Fraction Arithmetic: Step-by-Step Solutions

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Hey guys! Let's dive into some fraction fun! I'm here to help you understand how to add and subtract fractions, breaking down each step so you can ace your math problems. We'll go through the calculations for the given examples, explaining everything in detail. No worries, it's easier than it looks! Let's get started. Remember, the key to mastering fractions is understanding the fundamentals, and with a little practice, you'll be a pro in no time. So, grab your notebooks, and let's start solving some fraction problems! We will focus on clear, step-by-step instructions. Also, the answers will be clearly explained, so you can easily follow along and learn the process. Get ready to boost your fraction skills! Let's start with some basic definitions to set the stage for our fraction adventures. A fraction represents a part of a whole. It's written as two numbers stacked on top of each other, separated by a line. The top number is the numerator (how many parts we have), and the bottom number is the denominator (how many parts the whole is divided into). For example, in the fraction 1/2, the numerator is 1, and the denominator is 2. This means we have one part out of a whole that has been split into two equal parts. Understanding these components is critical to grasping how to work with fractions. Now, let's look at adding and subtracting fractions. When the fractions have the same denominator, it's pretty simple: just add or subtract the numerators and keep the denominator the same. But what happens when the denominators are different? That’s where finding a common denominator comes in handy. It’s like finding a universal language so all the fractions can chat easily. We will find this in action soon, I promise.

1. 5/6 + 2/7

To add these fractions, we need a common denominator. Let's find the least common multiple (LCM) of 6 and 7. Since 6 and 7 don't share any common factors other than 1, their LCM is simply their product. Therefore, 6 * 7 = 42. So, our common denominator is 42.

  • Step 1: Convert the first fraction: To change 5/6 to a fraction with a denominator of 42, we multiply both the numerator and the denominator by 7. (5 * 7) / (6 * 7) = 35/42.
  • Step 2: Convert the second fraction: To change 2/7 to a fraction with a denominator of 42, we multiply both the numerator and the denominator by 6. (2 * 6) / (7 * 6) = 12/42.
  • Step 3: Add the fractions: Now we have 35/42 + 12/42. Add the numerators and keep the denominator the same: (35 + 12) / 42 = 47/42.
  • Step 4: Simplify (if possible): This fraction is improper (the numerator is larger than the denominator). We can convert it to a mixed number. 47 divided by 42 is 1 with a remainder of 5. So, the answer is 1 5/42. Great job guys!

2. 11/15 - 3/5

Let's find the common denominator of 15 and 5. The LCM of 15 and 5 is 15 (since 5 goes into 15). So, our common denominator is 15.

  • Step 1: Convert the fractions: 11/15 already has the denominator we want.
  • Step 2: Convert the second fraction: To change 3/5 to a fraction with a denominator of 15, we multiply both the numerator and the denominator by 3. (3 * 3) / (5 * 3) = 9/15.
  • Step 3: Subtract the fractions: Now we have 11/15 - 9/15. Subtract the numerators and keep the denominator the same: (11 - 9) / 15 = 2/15.
  • Step 4: Simplify (if possible): This fraction is already in its simplest form. The answer is 2/15. Nice work!

3. 15/16 - 3/4

To solve this, we need a common denominator for 16 and 4. The LCM of 16 and 4 is 16. So, our common denominator is 16.

  • Step 1: Convert the fractions: 15/16 already has the desired denominator.
  • Step 2: Convert the second fraction: To change 3/4 to a fraction with a denominator of 16, we multiply both the numerator and the denominator by 4: (3 * 4) / (4 * 4) = 12/16.
  • Step 3: Subtract the fractions: Now we have 15/16 - 12/16. Subtract the numerators and keep the denominator: (15 - 12) / 16 = 3/16.
  • Step 4: Simplify (if possible): The fraction 3/16 is already simplified. The result is 3/16. Awesome!

4. 3/20 + 7/15

Let’s find the common denominator for 20 and 15. The LCM of 20 and 15 is 60. So, our common denominator is 60.

  • Step 1: Convert the first fraction: To change 3/20 to a fraction with a denominator of 60, we multiply the numerator and the denominator by 3. (3 * 3) / (20 * 3) = 9/60.
  • Step 2: Convert the second fraction: To change 7/15 to a fraction with a denominator of 60, multiply the numerator and the denominator by 4: (7 * 4) / (15 * 4) = 28/60.
  • Step 3: Add the fractions: Now we have 9/60 + 28/60. Add the numerators and keep the denominator: (9 + 28) / 60 = 37/60.
  • Step 4: Simplify (if possible): The fraction 37/60 is already in its simplest form. So, the answer is 37/60. Good job, guys!

5. 13/16 - 7/12

We need to find a common denominator for 16 and 12. The LCM of 16 and 12 is 48. Hence, our common denominator is 48.

  • Step 1: Convert the first fraction: To change 13/16 to a fraction with a denominator of 48, we multiply the numerator and the denominator by 3. (13 * 3) / (16 * 3) = 39/48.
  • Step 2: Convert the second fraction: To change 7/12 to a fraction with a denominator of 48, we multiply the numerator and the denominator by 4. (7 * 4) / (12 * 4) = 28/48.
  • Step 3: Subtract the fractions: Now we have 39/48 - 28/48. Subtract the numerators and keep the denominator: (39 - 28) / 48 = 11/48.
  • Step 4: Simplify (if possible): The fraction 11/48 is already simplified. The result is 11/48. Fantastic!

6. 9/14 + 2/21

For 14 and 21, the LCM is 42. So, the common denominator is 42.

  • Step 1: Convert the first fraction: To change 9/14 to a fraction with a denominator of 42, multiply the numerator and the denominator by 3. (9 * 3) / (14 * 3) = 27/42.
  • Step 2: Convert the second fraction: To change 2/21 to a fraction with a denominator of 42, multiply the numerator and the denominator by 2. (2 * 2) / (21 * 2) = 4/42.
  • Step 3: Add the fractions: Now we have 27/42 + 4/42. Add the numerators and keep the denominator: (27 + 4) / 42 = 31/42.
  • Step 4: Simplify (if possible): The fraction 31/42 is already in simplest form. The result is 31/42. Great work!

7. 19/36 - 11/48

Let’s find a common denominator for 36 and 48. The LCM of 36 and 48 is 144. Our common denominator is 144.

  • Step 1: Convert the first fraction: To change 19/36 to a fraction with a denominator of 144, multiply the numerator and the denominator by 4. (19 * 4) / (36 * 4) = 76/144.
  • Step 2: Convert the second fraction: To change 11/48 to a fraction with a denominator of 144, multiply the numerator and the denominator by 3. (11 * 3) / (48 * 3) = 33/144.
  • Step 3: Subtract the fractions: Now we have 76/144 - 33/144. Subtract the numerators and keep the denominator: (76 - 33) / 144 = 43/144.
  • Step 4: Simplify (if possible): The fraction 43/144 is already in simplest form. The answer is 43/144. Keep it up, guys!

8. 3 7/9 + 5 1/6

Let's add these mixed numbers! First, convert them to improper fractions.

  • Step 1: Convert the first mixed number: 3 7/9 = (3 * 9 + 7) / 9 = 34/9.
  • Step 2: Convert the second mixed number: 5 1/6 = (5 * 6 + 1) / 6 = 31/6.
  • Step 3: Find the common denominator: The LCM of 9 and 6 is 18. Our common denominator is 18.
  • Step 4: Convert the first fraction: To change 34/9 to a fraction with a denominator of 18, multiply by 2. (34 * 2) / (9 * 2) = 68/18.
  • Step 5: Convert the second fraction: To change 31/6 to a fraction with a denominator of 18, multiply by 3. (31 * 3) / (6 * 3) = 93/18.
  • Step 6: Add the fractions: Now we have 68/18 + 93/18. Add the numerators and keep the denominator: (68 + 93) / 18 = 161/18.
  • Step 7: Simplify (if possible): Convert this improper fraction back to a mixed number. 161 divided by 18 is 8 with a remainder of 17. So, the answer is 8 17/18. Excellent job!

9. 3 + 4 1/7

  • Step 1: Rewrite the whole number as a fraction: 3 can be written as 3/1.
  • Step 2: Convert the mixed number to an improper fraction: 4 1/7 = (4 * 7 + 1) / 7 = 29/7.
  • Step 3: Find a common denominator: The LCM of 1 and 7 is 7. Our common denominator is 7.
  • Step 4: Convert 3/1: Multiply the numerator and the denominator by 7. (3 * 7) / (1 * 7) = 21/7.
  • Step 5: Add the fractions: Now we have 21/7 + 29/7. Add the numerators and keep the denominator: (21 + 29) / 7 = 50/7.
  • Step 6: Simplify (if possible): Convert this improper fraction back to a mixed number. 50 divided by 7 is 7 with a remainder of 1. The result is 7 1/7. Awesome!

10. 3 1/8 + 3/4

  • Step 1: Convert the mixed number to an improper fraction: 3 1/8 = (3 * 8 + 1) / 8 = 25/8.
  • Step 2: Find the common denominator: The LCM of 8 and 4 is 8. Our common denominator is 8.
  • Step 3: Convert the fractions: 25/8 already has the desired denominator.
  • Step 4: Convert 3/4: Multiply the numerator and the denominator by 2. (3 * 2) / (4 * 2) = 6/8.
  • Step 5: Add the fractions: Now we have 25/8 + 6/8. Add the numerators and keep the denominator: (25 + 6) / 8 = 31/8.
  • Step 6: Simplify (if possible): Convert this improper fraction back to a mixed number. 31 divided by 8 is 3 with a remainder of 7. The final answer is 3 7/8. Great work, everyone! You've successfully navigated through fraction arithmetic.

Hopefully, these detailed explanations help you understand how to solve fraction problems. Keep practicing, and you'll get better with each problem. If you need more help, feel free to ask. Keep up the awesome work, and happy calculating!