Problem 102 - Algebra 1: Concepts and Skills

Question

Write three equivalent fractions for the given fraction. $$ \frac{5}{32} $$

Step-by-Step Solution

Answer

The three equivalent fractions to the given fraction \(\frac{5}{32}\) are \(\frac{10}{64}\), \(\frac{15}{96}\), and \(\frac{20}{128}\).

Step 1: Identify the Given Fraction

Identify the given fraction. The given fraction is \(\frac{5}{32}\).

Step 2: Create the First Equivalent Fraction

Find a number to multiply both the numerator and the denominator of the given fraction by to create an equivalent fraction. For example, multiply both terms of the given fraction \(\frac{5}{32}\) by 2, resulting in \(\frac{10}{64}\). This constitutes the first equivalent fraction.

Step 3: Create the Second Equivalent Fraction

Find a different number to multiply both the numerator and the denominator of the original fraction by. For instance, multiply both terms of the given fraction \(\frac{5}{32}\) by 3, yielding \(\frac{15}{96}\). This is the second equivalent fraction.

Step 4: Create the Third Equivalent Fraction

Find yet another number to multiply the numerator and the denominator of the original fraction by. An example might be multiplying both terms of the given fraction \(\frac{5}{32}\) by 4, providing \(\frac{20}{128}\). This gives the third and final equivalent fraction.

Key Concepts

Fraction multiplication

Understanding fraction multiplication is crucial when creating equivalent fractions. Multiplication of fractions involves multiplying the numerators together and then the denominators, essentially creating a new fraction that represents the product of the two. For instance, when we take the fraction \( \frac{a}{b} \) and multiply it by a factor of c (assuming c is a non-zero whole number), we apply the multiplication to both the numerator and denominator, like so: \( \frac{a \times c}{b \times c} \).

This process neither changes the value of the fraction nor its meaning. Instead, it produces an equivalent fraction, a new way to represent the same quantity. When doing homework or working on exercises involving fraction multiplication, remember the principle that multiplying both the numerator and the denominator by the same number gives a fraction that is equivalent to the original. This concept is at the heart of many mathematical operations and problems including ratios, proportions, and algebraic expressions.

Numerator and denominator

The numerator and denominator are the two components that make up a fraction. The numerator, located above the fraction bar, indicates the number of equal parts being considered, while the denominator, below the fraction bar, tells us how many of those equal parts make up a whole.

For instance, in the fraction \( \frac{5}{32} \), 5 is the numerator and 32 is the denominator. This means you have 5 parts out of a total of 32 equal parts that make a whole. To create equivalent fractions, we manipulate these two numbers while maintaining the fraction's original value. Equivalence in fractions happens when two or more fractions represent the same quantity, even if they look different numerically. Multiplying or dividing both the numerator and the denominator by the same non-zero number will retain the value of the fraction.

Creating equivalent fractions

Creating equivalent fractions is a fundamental concept in understanding fractions and their properties. An equivalent fraction maintains the same value as the original fraction, even though its numerator and denominator may look different. To create an equivalent fraction, you multiply or divide both the numerator and the denominator by the same non-zero whole number.

Let's consider the fraction \( \frac{5}{32} \). To generate equivalent fractions, we can multiply both the numerator (5) and the denominator (32) by the same whole numbers. For example:

• Multiplying by 2 gives \( \frac{5 \times 2}{32 \times 2} = \frac{10}{64} \),
• Multiplying by 3 gives \( \frac{5 \times 3}{32 \times 3} = \frac{15}{96} \),
• Multiplying by 4 gives \( \frac{5 \times 4}{32 \times 4} = \frac{20}{128} \).

Each of these fractions—\( \frac{10}{64} \), \( \frac{15}{96} \), and \( \frac{20}{128} \)—are equivalent to the original \( \frac{5}{32} \). The idea is to maintain the balance between the numerator and denominator through consistent multiplication or division, thereby keeping the overall value of the fraction the same.