Problem 39 - Elementary Algebra

Question

Simplify. $$ 316 \div(512-12+23) \cdot 4 $$

Step-by-Step Solution

Answer

The expression simplifies approximately to 2.4168.

Step 1: Simplify the Parentheses

First, solve the expression inside the parentheses: \(512 - 12 + 23\).Calculate \(512 - 12\):\[512 - 12 = 500\]Then add 23:\[500 + 23 = 523\]So, \(512 - 12 + 23 = 523\).

Step 2: Perform Division

Now, divide 316 by the result from Step 1, which is 523.Calculate \(316 \div 523\):\[316 \div 523 \approx 0.604198\]If approximate values are acceptable or further accuracy is not mandated, use this figure. Otherwise, further decimals or fractions may be used.

Step 3: Multiply by 4

Next, multiply the outcome from Step 2 by 4.Multiply the division result by 4:\[0.604198 \times 4 \approx 2.416792\]So the final expression simplifies approximately to 2.4168 when rounded to four decimal places.

Key Concepts

Order of Operations

Understanding the Order of Operations is essential to correctly solving mathematical expressions like the one given in this exercise. You may have heard of the acronym PEMDAS, which helps you remember the correct order:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

It's important to solve these operations in the specified order to avoid mistakes. In our example, the first step is solving the operations inside the parentheses. After simplifying the parentheses, you proceed with division, and only then move on to multiplication. By following this order, you ensure that your calculations are correct and accurate.
Skipping or mixing the sequence can lead to drastically different (and incorrect) results. Always stick to the order of operations to maintain the integrity of your solution.

Arithmetic Operations

Arithmetic Operations involve basic calculations we do with numbers, namely, addition, subtraction, multiplication, and division. These operations form the foundation of almost all mathematical tasks.
The problem in question contains multiple arithmetic operations. Let's break them down:

• Addition: We add numbers together to get their sum. For example, in the expression inside the parentheses, after subtracting 12 from 512, you add 23 to 500 to get 523.
  
• Subtraction: This operation involves finding the difference between numbers. As seen, 512 minus 12 equals 500.
• Division: Here, you divide one number by another, as done when dividing 316 by 523.
• Multiplication: It's a process of scaling one number by another, like multiplying the result of the division by 4.

Each operation affects the outcome and must be executed carefully to ensure the final result is correct. By practicing these operations, you enhance your skill and speed in solving mathematical expressions.

Mathematical Expressions

Mathematical Expressions are a combination of numbers, variables, and arithmetic operations that represent a particular value. They can be as simple as a basic addition problem or as complex as the given problem.
A clear understanding of expressions allows for accurate simplification and solution finding. Take note of how each part—such as parentheses and operators (like division and multiplication)—fits into the whole.
Breaking down the original expression, we start with using parentheses to indicate a prioritized computation:

• The given expression, initially, creates a group instruction for what to calculate first.
• Next, the proper use of operators specifies the sequence in which calculations must occur outside of parentheses (following PEMDAS).

Being proficient with expressions lets you approach complex problems systematically, ensuring consistency and reducing errors in your calculations. It's this structured approach that enhances your overall mathematical competence.