Problem 55 - Prealgebra

Question

Divide the decimals. \(\frac{-1.634}{-8.6}\)

Step-by-Step Solution

Answer

0.19

Step 1: Remove Negative Signs

Starting with the division of two negative numbers: \[\frac{-1.634}{-8.6}\]we know the result of dividing two negative numbers will be positive because a negative divided by a negative is positive. Therefore, we can focus on dividing the positive numbers, i.e., we compute \[\frac{1.634}{8.6}.\]

Step 2: Express Dividends and Divisors as Whole Numbers

To avoid fractions in decimals, convert both the dividend and divisor to whole numbers. Since 1.634 has three decimal places and 8.6 has one, we multiply both numbers by 1000 to eliminate decimal points:\[1.634 \times 1000 = 1634\]\[8.6 \times 100 = 8600\]

Step 3: Perform Division

Now divide the whole numbers obtained:\[\frac{1634}{8600}\]To simplify this example, we'll perform long division or simplify using greatest common divisors if comfort allows. The division results in approximately 0.19 when you calculate 1634 divided by 8600.

Step 4: Round (if necessary)

In this problem, if precision to the hundredth is necessary, then round the final result, which in this case remains 0.19 because the third decimal place is not 5 or above.

Key Concepts

Negative Numbers

Understanding negative numbers is crucial. They are the opposites of positive numbers and reflect values less than zero. When you multiply or divide two negative numbers, the negatives cancel out, resulting in a positive number. This is a fundamental rule in mathematics:

• Negative × Negative = Positive
• Negative ÷ Negative = Positive

In our given exercise \[\frac{-1.634}{-8.6}\]we begin by removing the negative signs because two negative numbers divided yield a positive number. Hence, we proceed with \[\frac{1.634}{8.6}\].Understanding this concept simplifies our approach, allowing us to focus on the numerical division itself.

Long Division

Long division is a methodical way of dividing larger numbers by hand. Let's explore how to effectively use this technique:

• First, align the numbers based on place value.
• Divide the dividend by the divisor starting from the highest place value.
• Note how many times the divisor fits into the portion of the dividend considered.
• Write the result (quotient) above the division line.
• Subtract the product of the divisor and quotient from the considered portion of the dividend.
• Bring down the next number and repeat the process until completion.

In our example \[\frac{1634}{8600}\],you would divide as detailed above, ultimately arriving at a quotient of 0.19. This kind of practice improves accuracy and understanding of division with complex numbers.

Rounding Decimals

Rounding decimals ensures precision by altering the number to a specified place value, often simplifying calculations. When rounding:

• Identify the decimal place you need to round to.
• Look one place to the right of this—the rounding digit.
• If this digit is 5 or greater, round up by adding 1 to your target place.
• If this digit is less than 5, keep the target place the same.
• Trim all digits beyond the chosen place value.

In our exercise, the division resulted in approximately 0.19. There was no need to round further because rounding to the nearest hundredth was already complete. Understanding rounding helps in maintaining appropriate levels of precision in your answers.