Problem 43 - Beginning and Intermediate Algebra

Question

Write each fraction as an equivalent fraction with the given denominator. See Example 6 . \(\frac{2}{9}\) with a denominator of 18

Step-by-Step Solution

Answer

The equivalent fraction is \(\frac{4}{18}\).

Step 1: Understand the Problem

We are given the fraction \(\frac{2}{9}\) and we need to find an equivalent fraction with the denominator 18.

Step 2: Determine the Multiplicative Factor

To find an equivalent fraction, determine how many times the original denominator (9) fits into the new denominator (18). We do this by dividing the new denominator by the original denominator: \(\frac{18}{9} = 2\). This tells us we need to multiply both the numerator and the denominator of the original fraction by 2.

Step 3: Multiply the Numerator and Denominator

Multiply the numerator and the denominator of the original fraction by the multiplicative factor found in Step 2 (which is 2).Compute: \(2 \times 2 = 4\) for the numerator, and \(9 \times 2 = 18\) for the denominator. Thus, \(\frac{2}{9} = \frac{4}{18}\).

Step 4: Write the Final Equivalent Fraction

With the calculations complete, the equivalent fraction of \(\frac{2}{9}\) with a denominator of 18 is \(\frac{4}{18}\).

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. They consist of two numbers: a numerator and a denominator. The numerator is written above a line, while the denominator is below it. For example, in the fraction \(\frac{2}{9}\), 2 is the numerator, indicating how many parts we have, and 9 is the denominator, showing into how many parts the whole is divided.

When working with fractions, it's important to understand that the numerator and the denominator are related. Changing either one can change the overall value of the fraction. Fractions are essential in expressing quantities between whole numbers, comparing sizes, and understanding ratios.

• Numerator: Shows the number of parts you have.
• Denominator: Indicates the total number of equal parts the whole is divided into.

The Role of the Denominator

The denominator in a fraction plays a crucial role as it represents the total number of equal parts the whole is divided into. When we talk about making equivalent fractions, this means altering the numerator and denominator in such a way that the overall value of the fraction remains unchanged.

In our exercise, the task was to change the denominator from 9 to 18. To maintain equivalency, both the numerator and the denominator must be multiplied by the same number (the multiplicative factor). By understanding how the denominator works, you can manipulate fractions without affecting their value. The key is to ensure the scale of division stays proportional.

Using a Multiplicative Factor

A multiplicative factor is what you multiply both the numerator and the denominator by to transform a fraction. To find this factor, divide the new denominator by the original denominator.

For the fraction \(\frac{2}{9}\) needing a new denominator of 18, the equation \(\frac{18}{9} = 2\) tells us that 2 is our multiplicative factor. This means we multiply both 2 (numerator) and 9 (denominator) by 2 to achieve an equivalent fraction with the denominator 18.

• Find the multiplicative factor: Divide the new denominator by the old denominator.
• Apply the factor to both numerator and denominator to obtain the new equivalent fraction.

Thus, multiplying both parts of \(\frac{2}{9}\) by 2 gives \(\frac{4}{18}\), proving they are equivalent as they represent the same portion of a whole.