Factoring And Simplifying Fractions: Step-by-Step Guide
Hey guys! Today, we're diving deep into the world of algebra to tackle a common challenge: factoring out common factors and simplifying fractions. This is a super important skill that will help you in so many areas of math, from basic arithmetic to more advanced calculus. We're going to break down four different problems step-by-step, so you can master this technique. Let's get started!
1) Factoring and Simplifying: (35 * 28 + 35 * 32) / (60 * 15)
Let's start with our first fraction: (35 * 28 + 35 * 32) / (60 * 15). The key here is to identify the common factor in the numerator. In this case, we see that 35 appears in both terms. Factoring out common factors helps us simplify complex expressions and make them easier to work with. Think of it like decluttering your math problem!
Step 1: Factor out the common factor in the numerator.
In the numerator, we have 35 * 28 + 35 * 32. We can factor out 35 from both terms:
35 * (28 + 32)
This simplifies to:
35 * 60
Step 2: Rewrite the fraction.
Now, let’s rewrite our fraction with the factored numerator:
(35 * 60) / (60 * 15)
Step 3: Simplify by canceling common factors.
We see that 60 is a common factor in both the numerator and the denominator, so we can cancel it out:
(35 * 60) / (60 * 15) = 35 / 15
Now, both 35 and 15 are divisible by 5. Let's divide both by 5:
35 / 15 = (35 ÷ 5) / (15 ÷ 5) = 7 / 3
So, the simplified fraction is 7 / 3. Remember, guys, simplifying is all about finding those common pieces and making the problem smaller and easier to handle.
2) Factoring and Simplifying: (57 * 62 – 57 * 34) / (28 - 19)
Moving on to our second problem: (57 * 62 – 57 * 34) / (28 - 19). This one involves subtraction in the numerator and the denominator, but the principle of factoring out common factors remains the same.
Step 1: Factor out the common factor in the numerator.
In the numerator, we have 57 * 62 – 57 * 34. The common factor here is 57. Factoring it out, we get:
57 * (62 – 34)
This simplifies to:
57 * 28
Step 2: Simplify the denominator.
In the denominator, we have 28 - 19, which equals:
9
Step 3: Rewrite the fraction.
Now, let's rewrite the fraction:
(57 * 28) / 9
Step 4: Simplify by canceling common factors.
Here, we need to look for common factors between 57 * 28 and 9. We know that 57 = 3 * 19 and 9 = 3 * 3. So, we can divide 57 by 3 to get 19 and divide 9 by 3 to get 3:
(57 * 28) / 9 = (19 * 28) / 3
Since 19 and 28 don't share any common factors with 3, we can't simplify further. Thus, the simplified fraction is (19 * 28) / 3, which equals 532 / 3. Great job so far, guys! You're getting the hang of this.
3) Factoring and Simplifying: (48 * 31 + 17 * 24) / (16 * 100)
Let's tackle the third fraction: (48 * 31 + 17 * 24) / (16 * 100). This one looks a bit trickier, but don’t worry! We'll break it down step by step. The important thing is to remember our factoring rules and to look for common elements.
Step 1: Factor out the common factor in the numerator.
In the numerator, we have 48 * 31 + 17 * 24. At first glance, it might not be obvious, but notice that 48 and 24 share a common factor. Specifically, 24 is a factor of 48 (48 = 2 * 24). So, we can rewrite the numerator as:
(2 * 24 * 31) + (17 * 24)
Now, we can factor out 24:
24 * (2 * 31 + 17)
This simplifies to:
24 * (62 + 17) = 24 * 79
Step 2: Rewrite the fraction.
Now, let’s rewrite our fraction:
(24 * 79) / (16 * 100)
Step 3: Simplify by canceling common factors.
We can simplify this fraction by finding common factors between the numerator and the denominator. We know that 24 = 8 * 3 and 16 = 8 * 2, so we can divide both 24 and 16 by 8:
(24 * 79) / (16 * 100) = (3 * 79) / (2 * 100)
This gives us:
237 / 200
Since 237 and 200 don't share any common factors, we can't simplify further. So, the simplified fraction is 237 / 200. You're doing amazing, guys! Just one more to go!
4) Factoring and Simplifying: (17 * 96 - 17 * 24) / (48 * 85)
Finally, let's tackle our last problem: (17 * 96 - 17 * 24) / (48 * 85). By now, you're practically experts at this! Remember to look for those common factors and simplify step by step.
Step 1: Factor out the common factor in the numerator.
In the numerator, we have 17 * 96 - 17 * 24. The common factor here is 17. Factoring it out, we get:
17 * (96 - 24)
This simplifies to:
17 * 72
Step 2: Rewrite the fraction.
Let's rewrite the fraction:
(17 * 72) / (48 * 85)
Step 3: Simplify by canceling common factors.
Now, we need to find common factors between the numerator and the denominator. Notice that 72 and 48 share a common factor. We know that 72 = 24 * 3 and 48 = 24 * 2, so we can divide both 72 and 48 by 24:
(17 * 72) / (48 * 85) = (17 * 3) / (2 * 85)
Also, 85 = 17 * 5, so we can cancel out 17:
(17 * 3) / (2 * 85) = 3 / (2 * 5)
This simplifies to:
3 / 10
So, the simplified fraction is 3 / 10. Fantastic job, guys! You've successfully factored and simplified all four fractions.
Conclusion
Factoring out common factors and simplifying fractions is a fundamental skill in algebra. By breaking down each problem into manageable steps, identifying common factors, and simplifying, you can tackle even the most complex fractions. Remember, practice makes perfect! Keep working on these types of problems, and you'll become a pro in no time. You've got this, guys! Keep up the great work, and I'll see you in the next math adventure!