3.3 Multiplication & Division of Decimals

3.3 Multiplication & Division of Decimals\n

Multiplying and dividing decimals are foundational operations on the ACCUPLACER Arithmetic test. Unlike addition and subtraction, multiplication does not require aligning decimal points before calculating. Division, on the other hand, requires transforming decimal divisors into whole numbers before executing standard long division.\n ---\n

Multiplication of Decimals\n

To multiply decimal numbers, execute standard integer multiplication, then position the decimal point in the product based on place value summation.\n

The Standard Decimal Multiplication Algorithm\n1. Ignore Decimal Points Initially: Align the factors by their rightmost digits, ignoring decimal points.\n2. Multiply as Whole Numbers: Compute the product using standard whole-number multiplication algorithms.\n3. Count Factor Decimal Places: Count the number of digits to the right of the decimal point in both factors and sum them together.\n4. Place the Decimal Point: Starting from the rightmost digit of the product, move left the total number of places calculated in Step 3 and insert the decimal point.\n5. Annex Leading Zeros if Necessary: If the product contains fewer digits than the required decimal places, insert zeros to the left of the product digits before placing the decimal point.\n

Multiplication Decimal Place Reference Table\n

| Factor 1 | Decimal Places | Factor 2 | Decimal Places | Sum of Places | Raw Product | Final Decimal Product |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| $3.4$ | 1 | $1.2$ | 1 | $1 + 1 = 2$ | $408$ | $4.08$ |\n| $0.45$ | 2 | $0.8$ | 1 | $2 + 1 = 3$ | $360$ | $0.360 = 0.36$ |\n| $0.125$ | 3 | $0.04$ | 2 | $3 + 2 = 5$ | $500$ | $0.00500 = 0.005$ |\n| $0.006$ | 3 | $0.007$ | 3 | $3 + 3 = 6$ | $42$ | $0.000042$ |\n

Step-by-Step Worked Multiplication Examples\n

Example 1: Standard Decimal Multiplication\nMultiply 3.45 by 2.8.\n

• Step 1: Count decimal places in factors:\n - $3.45$ has 2 decimal places.\n - $2.8$ has 1 decimal place.\n - Total required decimal places = $2 + 1 = \mathbf{3}$.\n
• Step 2: Multiply whole numbers $345 \times 28$:\n $$\n \begin{array}{r}\n 345 \ \times 28 \ \hline\n 2760 \
  • 6900 \ \hline\n \mathbf{9660}\n \end{array}\n $$\n
• Step 3: Place decimal point 3 positions from the right:\n - $9660 \rightarrow \mathbf{9.660} = \mathbf{9.66}$\n

Example 2: Annexing Leading Zeros in Product\nMultiply 0.042 by 0.15.\n

• Factor decimal places: $0.042$ (3 places) + $0.15$ (2 places) = 5 total decimal places.\n- Multiply whole numbers: $42 \times 15 = 630$.\n- Since $630$ has only 3 digits, annex 2 leading zeros to reach 5 places:\n - $630 \rightarrow \mathbf{0.00630} = \mathbf{0.0063}$\n ---\n

Division of Decimals\n

Decimal division falls into two structural categories: dividing a decimal by a whole number, and dividing a decimal by another decimal.\n

Case 1: Dividing a Decimal by a Whole Number\nWhen the divisor is already a whole number, division is straightforward:\n1. Place the decimal point in the quotient area directly above the decimal point in the dividend.\n2. Divide as you would with whole numbers.\n

Worked Example: $18.36 \div 6$\n- Set up long division: $6 \overline{) 18.36}$.\n- Place decimal point in quotient directly above: $6 \overline{) 18.36} \rightarrow \text{Quotient decimal is } 3.\text{__}$.\n- Execute division: $18 \div 6 = 3$, bring down 3 ($3 < 6 \rightarrow 0$), bring down 6 ($36 \div 6 = 6$).\n- Quotient: 3.06\n

Case 2: Dividing by a Decimal Divisor\nWhen the divisor contains a decimal point, you cannot divide directly. You must transform the divisor into a whole number using the fundamental property of fractions (multiplying divisor and dividend by equal powers of 10).\n

The Divisor Transformation Rule\n1. Count Divisor Decimal Places: Determine how many places the decimal point must shift to the right to make the divisor a whole number.\n2. Shift Dividend Decimal Point: Shift the decimal point in the dividend to the right by the exact same number of places. Annex trailing zeros to the dividend if necessary.\n3. Place Quotient Decimal Point: Place the decimal point in the quotient directly above the new position of the dividend decimal point.\n4. Divide: Perform long division.\n

Step-by-Step Worked Division Examples\n

Example 1: Standard Decimal Division\nEvaluate 15.12 ÷ 0.36.\n

• Step 1: Divisor is $0.36$. Move decimal 2 places right to make it $36$.\n- Step 2: Dividend is $15.12$. Move decimal 2 places right to make it $1512$.\n- Step 3: Problem transforms into whole number division: 1512 ÷ 36.\n- Step 4: Perform long division:\n $$\n \begin{array}{r}\n 42 \ 36 \overline{) 1512} \
  • 144\phantom{0} \ \hline\n 72 \
  • 72 \ \hline\n 0\n \end{array}\n $$\n- Quotient: 42\n

Example 2: Annexing Zeros in Dividend During Division\nEvaluate 9.1 ÷ 0.014.\n

• Divisor $0.014$ requires shifting decimal 3 places right to become 14.\n- Dividend $9.1$ shifted 3 places right becomes 9100 (annexing two zeros).\n- Transform problem to 9100 ÷ 14:\n $9100 \div 14 = 650$\n- Quotient: 650\n ---\n

Terminating vs. Repeating Decimals & Quotient Rounding\n

When dividing numbers, the division process will produce either a terminating decimal or a repeating decimal.\n

• Terminating Decimal: Division reaches a remainder of zero (e.g., $\frac{3}{8} = 0.375$).\n- Repeating Decimal: Division enters an infinite loop of repeating remainders (e.g., $\frac{5}{11} = 0.454545...$). A repeating digit block is written with an overbar: $0.\overline{45}$.\n

Rounding Division Quotients\nIf an ACCUPLACER question requests a quotient rounded to a specific place value (e.g., nearest hundredth), carry the long division out to one extra decimal place (thousandths), then apply standard rounding rules.\n

Worked Rounding Division Example\nDivide 7 by 11, rounded to the nearest hundredth.\n

• Perform long division to thousandths place:\n $7 \div 11 = 0.6363...$\n- Thousandths digit is 6 (the test digit for rounding hundredths).\n- Since $6 \ge 5$, round hundredths digit 3 up to 4.\n- Rounded Quotient: 0.64\n ---\n

ACCUPLACER Exam Traps & Common Errors\n

[!WARNING]\n> Trap 1: Aligning Decimal Points in Multiplication\n> Aligning decimal points vertically when setting up multiplication is unnecessary and leads to calculation errors. Only line up rightmost digits when multiplying.\n [!WARNING]\n> Trap 2: Unequal Decimal Point Shifts in Division\n> If you shift the divisor decimal point 2 places right, you MUST shift the dividend decimal point 2 places right. Shifting $4.2 \div 0.07$ into $4.2 \div 7 = 0.6$ instead of $420 \div 7 = 60$ is a frequent mistake.\n [!WARNING]\n> Trap 3: Dropping Trailing Zeros Before Counting Decimal Places\n> In $0.25 \times 0.4$, raw product is $100$. With $2 + 1 = 3$ decimal places, the result is $0.100 = 0.1$. If you drop the zero from $0.4$ prior to counting places, you will misplace the decimal point.