Problem 10 - Fundamentals of Mathematics

Question

Determine the value of each of the following. \(86+[14 \div(10-8)]\)

Step-by-Step Solution

Answer

The value of the expression is 93.

Step 1: Identify the Innermost Operation

In the expression \(86 + [14 \div (10 - 8)]\), we first identify the innermost operation within the parentheses, which is \(10 - 8\).

Step 2: Perform the Operation Inside Parentheses

Calculate the expression inside the parentheses: \(10 - 8 = 2\). The expression now becomes \(86 + [14 \div 2]\).

Step 3: Division Within the Brackets

Perform the division inside the brackets: \(14 \div 2 = 7\). The expression now simplifies to \(86 + 7\).

Step 4: Final Addition

Complete the final operation by adding \(86 + 7 = 93\).

Key Concepts

Understanding Parentheses

Parentheses play a crucial role in determining the order in which calculations are performed in a mathematical expression. They indicate what operation should be completed first before moving on to others. In the expression \(86 + [14 \div (10 - 8)]\), the operations inside the parentheses must be calculated first. This is because the operations within parentheses take precedence over others, known as the "order of operations" rule.

Key steps to remember with parentheses include:

• Identify and solve the operations inside the parentheses first before moving on to others outside of them.
• Always follow the sequence of operations inside nested parentheses, if applicable, from the innermost to the outermost.

In this example, the calculation \(10 - 8\) should be handled before any division or addition, resulting in a value of 2 for the parentheses content.

Mastering Division

Division is another important mathematical operation that needs to be understood clearly. In any expression, division should be performed after all calculations inside parentheses are complete, but before addition or subtraction. Referring again to our example, after simplifying the content inside the parentheses, we have the expression \(86 + [14 \div 2]\).

Here's how you approach division:

• Once any operation within parentheses has been simplified, move to the division, even if it appears within bracketed expressions, such as \([14 \div 2]\).
• Calculate the quotient, which will further simplify the expression. Here, dividing 14 by 2 gives us 7.

In this particular problem, understanding that division follows any parenthetical simplification helps move the problem-solving process forward smoothly.

Grasping Addition

Addition comes after the calculations of parentheses and division. This is usually one of the final steps in evaluating a mathematical expression using the order of operations. Returning to the expression \(86 + 7\), all previous calculations have been completed, and the operation that remains is simply adding the results together to reach the final answer.

Tips for executing addition in this context:

• Add or combine the remaining simplified numbers from previous operations.
• Ensure that all prior steps—including operations within parentheses and any division—have been accurately completed to maintain the correctness of the addition.
• The sum calculated from an expression like \(86 + 7\) is straightforward once all previous operations are correctly executed, resulting in a final value, in this case, 93.

By understanding and following the proper sequence of operations, addition, as the final operation, becomes a simple matter.