Problem 73 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator 30. $$\frac{3}{5}$$

Step-by-Step Solution

Answer

The equivalent fraction is \(\frac{18}{30}\).

Step 1: Identify the Denominator

The original fraction has a denominator of 5. We need to find an equivalent fraction that has a denominator of 30 instead.

Step 2: Determine the Multiplication Factor

To convert the denominator from 5 to 30, you need to determine what number you multiply 5 by to get 30. Since 30 divided by 5 is 6, the multiplication factor is 6.

Step 3: Multiply the Numerator and Denominator

Multiply both the numerator and the denominator of the original fraction \(\frac{3}{5}\) by the multiplication factor (6): \[ \frac{3 \times 6}{5 \times 6} = \frac{18}{30} \]

Step 4: Verify the Result

Double-check the calculation to ensure accuracy: The original fraction \( \frac{3}{5} \) becomes \( \frac{18}{30} \), which is equivalent since both numerator and denominator were multiplied by the same number (6).

Key Concepts

Understanding Fractions

Fractions are a way to express numbers that are not whole, representing a part of a whole. A fraction consists of two parts:

• Numerator: The top number of the fraction, indicating how many parts are considered.
• Denominator: The bottom number, indicating the total number of equal parts the whole is divided into.

Fractions can be used in everyday life when dividing items or quantities into smaller parts. For example, if you have a pie and cut it into 5 equal slices, each slice represents a fraction of the whole pie, specifically \( \frac{1}{5} \).
Understanding equivalent fractions is crucial because it allows us to express the same value in different forms. Equivalent fractions are different fractions that represent the same value or proportion of the whole. For instance, \( \frac{1}{2} \) is equivalent to \( \frac{2}{4} \) because both equal half of something. Recognizing this helps us perform calculations and comparisons more easily.

The Role of the Denominator

The denominator is a critical part of a fraction since it tells us into how many equal parts the whole is divided. In order to change a fraction to have a specific denominator without altering its value, we create an equivalent fraction.
In the given exercise, we needed to convert the denominator of \( \frac{3}{5} \) to 30. By doing this, we align both fractions to have the same baseline, making it easier to compare them directly or use them in calculations together.
It's important to remember that we cannot change the denominator of a fraction by itself without affecting the numerator similarly, or else the value of the fraction will be altered. This leads us to the next concept: the multiplication factor.

Finding the Multiplication Factor

A multiplication factor is a number you multiply both the numerator and denominator of a fraction by to create an equivalent fraction. This factor is essential for maintaining the same value of the fraction when changing its denominator.

In the exercise, once we identified that the denominator needed to be changed from 5 to 30, we determined the multiplication factor by dividing the new denominator by the old denominator: \( \frac{30}{5} = 6 \).
This means both numbers in the fraction \( \frac{3}{5} \) are multiplied by 6, resulting in an equivalent fraction: \( \frac{18}{30} \).

This demonstrates how multiplication factors allow us to adjust fractions dynamically while preserving their value, an essential skill in various mathematical topics like addition and subtraction of fractions with different denominators.