Least Common Denominator Practice Problems

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Least Common Denominator Practice Problems

Hey guys! Let's break down how to find the least common denominator (LCD) for these fractions. Finding the LCD is super important when you want to add or subtract fractions. It's all about finding the smallest number that each denominator can divide into evenly. Let's get started!

Understanding Least Common Denominator (LCD)

Before we jump into solving these problems, let's have a short discussion on what LCD means. The Least Common Denominator is the smallest common multiple of the denominators of a given set of fractions. To find it, you usually look for the Least Common Multiple (LCM) of the denominators. Factoring each denominator into its prime factors is a usual way to find the LCM. After this, you take the highest power of each prime factor that appears in any of the factorizations and multiply them together to get the LCM, which in turn becomes your LCD. Why do we need it? Well, it allows us to easily add or subtract fractions because, with a common denominator, you can simply add or subtract the numerators. Understanding the LCD streamlines fraction operations and makes more complex calculations much more manageable. It's a fundamental concept, so let's solidify it as we work through these examples!

Problems and Solutions

1) 2/25 and 11/5

To find the least common denominator (LCD) of 2/25 and 11/5, we need to determine the least common multiple (LCM) of their denominators, which are 25 and 5.

  • List the multiples of each denominator:
    • Multiples of 5: 5, 10, 15, 20, 25, 30, ...
    • Multiples of 25: 25, 50, 75, ...

The smallest multiple that both denominators share is 25. Therefore, the least common denominator (LCD) of 2/25 and 11/5 is 25. This means we can easily work with these fractions by converting 11/5 to have a denominator of 25, making it 55/25, and then perform any required operations with 2/25.

2) 7/19 and 3/31

To find the least common denominator (LCD) of 7/19 and 3/31, we need to find the least common multiple (LCM) of the denominators 19 and 31. Since both 19 and 31 are prime numbers, they do not have any common factors other than 1.

When the denominators are prime numbers, the LCM is simply their product:

LCM(19, 31) = 19 × 31 = 589

Therefore, the least common denominator (LCD) of 7/19 and 3/31 is 589. To perform any operations such as addition or subtraction, you would convert both fractions to have this denominator.

3) 3/4 and 4/40

Alright, let's figure out the least common denominator (LCD) for 3/4 and 4/40. We need to find the least common multiple (LCM) of the denominators 4 and 40.

  • Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, ...
  • Multiples of 40: 40, 80, 120, ...

From the lists, we can see that the smallest multiple that both 4 and 40 share is 40. Therefore, the least common denominator (LCD) of 3/4 and 4/40 is 40. To work with these fractions, you'd convert 3/4 to have a denominator of 40, making it 30/40, and then proceed with any required operations along with 4/40. This common denominator simplifies adding or subtracting these fractions.

4) 2/42 and 9/13

Let's find the least common denominator (LCD) of 2/42 and 9/13. To do this, we need to determine the least common multiple (LCM) of the denominators 42 and 13.

First, find the prime factorization of each denominator:

  • 42 = 2 × 3 × 7
  • 13 = 13 (since 13 is a prime number)

Now, to find the LCM, we take the highest power of each prime factor that appears in either factorization:

LCM(42, 13) = 2 × 3 × 7 × 13 = 546

Therefore, the least common denominator (LCD) of 2/42 and 9/13 is 546. To perform addition or subtraction, you would convert both fractions to have this denominator.

5) 13/17 and 1/69

Okay, let's tackle the least common denominator (LCD) for 13/17 and 1/69. We need to find the least common multiple (LCM) of the denominators 17 and 69.

  • First, find the prime factorization of each denominator:
    • 17 = 17 (since 17 is a prime number)
    • 69 = 3 × 17

To find the LCM, we take the highest power of each prime factor that appears in either factorization:

LCM(17, 69) = 3 × 17 = 51

Wait a minute! There's a small error here. The correct LCM calculation should be:

LCM(17, 69) = 3 * 17 = 69

Therefore, the least common denominator (LCD) of 13/17 and 1/69 is 69. To add or subtract these fractions, you'd convert 13/17 to have a denominator of 69, making it 39/69, and then proceed with any required operations along with 1/69. Having this common denominator makes the calculations much easier.

6) 23/20 and 4/25

Let's figure out the least common denominator (LCD) for 23/20 and 4/25. We need to find the least common multiple (LCM) of the denominators 20 and 25.

First, find the prime factorization of each denominator:

  • 20 = 2 × 2 × 5 = 2^2 × 5
  • 25 = 5 × 5 = 5^2

To find the LCM, take the highest power of each prime factor that appears in either factorization:

LCM(20, 25) = 2^2 × 5^2 = 4 × 25 = 100

Therefore, the least common denominator (LCD) of 23/20 and 4/25 is 100. To work with these fractions, you would convert both to have a denominator of 100. This makes calculations like addition or subtraction much simpler.

7) 36/100 and 139/100

Alright, what is the least common denominator (LCD) for 36/100 and 139/100? Here, we need to find the least common multiple (LCM) of the denominators 100 and 100.

Since both fractions already have the same denominator, the least common denominator (LCD) is simply that common denominator.

Therefore, the least common denominator (LCD) of 36/100 and 139/100 is 100. No conversion is needed here since both fractions already share the same denominator. You can directly add or subtract the numerators.

8) 1/32 and 223/96

Let's find the least common denominator (LCD) for 1/32 and 223/96. We need to find the least common multiple (LCM) of the denominators 32 and 96.

  • First, find the prime factorization of each denominator:
    • 32 = 2 × 2 × 2 × 2 × 2 = 2^5
    • 96 = 2 × 2 × 2 × 2 × 2 × 3 = 2^5 × 3

To find the LCM, we take the highest power of each prime factor that appears in either factorization:

LCM(32, 96) = 2^5 × 3 = 32 × 3 = 96

Therefore, the least common denominator (LCD) of 1/32 and 223/96 is 96. To perform operations, you would convert 1/32 to have a denominator of 96, making it 3/96, and then proceed with any required operations along with 223/96. This common denominator simplifies the math.

9) 5/70 and 7/140

Okay, let's determine the least common denominator (LCD) for 5/70 and 7/140. We need to find the least common multiple (LCM) of the denominators 70 and 140.

  • First, find the prime factorization of each denominator:
    • 70 = 2 × 5 × 7
    • 140 = 2 × 2 × 5 × 7 = 2^2 × 5 × 7

To find the LCM, we take the highest power of each prime factor that appears in either factorization:

LCM(70, 140) = 2^2 × 5 × 7 = 4 × 5 × 7 = 140

Therefore, the least common denominator (LCD) of 5/70 and 7/140 is 140. To work with these fractions, you'd convert 5/70 to have a denominator of 140, making it 10/140, and then perform any required operations along with 7/140. This common denominator makes the calculations smoother.

Wrapping Up

Alright guys, mastering the least common denominator will seriously level up your fraction game. Keep practicing, and you'll become a pro in no time! You got this!