Problem 24 - Prealgebra

Question

Multiply each of the following. Be sure all answers are written in lowest terms. $$\frac{135}{16} \cdot \frac{2}{45}$$

Step-by-Step Solution

Answer

The product is \( \frac{3}{8} \).

Step 1: Write the Multiplication Formula

The expression we need to multiply is \( \frac{135}{16} \cdot \frac{2}{45} \). Multiplying fractions involves multiplying the numerators together and the denominators together: \( \frac{135 \times 2}{16 \times 45} \).

Step 2: Perform Numerator and Denominator Multiplication

Multiply the numerators: \( 135 \times 2 = 270 \). Multiply the denominators: \( 16 \times 45 = 720 \). This gives us \( \frac{270}{720} \).

Step 3: Simplify the Fraction

Find the greatest common divisor (GCD) of 270 and 720 to simplify the fraction. The GCD is 90, so divide both the numerator and the denominator by 90.\[ \frac{270 \div 90}{720 \div 90} = \frac{3}{8} \].

Step 4: Confirm Simplification to Lowest Terms

Check if \( \frac{3}{8} \) is in its lowest terms. Since 3 and 8 have no common factors other than 1, \( \frac{3}{8} \) is already in its lowest terms.

Key Concepts

Simplifying Fractions

When you're working with fractions, simplifying them means finding an equivalent fraction where the numerator and the denominator are as small as possible. You achieve this by dividing both the top and bottom numbers by their greatest common divisor (GCD). For instance, to simplify \( \frac{270}{720} \), we first find the GCD, which is 90 in this case. By dividing both the numerator and the denominator by this number, you get \( \frac{3}{8} \). This final fraction cannot be further reduced, meaning it’s in its simplest form. Simplifying fractions makes them easier to understand and use.

Greatest Common Divisor

Finding the greatest common divisor (GCD) is crucial for breaking down and understanding fractions. The GCD is the largest number that can evenly divide two numbers without leaving a remainder.
To find the GCD, you can:

• List out all the factors of each number and choose the largest number that appears in both lists.
• Use the Euclidean algorithm, which involves repeated division.

For example, when simplifying \( \frac{270}{720} \), the GCD is 90. This means both numbers can be divided by 90 to reduce the fraction.

Step-by-Step Solution

Approaching fraction multiplication in a structured, step-by-step way ensures you won’t miss any important details. Begin by identifying the fractions you wish to multiply. Here, it's \( \frac{135}{16} \) and \( \frac{2}{45} \).

• **Step 1:** Write down the basic multiplication formula for fractions, where you multiply the numerators together, and then multiply the denominators together.
• **Step 2:** Carry out these multiplications. For the numerators, multiply 135 by 2 to get 270. For the denominators, multiply 16 by 45 to get 720. You now have \( \frac{270}{720} \).
• **Step 3:** To simplify, find the GCD of the numerator and denominator. In this instance, it is 90.
• **Step 4:** Divide both 270 and 720 by 90 to simplify the fraction to \( \frac{3}{8} \).

Following these steps ensures you simplify correctly and avoid mistakes, leading to the simplest, most understandable answer.