3.2 Addition & Subtraction of Decimals

3.2 Addition & Subtraction of Decimals

Adding and subtracting decimal numbers requires the exact same column-by-column regrouping rules as whole number arithmetic, with one critical prerequisite: you must line up the decimal points vertically. Aligning decimal points ensures that digits of equal place value—tenths with tenths, hundredths with hundredths, ones with ones—are added or subtracted together.

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The Core Rule: Vertical Alignment of Decimal Points

When adding or subtracting whole numbers, numbers are naturally right-justified because the ones place is always the rightmost digit. With decimal numbers, right-justification fails because numbers can have different numbers of digits after the decimal point.

Alignment Comparison

Consider adding 14.2, 3.85, and 0.174:

• INCORRECT (Right-Justified):

  $$\begin{array}{r} 14.2 \\ 3.85 \\ + 0.174 \\ \hline \text{Wrong Column Sums} \end{array}$$

  Error: Adds 2 (tenths) to 5 (hundredths) to 4 (thousandths).

• CORRECT (Decimal Points Aligned Vertically):

  $$\begin{array}{rl} 14.200 & (\text{annexed zeros}) \\ 3.850 & (\text{annexed zero}) \\ + 0.174 & (\text{aligned}) \\ \hline \mathbf{18.224} & (\text{decimal drops straight down}) \end{array}$$

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Annexing Trailing Zeros for Subtraction

Annexing zeros means adding zeros to the right of the last digit in a decimal fraction. Because $0.7 = 0.70 = 0.700$, annexing zeros never alters the value of a number, but it is indispensable when subtracting a decimal with more decimal places from one with fewer decimal places.

Step-by-Step Subtraction with Annexed Zeros

Subtract 9.175 from 25.4.

• Step 1: Write the top number (25.4) and annex two zeros to match the 3 decimal places of 9.175: $25.400 - 9.175$

• Step 2: Line up by decimal point and set up column borrowing:

  $$\begin{array}{rl} & \quad 1 \; 15 \cdot 3 \; 9 \; 10 \\ & \quad \not{2} \; \not{5} \cdot \not{4} \; \not{0} \; \not{0} \\ - & \quad 0 \; 9 \cdot 1 \; 7 \; 5 \\ \hline & \quad \mathbf{1 \; 6 \cdot 2 \; 2 \; 5} \end{array}$$

• Step 3: Verify place values column by column:

  • Thousandths: Borrow from tenths to make $10 - 5 = 5$.
  • Hundredths: $9 - 7 = 2$.
  • Tenths: $3 - 1 = 2$.
  • Ones: Borrow 1 ten to make $15 - 9 = 6$.
  • Tens: $1 - 0 = 1$.
  • Result: 16.225

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Regrouping Across Multiple Zeros

When subtracting from a number containing consecutive zeros (such as whole dollar amounts like $\text{\textdollar}100.00$), regrouping must cascade leftward to the first non-zero digit.

Step-by-Step Worked Example: Subtracting from Round Numbers

Calculate $100.00 - $37.84.

1. Align decimal points vertically:

   $$\begin{array}{r} 100.00 \\ - \phantom{0}37.84 \\ \hline \end{array}$$

2. Cascade borrowing from the hundreds place:

   • The 1 in the hundreds place becomes 0.
   • The tens place becomes 9.
   • The ones place becomes 9.
   • The tenths place becomes 9.
   • The hundredths place becomes 10.

3. Perform column subtraction from right to left:

   • Hundredths: $10 - 4 = 6$
   • Tenths: $9 - 8 = 1$
   • Ones: $9 - 7 = 2$
   • Tens: $9 - 3 = 6$
   • Hundreds: $0 - 0 = 0$

4. Place the decimal point straight down: $\text{Result: } \mathbf{62.16}$

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Practical Applications: Perimeter, Distance & Money

Decimal addition and subtraction are used in geometric measurements, inventory tracking, and personal finance calculations.

Example 1: Calculating Geometric Perimeter

Find the perimeter of an irregular quadrilateral with side lengths of 14.35 cm, 9.8 cm, 12.075 cm, and 16.4 cm.

• Recall Formula: $\text{Perimeter} = \text{Side}_1 + \text{Side}_2 + \text{Side}_3 + \text{Side}_4$

• Step 1: Line up decimal points vertically and annex zeros:

  $$\begin{array}{r} 14.350 \\ 09.800 \\ 12.075 \\ + 16.400 \\ \hline \end{array}$$

• Step 2: Add column by column:

  • Thousandths column: $0 + 0 + 5 + 0 = 5$
  • Hundredths column: $5 + 0 + 7 + 0 = 12$ (write 2, carry 1)
  • Tenths column: $1 \text{ (carried)} + 3 + 8 + 0 + 4 = 16$ (write 6, carry 1)
  • Ones column: $1 \text{ (carried)} + 4 + 9 + 2 + 6 = 22$ (write 2, carry 2)
  • Tens column: $2 \text{ (carried)} + 1 + 0 + 1 + 1 = 5$

• Final Answer: The perimeter is 52.625 cm.

Example 2: Financial Account Balancing

At the start of the week, Alex had a checking account balance of $524.10. During the week, he made a cash deposit of $145.50, paid an electric bill of $89.75, bought groceries for $64.38, and received a refund check of $22.15. What is his ending account balance?

• Step 1: Add all credits (deposits and refunds) to initial balance:

  $$\begin{array}{rl} 524.10 & (\text{starting balance}) \\ 145.50 & (\text{deposit}) \\ + 22.15 & (\text{refund}) \\ \hline \mathbf{691.75} & (\text{total funds}) \end{array}$$

• Step 2: Add all debits (expenses):

  $$\begin{array}{rl} 89.75 & (\text{electric bill}) \\ + 64.38 & (\text{groceries}) \\ \hline \mathbf{154.13} & (\text{total debits}) \end{array}$$

• Step 3: Subtract total debits from total funds:

  $$\begin{array}{r} 691.75 \\ - 154.13 \\ \hline \mathbf{537.62} \end{array}$$

• Final Answer: Alex's ending account balance is $537.62.

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ACCUPLACER Exam Traps & Common Errors