Problem 63 - Prealgebra

Question

Add or subtract the decimals, as indicated. \(-96.1+(-9.65)\)

Step-by-Step Solution

Answer

The sum is -105.75.

Step 1: Identify the Numbers to Combine

We need to add two decimal numbers: -96.1 and -9.65. Since both numbers are negative, their absolute values will be added together, and the result will also be negative.

Step 2: Align the Decimal Points

Align the numbers vertically with their decimal points directly above each other to ensure correct addition:\[\begin{array}{r}-96.10 \-09.65 \\hline\end{array}\]

Step 3: Add the Absolute Values

Add the absolute values of the numbers as if they were positive: 1. Start from the rightmost digits: 0 + 5 = 5. 2. Second column from the right: 10 + 6 = 16, place 6 and carry over 1. 3. In the next column: 6 + 9 + 1 (carry) = 16, place 6 and carry over 1. 4. For the leftmost column: 9 + 0 + 1 (carry) = 10, place 0 and carry over 1. This results in 105.75.

Step 4: Apply the Negative Sign

Since both original numbers are negative, the final result of the addition will also be negative. Therefore, the result is -105.75.

Key Concepts

Negative Numbers

Negative numbers represent values less than zero and are depicted with a minus sign (-) in front. In mathematics, understanding how to work with negative numbers is crucial, especially when performing addition and subtraction.
For instance, when you add two negative decimal numbers like -96.1 and -9.65, you are actually combining their amounts in the negative direction.

• Both numbers being negative means you add their distances from zero and retain the negative sign.
• This is similar to moving further to the left on the number line for each number you add.

Thus, the sum -105.75 reflects the continued movement further away from zero in the negative territory.

Place Value

Place value refers to the importance of the position of a digit in a number. This understanding is vital for performing operations like addition and subtraction, especially with decimals.
Each digit in a decimal number stands for a certain place value, whether it's in the hundreds, tens, units, tenths, hundredths, etc.
When aligning decimal numbers like -96.10 and -9.65 for operation, it's important to ensure that their decimal points are lined up correctly.

• This alignment ensures that you add or subtract digits of the same place value.
• Misalignment can lead to incorrect results as digits representing different values would be incorrectly combined.

Remember, the decimal system extends to the right as well, including tenths, hundredths, and so on, each having a smaller place value than the previous.

Absolute Value

The absolute value of a number is its distance from zero on the number line, without regard to direction. In simpler terms, it is the non-negative value of the number.
When adding negative numbers like -96.1 and -9.65, we utilize their absolute values to find the sum: 96.1 and 9.65.

• These absolute values are treated as positive numbers when adding them together.
• This approach helps in focusing on the magnitude of the numbers, simplifying addition.

Once their absolute values are added, you reapply the negative sign to the result because both original numbers were negative, leading us back to the net negative result of -105.75.

Carry-over in Addition

Carry-over is a crucial concept in addition, particularly when numbers in a column sum to a value greater than 9. In the given problem, as you add decimal numbers, you encounter carry-overs due to such sums.
As we add the numbers 0.6 and 0.9, the resultant value, 1.5, means that 5 stays in the column and 1 is carried over to the next place value.

• This carry-over process ensures that each place value does not exceed its maximum value of 9.
• For subsequent columns, we add values including the carried-over digit to maintain accuracy.

Carry-over maintains the integrity of the calculation across all digits and ensures that, ultimately, the sum of 105.75 (in absolute terms) correctly represents the total value being combined.