Problem 12 - Prealgebra

Question

Find each of the following sums. (Add.) $$27.0179+7.89+0.009$$

Step-by-Step Solution

Answer

The sum is 34.9169.

Step 1: Align the numbers by their decimal points

First, write the numbers in a column ensuring the decimal points are aligned:\[\begin{array}{l} 27.0179 \ 7.89 \ 0.009 \\hlinediff\end{array}\]

Step 2: Add zeros if necessary

To make sure every number has the same number of decimals for easier addition, add zeros to the numbers as placeholders:\[\begin{array}{l} 27.0179 \ 7.8900 \ 0.0090 \\hlinediff\end{array}\]

Step 3: Start adding from the rightmost digit

Begin adding from the rightmost column and move left, carrying over if necessary:- From the thousandths column: \(9 + 0 + 0 = 9\)- From the hundredths column: \(7 + 9 + 0 = 16\) (write 6, carry over 1)- From the tenths column: \(1 + 8 + 0 + 1 = 10\) (write 0, carry over 1)- From the units column: \(7 + 0 + 1 = 8\)- From the tens column: \(2 + 7 = 9\)This gives us:\[\begin{array}{l} 27.0179 \ 7.8900 \ 0.0090 \\hline 34.9169\end{array}\]

Step 4: Write the final answer

The sum of the numbers is written as \(34.9169\).

Key Concepts

Understanding Place Value

Place value is a fundamental concept in mathematics that refers to the value of each digit in a number based on its position. In a number like 27.0179, each digit is in a specific column, such as the tens, units, tenths, hundredths, thousandths, and so on. The digit 7 in the number 27.0179 is in the tens place, so it represents 70. Meanwhile, the 9 in 27.0179 is in the ten-thousandths place, representing 0.0009. Knowing where each digit stands is important for performing operations like addition. Place value helps us understand how to align numbers properly by their decimals, and guides us on how to handle each digit during calculations. Using place value, children learn to add digits starting from the smallest place value (on the right) moving left.

Aligning Decimals for Addition

Aligning decimals is crucial for the process of adding decimal numbers. This means writing numbers in a column so that all decimal points are one under the other. This ensures that like terms, or digits of the same place value, are added together effectively. In our example, the numbers 27.0179, 7.89, and 0.009 need to be written such that their decimal points form a vertical line.

Here's how to visualize it:

• 27.0179
• 7.8900
• 0.0090

Notice that zeros can be added at the end of decimal numbers to ensure that all numbers have the same length for easier computation. This step is crucial because misalignment can lead to incorrect answers as it mixes different place values, which can significantly affect the sum.

Carrying Over in Addition

Carrying over occurs when the sum of a column of digits exceeds 9, in which case ten 'carries over' to the next left column. Let's take our example: when you add the hundredths column, 7 and 9 combine to make 16. Here, 6 remains in the hundredths column, while 1 is carried over to the tenths column.

• From the tenths column, we find: 1 (from 27.0179) + 8 (from 7.89) + 0 (from 0.009) + 1 (carried over) equals 10.

The tenths thus become 0, and 1 is again carried to the units column. This systematic way of adding ensures that all numbers, including those carried over, are correctly placed and calculated, aiding to obtain the accurate result.

Adding Decimals Together

Adding decimals follows the basic principle of addition, with the special attention paid to aligning decimal points. Here's a quick review:

• Align the decimal points of all numbers.
• Ensure every number has the same number of decimal places by adding zeros as placeholders if necessary.
• Begin adding from the rightmost side of the numbers, moving left, and carry over when needed.

With our example: starting from the thousandths column and moving leftward, remember to add each column, considering any carry overs. The addition resolves into 9, 16, 10, 8, and finally 9, resulting in the total 34.9169. This understanding ensures clarity when working with decimal numbers, translating an otherwise complex procedure into an approachable sequence.