Find The Least Common Multiple (LCM) To Determine The Least Common Denominator For $\frac{2}{3}$, $\frac{5}{6}$, And $\frac{9}{4}$.A. 60 B. 18 C. 9 D. 12

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Introduction

In mathematics, the least common multiple (LCM) is a fundamental concept used to find the smallest multiple that is common to two or more numbers. When dealing with fractions, finding the least common denominator (LCD) is crucial to add, subtract, multiply, or divide them. In this article, we will explore how to find the LCM to determine the least common denominator for the fractions 23\frac{2}{3}, 56\frac{5}{6}, and 94\frac{9}{4}.

Understanding the Concept of LCM

The LCM of two or more numbers is the smallest number that is a multiple of each of the given numbers. For example, the LCM of 12 and 15 is 60, because 60 is the smallest number that can be divided evenly by both 12 and 15.

Finding the LCM of Multiple Fractions

To find the LCM of multiple fractions, we need to first find the LCM of the denominators. The denominators of the given fractions are 3, 6, and 4.

Step 1: Find the LCM of the Denominators

To find the LCM of 3, 6, and 4, we need to list the multiples of each number and find the smallest number that is common to all.

  • Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60
  • Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
  • Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60

From the list, we can see that the smallest number that is common to all is 60.

Step 2: Find the LCM of the Numerators

Since the fractions are already in their simplest form, we can proceed to find the LCM of the denominators.

Step 3: Find the LCD

The LCD is the LCM of the denominators. In this case, the LCD is 60.

Step 4: Rewrite the Fractions with the LCD

To rewrite the fractions with the LCD, we need to multiply the numerator and denominator of each fraction by the necessary multiples to get the LCD.

  • 23=2×203×20=4060\frac{2}{3} = \frac{2 \times 20}{3 \times 20} = \frac{40}{60}
  • 56=5×106×10=5060\frac{5}{6} = \frac{5 \times 10}{6 \times 10} = \frac{50}{60}
  • 94=9×154×15=13560\frac{9}{4} = \frac{9 \times 15}{4 \times 15} = \frac{135}{60}

Step 5: Add, Subtract, Multiply, or Divide the Fractions

Now that the fractions have the same denominator, we can add, subtract, multiply, or divide them.

Conclusion

In conclusion, the LCM of the denominators of the fractions 23\frac{2}{3}, 56\frac{5}{6}, and 94\frac{9}{4} is 60. Therefore, the least common denominator (LCD) is also 60.

Answer

The correct answer is A. 60.

Additional Tips and Resources

  • To find the LCM of two or more numbers, you can use the following methods:
    • List the multiples of each number and find the smallest number that is common to all.
    • Use the prime factorization method to find the LCM.
  • To find the LCD of multiple fractions, you can use the following methods:
    • Find the LCM of the denominators.
    • Rewrite the fractions with the LCD.
  • For more information on finding the LCM and LCD, you can refer to the following resources:
    • Khan Academy: Least Common Multiple (LCM)
    • Mathway: Least Common Multiple (LCM)
    • Wolfram Alpha: Least Common Multiple (LCM)

Final Thoughts

Frequently Asked Questions

Q: What is the least common multiple (LCM) and least common denominator (LCD)?

A: The LCM is the smallest number that is a multiple of two or more numbers. The LCD is the LCM of the denominators of multiple fractions.

Q: How do I find the LCM of two or more numbers?

A: To find the LCM of two or more numbers, you can use the following methods:

  • List the multiples of each number and find the smallest number that is common to all.
  • Use the prime factorization method to find the LCM.

Q: How do I find the LCD of multiple fractions?

A: To find the LCD of multiple fractions, you can use the following methods:

  • Find the LCM of the denominators.
  • Rewrite the fractions with the LCD.

Q: What is the difference between the LCM and LCD?

A: The LCM is the smallest number that is a multiple of two or more numbers, while the LCD is the LCM of the denominators of multiple fractions.

Q: How do I add, subtract, multiply, or divide fractions with different denominators?

A: To add, subtract, multiply, or divide fractions with different denominators, you need to find the LCD and rewrite the fractions with the LCD.

Q: What is the importance of finding the LCM and LCD?

A: Finding the LCM and LCD is essential in mathematics, particularly when dealing with fractions. It helps you to perform various operations on fractions and solve mathematical problems.

Q: Can I use a calculator to find the LCM and LCD?

A: Yes, you can use a calculator to find the LCM and LCD. However, it's essential to understand the concept and method behind finding the LCM and LCD.

Q: How do I find the LCM of a fraction with a variable?

A: To find the LCM of a fraction with a variable, you need to find the LCM of the coefficients and the variable.

Q: Can I find the LCM and LCD of negative numbers?

A: Yes, you can find the LCM and LCD of negative numbers. The LCM and LCD of negative numbers are the same as the LCM and LCD of their absolute values.

Q: How do I find the LCM and LCD of decimals?

A: To find the LCM and LCD of decimals, you need to convert the decimals to fractions and then find the LCM and LCD.

Q: Can I find the LCM and LCD of fractions with different signs?

A: Yes, you can find the LCM and LCD of fractions with different signs. The LCM and LCD of fractions with different signs are the same as the LCM and LCD of their absolute values.

Q: How do I find the LCM and LCD of fractions with exponents?

A: To find the LCM and LCD of fractions with exponents, you need to find the LCM and LCD of the coefficients and the exponents.

Q: Can I find the LCM and LCD of complex numbers?

A: Yes, you can find the LCM and LCD of complex numbers. The LCM and LCD of complex numbers are the same as the LCM and LCD of their real and imaginary parts.

Conclusion

Finding the LCM and LCD is an essential skill in mathematics, particularly when dealing with fractions. By understanding the concept and method behind finding the LCM and LCD, you can perform various operations on fractions and solve mathematical problems. Remember to always list the multiples of each number and find the smallest number that is common to all, or use the prime factorization method to find the LCM. With practice and patience, you can become proficient in finding the LCM and LCD and apply this skill to various mathematical problems.

Additional Tips and Resources

  • To find the LCM and LCD, you can use online calculators or software, such as Wolfram Alpha or Mathway.
  • You can also use the following formulas to find the LCM and LCD:
    • LCM(a, b) = (a * b) / GCD(a, b)
    • LCD(a, b) = LCM(a, b)
  • For more information on finding the LCM and LCD, you can refer to the following resources:
    • Khan Academy: Least Common Multiple (LCM)
    • Mathway: Least Common Multiple (LCM)
    • Wolfram Alpha: Least Common Multiple (LCM)

Final Thoughts

Finding the LCM and LCD is an essential skill in mathematics, particularly when dealing with fractions. By understanding the concept and method behind finding the LCM and LCD, you can perform various operations on fractions and solve mathematical problems. Remember to always list the multiples of each number and find the smallest number that is common to all, or use the prime factorization method to find the LCM. With practice and patience, you can become proficient in finding the LCM and LCD and apply this skill to various mathematical problems.