Problem 72 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator 30. $$\frac{3}{10}$$

Step-by-Step Solution

Answer

The equivalent fraction is \( \frac{9}{30} \).

Step 1: Understand the Problem

We need to rewrite the fraction \( \frac{3}{10} \) so that its denominator is 30. We will find an equivalent fraction with this new denominator.

Step 2: Find the Conversion Factor

To convert the denominator from 10 to 30, we need to determine what number we multiply 10 by to get 30. \( 10 \times 3 = 30 \), so the conversion factor is 3.

Step 3: Multiply Numerator and Denominator

Using the conversion factor from Step 2, multiply both the numerator and denominator of the original fraction by 3. This gives us: \[ \frac{3}{10} \times \frac{3}{3} = \frac{9}{30} \].

Step 4: Simplify and Verify

Check that the new fraction \( \frac{9}{30} \) is correctly simplified from \( \frac{3}{10} \). Since the conversion factor is consistent, \( \frac{9}{30} \) is indeed the equivalent fraction with a denominator of 30.

Key Concepts

Understanding Fraction Conversion

Fraction conversion involves changing the form of a fraction to an equivalent fraction, which essentially represents the same value but may look different. This concept is crucial when dealing with operations like addition or comparison of fractions with different denominators.
To perform fraction conversion, a key step is identifying the conversion factor. This is the number by which you multiply the numerator and denominator to get an equivalent fraction with the desired denominator.

• Identify the current and new denominators.
• Calculate the conversion factor by dividing the new denominator by the current denominator.
• Apply this factor to both the numerator and denominator to get an equivalent fraction with the desired denominator.

By understanding fraction conversion, you can seamlessly change fractions to have common denominators, enabling easier arithmetic operations and comparisons.

The Role of the Denominator in Fractions

The denominator in a fraction indicates the total number of equal parts the whole is divided into. For example, in the fraction \(\frac{3}{10}\), the denominator is 10, meaning the whole is divided into 10 parts, and we have 3 of those parts.
When converting fractions, changing the denominator means altering the number of parts without changing the overall size represented by the fraction.

• Identify the number by which the current denominator must be multiplied to convert it to the new denominator.
• Ensure the numerator is also adjusted proportionally, often by using the same conversion factor.
• Check that the fraction remains equivalent, meaning it represents the same proportion of the whole as before.

Understanding the denominator's role helps in executing precise conversions and maintaining the fraction's value unchanged.

Importance of Simplification

Simplification of fractions means reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. It's a process that makes fractions easier to work with and understand.
Simplification involves the following steps:

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and numerator by this GCD.
• Ensure the resulting fraction cannot be further simplified.

While the fraction \(\frac{9}{30}\) simplifies to \(\frac{3}{10}\), in some cases, fractions converted to new denominators need simplification to retain clarity and simplicity. Simplification does not change the value a fraction represents; it only presents it in a neater, more standard form.