Problem 56 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. $$30+(-40+20)+50$$

Step-by-Step Solution

Answer

The simplified expression is 60.

Step 1: Apply Parentheses

Look at the expression inside the parentheses first: \(-40 + 20\). Evaluate this part first before considering the rest of the expression. Calculate: \(-40 + 20 = -20\). Now, rewrite the original expression using this value: \(30 + (-20) + 50\).

Step 2: Simplify Additions

Evaluate the expression from left to right, resolving the addition: First, calculate \(30 + (-20)\). This is equivalent to subtracting 20 from 30: \(30 - 20 = 10\). Now, rewrite the expression: \(10 + 50\).

Step 3: Final Addition

Finally, perform the last addition in the expression: \(10 + 50 = 60\). The simplified result of the original expression is \(60\).

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making a complex mathematical phrase as simple as possible, without changing its value. This often involves performing operations in a specific order and combining like terms.
When simplifying, always look for terms that can be combined or operations that can be performed immediately. An important part of this process is identifying operations within expressions that can be resolved to simplify the expression further.

• You break down the expression step by step.
• Solve one operation at a time.
• Ensure you follow the correct order of operations to get the accurate result.

Expressing any problem in a simpler form makes it easier to understand and solve. Simplifying helps in reducing computational complexity and avoids errors in further calculations.

Parentheses

Parentheses in mathematics are used to group parts of an expression that should be evaluated first, according to the order of operations. This grouping helps to clarify which operations should be carried out before others, preventing confusion.
The pivotal role of parentheses is demonstrated in the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Parentheses top this list, giving them the highest priority.

• Evaluate operations within parentheses first.
• Once simplified, eliminate the parentheses and integrate the simplified term into the main expression.
• If there are nested parentheses, evaluate the innermost pair first and work outwards.

This way, parentheses ensure that the expression is calculated in the intended order, maintaining mathematical accuracy.

Addition Rules

Addition rules in math help us understand how to efficiently add numbers, including both positive and negative numbers. When dealing with addition, you should consider the signs of the numbers involved.
For example, when adding a positive and a negative number, effectively subtract the smaller number from the larger and take the sign of the number with the larger absolute value. In essence, you calculate the difference and not the straightforward sum.

• When adding two positive numbers, simply combine them.
• When adding two negative numbers, combine them and retain the negative sign.
• When combining positive and negative numbers, subtract the smaller absolute value from the larger and take the sign of the number with the larger absolute value.

These rules ensure calculations are accurately executed and prevent errors when evaluating expressions that involve various numbers.