Problem 81 - Algebra 1: Concepts and Skills

Question

Write the answer as a fraction or as a mixed number in simplest form. $$ \frac{7}{22} \cdot \frac{22}{7} $$

Step-by-Step Solution

Answer

The answer is 1.

Step 1: Set Up the Multiplication

Arrange the fractions to be multiplied. We have \( \frac{7}{22} \) times \( \frac{22}{7} \).

Step 2: Multiply the Numerators Together

The numerator of the result of the multiplication is found by multiplying the numerators of the fractions. So, \( 7 \times 22 = 154 \).

Step 3: Multiply the Denominators Together

The denominator of the result of the multiplication is found by multiplying the denominators of the fractions. So, \( 22 \times 7 = 154 \).

Step 4: Simplify the Result

Our result is \( \frac{154}{154} \), therefore we can simplify it to 1 as there are 154 items in the numerator and the denominator.

Key Concepts

Simplifying Fractions

Fractions can often be reduced to simpler forms by dividing both the numerator and the denominator by their greatest common divisor (GCD). This process is called simplifying fractions.
For instance, consider the fraction \( \frac{6}{8} \). The GCD of 6 and 8 is 2, so we divide both by 2 to get \( \frac{3}{4} \).
Simplifying is crucial because it makes fractions easier to understand and compare.

• Check if the numerator and the denominator share any common factors.
• Divide by the greatest factor they share to simplify.
• Keep doing this until no more common factors exist.

In the case of our exercise, the fraction \( \frac{154}{154} \) simplifies down to 1. This is because any number divided by itself equals 1, representing a complete whole.

Numerators and Denominators

Understanding numerators and denominators is key to working with fractions. A fraction represents a division of one number (numerator) by another (denominator).
The numerator is the top number and indicates how many parts are considered, while the denominator is the bottom number and shows into how many parts the whole is divided.

• In \( \frac{7}{22} \), 7 is the numerator, and 22 is the denominator.
• The combination of numerators and denominators determines the value of the fraction.
• Fractions are expressions of division, where you divide the numerator by the denominator.

Thus, manipulating these values allows us to perform operations like simplification and multiplication. In the exercise, multiplying \( 7 \) by \( 22 \) and \( 22 \) by \( 7 \) highlights these roles particularly well.

Multiplication of Fractions

When multiplying fractions, the process involves a straightforward method: multiply the numerators together and do the same with the denominators.
This operation is simpler than addition or subtraction of fractions since you don't need a common denominator. Here's how you can do it:

• Multiply the numerators: This gives the new numerator. For example, with \( \frac{7}{22} \) and \( \frac{22}{7} \), multiply 7 by 22 to get 154.
• Multiply the denominators: This provides the new denominator. In our example, 22 times 7 also results in 154.
• Simplify the fraction if possible. In our case, \( \frac{154}{154} = 1 \).

Multiplying fractions like these shows how two fractions can simplify directly to a whole number, reflecting a perfect reciprocal relationship in the exercise.