10.2 Sales Tax (HST), Discounts & Purchasing Power

10.2 Sales Tax (HST), Discounts & Purchasing Power

Quick Summary: Consumer math on the Ontario MPT centers on calculating Harmonized Sales Tax (HST = 13%), applying trade and promotional discounts, evaluating sequential (stacked) percentage changes, performing unit price comparisons, and calculating the erosion of purchasing power due to inflation. Understanding pre-tax vs. post-tax pricing multipliers ($1.13$) and avoiding common stacked discount errors ($20% + 10% \neq 30%$) are critical skills for Ontario educators.

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Ontario Harmonized Sales Tax (HST = 13%)

In Ontario, retail transactions are subject to the Harmonized Sales Tax (HST), a single blended sales tax rate that replaced the separate federal Goods and Services Tax (GST) and provincial Retail Sales Tax (PST) in 2010.

HST Composition & Multipliers

$\text{Ontario HST Rate} = 5\% \text{ (Federal GST)} + 8\% \text{ (Provincial PST)} = 13\% \text{ Total}$

When solving tax problems on the MPT, convert percentages directly to decimal multipliers:

1. Calculating Tax Amount: $\text{HST Amount} = \text{Pre-Tax Price} \times 0.13$

2. Calculating Total Cost (Price + Tax): $\text{Total Cost} = \text{Pre-Tax Price} \times (1 + 0.13) = \text{Pre-Tax Price} \times 1.13$

3. Working Backwards (Extracting Pre-Tax Price from Total Receipt): If given the total cost including 13% HST, divide by $1.13$ to find the pre-tax price: $\text{Pre-Tax Price} = \frac{\text{Total Cost}}{1.13}$ $\text{HST Amount} = \text{Total Cost} - \text{Pre-Tax Price} = \text{Total Cost} \times \left(1 - \frac{1}{1.13}\right)$

Ontario Point-of-Sale Exemption Note: Certain items in Ontario qualify for a point-of-sale rebate on the 8% provincial portion of the HST, meaning consumers pay only the 5% GST portion. Qualifying items include basic groceries (milk, bread, fresh produce), prescription drugs, qualifying children's clothing/footwear, and printed books.

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Single & Sequential (Stacked) Discounts

Discounts reduce the original list price of an item. On the MPT, sales tax is always applied to the discounted sale price, never the original pre-discount price!

Single Discount Calculation

For an item with original price $P$ and discount rate $d$ (expressed as a decimal):

$\text{Discount Amount} = P \times d$

$\text{Sale Price} = P - (P \times d) = P \times (1 - d)$

Sequential (Stacked) Discounts

Retailers often offer stacked discounts, such as "Take 20% off the original price, plus an extra 10% off at checkout with a coupon."

CRITICAL WARNING — THE ADDITION PITFALL:
Never add discount percentages together! A 20% discount followed by a 10% discount is NOT a 30% discount ($20% + 10% \neq 30%$). The second discount applies only to the already-reduced price, not the original price!

graph LR
    P0["Original Price: P"] -->| Apply 20% Off | P1["Intermediate Price:<br/>P × (1 - 0.20) = 0.80 P"]
    P1 -->| Apply 10% Coupon | P2["Final Discounted Price:<br/>0.80 P × (1 - 0.10) = 0.72 P"]
    P2 -->| Add 13% HST | FINAL["Total Receipt Cost:<br/>0.72 P × 1.13 = 0.8136 P"]

General Multiplier Formula for Stacked Discounts & HST

For an original price $P$, sequential discount rates $d_1$ and $d_2$, and 13% HST:

$\text{Final Discounted Price} = P \times (1 - d_1) \times (1 - d_2)$

$\text{Total Cost with HST} = P \times (1 - d_1) \times (1 - d_2) \times 1.13$

$\text{Effective Single Discount Rate} = 1 - (1 - d_1)(1 - d_2)$

Example: For sequential discounts of 20% ($d_1 = 0.20$) and 10% ($d_2 = 0.10$): $\text{Multiplier} = (1 - 0.20)(1 - 0.10) = 0.80 \times 0.90 = 0.72$ This corresponds to an effective single discount of $28%$ ($1 - 0.72 = 0.28$), not $30%$!

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Unit Pricing Comparison (Best Buy Analysis)

Unit pricing standardizes product costs to allow consumers to compare value across different package sizes and brands.

Unit Price Formula

$\text{Unit Price} = \frac{\text{Total Price of Package}}{\text{Total Quantity / Mass / Volume}}$

Common standard unit bases in Ontario grocery pricing include:

• Dollars per 100 grams ($$/100\text{ g}$) or per kilogram ($$/\text{kg}$)
• Dollars per 100 milliliters ($$/100\text{ mL}$) or per liter ($$/\text{L}$)
• Dollars per individual item or unit (e.g., $$/\text{roll}$ for paper towels)

Analysis: Option B ($$1.60/100\text{ g}$) represents the best buy because it offers the lowest cost per standardized unit.

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Purchasing Power & Inflation Impact

Inflation is the general increase in prices and fall in the purchasing value of money over time, measured in Canada by Statistics Canada using the Consumer Price Index (CPI).

Real Purchasing Power Equation

When prices rise by an annual inflation rate $i$ (expressed as a decimal), the real purchasing power of a fixed sum of money decreases.

$\text{Real Value of Money} = \frac{\text{Nominal Value}}{(1 + i)^t}$

Where:

• $\text{Nominal Value}$ is the face dollar amount.
• $i$ is the annual rate of inflation.
• $t$ is the number of years.

Real Interest Rate Approximation

The real rate of return on an investment or savings account accounts for inflation:

$\text{Real Interest Rate} \approx \text{Nominal Interest Rate} - \text{Inflation Rate}$

Example: If a savings account pays $3.5%$ nominal interest while inflation is $2.5%$, the real growth of purchasing power is only $3.5% - 2.5% = 1.0%$. If inflation exceeds the nominal interest rate (e.g., $2.0%$ interest vs $3.0%$ inflation), real purchasing power decreases ($-1.0%$ real return)!

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Worked Step-by-Step Problem Walkthroughs

Problem 1: Sequential Discount and HST Calculation

Problem Statement: A classroom projector listed at $800.00 is on clearance at an Ontario electronics store. The store advertises 25% off the list price. Additionally, educators receive a 10% promotional discount applied at the cash register.

1. Calculate the final discounted price of the projector before taxes.
2. Calculate the total receipt cost including Ontario's 13% HST.
3. Find the effective single discount percentage overall.

Step-by-Step Solution:

1. Calculate Price After First Discount (25% off): $\text{Price}_1 = \$800.00 \times (1 - 0.25) = \$800.00 \times 0.75 = \$600.00$

2. Calculate Price After Second Discount (10% off): $\text{Price}_2 = \$600.00 \times (1 - 0.10) = \$600.00 \times 0.90 = \$540.00$

3. Calculate Total Cost with 13% HST: $\text{Total Cost} = \$540.00 \times 1.13 = \$610.20$

4. Determine Effective Single Discount Percentage:

   • Total dollar discount = $$800.00 - $540.00 = $260.00$
   • Effective Discount Rate = $\frac{$260.00}{$800.00} = 0.325 = 32.5%$
   • Direct Multiplier Verification: $(1 - 0.25)(1 - 0.10) = 0.75 \times 0.90 = 0.675 \implies 1 - 0.675 = 0.325 = 32.5%$.

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Problem 2: Unit Pricing Best-Buy Analysis and Inflation Impact

Problem Statement:

1. A school cafeteria manager compares two bulk cheese options:
   • Supplier A: $7.50 \text{ for } 400\text{ g}$
   • Supplier B: $27.00 \text{ for } 1.5\text{ kg} \ (1,500\text{ g})$ Which supplier offers the lower cost per 100 grams?
2. If the cafeteria budget is $1,000 this year and annual food inflation is 4%, what will be the nominal cost next year to purchase the exact same quantity of food?

Step-by-Step Solution:

1. Unit Price Analysis (Cost per 100 g):

   • Supplier A: $\frac{$7.50}{400\text{ g}} \times 100\text{ g} = \frac{$7.50}{4} = $1.875 \approx $1.88 \text{ per } 100\text{ g}$
   • Supplier B: $\frac{$27.00}{1,500\text{ g}} \times 100\text{ g} = \frac{$27.00}{15} = $1.80 \text{ per } 100\text{ g}$
   • Conclusion: Supplier B is the best buy, saving $$0.075$ per 100 grams.

2. Inflation Impact Calculation:

   • Nominal cost next year = $\text{Current Budget} \times (1 + i) = $1,000 \times (1 + 0.04) = $1,040.00$.