3.4 Decimal Word Problems & Financial Calculations
Key Takeaways
- Unit price equals total cost divided by quantity (Unit Price = Total Cost / Quantity); comparing unit prices identifies the best value regardless of package size.
- Total purchase cost equals the sum of individual item subtotals (Quantity x Unit Price); cash change due equals amount tendered minus total purchase cost.
- Gross paycheck earnings equal regular wage pay (up to 40 hours) plus overtime pay (1.5 x hourly wage x hours over 40); net pay equals gross pay minus itemized decimal deductions.
- Multi-step decimal word problems require systematic step-by-step calculation pathways while preserving exact decimal precision throughout intermediate steps.
- In financial calculations, maintain precision to at least 3-4 decimal places during intermediate steps and round to the nearest cent ($0.01) only at the final step.
3.4 Decimal Word Problems & Financial Calculations\n
Real-world applications of arithmetic frequently involve decimal numbers, particularly in context of money, wage calculations, unit pricing, and scientific or technical measurements. Word problems on the ACCUPLACER Arithmetic exam assess your ability to translate verbal scenarios into multi-step mathematical operations while executing decimal arithmetic with absolute precision.\n ---\n
Unit Pricing and Best-Buy Comparisons\n
When purchasing goods sold in different package sizes and prices, consumers compare unit prices to determine which option offers the best financial value.\n
The Unit Price Formula\n\n
Comparative Unit Price Table\n
Consider comparing three options for purchasing olive oil:\n | Option | Package Size | Total Price | Unit Price Calculation | Unit Price per Ounce | Best Value Rank |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| Brand A | 12 oz bottle | $4.20 | $\frac{$4.20}{12}$ | $0.35$ / oz | 3rd (Most Expensive) |\n| Brand B | 20 oz bottle | $6.40 | $\frac{$6.40}{20}$ | $0.32$ / oz | 2nd |\n| Brand C | 32 oz bottle | $8.96 | $\frac{$8.96}{32}$ | $0.28$ / oz | 1st (Best Buy) |\n
Step-by-Step Worked Example: Best-Buy Analysis\n
A grocery store sells Brand A peanut butter in a 16-ounce jar for $4.32 and Brand B peanut butter in a 24-ounce jar for $6.00. Which brand is the better buy, and by how much per ounce?\n
- Step 1: Calculate Unit Price for Brand A:\n \n
- Step 2: Calculate Unit Price for Brand B:\n \n
- Step 3: Compare Unit Prices and calculate savings:\n \n
- Conclusion: Brand B is the better buy by $0.02 per ounce.\n ---\n
Total Purchase Cost, Subtotals & Cash Change\n
Retail transactions involve multiplying quantities by unit prices, summing subtotals, and determining cash change returned from tendered cash payments.\n
General Financial Transaction Procedure\n1. Itemized Subtotals: Multiply quantity by item price for each line item.\n2. Total Cost: Sum all itemized subtotals.\n3. Change Returned: Subtract Total Purchase Cost from Amount Tendered (Cash Paid).\n
Step-by-Step Worked Example: Multi-Item Register Transaction\n
David purchases 3 notebooks at $2.85 each, 4 gel pens at $1.45 each, and 1 desk lamp for $18.95. If he pays with a $50 bill, how much change should he receive?\n
- Step 1: Calculate line item subtotals:\n - Notebooks: $3 \times $2.85 = \mathbf{$8.55}$\n - Gel pens: $4 \times $1.45 = \mathbf{$5.80}$\n - Desk lamp: $1 \times $18.95 = \mathbf{$18.95}$\n
- Step 2: Calculate Total Purchase Cost:\n $$\n \begin{array}{rl}\n 8.55 \
5.80 \
- 18.95 \ \hline\n \mathbf{33.30} & (\text{Total Cost})\n \end{array}\n $$\n
- Step 3: Calculate Change Due from $50.00:\n $$\n \begin{array}{rl}\n 50.00 \
- 33.30 \ \hline\n \mathbf{16.70} & (\text{Change Due})\n \end{array}\n $$\n
- Final Answer: David receives $16.70 in change.\n ---\n
Wage and Paycheck Mathematics\n
Calculating gross pay involves computing regular hourly wages and overtime pay. Overtime is typically paid at "time-and-a-half" ($1.5 \times \text{regular hourly wage}$) for all hours worked beyond 40 hours per week.\n
Key Paycheck Formulas\n- Regular Pay = $\text{Regular Hours (up to 40)} \times \text{Hourly Rate}$\n- Overtime Rate = $1.5 \times \text{Hourly Rate}$\n- Overtime Pay = $\text{Overtime Hours (hours over 40)} \times \text{Overtime Rate}$\n- Gross Pay = $\text{Regular Pay} + \text{Overtime Pay}$\n- Net Pay (Take-Home Pay) = $\text{Gross Pay} - \text{Total Deductions}$\n
Step-by-Step Worked Paycheck Example\n
Samantha earns $18.50 per hour as a dental technician. Last week she worked 46 hours. Her mandatory payroll deductions (tax, insurance, retirement) totaled $142.80. What was Samantha's net pay for the week?\n
- Step 1: Determine regular and overtime hours:\n - Regular hours = 40\n - Overtime hours = $46 - 40 = \mathbf{6}$\n
- Step 2: Compute Regular Pay:\n \n
- Step 3: Compute Overtime Rate and Overtime Pay:\n - Overtime Rate = $1.5 \times $18.50 = \mathbf{$27.75 \text{ per hour}}$\n - Overtime Pay = $6 \times $27.75 = \mathbf{$166.50}$\n
- Step 4: Compute Gross Pay:\n \n
- Step 5: Compute Net Pay:\n \n
- Final Answer: Samantha's net pay is $763.70.\n ---\n
Multi-Step Utility & Fuel Efficiency Applications\n
Decimal word problems frequently combine division and multiplication in practical physical applications, such as calculating automobile fuel economy and travel costs.\n
Worked Example: Fuel Efficiency and Travel Cost\n
A driver traveled 418.5 miles on a single tank of gasoline, consuming 13.5 gallons of fuel. Gasoline costs $3.60 per gallon.\n1. What was the vehicle's fuel efficiency in miles per gallon (MPG)?\n2. What was the total cost of the fuel consumed on this trip?\n
- Part 1: Calculate Fuel Efficiency (MPG):\n \n - Shift decimal 1 place right: $4185 \div 135 = \mathbf{31 \text{ MPG}}$\n
- Part 2: Calculate Total Fuel Cost:\n \n
- Final Answers: 31 MPG and $48.60 total fuel cost.\n ---\n
ACCUPLACER Exam Traps & Common Errors\n
[!WARNING]\n> Trap 1: Inverted Unit Price Division\n> Dividing quantity by price (e.g., $16 / $4.32 = 3.70$ ounces per dollar) instead of price by quantity ($$4.32 / 16 = $0.27$ per ounce) is a classic trap. Unit price always has money in the numerator.\n [!WARNING]\n> Trap 2: Forgetting the 1.5 Overtime Multiplier\n> Calculating all 46 hours at the regular $18.50 rate ($46 \times 18.50 = $851.00$) forgets that hours over 40 earn $1.5 \times$ wage.\n [!WARNING]\n> Trap 3: Premature Rounding in Financial Steps\n> Rounding intermediate quotients to two decimal places early can introduce errors in cents. Maintain exact decimals until the final answer.
Brand A sells a 20-ounce jar of peanut butter for $4.20. Brand B sells a 32-ounce jar of peanut butter for $7.04. Which brand is the better buy and by how much per ounce?
An employee earns an hourly rate of $16.40. During a one-week pay period, the employee works 45 hours. If total taxes and deductions equal $128.50, what is the employee's net pay?
David buys 4 light bulbs at $3.75 each, 2 light switches at $4.20 each, and 1 roll of electrical tape for $2.85. If he pays with a $50 bill, how much change should he receive?
A motorist drove 418.5 miles on a single tank of gas and used 13.5 gallons. Gas costs $3.60 per gallon. What was the vehicle's fuel efficiency in miles per gallon, and how much did the trip cost in total gas expenditure?