Problem 80 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator 36. $$\frac{7}{18}$$

Step-by-Step Solution

Answer

\( \frac{7}{18} \) is equivalent to \( \frac{14}{36} \).

Step 1: Understand Equivalent Fractions

Recall that equivalent fractions are different fractions that express the same value or proportion. To find an equivalent fraction with a specific denominator, you must multiply both the numerator and the denominator by the same number.

Step 2: Determine Multiplier for Denominator

We need to determine what number to multiply the denominator, 18, by to get the new denominator, 36. This is done by dividing 36 by 18: \[ \frac{36}{18} = 2 \].Thus, we multiply the original denominator, 18, by 2 to get 36.

Step 3: Multiply Numerator by Multiplier

Since we multiply the denominator by 2, we must also multiply the numerator by 2 to maintain the equivalence. Multiply the numerator 7 by 2:\[ 7 \times 2 = 14 \].

Step 4: Write the Equivalent Fraction

After completing the multiplication, the equivalent fraction of \( \frac{7}{18} \) with a denominator of 36 is:\[ \frac{14}{36} \].

Key Concepts

Fractions

A fraction is a mathematical expression representing a part of a whole. It consists of two numbers separated by a slash. The number above the slash is the numerator, and the one below is the denominator. The numerator indicates how many parts we have, while the denominator tells us into how many equal parts the whole is divided. For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator. This fraction means that the whole is divided into 4 equal parts, and we have 3 of those parts. Fractions are incredibly useful in mathematics as they allow us to express numbers that are not whole, whether we're dealing with measurements, ratios, or probabilities.

Numerator and Denominator

The terms numerator and denominator are fundamental when working with fractions. Understanding these terms is essential to making sense of fractions.

• The **numerator** is the top number in a fraction. It represents how many parts of a whole are being considered.
• The **denominator** is the bottom number. It shows into how many equal parts the whole is divided.

For instance, in the fraction \( \frac{5}{9} \), the numerator 5 indicates that 5 parts are being considered out of a total of 9 parts, which is indicated by the denominator.

Multiplying Fractions

When it comes to multiplying fractions, the process is straightforward but requires careful attention. Here's a step-by-step explanation: First, you multiply the numerators of your fractions together. Then, you multiply the denominators. For example, if we have \( \frac{a}{b} \times \frac{c}{d} \), the multiplication process is:

• Multiply the numerators: \( a \times c \)
• Multiply the denominators: \( b \times d \)

This gives us a new fraction: \( \frac{a \times c}{b \times d} \). When finding an equivalent fraction, the idea is to multiply the numerator and denominator of the original fraction by the same number, ensuring the fraction's value does not change.Always remember, to maintain the equivalence of fractions, whatever mathematical operation is performed on the numerator must also be done on the denominator.