2.3 Percents, Percent Change, Interest, and Real-World Financial Math

Percents, Percent Change, Interest, and Real-World Financial Math

OpenExamPrep provides this quantitative reasoning review to help students master percent concepts and financial math tested on the TSIA2 Mathematics assessment. Quantitative questions frequently embed percents within practical contexts such as retail sales discounts, tiered commissions, price inflation, loan interest, and tuition financing.

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1. Percent Foundations and the Three Core Question Types

The word percent derives from the Latin per centum, meaning "per hundred" or "out of 100". A percent is simply a ratio with a fixed denominator of 100:

P% = P / 100 = 0.01 × P

Converting Between Representations

• Percent to Decimal: Remove the % symbol and divide by 100 (shift decimal point two places left): 42.5% = 0.425; 6% = 0.06; 0.8% = 0.008.
• Decimal to Percent: Multiply by 100 and append % (shift decimal point two places right): 0.65 = 65%; 0.045 = 4.5%; 1.32 = 132%.
• Fraction to Percent: Convert the fraction to a decimal by division, then convert the decimal to a percent: 5/8 = 0.625 = 62.5%.

The Fundamental Percent Equation: Part = Percent × Whole

Every elementary percent problem involves three variables: the Percent (expressed as a decimal), the Whole (the base), and the Part (the portion):

Part = Rate (in decimal form) × Whole

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2. Percent Change: Increase and Decrease

Percent change quantifies the relative difference between an original value and a new value relative to that starting point.

The Universal Percent Change Formula

Percent Change = [ (New Value - Original Value) / Original Value ] × 100%

Percent Change = (Amount of Change / Original Baseline) × 100%

• If the result is positive, it represents a percent increase.
• If the result is negative, it represents a percent decrease.

⚠️ Critical TSIA2 Rule: The Denominator is Always the Original

The most common error in percent change calculations is dividing by the new value instead of the original baseline value. Even when numbers decrease, you must divide the difference by the starting amount.

Multipliers for Quick Calculations

Instead of calculating the change and adding or subtracting it from the original, use decimal multipliers:

• Percent Increase of r%: Multiply by (1 + r)
  • An item increased by 15%: New Price = Original × 1.15
  • Tuition increased by 4.5%: New Tuition = Original × 1.045
• Percent Decrease of r%: Multiply by (1 - r)
  • An item discounted by 25%: Sale Price = Original × (1 - 0.25) = Original × 0.75
  • Enrollment decreased by 12%: New Enrollment = Original × 0.88

Sequential (Successive) Percentage Changes

Percentages cannot be added or subtracted directly across successive transactions because each percentage applies to a different base amount.

Example: An electronics item costs $200. The store increases the price by 20%, 
and next week offers a 20% discount on that new price. What is the final price?

Step 1: Apply 20% markup:
  New Price = $200 × (1 + 0.20) = $200 × 1.20 = $240

Step 2: Apply 20% discount to the $240 price:
  Final Price = $240 × (1 - 0.20) = $240 × 0.80 = $192

Analysis: The final price is $192, not $200! 
Because the 20% decrease applied to a larger base ($240), the dollar deduction 
($48) was greater than the dollar increase ($40), resulting in an overall 4% net loss.

For successive changes of r₁ and r₂, the compound multiplier is:

Final = Original × (1 ± r₁) × (1 ± r₂)

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3. Commercial Applications: Tax, Discounts, Tips, and Commissions

Sales Tax and Total Purchase Price

Sales tax is calculated on the net purchase price after all trade discounts or promotional coupons have been deducted:

Tax Amount = Net Subtotal × Tax Rate
Total Cost = Net Subtotal × (1 + Tax Rate)

Multi-Tiered Sales Commission

In compensation problems, salespeople often earn a fixed base salary plus a commission on sales exceeding a specified quota baseline:

Total Earnings = Base Salary + Commission Rate × (Total Sales - Quota)

If total sales do not exceed the quota, the commission on the excess is zero, and only the base salary is earned.

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4. Interest Calculations: Simple vs. Compound

Interest represents the financial cost of borrowing capital or the financial return on invested funds.

Simple Interest: Linear Accumulation

Simple interest calculates return strictly on the initial principal balance throughout the entire duration of the loan or investment:

I = P × r × t

A = P + I = P(1 + rt)

Where:

• I = Total interest earned or paid ($)
• P = Principal amount borrowed or invested ($)
• r = Annual interest rate expressed as a decimal (e.g., 6.5% = 0.065)
• t = Time measured strictly in years
• A = Total accumulated future balance ($)

⚠️ Time Conversion Rule: If time is given in months, you must divide by 12: t = months / 12. If given in days, divide by 365 (or 360 in standard commercial interest questions).

Worked Problem: A college student secures a short-term educational loan 
of $3,600 at a 6% simple annual interest rate for 9 months. What is the total 
amount required to pay off the loan at maturity?

Step 1: Identify variables:
  P = $3,600
  r = 6% = 0.06
  t = 9 months = 9/12 = 0.75 years

Step 2: Calculate simple interest (I):
  I = 3,600 × 0.06 × 0.75 = $162.00

Step 3: Calculate total payoff amount (A = P + I):
  A = 3,600 + 162 = $3,762.00

Compound Interest: Exponential Growth

Compound interest calculates interest on the original principal plus all accumulated interest from prior compounding periods:

A = P [ 1 + (r / n) ]^(n × t)

Where:

• n = Number of compounding intervals per calendar year:
  • Annually: n = 1
  • Semi-annually: n = 2
  • Quarterly: n = 4
  • Monthly: n = 12
  • Daily: n = 365

Comparison: Simple vs. Compound Interest on $5,000 at 8% Annual Rate

Over time, compounding creates an accelerating curve because each year's interest payment is calculated on a progressively larger base.

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5. Summary of Percent Formulas and Procedures

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6. TSIA2 Financial Math Traps & Exam Pitfalls

1. Adding Sequential Discounts: If a store offers "30% off clearance plus an extra 20% off at the register," the total discount is not 50%! It is 1 - (0.70 × 0.80) = 1 - 0.56 = 44%. The second discount only reduces the already discounted price.
2. Wrong Denominator in Percent Change: If a book originally priced at $80 is marked down to $60, the percent decrease is (80 - 60) / 80 = 20 / 80 = 25%. Dividing by the new price ($60) yields 20/60 = 33.3% (incorrect).
3. Failing to Convert Interest Duration to Years: In I = Prt, entering t = 8 for an 8-month loan treats the loan as lasting 8 years, exaggerating interest by a factor of 12. Always write 8 months as 8/12 = 2/3 of a year.
4. Confusing Percentage Points with Percent Change: If an interest rate climbs from 4% to 5%, that is an increase of 1 percentage point, but a percent increase of (5 - 4) / 4 = 1/4 = 25%.