Problem 56 - Prealgebra

Question

Find the value of each expression. $$70-(16 \div 2+21)$$

Step-by-Step Solution

Answer

41

Step 1: Simplify the Division

Start by solving the division inside the parentheses. Compute \(16 \div 2\). This results in 8. So, the expression inside the parentheses becomes \(8 + 21\).

Step 2: Simplify the Addition inside the Parentheses

Next, solve the addition inside the parentheses \(8 + 21\). This results in 29. Now the entire expression is simplified to \(70 - 29\).

Step 3: Subtract the Results

Finally, subtract the simplified result from 29 from 70. So, \(70 - 29 = 41\).

Key Concepts

Arithmetic Expressions and Their Importance

An arithmetic expression is a combination of numbers, operators, and sometimes parentheses, that represent a particular value or equation. These expressions often include operations such as addition, subtraction, multiplication, and division. Understanding how to evaluate arithmetic expressions correctly is essential, as it forms the foundation of many mathematical concepts and real-life problem-solving situations.
Evaluating an arithmetic expression typically requires the application of the order of operations. This rule dictates the sequence in which operations should be performed. For example:

• First, solve expressions inside parentheses.
• Second, carry out any exponents (which are often not present in simpler problems).
• Third, perform multiplication and division from left to right.
• Lastly, complete addition and subtraction from left to right.

In the given exercise, the focus is on simplifying an expression by following these steps. This ensures accurate results every time.

Understanding Division and Subtraction in Arithmetic

Division and subtraction are two fundamental operations in mathematics. They are used to break down numbers into parts or compare differences between values. It's crucial to understand how these operations fit into the order of operations so that you can evaluate expressions accurately.
In the expression given: \[70-(16 \div 2+21)\]
The division occurs before subtraction due to its place in the order of operations. We first compute \(16 \div 2\), yielding 8. This step breaks the number 16 into two equal parts of 8.
Next, the subtraction \(70 - 29\) comes after we perform further calculations inside the parentheses, ensuring that the expression is fully simplified before reaching this stage. Subtraction represents the removal or comparison in quantities, and it must be executed correctly as per the sequence.

Mathematical Problem-Solving Techniques

Mathematical problem-solving involves the strategic use of different rules and techniques to find solutions. In this context, understanding and correctly applying the order of operations is a critical skill. It allows for a structured approach to tackling arithmetic expressions.
To solve such problems efficiently:

• Start by identifying the operations and their order based on rules.
• Simplify expressions inside parentheses as a priority.
• Carry out multiplication and division next, followed by addition and subtraction.

In our exercise, following these strategies led to the accurate simplification and solution of the problem. It's clear that the prescribed order helps maintain coherence and minimize errors in calculations. Practice these strategies frequently to develop a more intuitive sense for tackling a wide range of mathematical expressions and equations.