Adding & Subtracting Decimals - Sixth Grade Mathematics
Adding and subtracting decimals requires precise vertical place value alignment anchored by the decimal point, ensuring that like units are combined and separated accurately.
The Decimal Alignment Principle
The fundamental rule when adding or subtracting decimals is that digits of identical place value must be aligned vertically. The decimal point serves as the visible boundary marker: all tenths must line up with tenths, hundredths with hundredths, and whole ones with ones.
CORRECT ALIGNMENT: INCORRECT ALIGNMENT:
4 2 . 8 5 4 2 . 8 5
+ 7 . 3 0 + 7 . 3
------------- ---------
5 0 . 1 5 4 3 . 5 8 (COMPLETELY WRONG!)
(Tenths over tenths) (Tenths added to hundredths!)
The Four-Step Operational Protocol
Executing addition and subtraction of decimals follows a four-step sequence.
+-------------------------------------------------------------------------+
| DECIMAL ADDITION & SUBTRACTION PROTOCOL |
| |
| Step 1: Write the numbers vertically, aligning the decimal points |
| in a straight vertical column. |
| Step 2: Annex placeholder zeros to the right of any shorter decimal |
| so that all numbers have the same number of decimal digits. |
| Step 3: Add or subtract the digits place-by-place, starting from the |
| extreme right and regrouping (carrying/borrowing) as needed. |
| Step 4: Bring the decimal point STRAIGHT DOWN into the answer line |
| directly below the other decimal points. |
+-------------------------------------------------------------------------+
Regrouping and Subtracting Across Zeros
Subtracting decimals often requires decomposing across zeros when subtracting a number with more decimal places from a number with fewer decimal places. Annexing placeholder zeros before beginning is mandatory.
Evaluate: 14.2 - 6.845
Step 1: Align and annex placeholder zeros:
1 4 . 2 0 0
- 0 6 . 8 4 5
---------------
Step 2: Decompose across zeros:
13 11 9 10
1 4 . 2 0 0
- 0 6 . 8 4 5
---------------
0 7 . 3 5 5
Step 3: Verification via inverse addition:
7.355 + 6.845 = 14.200 = 14.2. (Confirmed!)
Whole Numbers in Decimal Arithmetic
When adding or subtracting whole numbers with decimals, students often struggle with where to place the decimal point. Every whole number possesses an implicit decimal point situated at its immediate right. For example, the whole number 18 is written as 18.0 or 18.00.
Evaluate: 25 - 8.37
Write 25 with a decimal point and two zeros:
2 5 . 0 0
- 0 8 . 3 7
-------------
1 6 . 6 3
Checking Reasonableness with Mental Estimation
Before calculating exact answers, compute a rapid mental estimate by rounding to the nearest whole number. For 42.85 + 7.3, round 42.85 to 43 and 7.3 to 7. The estimated sum is 43 + 7 = 50. The exact calculated answer of 50.15 is extremely close to 50, providing high confidence that no alignment or calculation errors occurred.
Exercises
Exercise 1. Calculate the sum: 48.75 + 6.9 + 124.085. Show the vertical alignment with annexed placeholder zeros.
Exercise 2. Calculate the difference: 34 - 18.642. Show all decomposition steps clearly.
Exercise 3. Maya bought a book for $14.85, a notebook for $3.50, and a pen for $1.79. If she paid with a $50 bill, calculate the exact change she received.
Solutions
Solution to Exercise 1. Align the decimal points vertically and annex placeholder zeros so all terms have three decimal places: 048.750 + 006.900 + 124.085. Add from right to left: thousandths: 0 + 0 + 5 = 5. Hundredths: 5 + 0 + 8 = 13 (write 3, carry 1). Tenths: 1 + 7 + 9 + 0 = 17 (write 7, carry 1). Bring down decimal point. Ones: 1 + 8 + 6 + 4 = 19 (write 9, carry 1). Tens: 1 + 4 + 0 + 2 = 7. Hundreds: 0 + 0 + 1 = 1. The total sum is 179.735.
Solution to Exercise 2. Place a decimal point and three zeros after 34 to match three decimal places: 34.000 - 18.642. Decompose across the zeros: borrow 1 from 4 in the ones place (4 becomes 3), making 10 tenths, then borrow 1 from 10 tenths (becomes 9 tenths) to make 10 hundredths, then borrow 1 from 10 hundredths (becomes 9 hundredths) to make 10 thousandths. Now subtract: thousandths: 10 - 2 = 8. Hundredths: 9 - 4 = 5. Tenths: 9 - 6 = 3. Bring down decimal point. Ones: 13 - 8 = 5 (borrowed from 3 in tens place). Tens: 2 - 1 = 1. The difference is 15.358.
Solution to Exercise 3. First, calculate the total cost by adding the prices: 14.85 + 3.50 + 1.79. In the hundredths place: 5 + 0 + 9 = 14 (write 4, carry 1). In the tenths place: 1 + 8 + 5 + 7 = 21 (write 1, carry 2). Bring down decimal point. In the ones place: 2 + 4 + 3 + 1 = 10 (write 0, carry 1). In the tens place: 1 + 1 = 2. Total cost is $20.14. Next, subtract the total cost from $50.00: 50.00 - 20.14 = 29.86. Maya received exactly $29.86 in change.