Problem 71 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{8}{9} \cdot 18$$

Step-by-Step Solution

Answer

The product is 16.

Step 1: Set up the Multiplication

Start by writing the multiplication expression as a single fraction: \( \frac{8}{9} \cdot 18 \) can be rewritten as \( \frac{8 \cdot 18}{9} \). This allows us to multiply the numerator directly with the integer.

Step 2: Multiply the Numerator and Denominator

Multiply 8 by 18 to get the new numerator and keep the denominator as is: \( 8 \times 18 = 144 \), so the expression becomes \( \frac{144}{9} \).

Step 3: Simplify the Fraction

To simplify \( \frac{144}{9} \), we divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 144 and 9 is 9.

Step 4: Perform the Division

Divide the numerator and the denominator by 9: \( \frac{144 \div 9}{9 \div 9} = \frac{16}{1} \). This results in the simplified fraction, which is 16.

Key Concepts

Simplifying Fractions

Simplifying fractions is about making them as easy to understand as possible. It involves reducing a complex fraction to its simplest form. This means there are no common factors left in both the numerator and the denominator apart from 1.

Here's how you can simplify a fraction:

• Identify any common factors in the numerator and the denominator.
• Divide them by their common factor to get the simplest form.

For example, if you have the fraction \( \frac{144}{9} \), notice that both numbers can be divided evenly by 9. This is because 9 is a factor of both, making it the "key" to simplifying the fraction.

After dividing by this common factor, what you're left with is \( \frac{16}{1} \), which is the simplest form of the original fraction.

Greatest Common Divisor

The greatest common divisor (GCD) plays a critical role when you simplify fractions. It is the largest number that can divide both the numerator and the denominator without leaving a remainder.

To find the GCD:

• List out the factors of the numerator and the denominator.
• Identify the largest number that appears in both lists.

In our example \( \frac{144}{9} \), the factors of 144 include 1, 2, 3, 4, 6, 8, 9, ..., 144. Factors for 9 are 1, 3, 9. The largest common factor here is 9, making it the GCD.

Using the GCD makes simplifying fractions much more efficient, as it ensures no further division is needed once you've used the GCD.

Numerator and Denominator

Understanding the positions and roles of the numerator and denominator in a fraction is essential. The numerator is the top number, showing how many parts of the whole you have. The denominator is the bottom number, showing the total number of equal parts the whole is divided into.

When multiplying fractions or numbers with fractions, as in our example \( \frac{8}{9} \cdot 18 \), you multiply the numerator of the first fraction by the number (or numerator if dealing with multiple fractions) and keep the original denominator. This forms a new fraction which may need simplification.

Understanding the roles of numerator and denominator helps greatly in fraction multiplication, as it allows you to structure the problem correctly from the start.