5.3 Decimal Operations & Place-Value Conversions

5.3 Decimal Operations & Place-Value Conversions

Core Concept: Decimals provide a continuous, base-10 extension of whole numbers into fractional quantities. Mastering decimal operations on the ACCUPLACER requires strict place-value alignment in addition and subtraction, decimal-place counting in multiplication, divisor normalization in division, and fluent conversion between terminating/repeating decimals and rational fractions.

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1. Decimal Place Value Architecture & Comparison Protocols

Digits positioned to the right of the decimal point represent fractional quantities whose denominators are increasing powers of $10$ ($10^{-n}$).

+-------------------------------------------------------------------------------------------------------------+
|                                    DECIMAL PLACE VALUE POSITION CHART                                       |
+---------------+-------+-------------------+-------------------+-------------------+-------------------------+
| Place Name    | Ones  | Tenths            | Hundredths        | Thousandths       | Ten-Thousandths         |
| Exponent      | 10^0  | 10^-1             | 10^-2             | 10^-3             | 10^-4                   |
| Fraction Form | 1     | 1/10              | 1/100             | 1/1,000           | 1/10,000                |
| Decimal Value | 1.0   | 0.1               | 0.01              | 0.001             | 0.0001                  |
+---------------+-------+-------------------+-------------------+-------------------+-------------------------+

Comparing and Ordering Decimals

To compare decimals with varying numbers of decimal digits, pad trailing zeros so that all numbers have the same number of decimal places, then compare digit by digit from left to right.

• Worked Example: Order from least to greatest: $0.405, \; 0.45, \; 0.0495, \; 0.4005, \; 0.4$.
  1. Pad each number to four decimal places (matching $0.0495$ and $0.4005$):
     • $0.0495 \to 0.0495$
     • $0.4000 \to 0.4000$
     • $0.4005 \to 0.4005$
     • $0.4050 \to 0.4050$
     • $0.4500 \to 0.4500$
  2. Compare the four-digit integer values ($495 < 4000 < 4005 < 4050 < 4500$): $$\mathbf{0.0495 < 0.4 < 0.4005 < 0.405 < 0.45}$$

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2. Addition and Subtraction of Decimals

The golden rule for adding and subtracting decimals is vertical alignment of the decimal points. Trailing zeros must be explicitly written in the minuend during subtraction to ensure correct borrowing.

$$\begin{array}{rl} \phantom{-} 42.300 & \text{(Minuend padded with two trailing zeros)} \\ -\ 17.864 & \text{(Subtrahend)} \\ \hline \mathbf{24.436} & \text{(Difference)} \end{array}$$

• Borrowing Walkthrough:
  • In $42.300$, subtracting $4$ from $0$ requires borrowing from the tenths place ($3$).
  • Decompose $.300$ into $.2$ tenths, $9$ hundredths, and $10$ thousandths.
  • Thousandths: $10 - 4 = 6$
  • Hundredths: $9 - 6 = 3$
  • Tenths: Borrow $1$ from the units digit $2$ in $42$ $\implies 12 - 8 = 4$
  • Units: $1$ (from $41$) minus $7$ requires borrowing from $4$ $\implies 11 - 7 = 4$
  • Tens: $3 - 1 = 2$
  • Total: $24.436$.

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3. Multiplication of Decimals

When multiplying decimals, do not align decimal points. Follow this three-step algorithm:

1. Multiply the factors as if they were whole numbers (ignoring all decimal points).
2. Count the total number of decimal places across all factors ($d_{\text{total}} = d_1 + d_2$).
3. Place the decimal point in the product so that it has $d_{\text{total}}$ digits to the right of the decimal point, inserting leading zeros if necessary.

$$\begin{array}{rl} 3.45 & \text{(2 decimal places)} \\ \times\phantom{0} 0.072 & \text{(3 decimal places)} \\ \hline 690 & (345 \times 2) \\ +\ 24150 & (345 \times 70) \\ \hline 24840 & \text{(Integer product: } 345 \times 72 = 24,840\text{)} \\ \mathbf{0.24840} & \text{(Shift 5 places left: } 2 + 3 = 5\text{ decimal places)} \\ = \mathbf{0.2484} & \text{(Drop trailing zero)} \end{array}$$

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4. Division of Decimals (Normalizing the Divisor)

Division by a decimal requires transforming the divisor into a whole integer before executing the division algorithm:

+-------------------------------------------------------------------------------------------------------+
|                                     DECIMAL DIVISION PROCEDURE                                        |
+--------+--------------------------+-------------------------------------------------------------------+
| Step 1 | Normalize Divisor        | Shift the divisor's decimal point k places right until it is an   |
|        |                          | integer.                                                          |
+--------+--------------------------+-------------------------------------------------------------------+
| Step 2 | Balance Dividend         | Shift the dividend's decimal point the exact same k places to the |
|        |                          | right (appending zeros if the dividend runs out of digits).       |
+--------+--------------------------+-------------------------------------------------------------------+
| Step 3 | Position Quotient Point  | Place a decimal point in the quotient directly above the new      |
|        |                          | decimal point in the dividend.                                    |
+--------+--------------------------+-------------------------------------------------------------------+
| Step 4 | Execute Long Division    | Divide as with standard whole numbers.                            |
+--------+--------------------------+-------------------------------------------------------------------+

• Worked Example: Calculate $15.75 \div 0.035$.
  1. The divisor $0.035$ has $3$ decimal places. Multiply both divisor and dividend by $1,000$ (shift $3$ places right): $$0.035 \times 1,000 = 35$$ $$15.75 \times 1,000 = 15,750$$
  2. Compute $15,750 \div 35$: $$157 \div 35 = 4 \text{ (since } 4 \times 35 = 140\text{)}, \quad 157 - 140 = 17$$ $$175 \div 35 = 5 \text{ (since } 5 \times 35 = 175\text{)}, \quad 175 - 175 = 0$$ $$0 \div 35 = 0$$
  3. Quotient: $450$.

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5. Terminating vs. Repeating Decimals & Fraction Conversions

The Prime Factor Criterion for Terminating Decimals

A simplified rational fraction $\frac{p}{q}$ in lowest terms produces a terminating decimal if and only if the prime factorization of the denominator $q$ contains no prime factors other than $2$ and/or $5$ ($q = 2^a \times 5^b$). If any other prime factor exists (such as $3, 7, 11, 13$), the fraction produces an infinitely repeating decimal.

• $\frac{7}{40} = \frac{7}{2^3 \times 5^1} \implies$ Terminating ($0.175$).
• $\frac{5}{24} = \frac{5}{2^3 \times 3^1} \implies$ Repeating ($0.208333\dots = 0.208\bar{3}$) due to prime factor $3$.

Converting Decimals to Fractions

• Terminating Decimals: Write the digits over $10^n$ (where $n$ is the number of decimal digits) and reduce by dividing by the GCF: $$0.375 = \frac{375}{1,000} = \frac{375 \div 125}{1,000 \div 125} = \mathbf{\frac{3}{8}}$$
• Repeating Decimals (Algebraic Subtraction Method): Convert $0.\overline{54} = 0.545454\dots$ to a fraction:
  1. Let $x = 0.\overline{54}$.
  2. Multiply by $10^2 = 100$ (since the repeating block has 2 digits): $100x = 54.\overline{54}$.
  3. Subtract the original equation: $$\begin{array}{rl} 100x &= 54.545454\dots \\ -\phantom{00} x &= \phantom{0}0.545454\dots \\ \hline 99x &= 54 \end{array}$$
  4. Solve for $x$ and reduce: $$x = \frac{54}{99} = \frac{54 \div 9}{99 \div 9} = \mathbf{\frac{6}{11}}$$

Essential Decimal-to-Fraction Reference Table

+-------------------------------------------------------------------------------------------------------+
|                                 STANDARD FRACTION-DECIMAL CONVERSIONS                                 |
+-------------------+-------------------+-------------------+-------------------+-----------------------+
| Fraction          | Decimal Value     | Fraction          | Decimal Value     | Fraction   | Decimal  |
+-------------------+-------------------+-------------------+-------------------+------------+----------+
| 1/2               | 0.5               | 1/5               | 0.2               | 1/8        | 0.125    |
| 1/3               | 0.333... (0.3bar) | 2/5               | 0.4               | 3/8        | 0.375    |
| 2/3               | 0.666... (0.6bar) | 3/5               | 0.6               | 5/8        | 0.625    |
| 1/4               | 0.25              | 4/5               | 0.8               | 7/8        | 0.875    |
| 3/4               | 0.75              | 1/6               | 0.166... (0.16bar)| 1/10       | 0.100    |
+-------------------+-------------------+-------------------+-------------------+------------+----------+