Problem 74 - Algebra A Combined Function

Question

Perform each indicated operation. Write all results in lowest terms. $$ \frac{3}{7} \cdot \frac{12}{17} $$

Step-by-Step Solution

Answer

The result is \( \frac{36}{119} \).

Step 1: Multiply the Numerators

First, multiply the numerators of the fractions: \[ 3 \times 12 = 36 \]

Step 2: Multiply the Denominators

Next, multiply the denominators of the fractions: \[ 7 \times 17 = 119 \]

Step 3: Form the New Fraction

Combine the results of the numerators and denominators to form a new fraction: \[ \frac{36}{119} \]

Step 4: Simplify the Fraction

Check to see if \( \frac{36}{119} \) can be simplified. The greatest common divisor of 36 and 119 is 1, meaning it is already in its simplest form.

Key Concepts

Understanding Numerators

In the context of fractions, the numerator is the number on top. It tells you how many parts you have out of a whole. Imagine a fraction as a pizza divided into equal slices. The numerator counts how many slices you get. For example, if we look at the fraction \(\frac{3}{7}\), the number 3 is the numerator. It indicates you have 3 out of a total 7 parts. When multiplying fractions, like \(\frac{3}{7} \times \frac{12}{17}\), the rule is to multiply the numerators together. So, you multiply 3 (from the first fraction) and 12 (from the second fraction). The product, 36, becomes the numerator of the new fraction. This simple rule is always true for multiplying any fractions, making it easy to remember.

Demystifying Denominators

The denominator is the number on the bottom of a fraction. It tells you how many total equal parts the whole is divided into. Using the pizza example again, if your pizza is cut into 7 slices, and you have 3 slices, your pizza fraction is \(\frac{3}{7}\). Here, 7 is the denominator. When multiplying fractions, just like with the numerators, multiply the denominators of each fraction. Taking our example, \(7 \times 17 = 119\), where 119 becomes the denominator of the resulting fraction. This step ensures you know the total parts involved in the new fraction formed.

Finding the Greatest Common Divisor

When simplifying fractions, we look for the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that can evenly divide both the numerator and denominator without leaving a remainder. To simplify a fraction, divide both the numerator and denominator by their GCD. This reduces the fraction to its simplest form without changing its value. In our example with \(\frac{36}{119}\), we find the GCD is 1. Since it’s 1, it shows us that 36 and 119 have no common factors other than 1, and the fraction is already in its simplest form. Identifying the GCD is essential, especially when you can simplify and further reduce the fractions.

Understanding Simplest Form

Simplest form means a fraction is expressed with the smallest possible whole numbers, and the greatest common divisor of the numerator and the denominator is 1. When a fraction cannot be further reduced, it is in its simplest form. Simplifying fractions is like cleaning up; you aim to make it as neat and straightforward as possible. In calculations involving fractions, expressing results in simplest form is considered good practice. For the fraction \(\frac{36}{119}\), since the GCD of 36 and 119 is 1, it confirms that \(\frac{36}{119}\) is already tidy and cannot be simplified further. Understanding simplest form ensures clarity and accuracy in mathematical communication.