Problem 99 - Algebra 1: Concepts and Skills

Question

Write three equivalent fractions for the given fraction. $$ \frac{1}{8} $$

Step-by-Step Solution

Answer

Three equivalent fractions for \( \frac{1}{8} \) are \( \frac{2}{16} \), \( \frac{3}{24} \), and \( \frac{4}{32} \).

Step 1: Multiply the Numerator and Denominator by the Same Non-Zero Number

Let's multiply the numerator and denominator of \( \frac{1}{8} \) by 2. This results in the equivalent fraction \( \frac{2}{16} \).

Step 2: Multiply the Numerator and Denominator by Another Non-Zero Number

Next, multiply the numerator and denominator of \( \frac{1}{8} \) by 3. This results in another equivalent fraction \( \frac{3}{24} \).

Step 3: Multiply the Numerator and Denominator by a Third Non-Zero Number

Finally, multiply the numerator and denominator of \( \frac{1}{8} \) by 4. This results in the third equivalent fraction \( \frac{4}{32} \).

Key Concepts

Fractions

A fraction represents a part of a whole or a division of a quantity. It is expressed as two numbers separated by a slash. The number above the line is called the numerator, and the number below the line is the denominator. Fractions are used in various scenarios, such as when dividing an object into equal parts or when needing to express parts of a total. Understanding fractions is essential because they serve as a foundation for more complex arithmetic and mathematical concepts, such as ratios and decimals. Equivalent fractions are fractions that represent the same value even though they may look different. This happens when both the numerator and the denominator of the fraction are multiplied by the same non-zero number. For example, multiplying both the numerator and the denominator of \( \frac{1}{8} \) by 2 gives \( \frac{2}{16} \); by 3 gives \( \frac{3}{24} \); and by 4 gives \( \frac{4}{32} \). All these fractions are equivalent.

Numerator and Denominator

The numerator and denominator are fundamental components of a fraction. The numerator, the top part of a fraction, indicates how many parts we have or are considering. The denominator, the bottom part, tells us the total number of equal parts the whole is divided into. For instance, in the fraction \( \frac{1}{8} \), the numerator is 1, meaning we have one part of something, and the denominator is 8, meaning the whole is divided into eight equal parts. This can be thought of as having one slice of a pizza that was originally cut into eight slices. When creating equivalent fractions, both the numerator and the denominator must be multiplied by the same non-zero number to maintain the value of the fraction. This ensures that each new fraction is just a different representation of the same quantity. It’s important never to multiply only one of them, as that changes the value of the fraction entirely.

Multiplication in Fractions

One of the key operations in working with fractions is multiplication, which can be used to find equivalent fractions. To create equivalent fractions, you multiply both the numerator and the denominator by the same non-zero number. This process changes the appearance of the fraction but not its actual value. For example, starting with \( \frac{1}{8} \), multiplying the numerator and denominator by 2 leads to \( \frac{2}{16} \). If you use 3, it becomes \( \frac{3}{24} \), and using 4 makes it \( \frac{4}{32} \). Each resulting fraction is equivalent to \( \frac{1}{8} \). It’s helpful to remember that when multiplying fractions, the operation applies to both components (numerator and denominator) uniformly. This consistency ensures that the fraction’s value remains unchanged, and thus helps us in identifying or creating equivalent fractions. Understanding this concept is critical for working with fractions in different mathematical contexts.