Problem 11 - Fundamentals of Mathematics

Question

Find the value of \(\frac{-5(2-6)-4(-8-1)}{2(3-10)-9(-2)}\).

Step-by-Step Solution

Answer

The value of the expression is 14.

Step 1: Evaluate the Numerator Inside Parentheses

First, solve inside each set of parentheses in the numerator.- For the first set of parentheses: \(2 - 6 = -4\).- For the second set of parentheses: \(-8 - 1 = -9\).

Step 2: Calculate the Numerator

Now that we have solved the inner operations, substitute back into the equation:- Multiply in the first term: \(-5 \times (-4) = 20\).- Multiply in the second term: \(-4 \times (-9) = 36\).- Sum the results: \(20 + 36 = 56\).Thus, the numerator is 56.

Step 3: Evaluate the Denominator Inside Parentheses

First, solve inside each set of parentheses in the denominator.- For the first set of parentheses: \(3 - 10 = -7\).- For the second set of parentheses: \(-2 = -2\) (There's no operation needed here).

Step 4: Calculate the Denominator

Now that we have solved the inner operations, substitute back into the equation:- Multiply in the first term: \(2 \times (-7) = -14\).- Multiply in the second term: \(-9 \times (-2) = 18\).- Sum the results: \(-14 + 18 = 4\).Thus, the denominator is 4.

Step 5: Compute the Final Value

Now divide the calculated numerator by the denominator:\[ \frac{56}{4} = 14 \]The value of the entire expression is 14.

Key Concepts

Arithmetic Operations

Arithmetic operations are the basic building blocks used in mathematics, allowing us to perform calculations necessary for a vast array of problems. These operations include addition, subtraction, multiplication, and division. Each operation serves a different purpose. Addition combines values, while subtraction finds the difference between them. Multiplication helps us find the total when one number is repeated several times, and division breaks down a number into smaller, equal parts.

Understanding these operations is crucial, as they form the foundation for more complex mathematical concepts. When solving problems like the exercise we have on hand, these operations help us simplify expressions step by step, leading to a final solution.

Numerical Evaluation

Numerical evaluation involves carrying out all arithmetic operations in a mathematical expression to arrive at a single numerical value. It's an orderly process of calculation, where each part of an expression is simplified step by step.

For instance, in the given exercise, we first evaluate the expressions inside parentheses and then address any multiplication or addition accordingly. By following this sequence, we refine the expression down to its simplest form, ultimately arriving at one final value that represents the entire expression.

This process not only aids in finding concrete answers but also hones the skill of following systematic methods to solve problems efficiently.

Problem Solving

Problem solving in mathematics is essentially about understanding and finding solutions to various numerical challenges or equations. It involves diagnosing the problem, choosing appropriate arithmetic operations and strategies, and executing them step by step.

In our exercise, we identified the need to simplify both the numerator and the denominator separately as part of the solution process. This approach allows the problem to be broken down into manageable parts, making it easier to solve. With each part tackled logically, the solution emerges in a clear and organized manner.

Effective problem solving is a key skill that enhances logical thinking and is applicable beyond mathematics, in everyday decision-making and critical thinking scenarios.

Parentheses Operations

Operations within parentheses are a critical aspect of solving mathematical expressions. Parentheses help prioritize which part of the expression should be calculated first, following the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In our specific exercise, the operations inside the parentheses were tackled first. This step is vital as it lays the groundwork for the subsequent arithmetic operations. Ignoring parentheses can lead to incorrect results since it disrupts the intended sequence and priority of operations.

Understanding how to deal with parentheses effectively ensures that we process each part of a mathematical expression accurately, leading to the correct final answer.