Problem 33 - Algebra and Trigonometry

Question

Perform the indicated operations. \(\frac{2}{3}\left(6-\frac{3}{2}\right)\)

Step-by-Step Solution

Answer

The result is 3.

Step 1: Simplify Inside the Parentheses

Let's start by simplifying the expression inside the parentheses. We have \(6 - \frac{3}{2}\). Convert the number 6 to a fraction with a denominator of 2: \(\frac{12}{2} - \frac{3}{2}\). Subtract the fractions: \(\frac{12 - 3}{2} = \frac{9}{2}\).

Step 2: Multiply the Fraction

Now multiply the simplified expression by \(\frac{2}{3}\). The operation is \(\frac{2}{3} \times \frac{9}{2}\). Multiply the numerators and multiply the denominators: \(\frac{2 \cdot 9}{3 \cdot 2} = \frac{18}{6}\).

Step 3: Simplify the Resulting Fraction

Simplify the fraction \(\frac{18}{6}\) by dividing the numerator and the denominator by their greatest common divisor, which is 6: \(\frac{18 \div 6}{6 \div 6} = \frac{3}{1} = 3\).

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. This is done by finding a common factor of both the numerator and the denominator and dividing them by it. For example, in the exercise, we ended up with the fraction \( \frac{18}{6} \).

• The common factor for both 18 and 6 is 6.
• By dividing both 18 and 6 by this number, we simplify the fraction to \( \frac{3}{1} \), or simply 3.

Simplifying fractions is an important skill because it helps to make the numbers easier to work with. It also reveals the 'true' size or value of the fraction, making operations like addition or multiplication much clearer and more straightforward.

Fraction Multiplication

Multiplying fractions involves multiplying the numerators together and the denominators together. This may sound complicated, but it's actually one of the easier operations to perform with fractions. In our example, we had to multiply \( \frac{2}{3} \) by \( \frac{9}{2} \).

• First, multiply the numerators: \( 2 \times 9 = 18 \).
• Then, multiply the denominators: \( 3 \times 2 = 6 \).
• Place the result in fraction form: \( \frac{18}{6} \).

Always check if you can simplify afterwards, as reducing the result makes it easier to interpret and use in further calculations.

Arithmetic Operations

Arithmetic operations, such as addition, subtraction, multiplication, and division, can be performed on fractions the same way they are on whole numbers, but with a few extra steps. Fractions must be managed correctly for accurate results. Specifically, every operation has its own set of rules when it comes to fractions.

• Addition/Subtraction: Ensure a common denominator before performing these operations.
• Multiplication: Multiply the numerators and denominators directly without requiring common denominators.
• Division: Multiply by the reciprocal of the second fraction.

Understanding these differences is crucial for accurately performing operations. Following the correct rules results in precise and simplified answers.