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Chapter 5: Problem 37

write as equivalent expressions with the LCD. $$\frac{7}{12 x} \text { and } \frac{3}{4}$$

Short Answer

Expert verified
Question: Write the fractions $\frac{7}{12x}$ and $\frac{3}{4}$ as equivalent expressions with the Least Common Denominator. Answer: The equivalent expressions with the Least Common Denominator are $\frac{7}{12x}$ and $\frac{9x}{12x}$.

Step by step solution

01

Find the Least Common Denominator (LCD)

To find the LCD of the denominators \(12x\) and \(4\), we need to find the Least Common Multiple (LCM) of \(12\) and \(4\) and the highest power/degree of the variable \(x\). For the LCM of \(12\) and \(4\), we need to break down each number into its prime factors: $$12 = 2^2 \cdot 3$$ $$4 = 2^2$$ We take the highest power of each prime factor and multiply them together: $$\text{LCM}(12, 4) = 2^2 \cdot 3 = 12$$ Now we add the variable part \(x\), so the LCD is \(12x\).
02

Write equivalent fractions with LCD

Now that we have the LCD \(12x\), let's write the equivalent expressions for each fraction: For the first fraction, the denominator is already \(12x\), so the equivalent fraction is: $$\frac{7}{12x}$$ For the second fraction, we need to multiply both the numerator and denominator by \(3x\) (since \(4 \cdot 3x =12x\)): $$\frac{3}{4} \cdot \frac{3x}{3x} = \frac{9x}{12x}$$ The equivalent expressions with the LCD are: $$\frac{7}{12x} \text{ and } \frac{9x}{12x}$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Least Common Multiple
When working with fractions, finding a common denominator is crucial to perform addition or subtraction. The Least Common Multiple (LCM) helps us find this common ground. If you consider the numbers 12 and 4, you need their prime factors to determine the LCM.
Break down each number:
  • 12 becomes \(2^2 \times 3\)
  • 4 is \(2^2\)
Identify the highest power of each prime number appearing in these factorizations:
  • The highest power of 2 is \(2^2\)
  • The power of 3 is \(3^1\)
Multiply these together to get the LCM: \(2^2 \times 3 = 12\).
Since one of the denominators is \(12x\), we keep the variable part \(x\) too, making the least common denominator \(12x\). This LCD provides a shared foundation for the fractions, allowing you to express them with a common denominator.
Exploring Prime Factorization
Prime factorization is the process of breaking down a number into its basic building blocks, prime numbers. It's like finding the DNA of numbers! For the numbers 12 and 4, we aim to express them as products of prime numbers to find an LCM.
Let's see how it's done:
  • For 12:
    Start by dividing by the smallest prime number, 2. 12 梅 2 = 6, and continue with 6 梅 2 = 3. Then, 3 is a prime number. So, \(12 = 2^2 \times 3\).
  • For 4:
    4 梅 2 = 2 with 2 梅 2 = 1, leading to \(4 = 2^2\).
By comparing these factorizations, you see which prime factors are essential for the LCM. It helps you elegantly solve problems involving fractions with different denominators.
Mastering Multiplying Fractions
Multiplying fractions is simpler than it might seem. It's all about multiplying the numerators together and the denominators together. However, when aligning fractions to a common denominator, like 4 and 12x, it's a bit more strategic.
Here's how it鈥檚 applied:
  • Original Fraction: \(\frac{3}{4}\)
  • Target Denominator: \(12x\), find out what the original fraction's denominator needs. In this case, multiply by \(3x\) to transform \(4\) into \(12x\).
  • Apply Multiplication: Multiply both numerator and denominator by \(3x\). This isn鈥檛 just arithmetic; it's about maintaining the fraction's value: \(\frac{3}{4} \times \frac{3x}{3x} = \frac{9x}{12x}\).
This makes \(\frac{9x}{12x}\) equivalent to the original \(\frac{3}{4}\) but harmonized under a common denominator.
Understanding this process enables clearer, consistent fractions when working with multiple terms.

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