Problem 19 - Fundamentals of Mathematics

Question

Find each value. \(\frac{-42}{-7}\)

Step-by-Step Solution

Answer

The value is 6.

Step 1: Analyze the Numerator and Denominator

The given expression is \( \frac{-42}{-7} \). Here, both the numerator (-42) and the denominator (-7) are negative numbers.

Step 2: Simplify the Signs

When dividing two negative numbers, the negatives cancel each other out, resulting in a positive number. Thus, \( \frac{-42}{-7} = \frac{42}{7} \).

Step 3: Perform the Division

Now, divide the positive numbers. Calculate \( \frac{42}{7} \), which equals 6, since 42 divided by 7 is 6.

Key Concepts

Understanding Negative Numbers

When we encounter negative numbers, we are dealing with values that are less than zero. These numbers are often found in different real-world scenarios such as debts, temperatures below zero, or distances below sea level. In mathematics, negative numbers are represented with a minus sign (-) before the number.
When performing operations with negative numbers, it’s crucial to pay attention to their sign, as it can change the outcome significantly. For instance:

• Addition of two negative numbers results in a more negative number: For example, (-3) + (-2) = -5.
• Subtraction involving negative numbers can be tricky, particularly because it might require adding a positive number: For example, 5 - (-3) = 8, because subtracting a negative is the same as adding a positive.

Negative numbers can initially seem confusing, but with practice, you can comfortably manage them in various mathematical contexts.

Simplifying Signs When Dividing

One might encounter scenarios where both numbers have negative signs. Simplifying signs is a crucial step when dealing with such situations.
Let’s use the expression \(\frac{-42}{-7}\) as an example. Here, both the numerator and the denominator are negative. When you divide numbers with the same sign, whether both positive or both negative, the result is always positive.
To simplify, take note of the signs:

• \(\frac{-}{-} = +\)
• \(\frac{-42}{-7}\) becomes \(\frac{42}{7}\)

This simplification is because a negative divided by a negative is similar to a negative times a negative, which results in a positive.

Basic Arithmetic Operations: Division

Division is one of the core arithmetic operations, alongside addition, subtraction, and multiplication. It involves determining how many times a divisor fits into a dividend. When you divide, you are essentially splitting the dividend into equal parts.
Let's illustrate this with our earlier problem: \(\frac{42}{7}\). Here:

• The dividend is 42, the number to be divided.
• The divisor is 7, the number you are dividing by.

The operation tells you how many times 7 fits into 42 evenly. Dividing gives us the result of 6, because:

• 7 multiplied by 6 is 42
• Therefore, \(\frac{42}{7} = 6\)

This example is straightforward, but division can be applied in various complex scenarios, highlighting the importance of mastering these basic operations early on.