Problem 81 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator \(15 x\). $$\frac{4}{5}$$

Step-by-Step Solution

Answer

The equivalent fraction is \( \frac{12x}{15x} \)."

Step 1: Identify the Missing Factor

To convert the fraction \( \frac{4}{5} \) to have a denominator of \( 15x \), first identify the factor by which you need to multiply the current denominator (5) to get \(15x\). Divide \(15x\) by 5 to find the missing factor: \( \frac{15x}{5} = 3x \).

Step 2: Multiply Numerator and Denominator

Multiply both the numerator and the denominator of \( \frac{4}{5} \) by the missing factor (\(3x\)). This ensures that the fraction remains equivalent. So: \[ \frac{4}{5} \times \frac{3x}{3x} = \frac{4 \times 3x}{5 \times 3x} = \frac{12x}{15x} \]

Step 3: Verify the Calculation

Double-check the multiplication to ensure accuracy: The numerator is \(4 \times 3x = 12x\) and the denominator is \(5 \times 3x = 15x\). Therefore, the equivalent fraction is \( \frac{12x}{15x} \).

Key Concepts

Fractions

Fractions represent parts of a whole or a division of quantities. They consist of a numerator and a denominator.

• The numerator is the top number which indicates how many parts we have.
• The denominator is the bottom number which signifies the total number of equal parts the whole is divided into.

Fractions can express simple ratios, like having 3 slices out of a pizza cut into 4 equal parts. When working with fractions, one fundamental skill is creating equivalent fractions. Equivalent fractions are different fractions that represent the same value. For example, \( \frac{1}{2} \) is equivalent to \( \frac{2}{4} \). Achieving equivalent fractions involves multiplying or dividing both the numerator and denominator by the same non-zero number.
This keeps the overall value the same as you're just altering the partitioning of the parts, not the proportion.

Denominators

Working with denominators is essential when handling fractions. The denominator must remain consistent when converting a fraction to its equivalent form.
In our exercise, the fraction \( \frac{4}{5} \) needs to have its denominator changed to \( 15x \). This involves:

• Recognizing the need to adjust this bottom number to match desired conditions.
• Identifying the missing factor that converts the original denominator to the target one.

By dividing the new denominator by the original, \( \frac{15x}{5} \), we determine that multiplying by \( 3x \) changes the denominator from 5 to \( 15x \).
Simply applying this factor to both parts of the fraction ensures the conversion without altering the value.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, operations, and sometimes constants. In our problem, the terms like \( 15x \) are algebraic expressions. Here, x represents a variable, an unknown that can take any value or sometimes a specific value if known.
By understanding variables:

• We interpret how they mix with numerical parts.
• They enable flexibility in mathematical expressions and calculations.

When constructing equivalent fractions involving variables, as demonstrated in our exercise, the approach remains the same:
Multiply both numerator and denominator by the same algebraic expression (in this case, \( 3x \)). This operation keeps the fraction's value unchanged, using the interplay of operations and variable components.