Fraction Calculations: Step-by-Step Solutions

Hey guys! Let's break down how to calculate fractions with some easy examples. We’re going to tackle multiplication, division, and even squaring fractions. If you've ever felt a bit puzzled by fractions, don't worry – we'll go through it step by step.

a) Calculating 3/4 * 5/7

When we talk about multiplying fractions, it's pretty straightforward. You just multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, for 3/4 * 5/7, we multiply 3 by 5 to get the new numerator, and 4 by 7 to get the new denominator.

Let’s dive deeper. We have the fractions 3/4 and 5/7. The numerator of the first fraction is 3, and the numerator of the second fraction is 5. Multiplying these gives us 3 * 5 = 15. That’s our new numerator. Now, for the denominators, we have 4 and 7. Multiplying these gives us 4 * 7 = 28. That’s our new denominator. So, 3/4 * 5/7 equals 15/28. The key thing to remember here is the simplicity of the process: numerator times numerator, denominator times denominator. There's no need to find common denominators or do any extra steps – just multiply straight across. This makes fraction multiplication quite efficient once you get the hang of it. This method works for any two fractions you need to multiply, making it a fundamental skill in working with fractions.

So, here’s how it looks:

(3/4) * (5/7) = (3 * 5) / (4 * 7) = 15/28

b) Calculating 4/11 * 3/4

Now, let’s multiply 4/11 by 3/4. Same as before, we'll multiply the numerators and the denominators. This time, though, we have a little trick we can use to simplify things even further. Before we multiply, we can look for common factors between the numerators and denominators. This is called simplifying fractions, and it makes the final answer easier to work with.

In this case, we have 4 in the numerator of the first fraction and 4 in the denominator of the second fraction. They share a common factor of 4. So, we can divide both of these by 4, which turns them into 1. This simplification step is crucial because it reduces the size of the numbers we’re dealing with, making the multiplication easier and the final fraction simpler. After simplification, our problem looks like this: (1/11) * (3/1). Now, multiply the numerators (1 * 3 = 3) and the denominators (11 * 1 = 11), and we get 3/11. This technique of simplifying before multiplying is incredibly helpful, especially when dealing with larger numbers, as it prevents the need to simplify a large fraction at the end. Simplifying upfront ensures the final answer is in its simplest form, saving time and effort. Always keep an eye out for these opportunities to simplify; it's a hallmark of efficient fraction handling.

Here's the breakdown:

(4/11) * (3/4) = (4 * 3) / (11 * 4)

We can simplify by canceling out the 4s:

(1/11) * (3/1) = (1 * 3) / (11 * 1) = 3/11

c) Calculating 5/7 * 14/19

For 5/7 * 14/19, we’ll follow the same multiplication rule, but there's a simplification opportunity here too. Notice that 7 and 14 have a common factor. Can you spot it?

Just like in the previous example, we need to look for ways to simplify before we multiply. In this case, the denominator of the first fraction (7) and the numerator of the second fraction (14) share a common factor. Both numbers are divisible by 7. So, we can divide 7 by 7, which gives us 1, and divide 14 by 7, which gives us 2. This simplification step transforms our fractions into 5/1 and 2/19. Now, we multiply the numerators (5 * 2 = 10) and the denominators (1 * 19 = 19). This gives us the fraction 10/19. Simplifying before multiplying made the calculation much easier and avoided dealing with larger numbers. The ability to identify these common factors is a key skill in fraction manipulation. It not only simplifies the arithmetic but also reduces the risk of errors. By simplifying early on, we ensure that our final answer is in its most reduced form, which is a fundamental practice in mathematical problem-solving. This step-by-step simplification is a powerful tool for handling fractions confidently and accurately.

So, we simplify 7 and 14 by dividing both by 7:

(5/7) * (14/19) = (5/1) * (2/19) = (5 * 2) / (1 * 19) = 10/19

d) Calculating (3/5)^2

When we see (3/5)^2, it means we’re squaring the fraction, which is the same as multiplying the fraction by itself. So, (3/5)^2 is the same as 3/5 * 3/5.

Squaring a fraction involves multiplying the fraction by itself. In this instance, we're squaring 3/5, which means we need to calculate (3/5) * (3/5). To do this, we simply multiply the numerators together and the denominators together, just as we did with regular fraction multiplication. The numerator of the first fraction is 3, and the numerator of the second fraction is also 3, so we multiply 3 * 3, which equals 9. This will be the numerator of our result. For the denominators, we have 5 in both fractions, so we multiply 5 * 5, which equals 25. This will be the denominator of our result. Therefore, (3/5) * (3/5) equals 9/25. Squaring fractions is a common operation in various mathematical contexts, including geometry and algebra. The process is straightforward: square the numerator and square the denominator. This method is consistent and reliable, making it easy to handle squaring operations with any fraction. Understanding this concept is crucial for more advanced mathematical calculations involving fractions and exponents. Remember, squaring a fraction is just multiplying it by itself, and the same rules of multiplication apply.

Let's multiply the numerators and denominators:

(3/5)^2 = (3/5) * (3/5) = (3 * 3) / (5 * 5) = 9/25

e) Calculating 4/5 ÷ 3/7

Dividing fractions might seem tricky, but there’s a simple rule: to divide fractions, you flip the second fraction (the divisor) and multiply. So, 4/5 ÷ 3/7 becomes 4/5 * 7/3.

The division of fractions may initially appear complex, but it hinges on a straightforward rule: to divide by a fraction, you multiply by its reciprocal. The reciprocal of a fraction is simply the fraction flipped, where the numerator becomes the denominator, and the denominator becomes the numerator. In this case, we're dividing 4/5 by 3/7. To solve this, we first find the reciprocal of 3/7, which is 7/3. Then, we change the division operation to multiplication, making our problem 4/5 multiplied by 7/3. Now, we proceed with multiplication as we did in previous examples. We multiply the numerators (4 * 7 = 28) and then multiply the denominators (5 * 3 = 15). This gives us the fraction 28/15. This fraction is an example of an improper fraction, where the numerator is larger than the denominator. While 28/15 is a perfectly valid answer, it can also be converted to a mixed number if needed. The key takeaway here is the process of changing division into multiplication by the reciprocal. This method makes fraction division manageable and consistent, and it is a cornerstone of fraction arithmetic. Understanding and applying this rule allows for efficient and accurate division of fractions in various mathematical scenarios.

Now, multiply:

(4/5) ÷ (3/7) = (4/5) * (7/3) = (4 * 7) / (5 * 3) = 28/15

f) Calculating 15/8 ÷ 5/4

Let’s divide 15/8 by 5/4. Just like before, we flip the second fraction and multiply. So, 15/8 ÷ 5/4 becomes 15/8 * 4/5. And guess what? We can simplify before we multiply!

As we tackle the division of 15/8 by 5/4, we once again apply the rule of multiplying by the reciprocal. We start by flipping the second fraction, 5/4, to get its reciprocal, which is 4/5. This transforms our division problem into a multiplication problem: 15/8 multiplied by 4/5. Now, before we jump into multiplying the numerators and denominators, we take a moment to look for opportunities to simplify. Simplifying fractions before multiplying can significantly ease the calculation and ensure the final answer is in its simplest form. In this case, we notice that 15 and 5 share a common factor of 5, and 8 and 4 share a common factor of 4. We divide 15 by 5 to get 3, and we divide 5 by 5 to get 1. Similarly, we divide 4 by 4 to get 1, and we divide 8 by 4 to get 2. Our simplified problem now looks like this: 3/2 multiplied by 1/1. Multiplying the numerators (3 * 1 = 3) and the denominators (2 * 1 = 2) gives us the fraction 3/2. This fraction is in its simplest form, and we have successfully performed the division. The lesson here is clear: always be on the lookout for chances to simplify before multiplying. It's a practice that saves time, reduces the likelihood of errors, and helps build confidence in fraction manipulation.

We can simplify 15 and 5 by dividing both by 5, and 4 and 8 by dividing both by 4:

(15/8) ÷ (5/4) = (15/8) * (4/5) = (3/2) * (1/1) = (3 * 1) / (2 * 1) = 3/2

g) Calculating 4/9 ÷ 2

Now, let's divide 4/9 by 2. Remember, we can think of 2 as the fraction 2/1. So, this problem is really 4/9 ÷ 2/1. To divide, we flip the second fraction and multiply:

When dividing a fraction by a whole number, it’s helpful to think of the whole number as a fraction with a denominator of 1. In this scenario, we are dividing 4/9 by 2. We can rewrite 2 as 2/1, making our division problem 4/9 divided by 2/1. Now, we apply the rule for dividing fractions: multiply by the reciprocal. The reciprocal of 2/1 is 1/2. So, our problem becomes 4/9 multiplied by 1/2. Before multiplying, we should always check for opportunities to simplify. In this case, we see that the numerator of the first fraction (4) and the denominator of the second fraction (2) share a common factor of 2. We can divide 4 by 2 to get 2, and we divide 2 by 2 to get 1. Our simplified multiplication problem is now 2/9 multiplied by 1/1. Multiplying the numerators (2 * 1 = 2) and the denominators (9 * 1 = 9) gives us the fraction 2/9. This fraction is already in its simplest form, and we have successfully divided 4/9 by 2. This process highlights the importance of understanding that whole numbers can be expressed as fractions, and the technique of multiplying by the reciprocal is a universal approach to fraction division. Simplifying before multiplying makes the calculation cleaner and more straightforward, ensuring the final answer is in its reduced form.

(4/9) ÷ 2 = (4/9) ÷ (2/1) = (4/9) * (1/2)

We can simplify 4 and 2 by dividing both by 2:

(2/9) * (1/1) = (2 * 1) / (9 * 1) = 2/9

h) Calculating 6 ÷ 3/7

Finally, let’s divide 6 by 3/7. Again, we can think of 6 as 6/1. So, the problem is 6/1 ÷ 3/7. We flip the second fraction and multiply:

Dividing a whole number by a fraction may seem daunting, but it follows the same rules as fraction division. First, we express the whole number as a fraction. In this instance, we're dividing 6 by 3/7. We can rewrite 6 as 6/1, making our division problem 6/1 divided by 3/7. To divide fractions, we multiply by the reciprocal of the divisor. The reciprocal of 3/7 is 7/3. So, our problem transforms into 6/1 multiplied by 7/3. Before we proceed with multiplication, it's crucial to check for opportunities to simplify. We observe that the numerator of the first fraction (6) and the numerator of the second fraction (3) share a common factor of 3. We divide 6 by 3 to get 2, and we divide 3 by 3 to get 1. Our simplified problem now reads 2/1 multiplied by 7/1. Multiplying the numerators (2 * 7 = 14) and the denominators (1 * 1 = 1) gives us the fraction 14/1, which simplifies to 14. This means that 6 divided by 3/7 equals 14. The process of converting the whole number into a fraction and then applying the rule of multiplying by the reciprocal is fundamental in fraction division. Simplification, as always, is a key step in making the calculation more manageable and accurate. By following these steps, we can confidently divide whole numbers by fractions.

(6) ÷ (3/7) = (6/1) ÷ (3/7) = (6/1) * (7/3)

We can simplify 6 and 3 by dividing both by 3:

(2/1) * (7/1) = (2 * 7) / (1 * 1) = 14/1 = 14

Conclusion

So there you have it! We've walked through multiplying and dividing fractions, squaring them, and even simplifying along the way. Remember, the key is to take it step by step and look for opportunities to simplify. With a little practice, you’ll be a fraction master in no time! Keep practicing, and you'll find these calculations become second nature. Understanding fractions is a fundamental skill in mathematics, and mastering these basics will set you up for success in more advanced topics. Whether you're simplifying, multiplying, dividing, or squaring, remember the rules and look for ways to make the process easier. Happy calculating!