Problem 51 - Fundamentals of Mathematics

Question

For the following problems, determine the missing numerator or denominator. $$ \frac{4}{5}=\frac{?}{100} $$

Step-by-Step Solution

Answer

The missing numerator is 80.

Step 1: Set up the Proportion

We are given the fraction \( \frac{4}{5} \) and need to find the missing numerator in the equivalent fraction with a denominator of 100, \( \frac{?}{100} \). So, our equation is \( \frac{4}{5} = \frac{x}{100} \).

Step 2: Cross-Multiply

To solve for \( x \), we can cross-multiply the fractions. This means we multiply the numerator of one fraction by the denominator of the other fraction: \( 4 \times 100 = 5 \times x \)

Step 3: Simplify the Equation

Calculating the multiplication gives us: \( 400 = 5x \) This equation shows that 400 is equal to 5 times the unknown numerator \( x \).

Step 4: Solve for the Unknown Numerator

To find the value of \( x \), divide both sides of the equation by 5: \( x = \frac{400}{5} \) Perform the division to find \( x = 80 \).

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is a crucial technique used to solve equations involving proportions, particularly when you are dealing with fractions. It simplifies the process of finding unknown values, such as numerators or denominators, without needing to have a common denominator. In the example given, we have a proportion: \[ \frac{4}{5} = \frac{x}{100} \] To solve this, we use cross-multiplication by multiplying diagonally across the equal sign:

• Multiply the numerator of the first fraction by the denominator of the second fraction: \( 4 \times 100 \)
• Multiply the denominator of the first fraction by the numerator of the second fraction: \( 5 \times x \)

These calculations give us the equation:\[ 4 \times 100 = 5 \times x \] Thus leading to the equation: \( 400 = 5x \). With this equation, you can easily solve for \( x \) by performing further calculations, making cross-multiplication an effective and powerful strategy for solving for unknowns in proportions.

Managing Fractions in Proportions

Fractions often appear in everyday math problems, especially when dealing with proportions. A fraction represents a part of a whole, consisting of a numerator and a denominator. In our problem, both fractions \( \frac{4}{5} \) and \( \frac{x}{100} \) need to be equal. When working with these fractions in proportions:

• The left side of the equation includes \( \frac{4}{5} \), where 4 is parts of a whole divided into 5 equal parts.
• The right side \( \frac{x}{100} \) represents an unknown number of parts, \( x \), out of 100 equal parts.

To achieve balance, these two fractions must be equivalent through a method like cross-multiplication. Thus, understanding how fractions function is key to manipulating and solving proportional equations, where one fraction is effectively a scaled version of another.

Identifying the Numerator

The numerator is the top number in a fraction and indicates how many parts of the whole are represented. Solving the given problem requires finding the missing numerator \( x \) in the fraction \( \frac{x}{100} \). The proportion \( \frac{4}{5} = \frac{x}{100} \) shows that these two fractions are equivalent. To determine \( x \):

• Start by setting up the cross-multiplication equation: \( 4 \times 100 = 5 \times x \).
• Solve the equation to isolate the numerator by simplifying: \( 400 = 5x \).
• Then, divide both sides by 5 to find \( x \): \( x = \frac{400}{5}\).
• This results in \( x = 80 \), identifying the numerator successfully.

The ability to identify and solve for an unknown numerator in a fraction is a foundational aspect of working with proportions, aiding in developing mathematical fluency and problem-solving skills.