2.2 Decimal Operations, Place Value, & Practical Applications

2.2 Decimal Operations, Place Value, & Practical Applications

Decimals represent fractional quantities using the base-10 positional system. On the TABE 13&14 Mathematics examination, decimal competency is essential for solving applied workplace problems involving currency, hourly wages, payroll taxes, physical measurements, and unit pricing comparisons.

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Decimal Place Value System & Representations

Every position to the right of the decimal point represents a negative power of $10$ (a fractional part with a denominator of $10, 100, 1,000,$ etc.).

Reading and Writing Decimals

To write a decimal in word form:

1. Read the whole number part to the left of the decimal point.
2. Read the decimal point as the word "and".
3. Read the number to the right of the decimal point as a whole number, followed by the place name of the final digit.

Example: $347.2815$ is read as "Three hundred forty-seven and two thousand eight hundred fifteen ten-thousandths".

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Comparing, Ordering, & Rounding Decimals

Comparing Decimals Using Annexed Zeros

To compare decimals, align the decimal points and annex (pad) trailing zeros so all numbers have the same number of decimal places. Then compare digits from left to right.

Example: Compare $0.4$ and $0.085$.

• Annex zeros: $0.400$ vs. $0.085$.
• In the tenths place, $4 > 0$; therefore, $0.4 > 0.085$ (even though $85 > 4$ as whole numbers).

Rounding Decimals

1. Locate the specified rounding place.
2. Look at the digit immediately to the right:
   • If the digit is $5$ or greater, round up (add $1$ to the target digit).
   • If the digit is $4$ or less, round down (keep the target digit unchanged).
3. Drop all digits to the right of the target place (do not write trailing zeros after a rounded decimal unless specified for significant figures/cents).

Example: Round $18.4762$ to:

• Nearest Whole Number: $18.\mathbf{4}762 \to 4 \le 4 \implies \mathbf{18}$
• Nearest Tenth: $18.4\mathbf{7}62 \to 7 \ge 5 \implies \mathbf{18.5}$
• Nearest Hundredth (Cent): $18.47\mathbf{6}2 \to 6 \ge 5 \implies \mathbf{18.48}$
• Nearest Thousandth: $18.476\mathbf{2} \to 2 \le 4 \implies \mathbf{18.476}$

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Decimal Addition & Subtraction

[!IMPORTANT] The Golden Rule for Addition and Subtraction: Always line up the decimal points vertically so that tenths align with tenths, hundredths with hundredths, and ones with ones. Annex zeros to the right of shorter decimals to prevent subtraction errors.

Worked Example: Subtraction with Annexed Zeros

Problem: Compute $42.3 - 17.846$.

& 4 & 1 & 12 & 9 & 10 \\ & \not{4} & \not{2} . & \not{3} & \not{0} & \not{0} \\ - & 1 & 7 . & 8 & 4 & 6 \\ \hline & \mathbf{2} & \mathbf{4} . & \mathbf{4} & \mathbf{5} & \mathbf{4} \end{array}$$ - Annex two zeros to $42.3$ to make $42.300$. - Align decimal points and borrow across digits: $10 - 6 = 4$; $9 - 4 = 5$; $12 - 8 = 4$; $11 - 7 = 4$; $3 - 1 = 2$. - Bring the decimal point straight down: **$24.454$**. --- ## Decimal Multiplication Algorithm When multiplying decimals, do **NOT** line up decimal points. Follow this three-step algorithm: 1. Multiply the numbers as if they were whole integers (ignoring decimal points). 2. Count the **total number of decimal places** to the right of the decimal points across all factors. 3. Starting from the extreme right of the product, move the decimal point to the left by that total count. ### Worked Example: Decimal Multiplication **Problem:** Compute $3.45 \times 0.28$. $$\begin{aligned} &\phantom{\times} 3.45 \quad (2\text{ decimal places}) \\ \times &\phantom{3} 0.28 \quad (2\text{ decimal places}) \\ \hline &\phantom{0} 2760 \quad (345 \times 8) \\ + &\phantom{0} 6900 \quad (345 \times 20) \\ \hline &\phantom{0} \mathbf{0.9660} \quad (4\text{ decimal places} \implies \mathbf{0.966}) \end{aligned}$$ - Total decimal places needed: $2 + 2 = 4$ decimal places. - Count $4$ positions left from the right of $9660$: $0.9660 = \mathbf{0.966}$. --- ## Decimal Division Algorithm To divide when the divisor is a decimal: 1. **Make the divisor a whole number:** Shift the divisor's decimal point all the way to the right until it is an integer. Count how many places $k$ you shifted. 2. **Shift the dividend by the same amount:** Shift the dividend's decimal point $k$ places to the right (annexing zeros if necessary). 3. **Place the decimal point in the quotient:** Bring the decimal point directly straight up into the quotient space above the dividend. 4. Divide using standard whole-number long division. ``` 3 4 . 5 <-- Decimal placed straight up __________________ 1.4 ) 4 8 . 3 0 ==> 14 ) 4 8 3 . 0 4 2 --- 6 3 5 6 --- 7 0 7 0 --- 0 ``` **Problem:** $48.3 \div 1.4 = 483 \div 14 = \mathbf{34.5}$. --- ## Practical Workplace & Applied Math Scenarios ### 1. Hourly Wages and Overtime Pay Under standard workplace rules, overtime hours (exceeding $40$ hours per week) are compensated at "time-and-a-half" ($1.5 \times \text{base hourly wage}$). $$\text{Total Earnings} = (40 \times \text{Hourly Wage}) + (\text{Overtime Hours} \times 1.5 \times \text{Hourly Wage})$$ **Example:** An employee earns $\$18.40$ per hour and works $46.5$ hours in a week. - Regular earnings: $40 \times \$18.40 = \$736.00$ - Overtime wage rate: $1.5 \times \$18.40 = \$27.60$ per hour - Overtime hours: $46.5 - 40 = 6.5$ hours - Overtime earnings: $6.5 \times \$27.60 = \$179.40$ - Total gross pay: $\$736.00 + \$179.40 = \mathbf{\$915.40}$ ### 2. Unit Pricing & Best Value Analysis To determine the most economical purchase, calculate the unit cost by dividing total price by total quantity: $$\text{Unit Price} = \frac{\text{Total Cost}}{\text{Quantity / Volume}}$$ | Brand Option | Package Size | Package Cost | Unit Price Calculation | Unit Cost (per oz) | | :--- | :--- | :--- | :--- | :--- | | **Brand A** | $16\text{ oz}$ | $\$4.48$ | $\$4.48 \div 16$ | $\mathbf{\$0.280}$ | | **Brand B** | $24\text{ oz}$ | $\$6.24$ | $\$6.24 \div 24$ | $\mathbf{\$0.260}$ (Best Value) | | **Brand C** | $32\text{ oz}$ | $\$8.96$ | $\$8.96 \div 32$ | $\mathbf{\$0.280}$ |