3.4 Decimal Word Problems & Financial Calculations

3.4 Decimal Word Problems & Financial Calculations

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.

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Unit Pricing and Best-Buy Comparisons

When purchasing goods sold in different package sizes and prices, consumers compare unit prices to determine which option offers the best financial value.

The Unit Price Formula

$\text{Unit Price} = \frac{\text{Total Price or Cost}}{\text{Quantity (ounces, pounds, items)}}$

Comparative Unit Price Table

Consider comparing three options for purchasing olive oil:

Step-by-Step Worked Example: Best-Buy Analysis

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?

• Step 1: Calculate Unit Price for Brand A: $\text{Unit Price}_A = \frac{\$4.32}{16} = \mathbf{\$0.27 \text{ per ounce}}$

• Step 2: Calculate Unit Price for Brand B: $\text{Unit Price}_B = \frac{\$6.00}{24} = \mathbf{\$0.25 \text{ per ounce}}$

• Step 3: Compare Unit Prices and calculate savings: $\$0.27 - \$0.25 = \mathbf{\$0.02 \text{ per ounce}}$

• Conclusion: Brand B is the better buy by $0.02 per ounce.

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Total Purchase Cost, Subtotals & Cash Change

Retail transactions involve multiplying quantities by unit prices, summing subtotals, and determining cash change returned from tendered cash payments.

General Financial Transaction Procedure

1. Itemized Subtotals: Multiply quantity by item price for each line item.
2. Total Cost: Sum all itemized subtotals.
3. Change Returned: Subtract Total Purchase Cost from Amount Tendered (Cash Paid).

Step-by-Step Worked Example: Multi-Item Register Transaction

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?

• Step 1: Calculate line item subtotals:

  • Notebooks: $3 \times \text{\textdollar}2.85 = \mathbf{\text{\textdollar}8.55}$
  • Gel pens: $4 \times \text{\textdollar}1.45 = \mathbf{\text{\textdollar}5.80}$
  • Desk lamp: $1 \times \text{\textdollar}18.95 = \mathbf{\text{\textdollar}18.95}$

• Step 2: Calculate Total Purchase Cost:

  $$\begin{array}{rl} 8.55 \\ 5.80 \\ + 18.95 \\ \hline \mathbf{33.30} & (\text{Total Cost}) \end{array}$$

• Step 3: Calculate Change Due from $50.00:

  $$\begin{array}{rl} 50.00 \\ - 33.30 \\ \hline \mathbf{16.70} & (\text{Change Due}) \end{array}$$

• Final Answer: David receives $16.70 in change.

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Wage and Paycheck Mathematics

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.

Key Paycheck Formulas

• Regular Pay = $\text{Regular Hours (up to 40)} \times \text{Hourly Rate}$
• Overtime Rate = $1.5 \times \text{Hourly Rate}$
• Overtime Pay = $\text{Overtime Hours (hours over 40)} \times \text{Overtime Rate}$
• Gross Pay = $\text{Regular Pay} + \text{Overtime Pay}$
• Net Pay (Take-Home Pay) = $\text{Gross Pay} - \text{Total Deductions}$

Step-by-Step Worked Paycheck Example

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?

• Step 1: Determine regular and overtime hours:

  • Regular hours = 40
  • Overtime hours = $46 - 40 = \mathbf{6}$

• Step 2: Compute Regular Pay: $\text{Regular Pay} = 40 \times \$18.50 = \mathbf{\$740.00}$

• Step 3: Compute Overtime Rate and Overtime Pay:

  • Overtime Rate = $1.5 \times \text{\textdollar}18.50 = \mathbf{\text{\textdollar}27.75 \text{ per hour}}$
  • Overtime Pay = $6 \times \text{\textdollar}27.75 = \mathbf{\text{\textdollar}166.50}$

• Step 4: Compute Gross Pay: $\text{Gross Pay} = \$740.00 + \$166.50 = \mathbf{\$906.50}$

• Step 5: Compute Net Pay: $\text{Net Pay} = \$906.50 - \$142.80 = \mathbf{\$763.70}$

• Final Answer: Samantha's net pay is $763.70.

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Multi-Step Utility & Fuel Efficiency Applications

Decimal word problems frequently combine division and multiplication in practical physical applications, such as calculating automobile fuel economy and travel costs.

Worked Example: Fuel Efficiency and Travel Cost

A driver traveled 418.5 miles on a single tank of gasoline, consuming 13.5 gallons of fuel. Gasoline costs $3.60 per gallon.

1. What was the vehicle's fuel efficiency in miles per gallon (MPG)?
2. What was the total cost of the fuel consumed on this trip?

• Part 1: Calculate Fuel Efficiency (MPG): $\text{MPG} = \frac{\text{Miles Driven}}{\text{Gallons Consumed}} = \frac{418.5}{13.5}$

  • Shift decimal 1 place right: $4185 \div 135 = \mathbf{31 \text{ MPG}}$

• Part 2: Calculate Total Fuel Cost: $\text{Total Cost} = 13.5 \text{ gallons} \times \$3.60 / \text{gallon} = \mathbf{\$48.60}$

• Final Answers: 31 MPG and $48.60 total fuel cost.

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ACCUPLACER Exam Traps & Common Errors