3.2 Addition & Subtraction of Decimals

3.2 Addition & Subtraction of Decimals\n

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.\n ---\n

The Core Rule: Vertical Alignment of Decimal Points\n

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.\n

Alignment Comparison\n

Consider adding 14.2, 3.85, and 0.174:\n

• INCORRECT (Right-Justified):\n $$\n \begin{array}{r}\n 14.2 \ 3.85 \
  • 0.174 \ \hline\n \text{Wrong Column Sums}\n \end{array}\n $$\n Error: Adds 2 (tenths) to 5 (hundredths) to 4 (thousandths).\n
• CORRECT (Decimal Points Aligned Vertically):\n $$\n \begin{array}{rl}\n 14.200 & (\text{annexed zeros}) \ 3.850 & (\text{annexed zero}) \
  • 0.174 & (\text{aligned}) \ \hline\n \mathbf{18.224} & (\text{decimal drops straight down})\n \end{array}\n $$\n ---\n

Annexing Trailing Zeros for Subtraction\n

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.\n

Step-by-Step Subtraction with Annexed Zeros\n

Subtract 9.175 from 25.4.\n

• Step 1: Write the top number (25.4) and annex two zeros to match the 3 decimal places of 9.175:\n $25.400 - 9.175$\n
• Step 2: Line up by decimal point and set up column borrowing:\n $$\n \begin{array}{rl}\n & \quad 1 ; 15 \cdot 3 ; 9 ; 10 \ & \quad \not{2} ; \not{5} \cdot \not{4} ; \not{0} ; \not{0} \
  • & \quad 0 ; 9 \cdot 1 ; 7 ; 5 \ \hline\n & \quad \mathbf{1 ; 6 \cdot 2 ; 2 ; 5}\n \end{array}\n $$\n
• Step 3: Verify place values column by column:\n - Thousandths: Borrow from tenths to make $10 - 5 = 5$.\n - Hundredths: $9 - 7 = 2$.\n - Tenths: $3 - 1 = 2$.\n - Ones: Borrow 1 ten to make $15 - 9 = 6$.\n - Tens: $1 - 0 = 1$.\n - Result: 16.225\n ---\n

Regrouping Across Multiple Zeros\n

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

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

Calculate $100.00 - $37.84.\n

1. Align decimal points vertically:\n $$\n \begin{array}{r}\n 100.00 \
   • \phantom{0}37.84 \ \hline\n \end{array}\n $$\n
2. Cascade borrowing from the hundreds place:\n - The 1 in the hundreds place becomes 0.\n - The tens place becomes 9.\n - The ones place becomes 9.\n - The tenths place becomes 9.\n - The hundredths place becomes 10.\n
3. Perform column subtraction from right to left:\n - Hundredths: $10 - 4 = 6$\n - Tenths: $9 - 8 = 1$\n - Ones: $9 - 7 = 2$\n - Tens: $9 - 3 = 6$\n - Hundreds: $0 - 0 = 0$\n
4. Place the decimal point straight down:\n $\text{Result: } \mathbf{62.16}$\n ---\n

Practical Applications: Perimeter, Distance & Money\n

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

Example 1: Calculating Geometric Perimeter\n

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

• Recall Formula: $\text{Perimeter} = \text{Side}_1 + \text{Side}_2 + \text{Side}_3 + \text{Side}_4$\n- Step 1: Line up decimal points vertically and annex zeros:\n $$\n \begin{array}{r}\n 14.350 \ 09.800 \ 12.075 \
  • 16.400 \ \hline\n \end{array}\n $$\n
• Step 2: Add column by column:\n - Thousandths column: $0 + 0 + 5 + 0 = 5$\n - Hundredths column: $5 + 0 + 7 + 0 = 12$ (write 2, carry 1)\n - Tenths column: $1 \text{ (carried)} + 3 + 8 + 0 + 4 = 16$ (write 6, carry 1)\n - Ones column: $1 \text{ (carried)} + 4 + 9 + 2 + 6 = 22$ (write 2, carry 2)\n - Tens column: $2 \text{ (carried)} + 1 + 0 + 1 + 1 = 5$\n
• Final Answer: The perimeter is 52.625 cm.\n

Example 2: Financial Account Balancing\n

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?\n

• Step 1: Add all credits (deposits and refunds) to initial balance:\n $$\n \begin{array}{rl}\n 524.10 & (\text{starting balance}) \ 145.50 & (\text{deposit}) \
  • 22.15 & (\text{refund}) \ \hline\n \mathbf{691.75} & (\text{total funds})\n \end{array}\n $$\n
• Step 2: Add all debits (expenses):\n $$\n \begin{array}{rl}\n 89.75 & (\text{electric bill}) \
  • 64.38 & (\text{groceries}) \ \hline\n \mathbf{154.13} & (\text{total debits})\n \end{array}\n $$\n
• Step 3: Subtract total debits from total funds:\n $$\n \begin{array}{r}\n 691.75 \
  • 154.13 \ \hline\n \mathbf{537.62}\n \end{array}\n $$\n
• Final Answer: Alex's ending account balance is $537.62.\n ---\n

ACCUPLACER Exam Traps & Common Errors\n

[!WARNING]\n> Trap 1: The 'Subtracting Down' Borrowing Mistake\n> When subtracting $8.4 - 2.67$, students who fail to annex zeros write $8.40 - 2.67$ incorrectly by subtracting $7 - 0 = 7$ in the hundredths column, getting $5.77$ instead of borrowing ($10 - 7 = 3$) to get $5.73$.\n [!WARNING]\n> Trap 2: Misaligning Whole Numbers and Decimals\n> When adding a whole number to a decimal (e.g., $15 + 3.45$), remember that whole numbers have an unwritten decimal point at their far right ($15.00$). Adding $15 + 3.45$ as $0.15 + 3.45$ yields a catastrophic error.