Problem 3 - Prealgebra

Question

Find each of the following sums. (Add.) $$0.04+0.31+0.78$$

Step-by-Step Solution

Answer

The sum of 0.04, 0.31, and 0.78 is 1.13.

Step 1: Align the Numbers

Structure the numbers to be added so that the decimal points are aligned vertically. This ensures that each decimal place value is correctly summed:\[\begin{array}{r} 0.04 \+ 0.31 \+ 0.78 \\hlinedotdotdot\hline\end{array}\]

Step 2: Add the Thousandths

Start adding from the rightmost decimal place (thousandths place). In this case, the numbers all have no thousandths (i.e., a zero in the third decimal place), so the sum is 0. No carrying is necessary.

Step 3: Add the Hundredths

Move to the hundredths place and add the digits: 4 (from 0.04) + 1 (from 0.31) + 8 (from 0.78). This gives us 13. Write down 3 and carry over 1 to the tenths place.

Step 4: Add the Tenths with Carry Over

Now, add the tenths place digits: 0 (from 0.04) + 3 (from 0.31) + 7 (from 0.78), plus the carry over 1 from the previous step. This totals to 11. Write down 1 and carry over 1 to the units place.

Step 5: Add the Units Place with Carry Over

In the units place, add the carry over only, as the numbers being added are all decimals less than 1. The total is 1.

Step 6: Compile the Result

Write down the decimal places and the units place result to compile the full sum: 1 (units) + 1 (tenths) + 3 (hundredths). The resulting sum is 1.13.

Key Concepts

Aligning Decimals

When adding decimal numbers, the first step is to align the decimal points vertically. This alignment ensures that each digit in the same place value column is added correctly. Imagine stacking the numbers directly above one another like perfectly fitted building blocks, starting with their decimal points as the key alignment.

• When you line up decimal numbers, match the decimal points so that each specific place value aligns vertically.
• Ensures that, for instance, the tenths are above tenths, and hundredths are above hundredths.

Practically, you're structuring the numbers similarly to how you would for whole numbers, just with the added precision of the decimal point as your guide. Think of this like aligning columns in a spreadsheet: clarity comes from straight lines.

Place Value

Place value is fundamental when working with decimals. Each digit in a number has a specific position, or place, which gives it a value. Take decimals such as 0.04, 0.31, and 0.78. Each digit has a different weight depending on its position:

• The rightmost digit is in the hundredths or thousandths place. It illustrates very small quantities.
• The second digit from the right falls in the tenths place. It's worth more than the hundredths but less than the units.
• Finally, the units place stands on the far left for whole numbers, marking complete units.

Understanding place value allows us to see the precise size and importance of each part of the number. It's the language that defines how numbers communicate their specific quantities.

Carrying in Addition

Carrying is a crucial procedure when the sum of any column in your addition exceeds 9, even in decimals. Let's see how carrying works in our exercise:

• When summing the hundredths place, the result is 13. You write down the 3, the rightmost digit, and carry over the 1 to the next column, the tenths place.
• This carry-over process changes the next column's total by a whole unit. In our case in the tenths, the carry 1 is added to make that total 11 for the next column.
• Finally, any carry from tenths to units means incrementing units count by one as well.

The act of carrying ensures the correct answer, as it acknowledges the overflow from one decimal place, transferring its energy and influence, not to mention its value, to the immediate left. It maintains mathematical integrity across the universally known place value columns.