Problem 45 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \(26-2 \cdot\left\\{\frac{6+20}{13}\right\\}\)

Step-by-Step Solution

Answer

The value is 22.

Step 1: Simplify the Expression Inside the Parentheses

Start by simplifying the expression inside the curly brackets \(\left\{ \frac{6+20}{13} \right\}\). First, add the numbers in the numerator: \(6 + 20 = 26\).

Step 2: Divide the Numerator by the Denominator

Take the result from step 1, which is \(26\), and divide it by the denominator, which is \(13\): \(\frac{26}{13} = 2\).

Step 3: Simplify Using the Result from the Braces

Now, substitute the result of \(2\) from the braces back into the original expression: \(26 - 2 \cdot 2\).

Step 4: Perform the Multiplication

Calculate \(2 \cdot 2\), which gives \(4\).

Step 5: Perform the Subtraction

Finally, subtract the result from step 4 from \(26\): \(26 - 4 = 22\).

Step 6: Verify with a Calculator

Use a calculator to verify the steps: calculating \(26 - 2 \cdot \frac{26}{13}\) should confirm that the result is \(22\).

Key Concepts

Simplifying Expressions

When simplifying expressions, the goal is to perform operations in the correct sequence to reduce them to their simplest form. This can make complex problems much easier to solve and understand. During simplification, you should:

• Combine like terms whenever possible.
• Follow the established mathematical rules for arithmetic operations: addition, subtraction, multiplication, and division.
• Approach the expression step-by-step, focusing on one operation at a time to prevent mistakes.

In the original exercise, we began by simplifying the innermost operation within the curly brackets. By calculating \(6 + 20\), we reduced the expression inside the brackets to 26. Only once this was simplified did we move on to the next step, dividing this result by 13.

Arithmetic Operations

Arithmetic operations are the basic building blocks for mathematics and include addition, subtraction, multiplication, and division. Each operation has a specific role and is interconnected with others to solve more complex mathematical equations. Here’s a quick overview:

• Addition combines numbers into a larger number.
• Subtraction finds the difference between numbers.
• Multiplication is repeated addition of a number.
• Division splits a number into equal parts.

In our problem, after simplifying the expression inside the brackets, we performed division: \(\frac{26}{13} = 2\). This was followed by a multiplication \(2 \cdot 2\), leading to a subtraction as the final step: 26 - 4. The operation order is crucial; performing them in the wrong order can lead to incorrect results.

Use of Parentheses

Parentheses, braces, and brackets are essential for indicating which operations should be performed first in an expression. This is fundamental in following the order of operations, often recalled by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).By evaluating the expression within the parentheses or other grouping symbols first, it guides the sequence of calculations. In the original exercise, solving the equation within the curly brackets \(\frac{6 + 20}{13}\) first dramatically changed the outcome of the entire problem. Without these, the equation could be misinterpreted as involving operations in incorrect sequences, potentially leading to an incorrect result.By understanding and correctly employing parentheses, expressions become easier to solve and less error-prone. It ensures clarity in more complex expressions where the order of operations significantly impacts the result.