Multi-digit decimal operations Knowledge-Based Study

Study notes

1. Core Principle: Place Value Alignment

   When adding or subtracting multi-digit decimals, digits belonging to the same place value must be combined. The most reliable way to guarantee this alignment is to line up the decimal points vertically.

   Step-by-Step Procedure

   1. Write the numbers vertically, aligning their decimal points.
   2. Insert placeholder zeros to the right of the last digit in shorter decimals so that all numbers have the same number of decimal places.
   3. Add or subtract column by column from right to left, carrying or regrouping across place values as needed.
   4. Place the decimal point in the sum or difference directly below the aligned decimal points above.

   Worked Examples

   Addition: Calculate $43.75 + 8.682$.

   Pad $43.75$ with a zero to become $43.750$:

   \begin{array}{r@{\quad}l} 43.750 \\ +\;\; 8.682 \\ \hline 52.432 \end{array}

   Subtraction: Calculate $15.4 - 7.825$.

   Pad $15.4$ with two zeros to become $15.400$, then regroup across places:

   \begin{array}{r@{\quad}l} 15.400 \\ -\;\; 7.825 \\ \hline 7.575 \end{array}

   Core Principle: Whole Number Multiplication and Place-Value Scaling

   Unlike addition and subtraction, decimal points do not need to line up during multiplication. Decimals are multiplied as though they were whole numbers, and the total count of decimal places across all factors determines the placement of the decimal point in the product.

   Step-by-Step Procedure

   1. Ignore the decimal points and multiply the numbers as whole integers using standard multi-digit multiplication.
   2. Count the total number of decimal digits (digits to the right of the decimal point) in every factor being multiplied.
   3. Position the decimal point in the product by counting from right to left the total number of places identified in step 2.
   4. Prefix zeros if necessary if the product contains fewer digits than the required number of decimal places.

   Worked Example

   Calculate $3.45 \times 0.28$.

   First, multiply the values as whole numbers:

   $$345 \times 28 = 9660$$

   Next, determine the total number of decimal places:

   • $3.45$ has 2 decimal places.
   • $0.28$ has 2 decimal places.
   • Total decimal places required: $2 + 2 = 4$.

   Starting from the right of $9660$, count 4 places to the left:

   $$3.45 \times 0.28 = 0.9660 = 0.966$$

   Core Principle: Transforming the Divisor into a Whole Number

   Dividing by a decimal value is streamlined by scaling both the divisor and the dividend by an identical power of 10. This creates an equivalent division problem with an integer divisor.

   Step-by-Step Procedure

   1. Examine the divisor: Count how many decimal places it contains.
   2. Shift the decimal point: Move the decimal point to the right in both the divisor and the dividend by that same number of places (multiplying both by $10$, $100$, $1000$, etc.). Annex zeros to the dividend if needed.
   3. Place the decimal point in the quotient directly above the newly positioned decimal point in the dividend.
   4. Perform long division as with whole numbers, continuing until the remainder is zero or the desired level of precision is reached.

   Worked Examples

   Example 1: Calculate $7.56 \div 0.6$.

   The divisor $0.6$ has one decimal place. Multiply both numbers by $10$:

   $$7.56 \div 0.6 = 75.6 \div 6$$

   Divide $75.6$ by $6$:

   $$75.6 \div 6 = 12.6$$

   Example 2: Calculate $4.9 \div 0.14$.

   The divisor $0.14$ has two decimal places. Multiply both numbers by $100$:

   $$4.9 \times 100 = 490 \quad \text{and} \quad 0.14 \times 100 = 14$$

   Divide $490$ by $14$:

   $$490 \div 14 = 35$$

   Quick Reference Guide

   • Addition & Subtraction: Align decimal points vertically. Add trailing zeros to balance place values. Bring the decimal point straight down.
   • Multiplication: Do not align decimals. Multiply as whole numbers. Total decimal places in factors equals total decimal places in the product.
   • Division: Convert the divisor to a whole number by shifting the decimal point to the right. Shift the dividend's decimal point by the exact same count. Bring the decimal straight up into the quotient.

   Common Pitfalls to Avoid

   • Right-aligning without regard to decimals in addition/subtraction: Always align by the decimal point, never by the rightmost digit.
   • Lining up decimals in multiplication: Lining up decimals creates unnecessary trailing zeros and increases the likelihood of multiplication errors.
   • Unequal shifting in division: If the divisor moves 2 places to the right, the dividend must also move 2 places to the right, appending zeros if it runs out of digits.