Problem 60 - Prealgebra

Question

Add or subtract the decimals, as indicated. \(-0.12+6.973\)

Step-by-Step Solution

Answer

The sum is 6.853.

Step 1: Align the Numbers

Align the two decimal numbers vertically, ensuring that the decimal points are in the same column to accurately perform the addition.\[\begin{array}{r}-0.12 \+ 6.973 \\hline\end{array}\]

Step 2: Add the Numbers

Perform the addition starting from the rightmost column (smallest decimal value) and moving leftward. Add each column:- The thousandths column: \(3 + 0 = 3\)- The hundredths column: \(7 + 2 = 9\)- The tenths column: \(9 - 1 = 8\) (since we need to subtract because one number is negative)- The units place: \(6 + 0 = 6\)\[\begin{array}{c}\ 6.853 \\hline\end{array}\]

Step 3: Check the Sign

Determine the sign of the result. Since the larger absolute value is from 6.973 (positive), the result of the sum will be positive. The result is \(6.853\).

Key Concepts

Aligning Decimal Points

When adding or subtracting decimals, aligning the decimal points is crucial. This ensures that each place value is correctly matched, from tenths to hundredths to thousandths, and so on. Start by writing each number in a column format. Make sure the decimal points are directly under each other. This helps to ensure accuracy in your calculation. For instance, when aligning \(-0.12 + 6.973\), place the numbers so that the numbers to the right and left of the decimal point match. Not only does this make it easier to perform operations like addition or subtraction, but it also helps in visualizing the problem. Always include any necessary zeros to hold place values if one number has fewer digits after the decimal.

Column Addition

Once your decimal points are aligned, you're ready for column addition. Begin with the column farthest to the right, which represents the smallest value place, usually the thousandths or hundredths. Add each digit, column by column, moving left. If the sum in a column exceeds 9, carry over the additional value to the next left column. For example, in our problem, the thousandths column sums to 3, the hundredths to 9, and the tenths to 8 due to borrowing. It’s essential to handle carries correctly, as this affects the final result. Consistently verify each column sum to avoid errors.

Sign Determination

In calculations involving both positive and negative numbers, recognizing the sign of the answer is vital. When adding \(-0.12 + 6.973\), the decimals must be viewed through their absolute values. Here, 6.973 is larger than 0.12, meaning the final answer takes the sign of 6.973. Another tip is to subtract the smaller absolute value from the larger one when dealing with numbers of different signs. The result will carry the sign of the number with the larger absolute value. This rule ensures the correctness of the final outcome, so for our example, the result is positive: \(6.853\).