Money & Decimal Word Problems - Sixth Grade Mathematics

Solving real-world decimal and money problems requires translating narrative scenarios into multi-step arithmetic, managing currency conventions, and evaluating unit rates.

Financial Conventions and Currency Precision

In financial mathematics, calculations model dollars and cents. Currency requires adherence to specific formatting rules: Standard Rounding: Currency is rounded to the nearest hundredth (cent), two places past the decimal point. The Dollar Sign ($): Written immediately before the number without a space. Two Decimal Places Always: When writing currency amounts, always write two digits after the decimal point (write $14.50, never $14.5).

Financial Arithmetic Structure:
Subtotal  = Sum of all purchased goods
Sales Tax = Subtotal * Tax Rate
Total Due = Subtotal + Sales Tax
Change    = Amount Tendered - Total Due

Unit Pricing and Best Buys

A unit price is the cost of a single unit of item (such as price per ounce, price per pound, or price per liter). Unit pricing allows consumers to compare the true cost of items packaged in different quantities.

Unit Price Formula:
               Unit Price = Total Cost / Total Quantity

Example: Which is the better buy?
Brand A: 16 ounces of cereal for $3.84  --> $3.84 / 16 = $0.24 per ounce
Brand B: 24 ounces of cereal for $5.28  --> $5.28 / 24 = $0.22 per ounce
Conclusion: Brand B is the better buy because it costs $0.02 less per ounce!

Multi-Step Real-World Problem Architecture

Resolving multi-step decimal problems involves breaking the narrative into sequential goals.

+-------------------------------------------------------------------------+
|                  MULTI-STEP DECIMAL PROBLEM ARCHITECTURE                |
|                                                                         |
|   Phase 1: Determine Itemized Subtotals (Multiplication).               |
|            Multiply quantity by unit cost for each item.                |
|   Phase 2: Calculate Aggregate Subtotal (Addition).                     |
|            Sum all item subtotals.                                      |
|   Phase 3: Apply Deductions or Additions (Discounts / Taxes / Tips).    |
|            Subtract discounts or add taxes/tips to find total bill.     |
|   Phase 4: Determine Final Balance or Change (Subtraction).             |
|            Subtract total bill from payment amount.                     |
+-------------------------------------------------------------------------+

Worked Case Study:
Javier purchases 3 notebooks at $2.49 each, 2 pens at $1.15 each, and
1 backpack for $24.80. He has a coupon for $5.00 off the total purchase.
If Javier pays with two $20 bills, how much change should he receive?

1. Calculate item subtotals:
   - Notebooks: 3 * 2.49 = $7.47
   - Pens:      2 * 1.15 = $2.30
   - Backpack:  1 * 24.80 = $24.80

2. Calculate aggregate subtotal:
   7.47 + 2.30 + 24.80 = $34.57

3. Apply coupon deduction:
   34.57 - 5.00 = $29.57 (Total Due)

4. Calculate payment tendered and change:
   Javier pays with two $20 bills: 2 * 20 = $40.00.
   Change = 40.00 - 29.57 = $10.43.
   Javier receives exactly $10.43 in change.

Exercises

Exercise 1. A grocery store sells a 12-pack of sparkling water for $5.40 and an 18-pack of the same water for $7.56. Calculate the unit price per can for each package, and identify which pack is the better buy.

Exercise 2. Samantha worked 6.5 hours on Saturday and 4.25 hours on Sunday. If she earns $14.20 per hour, calculate her total gross earnings for the weekend.

Exercise 3. An Uber driver drove 184.6 miles on 7.1 gallons of gasoline. What was the vehicle's fuel efficiency in miles per gallon?

Solutions

Solution to Exercise 1. For the 12-pack: divide total cost by count: $5.40 / 12 = $0.45 per can. For the 18-pack: divide total cost by count: $7.56 / 18. Set up long division: 18 into 75 goes 4 times (4 * 18 = 72; 75 - 72 = 3). Bring down 6 to make 36: 18 into 36 goes 2 times (2 * 18 = 36; remainder 0). The unit price for the 18-pack is $0.42 per can. Comparing $0.45 and $0.42, the 18-pack is the better buy because each can costs $0.03 less.

Solution to Exercise 2. First, calculate total hours worked: 6.5 + 4.25 = 10.75 hours. Next, multiply total hours by hourly wage: 10.75 * 14.20. Multiply 1,075 by 142 as whole numbers: 1075 * 2 = 2150; 1075 * 40 = 43000; 1075 * 100 = 107500. Sum: 2150 + 43000 + 107500 = 152650. Count decimal places: 10.75 has 2 places, and 14.20 has 2 places (or 14.2 has 1 place). With 10.75 (2 places) * 14.2 (1 place), total decimal places = 3. Shifting 3 places in 152650 gives $152.65. Samantha earned $152.65 for the weekend.

Solution to Exercise 3. To find miles per gallon, divide total miles by total gallons: 184.6 / 7.1. Shift the decimal point 1 place right in both numbers to eliminate the decimal in the divisor: 1,846 / 71. Perform long division: 71 into 184 goes 2 times (2 * 71 = 142; 184 - 142 = 42). Bring down 6 to make 426: 71 into 426 goes exactly 6 times (6 * 71 = 426; remainder 0). The vehicle achieved exactly 26 miles per gallon.