Problem 51 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ \frac{4(8+1)-3(-2)}{-4-2} $$

Step-by-Step Solution

Answer

The value is -7.

Step 1: Simplify the Expression Inside Parentheses

First, identify and simplify the expression inside the parentheses: \( 8 + 1 = 9 \).Then the equation becomes: \[ \frac{4(9) - 3(-2)}{-4 - 2} \]

Step 2: Compute Multiplications

Multiply the numbers: \( 4(9) = 36 \) and \( 3(-2) = -6 \).Now the equation looks like:\[ \frac{36 + 6}{-4 - 2} \]

Step 3: Simplify the Numerator

Add the numbers in the numerator:\( 36 + 6 = 42 \).The equation now is:\[ \frac{42}{-4 - 2} \]

Step 4: Simplify the Denominator

Perform the subtraction in the denominator:\( -4 - 2 = -6 \).The equation becomes:\[ \frac{42}{-6} \]

Step 5: Divide the Numerator by the Denominator

Now divide the simplified numerator by the simplified denominator:\( \frac{42}{-6} = -7 \).

Key Concepts

Order of Operations

When faced with a complex arithmetic expression, it's essential to follow the correct sequence of steps to find the right answer. This sequence is known as the Order of Operations. A helpful way to remember this is with the acronym PEMDAS:

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

By following these steps, you ensure that you're solving the problem correctly. For example, in the expression \( \frac{4(8+1)-3(-2)}{-4-2} \), the first step is to solve the operations inside the parentheses: \( 8+1 \). After simplifying within the parentheses, you move on to multiplication and division. This structured approach is vital for arriving at the correct result.

Numerator and Denominator

Fractions are composed of two parts: the numerator and the denominator. The numerator is the top number, representing how many parts of the whole are being considered. The denominator is the bottom number, indicating the total number of equal parts the whole is divided into.

In our expression \( \frac{4(8+1)-3(-2)}{-4-2} \), the entire top portion \( 4(8+1)-3(-2) \) is the numerator, and \( -4-2 \) is the denominator. Understanding these components is crucial as each part must be simplified separately before combining the two in division. This clarity ensures that you accurately perform operations on each fraction part.

Multiplication and Division

Handling multiplication and division accurately is key in simplifying arithmetic expressions. Both operation types are of equal precedence in the order of operations, so they occur from left to right after any parentheses.

• Start with multiplication: Solve expressions like \( 4(9) \) and \( 3(-2) \).
• Proceed to division: After addressing the top and bottom of the fraction, complete the final division.

For instance, in our problem, after simplifying the numerator and denominator, we end up with \( \frac{42}{-6} \). This fraction requires division, which simplifies to \( -7 \). Careful attention to these operations prevents errors in solving mathematical problems.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This often includes completing any addition or subtraction in the numerator and denominator before dividing.

• Combine numbers in the numerator and denominator independently.
• Perform the final division step once all parts are simplified.

In our example, after combining \( 36 + 6 \) for the numerator and calculating \( -4 - 2 \) for the denominator, you get \( \frac{42}{-6} \). Simplifying this by dividing gives \( -7 \). Simple arithmetic manipulation like this makes working with fractions considerably easier, ensuring the final step is straightforward and accurate.