $\frac{268

2}{240

2} = \frac{134}{120}$.

The numbers are still even, so let's divide by 2 again: $\frac{134

2}{120

2} = \frac{67}{60}$.

Now we have $\frac{67}{60}$. Can we simplify further? 67 is a prime number (it's only divisible by 1 and itself), and 60 is not divisible by 67, so this fraction is in its simplest form.

However, it's an improper fraction (the numerator is bigger than the denominator). Let's convert it to a mixed number. How many times does 60 fit into 67? It fits in once. That's our whole number part.

How much is leftover? 67 - 60 = 7. So, we have 7 'sixtieths' leftover. That's our fraction part: $\frac{7}{60}$.

So, our final answer is $1 \frac{7}{60}$. You did it!

To recap, when dividing mixed fractions, remember to convert mixed numbers to improper fractions, use the "Keep, Change, Flip" rule, multiply the fractions, simplify the result, and convert back to a mixed number if necessary. With practice, you'll be dividing mixed fractions like a pro!

6. Adding Mixed Numbers: 4 rac{3}{8} + 2 rac{1}{2}

Let's dive into adding mixed numbers with the problem 4 rac{3}{8} + 2 rac{1}{2}. Adding mixed numbers can be done in a couple of ways, and we'll explore the most common method here, which involves converting to improper fractions. This approach is reliable and helps build a strong understanding of fraction operations.

As we've seen in previous examples, the first step when dealing with mixed numbers in addition (or subtraction, division, or multiplication) is to convert them into improper fractions. So, let's start with 4 rac{3}{8}.

To convert it, we need to figure out how many 'eighths' we have in total. The whole number part is 4, and each 'one' contains 8 'eighths' (because our denominator is 8). So, 4 'ones' contain 4 * 8 = 32 'eighths.' Then we add the 3 'eighths' from the fraction part: 32 + 3 = 35 'eighths.' This means 4 rac{3}{8} is equal to $\frac{35}{8}$.

Now let's convert the second mixed number, 2 rac{1}{2}. How many 'halves' do we have? The whole number part is 2, and each 'one' contains 2 'halves.' So, 2 'ones' contain 2 * 2 = 4 'halves.' Then we add the 1 'half' from the fraction part: 4 + 1 = 5 'halves.' So, 2 rac{1}{2} is the same as $\frac{5}{2}$.

Our problem now looks like this: $\frac{35}{8} + \frac{5}{2}$. We've successfully converted the mixed numbers to improper fractions. Great job!

Remember, to add fractions, they need to have a common denominator. That means we need to find the least common denominator (LCD) for 8 and 2. The LCD is the smallest number that both 8 and 2 divide into evenly. In this case, the LCD is 8, since 8 divides into itself and 2 divides into 8.

The first fraction, $\frac{35}{8}$, already has a denominator of 8, so we don't need to change it. That makes things easier!

We do need to convert the second fraction, $\frac{5}{2}$, so it has a denominator of 8. To get from 2 to 8, we need to multiply by 4. Remember, we must multiply both the numerator and the denominator by the same number to keep the fraction equivalent. So, we multiply both the top and bottom of $\frac{5}{2}$ by 4: $\frac{5 imes 4}{2 imes 4} = \frac{20}{8}$.

Now our problem looks like this: $\frac{35}{8} + \frac{20}{8}$. We have a common denominator, so we can add the fractions!

Adding fractions with a common denominator is simple: add the numerators and keep the denominator the same. So, 35 + 20 = 55. Our fraction becomes $\frac{55}{8}$.

We have our answer, but it's an improper fraction. Let's convert it back to a mixed number. How many times does 8 fit into 55? It fits in 6 times (because 8 * 6 = 48). That's our whole number part.

How much is leftover? 55 - 48 = 7. So, we have 7 'eighths' leftover. That's our fraction part: $\frac{7}{8}$.

Therefore, our final answer is $6 \frac{7}{8}$.

To summarize, when adding mixed numbers, we convert them to improper fractions, find the least common denominator, convert the fractions to have the LCD, add the numerators, and then convert the result back to a mixed number. Practice these steps, and you'll become a pro at adding mixed numbers!

7. Subtracting Fractions Again: 1 rac{1}{8} - rac{3}{10}

Let's revisit subtracting fractions with the problem 1 rac{1}{8} - rac{3}{10}. This one's a great review of all the steps we've learned so far, from converting mixed numbers to finding common denominators. So, grab your pencil and let's work through it!

The first thing we need to do is tackle the mixed number, 1 rac{1}{8}. As we've discussed, mixed numbers aren't the most convenient for subtraction, so we'll convert it to an improper fraction. We need to figure out how many 'eighths' we have in total.

We have one whole, which is made up of 8 'eighths' (since our denominator is 8). Then we add the 1 'eighth' from the fraction part, giving us 8 + 1 = 9 'eighths.' So, 1 rac{1}{8} is equal to $\frac{9}{8}$.

Now our problem is \frac{9}{8} - rac{3}{10}. We've gotten rid of the mixed number, which is a good start!

To subtract fractions, they need to have a common denominator. That means we need to find the least common denominator (LCD) for 8 and 10. The LCD is the smallest number that both 8 and 10 divide into evenly. If you're not sure right away, you can list out the multiples of each number until you find a common one. Multiples of 8: 8, 16, 24, 32, 40… Multiples of 10: 10, 20, 30, 40… Ah, we see that 40 is the smallest multiple they share, so our LCD is 40.

Now we need to convert both fractions so they have a denominator of 40. Let's start with $\frac{9}{8}$. To get from 8 to 40, we need to multiply by 5. Remember, whatever we do to the bottom, we must do to the top to keep the fraction equivalent. So, we multiply both the numerator and the denominator by 5: $\frac{9 imes 5}{8 imes 5} = \frac{45}{40}$.

Next, let's convert $\frac{3}{10}$. To get from 10 to 40, we need to multiply by 4. So, we multiply both the numerator and the denominator by 4: $\frac{3 imes 4}{10 imes 4} = \frac{12}{40}$.

Now our problem looks like this: $\frac{45}{40} - \frac{12}{40}$. We have a common denominator, so we're ready to subtract!

Subtracting fractions with a common denominator is easy: subtract the numerators and keep the denominator the same. So, 45 - 12 = 33. Our fraction becomes $\frac{33}{40}$.

Now, let's see if we can simplify our answer. Can we find a number that divides evenly into both 33 and 40? The factors of 33 are 1, 3, 11, and 33. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. The only factor they have in common is 1, which means our fraction is already in its simplest form.

Since our fraction $\frac{33}{40}$ is a proper fraction (the numerator is smaller than the denominator), we don't need to convert it to a mixed number. We're all done!

So, 1 rac{1}{8} - rac{3}{10} = \frac{33}{40}.

In summary, remember these key steps when subtracting fractions: convert mixed numbers to improper fractions, find the least common denominator, convert the fractions to have the LCD, subtract the numerators, and simplify your answer if possible. You're getting more and more confident with fractions!

8. A Single Fraction: $\frac{7}{12}$

Lastly, let's consider the fraction $\frac{7}{12}$. Sometimes, a problem is just presented as a single fraction, and there's not much to "solve" in the traditional sense. However, it's still valuable to analyze the fraction and understand its properties.

The first thing we can consider is whether the fraction can be simplified. Simplifying a fraction means finding a common factor (a number that divides evenly) for both the numerator (the top number) and the denominator (the bottom number), and then dividing both by that factor. This reduces the fraction to its simplest form.

In the case of $\frac{7}{12}$, let's look at the factors of 7 and 12. The factors of 7 are 1 and 7 (since 7 is a prime number, it's only divisible by 1 and itself). The factors of 12 are 1, 2, 3, 4, 6, and 12. The only factor that 7 and 12 have in common is 1.

When the only common factor between the numerator and denominator is 1, it means the fraction is already in its simplest form. We can't reduce it any further.

So, $\frac{7}{12}$ is in its simplest form. We can also think about what this fraction represents. It means that if we have something divided into 12 equal parts, we're considering 7 of those parts. For example, if we had a pizza cut into 12 slices, $\frac{7}{12}$ would represent 7 of those slices.

We could also express $\frac{7}{12}$ as a decimal or a percentage if needed. To convert it to a decimal, we would divide 7 by 12. This gives us approximately 0.5833.

To convert it to a percentage, we would multiply the decimal by 100. So, 0.5833 * 100 is approximately 58.33%. This means $\frac{7}{12}$ is roughly equivalent to 58.33%.

So, while there's no specific calculation to perform with $\frac{7}{12}$, understanding its simplest form, what it represents, and how to convert it to other forms (like decimals and percentages) is a key part of working with fractions.

Great job working through all these fraction problems! Remember, practice makes perfect, so keep tackling those fractions, and you'll become a true math whiz!