Problem 45 - Prealgebra

Question

Divide the decimals. \(\frac{-0.1056}{0.22}\)

Step-by-Step Solution

Answer

The result of dividing \(-0.1056\) by \(0.22\) is approximately \(-0.48\).

Step 1: Understand the Problem

The problem requires us to divide the decimal number \(-0.1056\) by \(0.22\). This involves basic division of two decimal numbers.

Step 2: Set up the Division

To divide \(-0.1056\) by \(0.22\), we can first move the decimal places to remove the decimals from the divisor. To do this, multiply both the dividend \(-0.1056\) and the divisor \(0.22\) by \(100\) to get \(-10.56\) and \(22\) respectively.

Step 3: Perform the Division

Now, divide \(-10.56\) by \(22\). This can be done by traditional long division or by using a calculator. Performing the division, \(-10.56 \div 22\), gives approximately \(-0.48\).

Step 4: Adjust the Sign

Remember that the division of a negative number by a positive number results in a negative number. Thus, the final answer is \(-0.48\).

Key Concepts

Negative Numbers

Negative numbers can sometimes be confusing, especially when handling operations like division. But it's quite simple when you know the rules. A negative number is a number less than zero, and they are represented with a minus sign. Negative numbers are often found in everyday situations like temperatures below zero or debts.

When dividing, it’s crucial to remember the signs of the numbers:

• A positive number divided by a positive number gives a positive result.
• A negative number divided by a positive number gives a negative result.
• A positive number divided by a negative number also leads to a negative result.
• A negative number divided by a negative number results in a positive result.

In the exercise provided, the division involves a negative dividend \(-0.1056\) and a positive divisor \(0.22\). Therefore, according to the rules, the final result must be negative.

Long Division

Long division is a method that helps us divide large numbers systematically. It might appear a bit tricky initially, but with practice, it’s a straightforward process. Long division involves:

• Dividing the dividend by the divisor.
• Multiplying the quotient by the divisor.
• Subtracting the result from the dividend.
• Bringing down the next digit of the dividend, if necessary, to repeat the process.

For example, when we perform \(-10.56 \div 22\), we break it down using these steps until we reach a quotient that doesn’t need further division. The remainder, if any, helps determine if the result is exact or if it needs to be considered as an approximation. In practice, long division may also incorporate decimals, and you might need to adjust for any remaining digits in the quotient.

Division by Decimals

Dividing by decimals might certainly seem complicated because of the positioning of decimal points. But here’s the trick: convert the divisor into a whole number. You can do this by moving the decimal point to the right.

This can be effectively demonstrated with the given problem:

• Given: \(-0.1056 \div 0.22\).
• Shift the decimal in both numbers to make the divisor a whole number: Multiply both \(-0.1056\) and \(0.22\) by 100, transforming them to \(-10.56\) and \(22\) respectively.
• This simplifies the division task, making it much more manageable using long division.

By turning the division of decimals into the division of whole numbers, we reduce the complexity and make calculations more precise. Ultimately, understanding this process is key to solving any division problem involving decimals seamlessly.