Problem 83 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The equivalent fraction is \(\frac{60}{15x}\).

Step 1: Understand the Current Denominator

The original fraction is \(\frac{4}{x}\), where the denominator is \(x\). We need to transform this fraction to have the denominator \(15x\).

Step 2: Determine the Multiplicative Factor

To change the denominator from \(x\) to \(15x\), you must multiply the denominator \(x\) by \(15\). This means you need to multiply both the numerator and the denominator of the fraction by \(15\) to maintain the equivalence of the fraction.

Step 3: Calculate the New Numerator and Denominator

Multiply the numerator by \(15\): \(4 \times 15 = 60\). Multiply the denominator by \(15\): \(x \times 15 = 15x\).

Step 4: Write the Equivalent Fraction

The equivalent fraction with denominator \(15x\) is \(\frac{60}{15x}\).

Key Concepts

Understanding Fractions

Fractions are numbers that represent a part of a whole. They consist of two key parts: the numerator and the denominator. The most common real-world example of a fraction is slicing a pizza. If you slice a pizza into 8 equal parts and eat 3 slices, then you have eaten \(\frac{3}{8}\) of the pizza.

Fractions are useful for expressing quantities that aren’t whole numbers. They allow you to precisely communicate how much of something you have or need, which is essential in math and everyday life. Equivalent fractions are different fractions that represent the same quantity. For example, \(\frac{1}{2}\) is equivalent to \(\frac{2}{4}\) or \(\frac{3}{6}\), and they all represent the same portion of a whole.

The Role of the Denominator

The denominator in a fraction is the bottom number. It tells you into how many equal parts the whole is divided. Going back to our pizza example, if you have \(\frac{3}{8}\), the number 8 in the denominator tells you the pizza was originally divided into 8 equal slices.

When working with fractions, adjusting the denominator to a common value is a crucial step in many mathematical calculations, such as adding or subtracting fractions. If two fractions have the same denominator, they are much easier to compare or combine.

• The denominator must never be zero, as dividing by zero is undefined.
• The denominator determines the size of the fractioned parts; a larger denominator means smaller individual parts.

Working with Multiplicative Factors

A multiplicative factor is a number you multiply by to change another number or expression. To adjust fractions to have new common denominators, we often use a multiplicative factor.

In the exercise provided, to change the denominator from \(x\) to \(15x\), the multiplicative factor used was 15. When you apply this factor:

• Multiply both the numerator and denominator by the same factor to get an equivalent fraction.
• This process keeps the value of the fraction unchanged while changing its form.

Using the same factor for both parts ensures the fractions remain equivalent, since you are essentially multiplying by \(\frac{15}{15}\), which equals one.

Understanding the Numerator

The numerator is the top number in a fraction. It indicates how many parts of the whole you have. In \(\frac{3}{8}\), the numerator 3 tells you that 3 slices of pizza are considered.

In equivalence exercises, like the one given, the numerator is often modified to keep the fraction equal in value. If a fraction's denominator changes by multiplying by a factor—like from \(x\) to \(15x\)—to find the equivalent fraction, the numerator is also multiplied by the same factor.

• The numerator is flexible and can take different forms depending on the context of the fraction.
• When a denominator is multiplied, the corresponding multiplication of the numerator ensures the fraction's value stays consistent.

Identify the pattern: What you do to the denominator, you must do to the numerator to maintain equivalence.