Problem 29 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ \frac{-45}{-5} $$

Step-by-Step Solution

Answer

The value is 9.

Step 1: Identify the Division

We start by recognizing the expression given in the problem. We need to divide \(-45\) by \(-5\). The expression is written as \(\frac{-45}{-5}\).

Step 2: Simplify the Division by Negatives

When dividing two negative numbers, the negatives cancel each other out. Therefore, \(\frac{-45}{-5}\) becomes \(\frac{45}{5}\).

Step 3: Perform the Division

Now, perform the division of the simplified expression. Divide 45 by 5, which results in 9.

Step 4: Verify with a Calculator

To ensure accuracy, use a calculator to check the division of \(\frac{-45}{-5}\). The calculator should confirm that the result is indeed 9.

Key Concepts

Negative Numbers

Understanding negative numbers is essential in mathematics. They often appear in division and need a thoughtful approach. Negative numbers are integers less than zero, represented with a minus sign (-).

When dividing negative numbers, remember a few key points:

• When a negative is divided by a negative, the result is positive because the two negatives cancel each other out.
• This is similar to multiplying negative numbers; two negatives make a positive outcome.

For instance, in the problem \( \frac{-45}{-5} \), we notice both numbers are negative. The division becomes \( \frac{45}{5} \) as the negatives cancel each other. This is an important rule and helps simplify calculations.

Simplification

Simplification is the process of reducing a complex expression into a simpler form. This often makes calculations easier and clearer. In division, simplifying involves canceling out terms where possible or reducing fractions to easier numbers.

For the expression \( \frac{-45}{-5} \), simplification plays a crucial role:

• First, the negative signs are canceled, changing the expression to \( \frac{45}{5} \).
• Next, actual division takes place. 45 divided by 5 simplifies to 9.

Learning to simplify is vital, especially with negative numbers, as it helps prevent mistakes and makes calculations quicker.

Calculator Verification

Calculator verification is crucial for ensuring accuracy in mathematics. After performing manual calculations, using a calculator helps confirm the results. It reduces errors and boosts confidence in the solution.

Here's how to use a calculator for verification:

• Perform the division operation just as you did manually: Enter \( \frac{-45}{-5} \)
• Check that the displayed result matches your manual calculation.

In this step, the calculator should show 9, verifying correct simplification and division. Using a calculator as a check-step when dealing with negative numbers minimizes the risks of calculation errors. It's a reliable way to confirm that every step was correctly followed and executed.