Problem 42 - Fundamentals of Mathematics

Question

For the following problems, determine the missing numerator or denominator. $$\frac{2}{3}=\frac{?}{9}$$

Step-by-Step Solution

Answer

The missing numerator is 6.

Step 1: Understand the Equation

We are given the equation \( \frac{2}{3} = \frac{?}{9} \). Our goal is to find the missing numerator in the second fraction that makes the two fractions equal.

Step 2: Cross-Multiply to Solve for the Unknown

To find the missing value, use cross multiplication: \( 2 \times 9 = 3 \times ? \). This simplifies to \( 18 = 3 \times ? \).

Step 3: Solve for the Missing Numerator

Divide both sides of the equation by 3 to find the missing value: \( ? = \frac{18}{3} \). Simplifying this gives \( ? = 6 \).

Step 4: Verify the Solution

Substitute 6 back into the original equation: \( \frac{2}{3} = \frac{6}{9} \). Both fractions simplify to the same value (\( \frac{2}{3} \)), confirming that the solution is correct.

Key Concepts

Cross-multiplication

Cross-multiplication is a technique used to solve equations where two fractions are set equal to one another, like the one in our exercise: \( \frac{2}{3} = \frac{?}{9} \). It is a powerful tool to find unknowns in fractions without altering the balance of the equation.
Here's how it works:

• When given the equation \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction and multiplying the numerator of the second by the denominator of the first.
• The equation becomes \( a \times d = b \times c \). By doing this, you "cross" the lines joining numerators and denominators, hence the term "cross-multiplication."

This results in a simpler equation that can be solved to find the missing number. In our specific example, we solved \( 2 \times 9 = 3 \times ? \), arriving at \( 18 = 3 \times ? \).
By doing this, we've created an equation involving only whole numbers, which is easier to solve.

Numerator and Denominator

Understanding the concepts of numerator and denominator is crucial for working with fractions effectively.
Here's what you need to know:

• The **numerator** is the top number of a fraction, representing how many parts of the whole are considered. In \( \frac{2}{3} \), \(2\) is the numerator.
• The **denominator** is the bottom number, indicating into how many parts the whole is divided. For \( \frac{2}{3} \), \(3\) is the denominator.

The basic idea of a fraction can be visualized as a pizza. For example, if you have a pizza sliced into three equal parts (the denominator), and you take two slices (the numerator), you have \( \frac{2}{3} \) of the pizza.
In our original exercise, we knew the denominator (9) of the second fraction, but not the numerator. Cross-multiplication helped us adjust both parts of the fractions so they stayed equivalent.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This process makes fractions easier to understand and compare.
Let's see how this works in practice:

• Take each of the numbers in the fraction and divide them by their greatest common divisor (GCD).
• For the fraction \( \frac{6}{9} \), both 6 and 9 can be divided by 3, their GCD.
• Dividing both by 3 gives us \( \frac{2}{3} \), which is in its simplest form.

In the context of our problem, after finding the missing numerator as 6, we checked that \( \frac{6}{9} \) simplifies to \( \frac{2}{3} \), confirming that our solution was correct.
By reducing fractions, you ensure clear and precise comparisons with other fractions, maintaining mathematical consistency.