Problem 52 - Prealgebra

Question

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6–9] $$(-500 \div 50) \div 10$$

Step-by-Step Solution

Answer

The simplified expression is \(-1\).

Step 1: Simplify the Innermost Division

Start by simplifying the innermost division inside the parentheses: \(-500 \div 50\). Divide \(-500\) by \(50\) to get \(-10\).

Step 2: Evaluate the Result

With the result from step 1, replace \((-500 \div 50)\) with \(-10\) in the original expression. The expression becomes \(-10 \div 10\).

Step 3: Perform the Final Division

Divide \(-10\) by \(10\). Perform the division to get \(-1\).

Key Concepts

Division Rules

When faced with a division problem, especially one with multiple divisions, understanding the rules can make the process smoother. Division is the process of finding out how many times one number can fit into another. The basic rule of division is to divide the dividend (the number being divided) by the divisor (the number you are dividing by). A helpful way to simplify is to perform one division at a time, starting from the innermost part of the expression if there are brackets.

In the expression \[(-500 \div 50) \div 10\]we first need to solve the division inside the parentheses. Here, -500 is the dividend and 50 is the divisor. Dividing -500 by 50 gives a quotient of -10. This division must be done before any others due to the order of operations, which dictates tackling expressions in brackets first.

• Calculate the division in parentheses: \(-500 \div 50 = -10\)
• Move to the next division step with the result: \(-10 \div 10\)
• The final result is: \(-1\)

Remember to also apply these rules when working through more complex expressions. Divisions should be done one at a time, always checking intermediate results before proceeding.

Negative Numbers

Negative numbers can sometimes be confusing when included in division and other operations. A negative number is less than zero and it is usually indicated by a minus sign. When dealing with division involving negative numbers, understanding how the signs affect the quotient is crucial.

With division, there are different rules for positive and negative numbers:

• When dividing two negative numbers, the quotient is positive. For instance, \(-50 \div -5\) equals \(10\).
• When dividing a positive number by a negative number, the quotient is negative. Example: \(50 \div -5 = -10\).
• When dividing a negative number by a positive number, the quotient is also negative, like in \(-500 \div 50 = -10\).

In our expression \((-500 \div 50) \div 10\), both divisions involve negative numbers. Since -500 is divided by 50, the result is a negative quotient, -10. Continuing with -10 divided by 10, you maintain the rule of dividing a negative by a positive, yielding -1. Each step follows logical rules assisting in clearer calculations.

Simplifying Expressions

Simplifying expressions means making them as neat and uncomplicated as possible while keeping the same value. This process typically involves performing arithmetic operations, applying rules of operations, and breaking down expressions into simpler forms.

To simplify complex expressions like \((-500 \div 50) \div 10\), first, identify parts that can be calculated. Inside out, or from brackets outward, works best:

• Handle any operations inside parentheses or brackets first. This is dictated by the order of operations, often remembered with the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Sometimes you need to combine similar terms or perform operations such as multiplication or division to make the numbers more manageable.
• Divide step-by-step to avoid errors and ensure clarity through each part, like converting \(-500 \div 50\) to \(-10\), then moving to the final division \(-10 \div 10\).

By working step-by-step, you can achieve a completed and simplified result efficiently. Thinking carefully through each part ensures solutions that are reliable and simple to share with others.