Problem 32 - Prealgebra

Question

Multiply each of the following. Be sure all answers are written in lowest terms. $$\frac{32}{27} \cdot \frac{72}{49} \cdot \frac{1}{40}$$

Step-by-Step Solution

Answer

The product is \( \frac{96}{2205} \).

Step 1: Identify the Fractions

The fractions that need to be multiplied together are \( \frac{32}{27} \), \( \frac{72}{49} \), and \( \frac{1}{40} \).

Step 2: Multiply Numerators

Multiply all the numerators together: \( 32 \times 72 \times 1 = 2304 \).

Step 3: Multiply Denominators

Multiply all the denominators together: \( 27 \times 49 \times 40 = 52920 \).

Step 4: Form the Initial Fraction

Combine the multiplied numerators and denominators to form a new fraction: \( \frac{2304}{52920} \).

Step 5: Simplify the Fraction

Find the greatest common divisor (GCD) of 2304 and 52920, which is 24. Divide both the numerator and the denominator by their GCD: \( \frac{2304 \div 24}{52920 \div 24} = \frac{96}{2205} \).

Step 6: Verify if the Fraction is in Lowest Terms

Check to see if \( \frac{96}{2205} \) can be simplified further. Since 96 and 2205 have no common factors other than 1, the fraction is in its lowest terms.

Key Concepts

Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form where the numerator (top number) and the denominator (bottom number) have no common factors other than 1.
When you simplify a fraction, you're looking for the largest number, known as the greatest common divisor (GCD), that divides both the numerator and the denominator evenly.
Once you find that number, divide both the numerator and the denominator by it.For example, in the fraction \(\frac{2304}{52920}\), you find the GCD, which is 24.
Dividing both the top and bottom by 24 simplifies the fraction to \(\frac{96}{2205}\).
This step is essential after performing operations such as multiplication of fractions to ensure you're working with the simplest possible form.

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept in simplifying fractions. It is the largest number that can evenly divide two numbers without leaving a remainder.
Finding the GCD helps reduce a fraction to its simplest form.
There are different methods to find the GCD, such as the Euclidean algorithm, but simply listing factors can work for smaller numbers too.Let's say you have the fractions after multiplication, \(2304\) and \(52920\).
To find their GCD, you could list all factors of each number or apply the Euclidean algorithm, which involves a series of division steps.Once the GCD is identified, both the numerator and the denominator are divided by this number to achieve the simplest form of the fraction as in our exercise.

Numerator and Denominator Multiplication

When multiplying fractions, the numerators are multiplied together and the denominators are multiplied together.
The process is straightforward:

• Take all the numerators from each fraction and multiply them to get the overall numerator.
• Similarly, multiply all the denominators to get the overall denominator.

In the exercise, for the fractions \(\frac{32}{27}\), \(\frac{72}{49}\), and \(\frac{1}{40}\), you multiply the numerators: \(32 \times 72 \times 1 = 2304\).
Then, multiply the denominators: \(27 \times 49 \times 40 = 52920\).By following these steps, you create a new fraction representing the product of the initial fractions: \(\frac{2304}{52920}\).
After forming this initial product, the next step is to simplify as explained in the previous sections.