Problem 81 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator 36. $$\frac{1}{4}$$

Step-by-Step Solution

Answer

\(\frac{1}{4}\) is equivalent to \(\frac{9}{36}\).

Step 1: Understand the Goal

We need to convert the fraction \(\frac{1}{4}\) into an equivalent fraction that has a denominator of 36.

Step 2: Find the Multiplicative Factor

To change the denominator from 4 to 36, determine what number 4 should be multiplied by to get 36. Calculate \(36 \div 4 = 9\). This means we must multiply both the numerator and the denominator of \(\frac{1}{4}\) by 9.

Step 3: Multiply Numerator and Denominator

Multiply the numerator and denominator of \(\frac{1}{4}\) by the multiplicative factor 9. This gives \(\frac{1 \times 9}{4 \times 9} = \frac{9}{36}\).

Step 4: Check Your Work

Verify that \(\frac{9}{36}\) has the same value as \(\frac{1}{4}\) by simplifying \(\frac{9}{36}\). Divide both the numerator and the denominator by their greatest common divisor, 9, which simplifies to \(\frac{1}{4}\). Thus, the conversion is correct.

Key Concepts

Understanding Fractions

Fractions are ways to express numbers as parts of a whole. Generally, they are represented as two numbers separated by a slash. The top number is called the numerator, and the bottom number is the denominator. Fractions could be expressed in different, but equivalent ways. For example, imagine slicing a pizza. If it's cut into four equal slices, one slice eaten could be represented as \(\frac{1}{4}\). However, by slicing each quarter into three smaller pieces, there would now be 12 pieces in total, and you’ve still eaten 3 of those pieces, \(\frac{3}{12}\).

• Numerator: Shows how many parts of the whole are being considered.
• Denominator: Indicates the total number of equal parts the whole is divided into.

Just like the pizza example, different fractions can represent the same ratio and thus be equivalent fractions.

The Role of the Denominator

In fractions, the denominator plays a crucial role by defining how many equal parts the whole is split into. When finding an equivalent fraction, the denominator changes while maintaining the original value of the fraction. For instance, changing the denominator from 4 to 36 requires finding an equivalent set of parts that could split the same whole into more pieces. Here’s how:

• Determine the required denominator, which is 36 in this case.
• Find a multiplier that scales the original denominator (4) to the target denominator (36).
• Apply this multiplicative factor to both numerator and denominator, ensuring the parts stay proportional.

This process allows for converting fractions while preserving their values, even if more pieces are created.

Using Multiplicative Factors for Equivalent Fractions

The concept of a multiplicative factor is fundamental when converting one fraction to another with a different denominator. A multiplicative factor is a number you multiply both the numerator and the denominator by to get a fraction that carries the same value as the original but meets the desired conditions, such as a specific denominator.When you want to turn \(\frac{1}{4}\) into a fraction with a denominator of 36, determining the multiplicative factor becomes essential. Here's how it works:

• Identify the original and target denominators, 4 and 36, respectively.
• Calculate the multiplicative factor by dividing the target denominator by the original denominator: \(36 \div 4 = 9\).
• Apply this factor, 9, to both the original numerator and denominator: \(\frac{1 \times 9}{4 \times 9} = \frac{9}{36}\).

By using this method, fractions can be easily adjusted to meet specific formatting requirements without altering their intrinsic value or ratio.