Problem 50 - Fundamentals of Mathematics

Question

For the following problems, determine the missing numerator or denominator. $$\frac{1}{8}=\frac{3}{?}$$

Step-by-Step Solution

Answer

The missing denominator is 24.

Step 1: Identify the Problem

We are given the equation \( \frac{1}{8} = \frac{3}{?} \) and need to find the missing denominator that makes both fractions equivalent.

Step 2: Set Up a Proportion

The two fractions are equal, which means they are in proportion. Hence, we can set up the proportion: \( \frac{1}{8} = \frac{3}{x} \), where \( x \) is the missing denominator.

Step 3: Cross-Multiply to Solve for the Unknown

Cross-multiplication helps us solve for \( x \) by multiplying the numerator of one fraction by the denominator of the other fraction: \( 1 \times x = 3 \times 8 \).

Step 4: Simplify the Equation

Perform the multiplication on the right side: \( 1 \times x = 24 \).

Step 5: Solve for x

Now, solve for \( x \) by dividing both sides by 1: \( x = 24 \).

Key Concepts

Cross-multiplication

Cross-multiplication is a mathematical technique used to solve equations involving two fractions set equal to each other. This method is powerful because it transforms a fraction problem into a simple multiplication problem.
Consider the proportion \( \frac{a}{b} = \frac{c}{d} \). Cross-multiplication tells us that \( a \times d = b \times c \). Thus, if we know three of these four values, we can find the fourth.

• This process helps eliminate the fractions, simplifying the equation considerably.
• Once the fractions are eliminated, the focus shifts to solving a simple algebraic equation.

This method is particularly useful when solving for either a missing numerator or denominator in a proportion, as demonstrated in the exercise. By cross-multiplying, we set up the equation \( 1 \times x = 3 \times 8 \), which simplifies to \( x = 24 \) upon solving.

Equivalent fractions

Equivalent fractions are fractions that represent the same value or part of a whole, even though they may look different. It is essential to grasp this concept to solve proportion problems effectively.
Take for instance the fractions \( \frac{1}{8} \) and \( \frac{3}{24} \). Even though they have different numerators and denominators, they are equivalent because they express the same quantity.

• To determine if two fractions are equivalent, you can cross-multiply and check whether the products are equal.
• Another method is to simplify or expand the fractions by multiplying or dividing both the numerator and the denominator by the same number.

A strong understanding of equivalent fractions aids in setting up and solving proportion problems by ensuring that you maintain equality between fractions when finding unknowns.

Numerator and denominator identification

Identifying the numerator and denominator is fundamental when dealing with fractions. The numerator is the top number, indicating how many parts of the whole are being considered. The denominator is the bottom number, showing the total parts the whole is divided into.
Correctly identifying these parts is crucial in solving problems such as \( \frac{1}{8} = \frac{3}{x} \). You need to know which part is missing or unknown.

• A numerator refers to the specific number of equal parts taken from a whole.
• A denominator signifies the total number of equal parts into which a whole is divided.

Clear understanding aids in correctly setting up equations and performing operations like cross-multiplication to find missing values. When you know what each part represents, you can adjust them to maintain the equation's balance and achieve accurate results.