Problem 20 - Prealgebra

Question

Find each of the following sums. (Add.) $$\begin{array}{c}987.654 \\\456.789 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The sum is 1444.443.

Step 1: Align Decimal Points

To add the numbers 987.654 and 456.789, start by writing them vertically, aligning the decimal points directly above each other.\[\begin{array}{r} 987.654 \+456.789 \\hline\end{array}\]

Step 2: Add the Thousandths Column

Begin by adding the digits in the thousandths column (the column furthest to the right). Add 4 and 9, which equals 13. Write down the 3 and carry over the 1.\[\begin{array}{r}1 \ 987.654 \+456.789 \\hline\end{array}\]

Step 3: Add the Hundredths Column

Next, add the digits in the hundredths column along with the carry-over 1. Add 5 and 8, which gives 13, plus 1 is 14. Write down the 4 and carry over the 1.\[\begin{array}{r}11 \ 987.654 \+456.789 \\hline\end{array}\]

Step 4: Add the Tenths Column

Now, add the digits in the tenths column along with the carry-over 1. Add 6 and 7 which is 13, plus 1 is 14. Write down the 4 and carry over the 1.\[\begin{array}{r}111 \ 987.654 \+456.789 \\hline\end{array}\]

Step 5: Add the Units Column

Continue with adding the digits in the units column along with the carry-over 1. Add 7 and 6, which equals 13, plus 1 is 14. Write down the 4 and carry over the 1.\[\begin{array}{r}1111 \ 987.654 \+456.789 \\hline\end{array}\]

Step 6: Add the Tens Column

Next, add the digits in the tens column plus the carry-over 1. Add 8 and 5, which equals 13, plus 1 is 14. Write down the 4 and carry over the 1.\[\begin{array}{r}11111 \ 987.654 \+456.789 \\hline\end{array}\]

Step 7: Add the Hundreds Column

Lastly, add the digits in the hundreds column along with the carry-over 1. Add 9 and 4, which equals 13, plus 1 is 14. Write down 14.\[\begin{array}{r} 1444.443 \\end{array}\]

Step 8: Write Final Answer

The final sum is:\[1444.443\]

Key Concepts

Understanding Place Value

Place value is a fundamental concept in decimal addition and arithmetic as a whole. Each digit in a number has a value depending on its position. Understanding place value helps us organize numbers, especially when dealing with decimals.

Consider the number 987.654. Each digit in this number has a specific place value:

• The '9' is in the hundreds place.
• The '8' is in the tens place.
• The '7' is in the units (or ones) place.
• The '6' is in the tenths place.
• The '5' is in the hundredths place.
• The '4' is in the thousandths place.

Recognizing and aligning digits according to their place values is crucial during the process of addition. This alignment ensures that each column holds the correct sum when digits are added together.

Mastering the Carry-Over Method

The carry-over method, also known as regrouping, is essential when adding numbers that yield sums greater than 9. It involves moving an extra value to the next column to maintain correctness.

For example, when adding 4 and 9 in the thousandths column, we receive 13. Here, we place the 3 in this column and carry over the 1 to the next column, adding to the next sum.

This method applies to each place value:

• In the tenths place, 6 and 7 sum to 13, and with the carried-over 1, this becomes 14. We place the 4 in the tenths column and carry over another 1.
• We keep repeating this method across every column to ensure that the solution is accurate.

Practicing the carry-over method can help make decimal addition more intuitive and error-free.

Step-by-Step Solutions: A Path to Accuracy

Step-by-step solutions provide a structured approach to solving complex problems, like decimal addition. This systematic approach ensures accuracy and aids students in understanding the process involved.

When adding decimals:

• Begin by aligning decimal points vertically to ensure correct place value addition.
• Start adding from the rightmost column (thousandths), moving leftwards.
• Carry over any extra value to the next left column when the sum exceeds 9.
• Continue adding each column systematically.
• Write down the final sum after adding all columns.

Step-by-step solutions help break down what can seem like a daunting task into manageable pieces, simplifying the learning process.

The Importance of Thousandths, Hundredths, and Tenths

In decimals, thousandths, hundredths, and tenths refer to the positions immediately to the right of the decimal point. Each of these positions contributes a unique value to the overall number, making them important in precise addition.

Here's a quick overview:

• Thousandths is the third position to the right of the decimal. The value affects the whole number by 1/1000th.
• Hundredths is the second position, affecting by 1/100th.
• Tenths is the first position, affecting by 1/10th.

When adding numbers like 987.654 and 456.789, accuracy in these positions ensures that even the smallest fractional differences are accounted for, leading to the precise sum of 1444.443.