Problem 18 - Elementary Algebra

Question

For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{4}{15 x+6} ; 12(5 x+2) $$

Step-by-Step Solution

Answer

Equivalent expression: \( \frac{16}{12(5x + 2)} \)

Step 1: Identify the original denominator

The original denominator in the expression is \(15x + 6\).

Step 2: Identify the new denominator

The new denominator given in the problem is \(12(5x + 2)\).

Step 3: Factor the original denominator

Factor \(15x + 6\) to its simplest form: \(15x + 6 = 3(5x + 2)\).

Step 4: Compare the original and new denominators

The original denominator after factoring is \(3(5x + 2)\), and the new denominator is \(12(5x + 2)\). Notice that the new denominator is a multiple of the original denominator.

Step 5: Determine the multiplier

Find the multiplier needed to convert the original denominator into the new denominator. To do this, divide the new denominator by the original denominator: \( \frac{12(5x + 2)}{3(5x + 2)} = 4 \). So, the multiplier is 4.

Step 6: Apply the multiplier to the numerator

Multiply the numerator of the original fraction by the determined multiplier: \(4 \times 4 = 16\).

Step 7: Write the final equivalent expression

The equivalent expression with the new denominator is \( \frac{16}{12(5x + 2)} \).

Key Concepts

Factoring

Factoring involves breaking down an expression into its simplest components, or factors.
This makes it easier to handle, especially when dealing with fractions.
In our exercise, the original denominator is \(15x + 6\). To simplify this denominator, we factor it as follows:
Notice that both terms, 15x and 6, have a common factor of 3. So, we can factor out the 3:
\[15x + 6 = 3(5x + 2)\] This shows us that 15x + 6 can be expressed as 3 times (5x + 2), which are its simplest components.

Denominator

The denominator is the bottom part of a fraction, which shows how many equal parts the whole is divided into.
It is crucial in determining the overall value of the fraction.
In our exercise, the initial denominator is given as \(15x + 6\). We later factor this to \(3(5x + 2)\). The new required denominator is \(12(5x + 2)\). Notice that the new denominator shares a common factor of \(5x + 2\) with the original.
This means the new denominator is just a scaled-up version of the original. Multiplying the numerator by the corresponding scale factor (in this case 4) will give us the equivalent fraction with the new denominator.
This ensures the overall value of the fraction remains unchanged.

Numerator

The numerator is the top part of a fraction, which indicates how many parts of the whole are being considered.
When we change the denominator of a fraction, we must adjust the numerator to maintain the fraction's value.
In our example, the initial numerator is 4.
After factoring and comparing the denominators, we found a scale factor of 4.
We thus multiply the numerator by this factor: \[4 \times 4 = 16\] The final equivalent fraction is then: \( \frac{16}{12(5x + 2)} \). This adjustment ensures that the fraction represents the same value, even with the new denominator.