Reduce Fractions: Divisibility Rules & Greatest Common Divisor

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Hey guys! Today, we're diving into the world of fractions and learning how to simplify them like pros. We'll be using divisibility rules and the greatest common divisor (GCD) to make those fractions nice and easy to work with. So, let's get started!

Understanding Divisibility Rules

Divisibility rules are your secret weapon when it comes to simplifying fractions. These rules help you quickly determine if a number can be divided evenly by another number, without actually doing the long division. Knowing these rules can save you a ton of time and effort. It's like having a mathematical superpower! Let's explore some of the most common divisibility rules:

  • Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). For example, the number 346 is divisible by 2 because the last digit is 6.
  • Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. Let's take 123 as an example. The sum of the digits is 1 + 2 + 3 = 6, and 6 is divisible by 3, so 123 is also divisible by 3.
  • Divisibility by 5: This one is super easy! A number is divisible by 5 if its last digit is either 0 or 5. So, 25, 130, and 555 are all divisible by 5.
  • Divisibility by 9: Similar to the rule for 3, a number is divisible by 9 if the sum of its digits is divisible by 9. For example, with the number 819, the sum of the digits is 8 + 1 + 9 = 18, which is divisible by 9, making 819 also divisible by 9.
  • Divisibility by 10: A number is divisible by 10 if its last digit is 0. Examples include 10, 100, 1000, and so on.

Mastering these divisibility rules is the first step in simplifying fractions efficiently. It allows you to quickly identify common factors between the numerator and the denominator, which is crucial for reducing fractions to their simplest form. Trust me, guys, these rules are your best friends in fraction simplification!

Simplifying Fractions Using Divisibility Rules: Part 1

Okay, let's put those divisibility rules to work! We're going to tackle the first set of fractions you gave us. Remember, the goal is to find common factors in the numerator (the top number) and the denominator (the bottom number) and divide both by those factors. This process reduces the fraction to its simplest form, making it easier to understand and use.

  1. 14/42:

    • Let’s look at 14/42. Applying the divisibility rules, you'll quickly notice that both 14 and 42 are even numbers, which means they are divisible by 2. So, we divide both the numerator and the denominator by 2: 14 ÷ 2 = 7 and 42 ÷ 2 = 21. Now our fraction looks like 7/21.
    • But we're not done yet! We can see that both 7 and 21 are divisible by 7. Dividing both by 7 gives us 7 ÷ 7 = 1 and 21 ÷ 7 = 3. So, the simplified fraction is 1/3. Awesome job!

  2. 18/39:

    • Moving on to 18/39, we can see that both numbers are divisible by 3. The sum of the digits in 18 (1 + 8 = 9) is divisible by 3, and the sum of the digits in 39 (3 + 9 = 12) is also divisible by 3. Dividing both by 3, we get 18 ÷ 3 = 6 and 39 ÷ 3 = 13. So, the simplified fraction is 6/13. Not too shabby!

  3. 24/51:

    • Next up is 24/51. Let's check for divisibility by 3. The sum of the digits in 24 (2 + 4 = 6) is divisible by 3, and the sum of the digits in 51 (5 + 1 = 6) is also divisible by 3. Dividing both by 3, we have 24 ÷ 3 = 8 and 51 ÷ 3 = 17. The simplified fraction is 8/17. You're getting the hang of this!

  4. 26/39:

    • Now, let's tackle 26/39. This one might not be as obvious, but if you know your multiplication facts, you'll recognize that both 26 and 39 are divisible by 13. Dividing both by 13, we get 26 ÷ 13 = 2 and 39 ÷ 13 = 3. So, the simplified fraction is 2/3. You're on fire!

  5. 24/51:

    • We see 24/51 again, and we already know from our previous work that both 24 and 51 are divisible by 3. So, let's divide both by 3. 24 ÷ 3 = 8 and 51 ÷ 3 = 17. The simplified fraction is 8/17. Practice makes perfect!

  6. 15/60:

    • For 15/60, we can see that both numbers are divisible by 5. Dividing both by 5, we get 15 ÷ 5 = 3 and 60 ÷ 5 = 12. This gives us 3/12. But hold on, we're not quite done! Both 3 and 12 are divisible by 3. Dividing both by 3 again, we get 3 ÷ 3 = 1 and 12 ÷ 3 = 4. So, the simplified fraction is 1/4. Keep it up!

  7. 35/60:

    • Let's simplify 35/60. Both numbers are divisible by 5. Dividing both by 5, we get 35 ÷ 5 = 7 and 60 ÷ 5 = 12. The simplified fraction is 7/12. Almost there!

  8. 108/100:

    • With 108/100, we can see that both numbers are even, so they are divisible by 2. Dividing both by 2, we get 108 ÷ 2 = 54 and 100 ÷ 2 = 50. This gives us 54/50. But wait, they're still even! Let's divide by 2 again: 54 ÷ 2 = 27 and 50 ÷ 2 = 25. The simplified fraction is 27/25. Fantastic!

You guys are doing an amazing job! By applying the divisibility rules, we've already simplified quite a few fractions. Now, let's keep going with the rest of the fractions in the first set.

Simplifying Fractions Using Divisibility Rules: Part 2

Alright, let's continue simplifying the remaining fractions from the first part of our list. Remember, we're using divisibility rules to find common factors between the numerator and denominator. This helps us reduce the fraction to its simplest form. Keep up the great work!

  1. 25/13:

    • Looking at 25/13, we need to determine if there are any common factors other than 1. The factors of 25 are 1, 5, and 25, while 13 is a prime number, meaning its only factors are 1 and 13. Since they don't share any common factors, the fraction 25/13 is already in its simplest form. Sometimes, guys, the fraction is already as simple as it can get!

  2. 81/5:

    • Now let's consider 81/5. The factors of 81 are 1, 3, 9, 27, and 81, while the factors of 5 are 1 and 5. Again, we don't find any common factors other than 1. Therefore, 81/5 is also in its simplest form. See? It's not always about reducing; sometimes it's about recognizing when you're already there.

  3. 30/2:

    • Next, we have 30/2. Both 30 and 2 are even numbers, making them divisible by 2. Dividing both by 2, we get 30 ÷ 2 = 15 and 2 ÷ 2 = 1. So, the simplified fraction is 15/1, which is simply 15. Awesome!

  4. 11/33:

    • Moving on to 11/33, we can see that both numbers are divisible by 11. Dividing both by 11, we get 11 ÷ 11 = 1 and 33 ÷ 11 = 3. So, the simplified fraction is 1/3. You're doing fantastic!

  5. 45/12:

    • For 45/12, we can check for divisibility by 3. The sum of the digits in 45 (4 + 5 = 9) is divisible by 3, and the sum of the digits in 12 (1 + 2 = 3) is also divisible by 3. Dividing both by 3, we get 45 ÷ 3 = 15 and 12 ÷ 3 = 4. Thus, the simplified fraction is 15/4. Great job!

  6. 50:

    • Wait a second... Just