Problem 99 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \frac{-24.24}{-0.8} $$

Step-by-Step Solution

Answer

The result of the operation is 30.5.

Step 1: Identify the Operation

The exercise requires performing division between two numbers: \(-24.24\) and \(-0.8\). Division by a negative number will change the sign, so start by rewriting our operation as \(rac{-24.24}{-0.8} = rac{24.24}{0.8}\).

Step 2: Simplify the Division

Simplify the division of 24.24 by 0.8 by converting it to an equivalent division operation that is easier to calculate: \(\frac{24.24}{0.8} = \frac{24.24 imes 10}{0.8 imes 10} = \frac{242.4}{8}\). Now, you'll divide 242.4 by 8.

Step 3: Perform the Division

Divide 242.4 by 8 using standard division:First, divide 24 by 8 which is 3.Next, drop down the 2 to make it 32, divided by 8 is 4.Drop down the 4 to get 4. Dividing 4 by 8 gives 0.5. Thus, \(\frac{242.4}{8} = 30.5\).

Step 4: Conclude and Verify

The division operation simplifies to \(24.24 ÷ 0.8 = 30.5\). Verify the answer by multiplying 30.5 by 0.8; you should retrieve the original number 24.24.

Key Concepts

Understanding Negative Numbers in Division

When dealing with division, especially involving negative numbers, it's important to remember some fundamental rules. Negative numbers can make arithmetic operations slightly different, and it's crucial to differentiate between negative and positive numbers. A negative sign indicates a number less than zero and can impact operations like addition, subtraction, multiplication, and division. For division, if both the dividend (the number being divided) and the divisor (the number dividing) are negative, as in the example \(-24.24 \div -0.8\), the negatives cancel each other out, resulting in a positive outcome:

• \(- \div - = +\)
• \(- \div + = -\)
• \(+ \div - = -\)
• \(+ \div + = +\)

Therefore, \(\frac{-24.24}{-0.8}\) simplifies to \(\frac{24.24}{0.8}\), yielding a positive result.

Simplifying Division for Easier Calculation

Simplifying division often involves making the numbers easier to work with while retaining the ratio. This step can make complex operations more straightforward and manageable. In the given equation, \(\frac{24.24}{0.8}\), directly dividing decimals might be tricky.To simplify, you can change the divisor into a whole number. Multiply both the numerator and the divisor by the same factor, such as 10, to eliminate the decimal. This transforms the division into a simpler format, like this:

• \(\frac{24.24 \times 10}{0.8 \times 10} = \frac{242.4}{8}\)

This transformation doesn't change the division's value but makes calculations smoother. Now, you can easily divide 242.4 by 8 using standard division rules, providing the result with less effort.

Verifying Division Through Arithmetic

Verification is a critical part of solving arithmetic problems. It not only confirms your result but also enhances understanding. After obtaining a result from a division operation, such as the simplified \(242.4 \div 8 = 30.5\), verify by reversing the operation. Multiply the quotient by the divisor to see if you retrieve the original number. Verification in this context looks like:

• Multiply \(30.5\) by \(0.8\) (our original divisor):
• \(30.5 \times 0.8 = 24.24\)

If the multiplication gives you the original number (24.24), then your division calculation is correct. This method reinforces learning and ensures accuracy, particularly when dealing with decimals and negative numbers.