Solving and Simplifying Complex Numbers: -5 1/3 - 3

In today's lesson, we will explore how to simplify the expression -5 1/3 - 3. We will break down each step of the process and ensure that we understand how to convert mixed numbers into improper fractions, find a common denominator, and ultimately simplify the expression.

Understanding the Problem and the Difference Between Equations and Expressions

Firstly, it's crucial to recognize the difference between an equation and an expression. An expression is a combination of numbers, variables, and operators without an equals sign. An equation, on the other hand, has an equals sign and requires solving for a variable.

The given problem -5 1/3 - 3 is an expression. While it is not an equation, we can still simplify and work with it. Simplification involves breaking down the problem into its simplest form without necessarily solving for any unknowns.

To simplify the expression, we will follow these steps:

Simplifying the Mixed Numbers

Let's start by converting the mixed numbers into improper fractions.

Step 1: Convert -5 1/3 to an Improper Fraction

1. Multiply the whole number by the denominator of the fractional part: 5 * 3 15.

2. Add the numerator of the fractional part to the result: 15 1 16.

3. The improper fraction is -16/3 since the original number was negative.

Step 2: Convert -3 1/4 to an Improper Fraction

1. Multiply the whole number by the denominator of the fractional part: 3 * 4 12.

2. Add the numerator of the fractional part to the result: 12 1 13.

3. The improper fraction is -13/4 since the original number was negative.

Finding a Common Denominator

To add or subtract fractions, we need to work with a common denominator. Here are the steps to find a common denominator for -16/3 and -13/4.

Step 1: Identify the Least Common Multiple (LCM)

The denominators are 3 and 4. The LCM of 3 and 4 is 12. Therefore, we will convert both fractions to have the denominator 12.

Step 2: Convert the Fractions

For -16/3:

1. Multiply both the numerator and the denominator by 4: -16 * 4 -64, 3 * 4 12.

2. The resulting fraction is -64/12.

For -13/4:

1. Multiply both the numerator and the denominator by 3: -13 * 3 -39, 4 * 3 12.

2. The resulting fraction is -39/12.

Adding the Numerators and Reducing the Fraction

Since we are subtracting, we need to perform -64/12 - (-39/12) which is equivalent to -64/12 39/12.

1. Add the numerators: -64 39 -25.

2. The result is -25/12.

Converting Back to Mixed Numbers

To convert -25/12 back to a mixed number, we perform the following steps:

Step 1: Perform Division

1. Divide the absolute value of the numerator by the denominator: 25 ÷ 12 2 remainder 1.

2. The quotient (2) becomes the whole number, and the remainder (1) becomes the numerator of the fractional part.

Step 2: Determine the Sign

Since the original number was negative, we keep the negative sign.

Therefore, the final answer is -25/12 or -2 1/12.

Conclusion

By following these steps, we have simplified the complex expression -5 1/3 - 3 to -2 1/12. This process not only demonstrates the importance of understanding the basics of fractions but also reinforces the skills necessary for working with more complicated algebraic expressions.

For further learning, additional resources and practice problems related to simplifying mixed numbers, working with improper fractions, finding common denominators, and performing operations with fractions can be found online or in educational textbooks.