Problem 43 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{3}{8} \cdot \frac{7}{11}$$

Step-by-Step Solution

Answer

The simplified form of \( \frac{3}{8} \cdot \frac{7}{11} \) is \( \frac{21}{88} \).

Step 1: Multiplication of Numerators and Denominators

Multiply the numerators together to get the new numerator. In this case, multiply 3 and 7. Also, multiply the denominators together to get the new denominator. In this case, multiply 8 and 11.

Step 2: Perform the Calculation

The multiplication in step 1 results in \( \frac{21}{88} \)

Step 3: Simplify the Fraction

The fraction \( \frac{21}{88} \) is already in its simplest form because the greatest common divisor (gcd) of 21 and 88 is 1. If the gcd had been greater than 1, we would divide both the numerator and the denominator by the gcd until it is 1 to simplify the fraction.

Key Concepts

Numerators and Denominators

When dealing with fractions, it's important to understand what a numerator and a denominator are. A fraction consists of two parts: the numerator, which is the top number, and the denominator, which is the bottom number. For example, in the fraction \( \frac{3}{8} \), 3 is the numerator and 8 is the denominator.

The numerator tells you how many parts you have, while the denominator tells you how many equal parts make up a whole. When multiplying fractions, you multiply the numerators together and the denominators together. Following this logic, when you multiply \( \frac{3}{8} \) by \( \frac{7}{11} \), you calculate the new numerator by multiplying 3 and 7, and the new denominator by multiplying 8 and 11.

Therefore, this results in:

• New numerator: 3 × 7 = 21
• New denominator: 8 × 11 = 88

Thus, the fraction becomes \( \frac{21}{88} \).

Simplifying Fractions

Once you've multiplied two fractions together, you need to check if the resulting fraction can be simplified. Simplifying a fraction means making it as compact as possible by ensuring the numerator and the denominator have no common factors besides 1.

To determine if a fraction can be simplified, find the greatest common divisor (GCD) of the numerator and the denominator. If the GCD is greater than 1, divide both the numerator and the denominator by it. For \( \frac{21}{88} \), the GCD is 1, which means that 21 and 88 have no other common factors apart from 1. This indicates that the fraction is already in its simplest form, or rather, it can’t be simplified further.

Having a fraction in its simplest form makes it easier to understand and work with in subsequent math problems, ensuring clarity and accuracy.

Lowest Terms Reduction

Reducing a fraction to its lowest terms is another way of saying 'simplifying the fraction'. When you express a fraction at its lowest terms, you've minimized both its numerator and its denominator to their smallest possible values while still having the same value as the original fraction.

Let’s take \( \frac{21}{88} \) again. Since the GCD of 21 and 88 is 1, there are no common factors to remove, meaning this fraction is already reduced to its lowest terms. If the GCD had been something else, you would remove common factors by dividing both the numerator and the denominator by this number.

Here are simple steps to check for lowest terms reduction:

• Find the GCD of the numerator and the denominator.
• Divide both by the GCD.
• If the result is a fraction whose numerator and denominator are coprime, you are at its lowest terms.

By reducing fractions, you make them cleaner and often easier to interpret or compare with other fractions.