Problem 46 - Beginning and Intermediate Algebra

Question

Write each fraction as an equivalent fraction with the given denominator. See Example 6 . \(\frac{4}{5}\) with a denominator of 25

Step-by-Step Solution

Answer

The equivalent fraction is \( \frac{20}{25} \).

Step 1: Identify the current denominator and the desired denominator

The given fraction is \( \frac{4}{5} \). The current denominator is 5 and the desired denominator is 25.

Step 2: Determine the multiplication factor

To convert the current denominator of 5 to the desired denominator of 25, we need to find a number that 5 can be multiplied by to equal 25. The multiplication factor is \( 25 \div 5 = 5 \).

Step 3: Multiply both numerator and denominator by the same factor

Multiply both the numerator and the denominator of the fraction \( \frac{4}{5} \) by the multiplication factor 5. This gives: \[ \frac{4 \times 5}{5 \times 5} = \frac{20}{25} \].

Step 4: Verify the equivalent fraction

Confirm that the equivalent fraction has the correct denominator. The fraction \( \frac{20}{25} \) maintains the same value as \( \frac{4}{5} \), but with the desired denominator of 25.

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. In a fraction, the top number is the numerator, and it tells you how many parts you have. The bottom number is the denominator, and it tells you how many parts make up a whole. For example, in the fraction \( \frac{4}{5} \), 4 is the numerator and 5 is the denominator, indicating that you have 4 out of 5 equal parts.
Understanding fractions is like understanding a pizza cut into slices. If a pizza is cut into 5 slices (each representing a part of the whole), and you have 4 slices, you have \( \frac{4}{5} \) of the pizza.

• The numerator counts the parts you have.
• The denominator tells you how many total parts the whole is divided into.

Fractions are used in everyday life, from measuring ingredients in a recipe to describing proportions in finances.

Denominator

The denominator of a fraction is crucial because it gives meaning to the fraction. It's the number below the line in a fraction, and it shows how many equal parts the whole is divided into. In our example \( \frac{4}{5} \), the denominator is 5.
Changing the denominator alters the fraction but not the amount it represents, if done correctly. When converting a fraction to one with a different denominator, the fraction's value remains constant. It's like changing how you describe the same length but in different units, such as feet or inches.
In order to change a fraction's denominator while keeping its value, you need to find an equivalent fraction. This is achieved by making sure the new denominator is a multiple of the original one, which allows you to maintain the proportion between the numerator and the denominator.

Multiplication Factor

The multiplication factor is an essential tool for converting fractions to equivalent fractions with different denominators. It is a number by which both the numerator and the denominator are multiplied to create a new equivalent fraction with the desired denominator.
To find the multiplication factor, divide the desired denominator by the current denominator. For example, to convert \( \frac{4}{5} \) to a fraction with a denominator of 25, calculate \( 25 \div 5 = 5 \). This tells us the fraction needs to be expanded by a factor of 5.
Here's how the multiplication factor works:

• Find the factor by dividing the new denominator by the original denominator.
• Multiply the numerator and the denominator by this factor.
• Check if the resulting fraction has the desired denominator and maintains the original value.

By using the multiplication factor, you ensure that the fractions remain equivalent, just expressed differently.