Problem 40 - 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: Understanding the Problem

The given problem is \( \frac{2}{3} = \frac{?}{9} \). We need to find the missing numerator, which is represented by \( ? \), in the fraction \( \frac{?}{9} \) such that both fractions are equivalent.

Step 2: Set Up the Proportion Equation

To find the unknown numerator, we set up an equation based on the given equivalent fractions: \( \frac{2}{3} = \frac{x}{9} \). Here, \( x \) represents the unknown numerator we are looking for.

Step 3: Cross-Multiply

Cross-multiply the fractions to create an equation involving \( x \):\[ 2 \times 9 = 3 \times x \] This can be simplified to: \[ 18 = 3x \]

Step 4: Solve for the Unknown

To find \( x \), divide both sides of the equation by 3:\[ x = \frac{18}{3} \] This simplifies to:\[ x = 6 \]

Step 5: Verify the Solution

Check that the solution is correct by substituting \( 6 \) back into the equation to see if the fractions are equivalent: \( \frac{2}{3} = \frac{6}{9} \). Both fractions reduce to \( \frac{2}{3} \), confirming that 6 is the correct numerator.

Key Concepts

Understanding the Numerator

A fraction is made up of two main parts: the numerator and the denominator. The numerator is the top number of the fraction. It illustrates how many parts of a whole are being considered. This is essential because it tells us exactly how much of the pie we are dealing with.
For instance, in the fraction \( \frac{2}{3} \), the number **2** is the numerator. It tells you that out of 3 equal parts, we are focusing on 2 parts.
In the exercise: \( \frac{2}{3} = \frac{?}{9} \), the challenge is to find the missing numerator to make the two fractions equivalent.

• The numerator helps in determining what portions of a whole are under observation.
• It plays a crucial role in making fractions equivalent by being appropriately scaled up or down.

To solve the problem correctly, understanding the role of the numerator is vital. It helps in keeping fractions balanced and equivalent when adjusted with the right denominator.

Understanding the Denominator

The denominator is the bottom number of a fraction. It shows how many equal parts make up a whole.
Think of it as dividing a pizza into equal slices. The denominator tells you how many slices there are.
In our case, in \( \frac{2}{3} \), **3** is the denominator, meaning the whole is split into 3 equal parts.

• It sets the base or the total number of parts the object or whole is divided into.
• Knowing the denominator helps compare different fractions to determine if they are equivalent.

For the exercise, the denominator of the fraction \( \frac{?}{9} \) is **9**. It indicates that the whole is now divided into 9 parts. By finding the correct numerator, the fraction can be equivalent to \( \frac{2}{3} \).
The denominator is central in setting up relationships between different fractions and plays a key role in understanding and solving such problems.

Cross-Multiplication Explained

Cross-multiplication is a technique used to solve equations involving two fractions. This technique helps in determining the unknown part of an equivalent fraction like our problem \( \frac{2}{3} = \frac{x}{9} \).
Here's how it works:
1. Multiply the numerator of one fraction by the denominator of the other fraction.2. Do the same for the other numerator and denominator pair.

• It essentially transforms a proportion into a simple equation.
• Helps easily solve for unknown numerators or denominators, ensuring the fractions remain equivalent.

In the exercise, we multiplied the top of the first fraction by the bottom of the second (2 * 9) and the bottom of the first by the top of the second (3 * x). This gave us the equation \(18 = 3x\). Solving this equation gives the value of \(x\), assuring the fractions are equivalent.
Cross-multiplication simplifies the process and is a powerful tool for solving fractional equations.