Select The Best Answer For The Question.8. Express The Fractions $\frac{3}{4}, \frac{7}{16},$ And $\frac{5}{8}$ With The LCD.A. $\frac{12}{16}, \frac{7}{16}, \frac{10}{16}$B. $\frac{3}{4}, \frac{2}{4}, \frac{3}{4}$C.

by ADMIN 217 views

Understanding the Concept of LCD

In mathematics, the least common denominator (LCD) is the smallest multiple that is common to the denominators of two or more fractions. It is essential to find the LCD when adding, subtracting, multiplying, or dividing fractions. In this article, we will focus on expressing the fractions 34,716,\frac{3}{4}, \frac{7}{16}, and 58\frac{5}{8} with the LCD.

Finding the LCD of the Given Fractions

To find the LCD, we need to identify the prime factors of each denominator. The prime factors of 4 are 2 x 2, the prime factors of 16 are 2 x 2 x 2 x 2, and the prime factors of 8 are 2 x 2 x 2.

Calculating the LCD

To find the LCD, we take the highest power of each prime factor that appears in any of the denominators. In this case, the highest power of 2 is 4 (from 16). Therefore, the LCD is 2 x 2 x 2 x 2 = 16.

Expressing the Fractions with the LCD

Now that we have found the LCD, we can express each fraction with the LCD by multiplying the numerator and denominator by the necessary factors.

  • For 34\frac{3}{4}, we multiply the numerator and denominator by 4 to get 3×44×4=1216\frac{3 \times 4}{4 \times 4} = \frac{12}{16}.
  • For 716\frac{7}{16}, the fraction is already in its simplest form with the denominator 16, so we can leave it as is: 716\frac{7}{16}.
  • For 58\frac{5}{8}, we multiply the numerator and denominator by 2 to get 5×28×2=1016\frac{5 \times 2}{8 \times 2} = \frac{10}{16}.

Comparing the Options

Now that we have expressed the fractions with the LCD, we can compare the options:

A. 1216,716,1016\frac{12}{16}, \frac{7}{16}, \frac{10}{16}

B. 34,24,34\frac{3}{4}, \frac{2}{4}, \frac{3}{4}

C. 34,716,58\frac{3}{4}, \frac{7}{16}, \frac{5}{8}

Conclusion

Based on our calculations, the correct answer is option A: 1216,716,1016\frac{12}{16}, \frac{7}{16}, \frac{10}{16}. This is because we have expressed each fraction with the LCD, which is 16.

Importance of Finding the LCD

Finding the LCD is crucial when working with fractions. It allows us to add, subtract, multiply, or divide fractions by ensuring that all the fractions have the same denominator. Without the LCD, it would be challenging to perform these operations, and the results would be incorrect.

Real-World Applications of LCD

The concept of LCD has numerous real-world applications. For example, in finance, finding the LCD is essential when calculating interest rates or investment returns. In engineering, it is used to calculate stress and strain on materials. In medicine, it is used to calculate medication dosages.

Tips for Finding the LCD

Here are some tips for finding the LCD:

  • Identify the prime factors of each denominator.
  • Take the highest power of each prime factor that appears in any of the denominators.
  • Multiply the numerators and denominators by the necessary factors to express each fraction with the LCD.

Common Mistakes to Avoid

Here are some common mistakes to avoid when finding the LCD:

  • Failing to identify the prime factors of each denominator.
  • Taking the lowest power of each prime factor instead of the highest power.
  • Not multiplying the numerators and denominators by the necessary factors.

Conclusion

In conclusion, finding the LCD is a crucial concept in mathematics that has numerous real-world applications. By understanding how to find the LCD, we can perform operations with fractions accurately and efficiently. Remember to identify the prime factors of each denominator, take the highest power of each prime factor, and multiply the numerators and denominators by the necessary factors to express each fraction with the LCD.

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

A: The least common denominator (LCD) is the smallest multiple that is common to the denominators of two or more fractions.

Q: Why is it necessary to find the LCD when working with fractions?

A: Finding the LCD is necessary when working with fractions because it allows us to add, subtract, multiply, or divide fractions by ensuring that all the fractions have the same denominator.

Q: How do I find the LCD of two or more fractions?

A: To find the LCD, you need to identify the prime factors of each denominator. Then, take the highest power of each prime factor that appears in any of the denominators. Multiply the numerators and denominators by the necessary factors to express each fraction with the LCD.

Q: What are some common mistakes to avoid when finding the LCD?

A: Some common mistakes to avoid when finding the LCD include:

  • Failing to identify the prime factors of each denominator.
  • Taking the lowest power of each prime factor instead of the highest power.
  • Not multiplying the numerators and denominators by the necessary factors.

Q: How do I express a fraction with a denominator that is not the LCD?

A: To express a fraction with a denominator that is not the LCD, you need to multiply the numerator and denominator by the necessary factors to get the LCD.

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

A: Yes, you can use a calculator to find the LCD. However, it is essential to understand the concept of LCD and how to find it manually to ensure accuracy.

Q: What are some real-world applications of the LCD?

A: The concept of LCD has numerous real-world applications, including:

  • Finance: Finding the LCD is essential when calculating interest rates or investment returns.
  • Engineering: It is used to calculate stress and strain on materials.
  • Medicine: It is used to calculate medication dosages.

Q: How do I determine if two fractions have the same LCD?

A: To determine if two fractions have the same LCD, you need to find the prime factors of each denominator and compare them. If the prime factors are the same, then the fractions have the same LCD.

Q: Can I use the LCD to add or subtract fractions?

A: Yes, you can use the LCD to add or subtract fractions. By expressing each fraction with the LCD, you can add or subtract the numerators while keeping the denominator the same.

Q: What are some tips for finding the LCD?

A: Some tips for finding the LCD include:

  • Identifying the prime factors of each denominator.
  • Taking the highest power of each prime factor that appears in any of the denominators.
  • Multiplying the numerators and denominators by the necessary factors to express each fraction with the LCD.

Q: Can I use the LCD to multiply or divide fractions?

A: Yes, you can use the LCD to multiply or divide fractions. By expressing each fraction with the LCD, you can multiply or divide the numerators while keeping the denominator the same.

Conclusion

In conclusion, the least common denominator (LCD) is a crucial concept in mathematics that has numerous real-world applications. By understanding how to find the LCD, you can perform operations with fractions accurately and efficiently. Remember to identify the prime factors of each denominator, take the highest power of each prime factor, and multiply the numerators and denominators by the necessary factors to express each fraction with the LCD.