B 508 Step By Step Guide To Solving Fraction Expressions After Simplifying
Hey guys! Let's dive into problem B 508 from our math textbook. We're going to tackle some fraction expressions, but there's a twist – we need to simplify the fractions before we do any adding or subtracting. This is a super important skill in math, because it makes working with fractions way easier and helps us avoid huge, messy numbers. Think of it like decluttering your workspace before starting a big project – it just makes everything smoother!
Why Simplify Fractions First?
Okay, so why is simplifying fractions beforehand such a big deal? Imagine trying to add fractions like 36/48 and 24/60 without simplifying them first. You'd have to find a common denominator for some pretty big numbers, and the calculations could get tricky. But if we simplify first, we can make our lives a whole lot easier. Simplifying fractions means reducing them to their simplest form, where the numerator (the top number) and the denominator (the bottom number) have no common factors other than 1. In other words, we're dividing both the numerator and the denominator by their greatest common factor (GCF). This process doesn't change the value of the fraction; it just represents it in a simpler way.
For example, let's say we have the fraction 12/18. Both 12 and 18 are divisible by 2, 3, and 6. The greatest common factor is 6. If we divide both the numerator and the denominator by 6, we get 2/3. Now, 2/3 is the simplified form of 12/18. It's much easier to work with 2/3 than 12/18, especially when we're adding or subtracting fractions. Simplifying fractions also helps us see the relationships between different fractions more clearly. When fractions are in their simplest form, it's easier to compare them, order them, and perform other operations. So, simplifying fractions is not just a matter of making the numbers smaller; it's about making the math easier and more intuitive.
Let's Break Down the Problems
Now, let's jump into the specific problems in B 508. We've got eight expressions to solve, each involving adding or subtracting fractions. Remember, our game plan is to simplify each fraction first, then find a common denominator if necessary, and finally perform the operation. We'll take it step by step, so you can see exactly how it's done. Let's get started!
1) 3/12 + 25/30
Okay, first up, we have 3/12 + 25/30. Let's simplify each fraction individually. For 3/12, both 3 and 12 are divisible by 3. So, we divide both by 3, which gives us 1/4. That's the simplified form of 3/12. Now, let's tackle 25/30. Both 25 and 30 are divisible by 5. Dividing both by 5, we get 5/6. So, 25/30 simplified is 5/6. Now our expression looks much friendlier: 1/4 + 5/6. To add these fractions, we need a common denominator. The least common multiple (LCM) of 4 and 6 is 12. So, we'll convert both fractions to have a denominator of 12. To get 1/4 to have a denominator of 12, we multiply both the numerator and the denominator by 3, giving us 3/12. To get 5/6 to have a denominator of 12, we multiply both the numerator and the denominator by 2, giving us 10/12. Now we have 3/12 + 10/12. Adding the numerators, we get 13/12. This is an improper fraction (the numerator is larger than the denominator), so we can convert it to a mixed number. 13 divided by 12 is 1 with a remainder of 1, so 13/12 is equal to 1 1/12. Therefore, 3/12 + 25/30 = 1 1/12.
2) 10/24 - 21/54
Next up, we have 10/24 - 21/54. Let's simplify these fractions. For 10/24, both 10 and 24 are divisible by 2. Dividing both by 2, we get 5/12. So, the simplified form of 10/24 is 5/12. Now, let's simplify 21/54. Both 21 and 54 are divisible by 3. Dividing both by 3, we get 7/18. So, 21/54 simplified is 7/18. Our expression now looks like this: 5/12 - 7/18. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 12 and 18 is 36. So, we'll convert both fractions to have a denominator of 36. To get 5/12 to have a denominator of 36, we multiply both the numerator and the denominator by 3, giving us 15/36. To get 7/18 to have a denominator of 36, we multiply both the numerator and the denominator by 2, giving us 14/36. Now we have 15/36 - 14/36. Subtracting the numerators, we get 1/36. Therefore, 10/24 - 21/54 = 1/36.
3) 30/54 + 22/30
Moving on to the third expression, we have 30/54 + 22/30. Time to simplify! Let's start with 30/54. Both 30 and 54 are divisible by 6. Dividing both by 6, we get 5/9. So, 30/54 simplifies to 5/9. Next, let's simplify 22/30. Both 22 and 30 are divisible by 2. Dividing both by 2, we get 11/15. So, 22/30 simplifies to 11/15. Now our expression looks like this: 5/9 + 11/15. To add these fractions, we need a common denominator. The least common multiple (LCM) of 9 and 15 is 45. So, we'll convert both fractions to have a denominator of 45. To get 5/9 to have a denominator of 45, we multiply both the numerator and the denominator by 5, giving us 25/45. To get 11/15 to have a denominator of 45, we multiply both the numerator and the denominator by 3, giving us 33/45. Now we have 25/45 + 33/45. Adding the numerators, we get 58/45. This is an improper fraction, so let's convert it to a mixed number. 58 divided by 45 is 1 with a remainder of 13, so 58/45 is equal to 1 13/45. Therefore, 30/54 + 22/30 = 1 13/45.
4) 28/40 - 10/75
Let's tackle the fourth expression: 28/40 - 10/75. First, we simplify. For 28/40, both 28 and 40 are divisible by 4. Dividing both by 4, we get 7/10. So, 28/40 simplifies to 7/10. Now, let's simplify 10/75. Both 10 and 75 are divisible by 5. Dividing both by 5, we get 2/15. So, 10/75 simplifies to 2/15. Our expression is now 7/10 - 2/15. To subtract, we need a common denominator. The least common multiple (LCM) of 10 and 15 is 30. So, we'll convert both fractions to have a denominator of 30. To get 7/10 to have a denominator of 30, we multiply both the numerator and the denominator by 3, giving us 21/30. To get 2/15 to have a denominator of 30, we multiply both the numerator and the denominator by 2, giving us 4/30. Now we have 21/30 - 4/30. Subtracting the numerators, we get 17/30. Therefore, 28/40 - 10/75 = 17/30.
5) 12/27 + 14/21
Moving on to expression number five: 12/27 + 14/21. Let's simplify these fractions. For 12/27, both 12 and 27 are divisible by 3. Dividing both by 3, we get 4/9. So, 12/27 simplifies to 4/9. Now, let's simplify 14/21. Both 14 and 21 are divisible by 7. Dividing both by 7, we get 2/3. So, 14/21 simplifies to 2/3. Our expression now looks like 4/9 + 2/3. To add these fractions, we need a common denominator. The least common multiple (LCM) of 9 and 3 is 9. So, we'll convert both fractions to have a denominator of 9. 4/9 already has a denominator of 9, so we don't need to change it. To get 2/3 to have a denominator of 9, we multiply both the numerator and the denominator by 3, giving us 6/9. Now we have 4/9 + 6/9. Adding the numerators, we get 10/9. This is an improper fraction, so let's convert it to a mixed number. 10 divided by 9 is 1 with a remainder of 1, so 10/9 is equal to 1 1/9. Therefore, 12/27 + 14/21 = 1 1/9.
6) 14/24 - 15/40
Now for the sixth expression: 14/24 - 15/40. Time to simplify! Let's start with 14/24. Both 14 and 24 are divisible by 2. Dividing both by 2, we get 7/12. So, 14/24 simplifies to 7/12. Next, let's simplify 15/40. Both 15 and 40 are divisible by 5. Dividing both by 5, we get 3/8. So, 15/40 simplifies to 3/8. Now our expression is 7/12 - 3/8. To subtract, we need a common denominator. The least common multiple (LCM) of 12 and 8 is 24. So, we'll convert both fractions to have a denominator of 24. To get 7/12 to have a denominator of 24, we multiply both the numerator and the denominator by 2, giving us 14/24. To get 3/8 to have a denominator of 24, we multiply both the numerator and the denominator by 3, giving us 9/24. Now we have 14/24 - 9/24. Subtracting the numerators, we get 5/24. Therefore, 14/24 - 15/40 = 5/24.
7) 12/18 - 5/60
Let's move on to the seventh expression: 12/18 - 5/60. Simplify first! For 12/18, both 12 and 18 are divisible by 6. Dividing both by 6, we get 2/3. So, 12/18 simplifies to 2/3. Now, let's simplify 5/60. Both 5 and 60 are divisible by 5. Dividing both by 5, we get 1/12. So, 5/60 simplifies to 1/12. Our expression now looks like 2/3 - 1/12. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 3 and 12 is 12. So, we'll convert both fractions to have a denominator of 12. To get 2/3 to have a denominator of 12, we multiply both the numerator and the denominator by 4, giving us 8/12. 1/12 already has a denominator of 12, so we don't need to change it. Now we have 8/12 - 1/12. Subtracting the numerators, we get 7/12. Therefore, 12/18 - 5/60 = 7/12.
8) 4/24 + 3/36
Last but not least, we have the eighth expression: 4/24 + 3/36. Time for the final simplification! Let's start with 4/24. Both 4 and 24 are divisible by 4. Dividing both by 4, we get 1/6. So, 4/24 simplifies to 1/6. Now, let's simplify 3/36. Both 3 and 36 are divisible by 3. Dividing both by 3, we get 1/12. So, 3/36 simplifies to 1/12. Our expression now looks like 1/6 + 1/12. To add these fractions, we need a common denominator. The least common multiple (LCM) of 6 and 12 is 12. So, we'll convert both fractions to have a denominator of 12. To get 1/6 to have a denominator of 12, we multiply both the numerator and the denominator by 2, giving us 2/12. 1/12 already has a denominator of 12, so we don't need to change it. Now we have 2/12 + 1/12. Adding the numerators, we get 3/12. But wait! We can simplify this fraction further. Both 3 and 12 are divisible by 3. Dividing both by 3, we get 1/4. Therefore, 4/24 + 3/36 = 1/4.
Wrapping Up
And there you have it! We've solved all eight expressions in problem B 508. Remember, the key to working with fractions is to simplify them first. It makes the rest of the calculations much easier and helps you avoid mistakes. Plus, it's a great way to show off your math skills! Keep practicing, guys, and you'll become fraction masters in no time! Simplifying fractions before adding or subtracting is like having a superpower in math – it makes everything smoother and more efficient.
