Problem 50 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{7}{4} \cdot \frac{6}{11}$$

Step-by-Step Solution

Answer

The multiplication of \( \frac{7}{4} \) and \( \frac{6}{11} \), reduced to its lowest term is \( \frac{21}{22} \)

Step 1: Understanding the problem

The problem is to multiply two fractions: \( \frac{7}{4} \) and \( \frac{6}{11} \) together and reduce the result to lowest terms.

Step 2: Multiply the fractions

Multiply the numerators to get the numerator of the answer and multiply the denominators to get the denominator of the answer. This will provide: \( \frac{7 * 6}{4 * 11} = \frac{42}{44} \).

Step 3: Reduce to lowest terms

The fraction \( \frac{42}{44} \) can be simplified to its lowest term by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2. This provides: \( \frac{42 ÷ 2}{44 ÷ 2} = \frac{21}{22} \).

Key Concepts

Simplifying Fractions

Simplifying fractions makes them easier to interpret and work with. The goal here is to reduce a fraction to its simplest form. Traditionally, a fraction is simplified by dividing the numerator and the denominator by their greatest common divisor (GCD).

• Identify the GCD of both the numerator and the denominator.
• Divide both terms of the fraction by the GCD.
• The result is a fraction in its simplest form, which means it is reduced to the smallest possible whole numbers representing the same ratio.

For example, when simplifying the fraction \( \frac{42}{44} \), the GCD is 2. So we divide both the numerator (42) and the denominator (44) by 2, resulting in the simplified fraction \( \frac{21}{22} \). This process does not change the value of the fraction, only its appearance.

Numerator and Denominator

A fraction consists of two parts: the numerator and the denominator. The numerator is the top number in a fraction and represents how many parts we have. The denominator is the bottom number and shows the total number of equal parts the whole is divided into.
Understanding these terms helps in performing operations like multiplication and division of fractions. In the expression \( \frac{7}{4} \), 7 is the numerator, and 4 is the denominator. Similarly, in \( \frac{6}{11} \), 6 is the numerator, and 11 is the denominator.

• When multiplying fractions, multiply the numerators together to get the new numerator.
• Multiply the denominators to form the new denominator.

So, when multiplying \( \frac{7}{4} \) by \( \frac{6}{11} \), the result is \( \frac{7 \times 6}{4 \times 11} = \frac{42}{44} \). This process highlights how each part of the fraction works together in calculations.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest whole number that divides each of them without leaving a remainder. It's a crucial concept when simplifying fractions because it allows us to minimize both the numerator and the denominator.
Here's how to find the GCD:

• List all the factors of each number.
• Identify the largest factor that appears in both lists.

For example, to find the GCD of 42 and 44, list their factors:

• Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
• Factors of 44: 1, 2, 4, 11, 22, 44

The largest common factor is 2, so the GCD of 42 and 44 is 2. This knowledge helps to simplify the fraction \( \frac{42}{44} \) down to \( \frac{21}{22} \) by dividing both numbers by the GCD.