Problem 12 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. [Examples 1–3] $$\left(3-\frac{3}{4}\right)\left(3+\frac{1}{3}\right)$$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{15}{2} \).

Step 1: Simplify Inside the Parentheses

Start by simplifying the expression within each parenthesis. For \( 3 - \frac{3}{4} \), convert 3 into a fraction: \( \frac{12}{4} \). Then calculate \( \frac{12}{4} - \frac{3}{4} = \frac{9}{4} \). For \( 3 + \frac{1}{3} \), convert 3 into a fraction: \( \frac{9}{3} \). Then calculate \( \frac{9}{3} + \frac{1}{3} = \frac{10}{3} \). This simplifies the expression to \( \left(\frac{9}{4}\right) \left(\frac{10}{3}\right) \).

Step 2: Multiply Fractions

Multiply the simplified fractions from Step 1. Multiply the numerators: \( 9 \times 10 = 90 \). Multiply the denominators: \( 4 \times 3 = 12 \). This results in the fraction \( \frac{90}{12} \).

Step 3: Simplify the Result

Simplify the fraction \( \frac{90}{12} \) by finding the greatest common divisor (GCD) of 90 and 12. The GCD is 6. Divide both numerator and denominator by 6: \( \frac{90}{12} = \frac{15}{2} \). This simplified fraction is the final result.

Key Concepts

Fraction Simplification

Simplifying fractions is all about making them easier to read and understand, while ensuring they still represent the same value. It's like having the neatest version of a sentence possible. To simplify a fraction, you divide the numerator and the denominator by their greatest common divisor (GCD), which we'll explore in detail later.
For example, if you have a fraction like \( \frac{9}{12} \), you want to find the biggest number that can divide both 9 and 12 evenly. That number is their GCD. Once found, you simply divide both parts of the fraction by this number, and voila! The fraction gets reduced or "simplified" without changing its value. So, \( \frac{9}{12} \) simplifies to \( \frac{3}{4} \).
Simplification helps in making calculations easier and helps in comparing fractions more directly. Remember, a simplified fraction is just a tidier way of expressing numbers.

Multiplying Fractions

When it comes to multiplying fractions, the process is quite straightforward. You multiply across the top and across the bottom. This means you multiply the numerators together and the denominators together. For instance, when multiplying \( \frac{2}{3} \) by \( \frac{4}{5} \), you multiply 2 and 4 for the numerator (which gives you 8), and 3 and 5 for the denominator (resulting in 15), leaving you with \( \frac{8}{15} \).
The multiplication doesn't require any common denominators, which makes it simpler than adding or subtracting fractions. After multiplying, you should always check if the resulting fraction can be simplified.

• Multiply the numerators to get the new numerator.
• Multiply the denominators to get the new denominator.
• Check if the result can be simplified further.

It’s that easy! This method allows you to quickly combine fractions and is an essential skill in any mathematical toolkit.

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is the largest number that can evenly divide two or more numbers. It's a key tool in fraction simplification and finding the neatest result from your fractions.
To find the GCD of two numbers, you can use a few methods. The simplest one is listing down all the divisors of each number and comparing them. For example, for the numbers 18 and 24, the divisors are:

• 18: 1, 2, 3, 6, 9, 18
• 24: 1, 2, 3, 4, 6, 8, 12, 24

Here, the largest common divisor is 6. Therefore, 6 is the GCD.
Understanding and finding the GCD helps simplify fractions efficiently, making calculations easier and results clearer. It's an important aspect of working with fractions and ensures your answers are as concise as possible.