Problem 25 - Algebra A Combined Function

Question

Perform the indicated operation. $$ \begin{array}{r} 45.02 \\ 3.006 \\ +\quad 8.405 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result of the operation is 56.431.

Step 1: Align the Decimal Points

First, align ALL numbers by their decimal points. Look at where the decimal point is in each number and place them in a vertical column:\[\begin{array}{r} 45.020 \ 3.006 \+ 8.405 \\hline\end{array}\] This ensures all numbers are added correctly based on place value.

Step 2: Add the Thousandths Place

Start adding from the rightmost side: the thousandths place (the third digit to the right of the decimal). Add: 0 (from 45.020) + 6 (from 3.006) + 5 (from 8.405) = 11. Write down 1 in the thousandths place and carry over 1 to the hundredths place:

Step 3: Add the Hundredths Place

Now add the hundredths place: 0 (from 45.020) + 0 (from 3.006) + 4 (from 8.405) + 1 (carry) = 5. Write down 5 in the hundredths place.

Step 4: Add the Tenths Place

Next, add the tenths place: 2 (from 45.020) + 0 (from 3.006) + 0 (from 8.405) = 2. Write down 2 in the tenths place.

Step 5: Add the Units Column

Add the units (ones) column: 5 (from 45.020) + 3 (from 3.006) + 8 (from 8.405) = 16. Write down 6 in the units place and carry over 1 to the tens place.

Step 6: Add the Tens Column

Finally, add the tens column: 4 (from 45.020) + 0 (from 3.006) + 0 (from 8.405) + 1 (carry) = 5. Write down 5 in the tens place.

Step 7: Write the Answer

Combine all the numbers from each column:\[\begin{array}{r}45.020 \ 3.006 \+ 8.405 \\hline56.431 \\end{array}\]The result is 56.431.

Key Concepts

Understanding Place Value

Place value is a fundamental concept in mathematics that refers to the value of a digit depending on its position within a number. Each position in a number represents a power of ten. This makes it easy to determine how large or small a number is in relation to others.

When adding decimals, understanding place value is crucial. For example, in the number 45.02:

• "4" is in the tens place, meaning it has a value of 40 (4 times 10).
• "5" is in the units (or ones) place, giving it a value of 5.
• "0" is the tenths place meaning it’s part of a fractional amount.
• "2" is in the hundredths place which is even smaller.

Keeping these values straight helps when performing operations like addition, as each digit must be aligned correctly.

Focus on the Thousandths Place

In decimal numbers, the thousandths place is the third position to the right of the decimal point. It is part of the fraction that explains how small portions of a whole number are divided.

In our example, the thousandths place is evident in the numbers:

• 45.020 (0 in thousandths)
• 3.006 (6 in thousandths)
• 8.405 (5 in thousandths)

When calculating, you start adding from the thousandths lest area furthest right numeral, progressing from smaller to larger units. It's important to remember that the thousandths place has only one thousandth of the value of the units place, meaning these numbers are very small, yet they count!

The Role of Carrying Over in Addition

Carrying over in addition helps manage the extra value created when numbers sum to more than nine in a decimal place. This process involves transferring excess value to the next left higher place value, maintaining the position of digits.

For example, in thousandths addition, if you have a result of 11:

• Write down 1 in the resulting thousandths place.
• Carry over 1 to the hundredths place.

This continues, making sure each place column is handled properly until the very end of addition.

Aligning Decimal Points for Accurate Addition

Aligning decimal points ensures that each digit is considered in its correct place value, allowing for a precise calculation. This alignment helps bring uniformity to addition problems, preventing problems with misalignment that could lead to errors.

To achieve this:

• Ensure every number is vertically aligned by their decimal points.
• Fill with zeroes where there aren't digits, such as adding zeroes to 45.02, resulting in 45.020 and making it easier to comprehend.

This step is crucial as it respects each number's place from the starting decimal point through the tenths, hundredths, and thousandths positions, aiding in a seamless addition process without value misplacement.