Problem 18 - Fundamentals of Mathematics

Question

Determine the missing numerator or denominator. $$ \frac{4}{5}=\frac{?}{40} $$

Step-by-Step Solution

Answer

The missing numerator is 32.

Step 1: Understand the Equation

We are given the equation \( \frac{4}{5} = \frac{x}{40} \). Our goal is to find the value of \( x \), which is the missing numerator in the second fraction.

Step 2: Cross Multiply

Since the two fractions are equal, we can cross multiply to find the missing numerator. This will look like: \( 4 \times 40 = 5 \times x \).

Step 3: Simplify the Multiplication

Calculate \( 4 \times 40 \), which equals 160. Now our equation is \( 160 = 5x \).

Step 4: Solve for the Missing Numerator

To isolate \( x \), divide both sides of the equation by 5. This gives us \( x = \frac{160}{5} \).

Step 5: Calculate the Result

Perform the division \( \frac{160}{5} \) to find \( x \). When you divide 160 by 5, you get 32.

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to solve equations where two fractions are set equal to each other. It's an efficient and straightforward method that provides a quick way to find unknown numerators or denominators. Here's how it works: when you have two fractions that are equal, like \( \frac{4}{5} = \frac{x}{40} \), you can "cross multiply" by multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa.

• For our example, this cross multiplication gives: \( 4 \times 40 \) and \( 5 \times x \).
• This results in the equation: \( 160 = 5x \).

The idea of cross multiplication stems from the property of equivalent fractions. It ensures that the products on both sides of the equation remain equal, helping us narrow down the missing component.

Numerator and Denominator

In any fraction, the numerator and the denominator play crucial roles in defining the fraction's value. The numerator is the number above the fraction bar, indicating how many parts of the whole we have. The denominator, on the other hand, is below the fraction bar and tells us how many parts the whole is divided into.

• In our exercise \( \frac{4}{5} = \frac{x}{40} \), "4" and "5" are the numerator and denominator of the first fraction, respectively.
• For the second fraction, "x" is the missing numerator and "40" is the denominator.

Understanding this structure is vital because it helps us set up equivalent equations correctly, like we did to apply cross multiplication. The relationship between the numerator and denominator informs us on how to make one fraction equivalent to another by scaling up or down.

Solving Fractional Equations

Solving fractional equations involves finding the value of unknown variables present in fractions. Using cross multiplication, we simplify and solve these equations more efficiently. Let's break down the steps illustrated in the exercise:

• Start with the equation \( \frac{4}{5} = \frac{x}{40} \) and apply cross multiplication to get \( 160 = 5x \).
• This simplification reduces our fractional equation to a simple algebra problem.
• To solve for \( x \), divide both sides of the equation by 5 to isolate the variable: \( x = \frac{160}{5} \).

Finally, perform the division to determine the value of \( x \): when \( 160 \) is divided by \( 5 \), the result is \( 32 \). Thus, in the original equation, \( x = 32 \) makes both fractions equivalent. This process of solving fractional equations by breaking them down using algebraic techniques ensures understanding and reinforces the fundamental principles of fractions.