Problem 80 - Prealgebra

Question

Write your answers as whole numbers, proper fractions, or mixed numbers. Find each product. (Multiply) $$\frac{7}{9} \cdot \frac{3}{14}$$

Step-by-Step Solution

Answer

The product is \( \frac{1}{6} \).

Step 1: Multiply the Numerators

First, multiply the numerators of the fractions. For \( \frac{7}{9} \) and \( \frac{3}{14} \), multiply 7 and 3: \( 7 \times 3 = 21 \).

Step 2: Multiply the Denominators

Next, multiply the denominators of the fractions. For \( \frac{7}{9} \) and \( \frac{3}{14} \), multiply 9 and 14: \( 9 \times 14 = 126 \).

Step 3: Form the New Fraction

Combine the results from the previous steps to form a new fraction: \( \frac{21}{126} \).

Step 4: Simplify the Fraction

To simplify \( \frac{21}{126} \), find the greatest common divisor (GCD) of 21 and 126, which is 21. Divide both the numerator and the denominator by their GCD: \( \frac{21 \div 21}{126 \div 21} = \frac{1}{6} \).

Key Concepts

Proper Fractions

A proper fraction is a type of fraction where the numerator is smaller than the denominator. This means the value of the fraction is less than one. Proper fractions are a common way to express small parts of a whole. For example, in the fraction \( \frac{3}{14} \), 3 is the numerator, and 14 is the denominator. As 3 is less than 14, it's considered a proper fraction.

• The numerator represents how many parts we have.
• The denominator represents the total number of equal parts in a whole.

These fractions are not to be confused with improper fractions, where the numerator is equal to or greater than the denominator. Proper fractions are particularly useful in applications where precise small portions, such as recipes or measurements, are needed.

Simplifying Fractions

Simplifying fractions is all about making the fraction as simple as possible by reducing it to its smallest form. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is the largest number that can evenly divide both.
For the fraction \( \frac{21}{126} \), let's simplify:

• Find the GCD of 21 and 126, which is 21.
• Divide both the numerator and the denominator by the GCD.
• So, \( \frac{21}{126} = \frac{21 \div 21}{126 \div 21} = \frac{1}{6} \).

By simplifying fractions, you preserve the value but make them easier to read and work with. Simplified fractions are needed in many areas of math and science to ensure calculations are manageable and results are clear.

Greatest Common Divisor

The greatest common divisor (GCD) is a fundamental concept in fraction simplification. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD is crucial for reducing fractions to their simplest form.
Let's find the GCD of 21 and 126 step-by-step:

• List the factors of 21: \(1, 3, 7, 21\).
• List the factors of 126: \(1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126\).
• Find the common factors: \(1, 3, 7, 21\).
• The greatest common factor from these is 21.

Using the GCD to simplify fractions is a key part of making fractions easier to handle in math problems. It's a handy tool for ensuring precision and simplicity in calculations.