Problem 74 - Fundamentals of Mathematics

Question

Perform each multiplication and division. $$ \frac{8}{9} \cdot \frac{3}{24} $$

Step-by-Step Solution

Answer

The result of the multiplication is \( \frac{1}{9} \).

Step 1: Understand the Problem

The problem asks to multiply the two fractions \( \frac{8}{9} \) and \( \frac{3}{24} \). We will handle this by multiplying the numerators and the denominators, then simplify the resulting fraction.

Step 2: Multiply the Numerators and Denominators

To multiply the fractions \( \frac{8}{9} \cdot \frac{3}{24} \), we first multiply the numerators: \( 8 \cdot 3 = 24 \) and then multiply the denominators: \( 9 \cdot 24 = 216 \). Therefore, the multiplied fraction is \( \frac{24}{216} \).

Step 3: Simplify the Fraction

To simplify \( \frac{24}{216} \), we need to find the greatest common divisor (GCD) of 24 and 216. The GCD is 24. Divide both the numerator and the denominator by 24: \( \frac{24 \div 24}{216 \div 24} = \frac{1}{9} \).

Step 4: Conclusion

After performing the multiplication and simplification, the final answer is \( \frac{1}{9} \).

Key Concepts

Numerators and Denominators

Fractions are made up of two key parts: the numerator, which is the top number, and the denominator, which is the bottom number. When you see a fraction like \( \frac{8}{9} \), "8" is the numerator, and "9" is the denominator.
The numerator represents how many parts you have, while the denominator shows how many parts make a whole. In fraction multiplication, you multiply the numerators together and the denominators together.
For example, multiplying \( \frac{8}{9} \times \frac{3}{24} \) means you multiply 8 and 3 for the numerator, and 9 and 24 for the denominator. This gives the product \( \frac{24}{216} \).

• Numerators: Multiply the top numbers
• Denominators: Multiply the bottom numbers

Simplifying Fractions

Once you have multiplied two fractions, the result might not be in its simplest form. Simplifying a fraction involves reducing it to the smallest possible equivalent fraction. For \( \frac{24}{216} \), you can simplify it to \( \frac{1}{9} \).
To do this, find a number that divides evenly into both the numerator and the denominator. This is where understanding the process of simplifying becomes important.
Here are the steps:

• Identify: Determine a common factor for the numerator and denominator.
• Divide: Use this common factor to divide both parts of the fraction.
• Check: Ensure the fraction cannot be reduced further.

This process ensures that the fraction is expressed as simply as possible, making it easier to understand and work with.

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept used in simplifying fractions. It's the largest number that divides both the numerator and the denominator without leaving a remainder.
In the case of \( \frac{24}{216} \), you seek the GCD of 24 and 216. That happens to be 24 in this instance.

• Dividing by the GCD: This reduces the fraction effectively and maintains its value.
• Efficient Simplification: Using the GCD directly simplifies the fraction in one step.

To simplify, divide both 24 and 216 by their GCD, 24, to get \( \frac{1}{9} \).

Being comfortable with finding the GCD will make fraction multiplication and simplification much more straightforward.