Problem 53 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ \frac{-3(4-2)+(-3)(-6)}{-4} $$

Step-by-Step Solution

Answer

The value is -3.

Step 1: Solve the Inner Parentheses

First, start with the expression inside the parentheses.\[ 4 - 2 = 2 \]

Step 2: Substitute and Simplify

Replace the expression inside the parentheses with its value and simplify.The expression becomes:\[ \frac{-3 \times 2 + (-3) \times (-6)}{-4} \]

Step 3: Perform Multiplications

Calculate the products:\[ -3 \times 2 = -6 \]\[ (-3) \times (-6) = 18 \]

Step 4: Simplify the Numerator

Substitute the products back into the expression and simplify the numerator.\[ -6 + 18 = 12 \]

Step 5: Finish the Division

Now divide the simplified numerator by \(-4\).\[ \frac{12}{-4} = -3 \]

Step 6: Verify with Calculator

Use a calculator to check the calculation:- Enter the original expression \( \frac{-3(4-2)+(-3)(-6)}{-4} \).- Confirm that the result is \(-3\).

Key Concepts

Arithmetic

Arithmetic serves as one of the fundamental building blocks of mathematics, encompassing basic operations like addition, subtraction, multiplication, and division. These operations rely on rules and procedures that govern how numbers interact with each other. Understanding arithmetic is crucial for solving a wide range of mathematical problems, from simple calculations to more complex algebraic expressions.

• Addition: Combining values to find a total.
• Subtraction: Finding the difference between values.
• Multiplication: Calculating the product of numbers.
• Division: Breaking numbers into equal parts.

One of the key elements of arithmetic is executing operations in the correct order, commonly referred to as the "Order of Operations." This dictates the sequence in which operations should be performed to ensure accurate results.

Parentheses

Parentheses, also known as brackets, play a vital role in mathematical expressions by altering the typical order of operations. They indicate which operations should be carried out first. In expressions with multiple operations, solving what's inside the parentheses first ensures the correct computation.
For example, consider the expression \[3 imes (2 + 4)\]. Without parentheses, the expression would be read and solved left to right. However, the parentheses signal that we first perform the addition inside, then multiply the result:

• Step 1: Solve inside parentheses: \(2 + 4 = 6\).
• Step 2: Perform the multiplication: \(3 imes 6 = 18\).

This principle is critical for achieving the correct outcome in mathematical calculations.

Division

Division is the arithmetic operation used to determine how many times one number is contained within another. It is akin to distributing a total amount into equal parts. The standard form is represented as \[ \frac{a}{b} = c \], meaning that "a divided by b gives you c."
In the order of operations, division is performed after parentheses and exponents and alongside multiplication, as they hold the same precedence level. When solving expressions with divisions, it's essential to ensure all numerator calculations are finalized first before division occurs.
For example, with an expression like \[\frac{12}{3}\]:

• Calculate the numerator (if needed) first: as in this case, 12 is already a constant;
• Divide 12 by 3 to get 4.

Recognizing and following these steps prevents errors, especially when numbers have different signs.

Negative Numbers

Negative numbers are values less than zero, represented with a minus sign (−). They bring complexity to arithmetic operations because you must carefully track their sign effects throughout calculations. Understanding how they interact in addition, subtraction, multiplication, and division is crucial.

• Add/Subtract Negative Numbers: Adding a negative number effectively subtracts that value, and subtracting a negative value involves adding its absolute equivalent.
• Multiply/Divide Negative Numbers: Two negatives make a positive. If either number (but not both) is negative, the result will be negative.

For instance, in the expression \((-3) imes (-6) = 18\), we see that multiplying two negative numbers provides a positive product. Conversely, \((-3) imes 6 = -18\) results in a negative product, reflecting the difference in sign. Mastering these rules allows for consistent, error-free computation in mathematics.