Adding Decimals to the Thousandths: Fifth Grade Mathematics

Adding decimals to the thousandths requires strict vertical alignment of place values, annexing zeros to maintain column consistency, and disciplined regrouping.

The Core Rule: Line Up the Decimal Points

The golden rule of decimal addition is to align the decimal points vertically before adding. When decimal points are aligned, ones sit directly over ones, tenths over tenths, hundredths over hundredths, and thousandths over thousandths.

   Vertical Decimal Addition Setup:

         1   1   1
         1 4 . 6 8 5
      +    7 . 4 3 8
      --------------
         2 2 . 1 2 3

The Role of Annexing Zeros

When adding decimals with varying lengths, annex zeros to the shorter decimal numbers so that all addends extend to the same number of decimal places. This prevents place value confusion.

Adding 4.8 and 3.275:

   Incorrect Setup (Right-aligned):    Correct Setup (Decimals aligned & zeros annexed):
         4 . 8                               4 . 8 0 0
      +  3 . 2 7 5                        +  3 . 2 7 5
      ------------                        ------------
         3 . 7 5 5  <-- WRONG!               8 . 0 7 5  <-- CORRECT!

Bringing Down the Decimal Point

In decimal addition, the decimal point in the sum drops straight down into the exact vertical line of the addends' decimal points. It does not move to the left or right.

Multi-Addend Decimal Problems

When adding three or more decimal numbers, you can add them simultaneously in a single column or group them using the Associative Property.

Adding Three Decimals:  12.45 + 8.092 + 5.6

          1  1
       1  2 . 4  5  0
          8 . 0  9  2
       +  5 . 6  0  0
       --------------
       2  6 . 1  4  2

Regrouping Across the Decimal Point

Regrouping from tenths into ones operates under the exact same base-ten rule as regrouping from tens into hundreds: 10 tenths equal 1 one. If the tenths column sums to 15, you write 5 in the tenths place and carry 1 to the ones place, placing the decimal point directly between them.

Exercises and Solutions

Exercise 1: Vertical Decimal Addition with Annexed Zeros. Calculate the sum of 35.7 and 9.486. Show the vertical setup with annexed zeros and describe each regrouping step.

Solution 1: Align the decimal points vertically and annex two zeros to 35.7 to make it 35.700. Write 9.486 below it. Add the thousandths column: 0 + 6 = 6. Add the hundredths column: 0 + 8 = 8. Add the tenths column: 7 + 4 = 11; write 1 in the tenths place and carry 1 to the ones place. Place the decimal point in the sum directly below the other decimal points. Add the ones column: 5 + 9 + 1 (carried) = 15; write 5 in the ones place and carry 1 to the tens place. Add the tens column: 3 + 1 (carried) = 4. The final sum is 45.186.

Exercise 2: Multi-Item Science Measurement. In a chemistry experiment, a student measures three liquid samples: 0.875 liters of water, 1.4 liters of saline solution, and 0.065 liters of dye. What is the total volume of liquid in the beaker?

Solution 2: Align the three decimal values vertically by decimal point and annex zeros: 0.875 + 1.400 + 0.065. Add the thousandths column: 5 + 0 + 5 = 10 (write 0, carry 1). Add the hundredths column: 7 + 0 + 6 + 1 = 14 (write 4, carry 1). Add the tenths column: 8 + 4 + 0 + 1 = 13 (write 3, carry 1). Place the decimal point. Add the ones column: 0 + 1 + 0 + 1 = 2. The total volume is 2.340 liters, which simplifies to 2.34 liters.