Study guide
2.3 Percent Applications, Markup, Discount, & Simple Interest
ACCUPLACER Quantitative Reasoning, Algebra, and Statistics Exam
Track your progress as you read
Finished sections get a check mark.
1 Chapter 1: Rational Numbers & Operations
1.1 Rational Number Arithmetic & Signed Numbers 1.2 Fractions, Decimals, & Mixed Numbers Operations 1.3 Absolute Value, Number Line, & Estimation
2 Chapter 2: Ratios, Proportions & Rates
2.1 Ratios, Unit Rates, & Dimensional Analysis 2.2 Proportions & Direct/Inverse Variation 2.3 Percent Applications, Markup, Discount, & Simple Interest
3 Chapter 3: Exponents & Scientific Notation
3.1 Integer Exponents & Laws of Exponents 3.2 Scientific Notation & Operations
4 Chapter 4: Algebraic Expressions
4.1 Evaluating & Translating Algebraic Expressions 4.2 Simplifying, Combining Like Terms, & Factoring
5 Chapter 5: Linear Equations & Inequalities
5.1 Solving Single-Variable Linear Equations 5.2 Solving & Graphing Linear Inequalities 5.3 Systems of Linear Equations
6 Chapter 6: Linear Applications & Graphing
6.1 Slope & Rate of Change 6.2 Forms of Linear Equations & Graphing 6.3 Linear Modeling & Inequality Graphs
7 Chapter 7: Probability & Set Theory
7.1 Fundamental Probability Concepts 7.2 Set Theory, Operations, & Venn Diagrams
8 Chapter 8: Descriptive Statistics & Data Displays
8.1 Measures of Center: Mean, Median, Mode, & Weighted Average 8.2 Measures of Spread: Range, IQR, Variance, & Standard Deviation 8.3 Data Displays: Histograms, Box Plots, & Scatter Plots
9 Chapter 9: Geometry Concepts & Measurement
9.1 2D Geometry: Perimeter, Area, & Circles 9.2 3D Geometry: Surface Area & Volume of Solids 9.3 Triangles, Angles, & the Pythagorean Theorem
2.3 Percent Applications, Markup, Discount, & Simple Interest
Key Takeaways Explain This Topic AI
The fundamental percent equation is Part = Percent (in decimal form) x Whole, where 'is' translates to equality and 'of' translates to multiplication.
Percent change is computed as (New Value - Original Value) / Original Value x 100%, always dividing by the baseline original value rather than the new value.
Successive percent changes (such as chained store discounts) are multiplicative rather than additive; a 20% discount followed by a 10% discount yields an effective 28% net reduction ((1 - 0.20)(1 - 0.10) = 0.72), not 30%.
Retail markup calculates price increase based on wholesale cost (Selling Price = Cost x (1 + m)), whereas gross profit margin calculates profit as a percentage of final selling revenue.
Simple interest follows the linear accumulation formula I = Prt (Principal x Annual Rate x Time in years), resulting in a total future balance of A = P(1 + rt).
Last updated: August 2026
Percent Fundamentals & Algebraic Translation
The word percent derives from the Latin per centum , meaning "out of one hundred" or "per hundredths". In all algebraic calculations, percentages must be converted to their decimal or fractional equivalents by dividing by 100 100 100 : p % = p 100 = 0.01 ⋅ p p\% = \frac{p}{100} = 0.01 \cdot p p % = 100 p = 0.01 ⋅ p
Examples : 45 % = 0.45 = 9 20 45\% = 0.45 = \frac{9}{20} 45% = 0.45 = 20 9 , 7.5 % = 0.075 = 3 40 \; 7.5\% = 0.075 = \frac{3}{40} 7.5% = 0.075 = 40 3 , 125 % = 1.25 = 5 4 \; 125\% = 1.25 = \frac{5}{4} 125% = 1.25 = 4 5 .
The Direct Translation Keyword Dictionary
Word problems on the ACCUPLACER test can be systematically translated into linear equations using standard verbal cues:
- Verbal Phrase Mathematical Operator Example Statement Translated Equation
- "is", "was", "equals", "represents" = = = (Equal sign) "18 is..." 18 = … 18 = \dots 18 = …
- "of" ⋅ \cdot ⋅ or × \times × (Multiplication) "...of 80..." ⋯ × 80 \dots \times 80 ⋯ × 80
- "what number", "what amount" x x x (Unknown variable) "What number is..." x = … x = \dots x = …
"what percent" p 100 \frac{p}{100} 100 p or p p p "...what percent of 50?" ( p 100 ) × 50 \left(\frac{p}{100}\right) \times 50 ( 100 p ) × 50
The Three Fundamental Percent Problem Types
Every single-step percent problem revolves around the core relationship: Part = Percent × Whole ( or A = P ⋅ W ) \text{Part} = \text{Percent} \times \text{Whole} \quad (\text{or } A = P \cdot W) Part = Percent × Whole ( or A = P ⋅ W )
Finding the Part ( A A A is unknown) : Problem : What is 35 % 35\% 35% of 160 160 160 ? Equation: x = 0.35 × 160 = 56 \text{Equation: } x = 0.35 \times 160 = 56 Equation: x = 0.35 × 160 = 56
Finding the Percent ( P P P is unknown) : Problem : What percent of 80 80 80 is 28 28 28 ? Equation: 28 = ( p 100 ) × 80 ⟹ 28 = 0.80 p ⟹ p = 28 0.80 = 35 % \text{Equation: } 28 = \left(\frac{p}{100}\right) \times 80 \implies 28 = 0.80p \implies p = \frac{28}{0.80} = 35\% Equation: 28 = ( 100 p ) × 80 ⟹ 28 = 0.80 p ⟹ p = 0.80 28 = 35%
Finding the Base/Whole ( W W W is unknown) : Problem : 42 42 42 is 60 % 60\% 60% of what number? Equation: 42 = 0.60 × W ⟹ W = 42 0.60 = 70 \text{Equation: } 42 = 0.60 \times W \implies W = \frac{42}{0.60} = 70 Equation: 42 = 0.60 × W ⟹ W = 0.60 42 = 70
Percent Change & Percent Error
Percent change measures the relative increase or decrease of a quantity compared to its initial baseline value .
General Percent Change Formula Percent Change = New Value − Original Value Original Value × 100 % = Δ Original × 100 % \text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\% = \frac{\Delta}{\text{Original}} \times 100\% Percent Change = Original Value New Value − Original Value × 100% = Original Δ × 100%
If New > Original \text{New} > \text{Original} New > Original , the result is positive, indicating a Percent Increase .
If New < Original \text{New} < \text{Original} New < Original , the result is negative, indicating a Percent Decrease .
- (Critical Rule: The denominator MUST always be the Original (initial) value, never the new value!)
- Worked Example 1: Calculating Percent Increase and Decrease
Percent Increase : A municipal transit authority raises the one-way bus fare from $ 2.50 \$2.50 $2.50 to $ 3.10 \$3.10 $3.10 . Calculate the percent increase in fare. Percent Increase = 3.10 − 2.50 2.50 × 100 % = 0.60 2.50 × 100 % = 0.24 × 100 % = 24 % \text{Percent Increase} = \frac{3.10 - 2.50}{2.50} \times 100\% = \frac{0.60}{2.50} \times 100\% = 0.24 \times 100\% = 24\% Percent Increase = 2.50 3.10 − 2.50 × 100% = 2.50 0.60 × 100% = 0.24 × 100% = 24%
Percent Decrease : A digital camera originally retailing for $ 450 \$450 $450 is marked down on clearance to $ 324 \$324 $324 . Calculate the percent discount. Percent Decrease = 324 − 450 450 × 100 % = − 126 450 × 100 % = − 0.28 × 100 % = − 28 % ( 28 % decrease ) \text{Percent Decrease} = \frac{324 - 450}{450} \times 100\% = \frac{-126}{450} \times 100\% = -0.28 \times 100\% = -28\% \; (28\%\text{ decrease}) Percent Decrease = 450 324 − 450 × 100% = 450 − 126 × 100% = − 0.28 × 100% = − 28% ( 28% decrease )
Commercial Applications: Discounts, Tax, Tips, & Multipliers
In practical arithmetic, using decimal multipliers is significantly faster and less prone to calculation error than performing separate multi-step percentage additions or subtractions.
Decimal Multipliers for Common Transactions
Discount of d % d\% d % : Sale Price = Original Price × ( 1 − d ) \text{Sale Price} = \text{Original Price} \times (1 - d) Sale Price = Original Price × ( 1 − d ) (Example: 25 % 25\% 25% off means you pay 1 − 0.25 = 0.75 1 - 0.25 = 0.75 1 − 0.25 = 0.75 of original)
Sales Tax of t % t\% t % : Total with Tax = Subtotal × ( 1 + t ) \text{Total with Tax} = \text{Subtotal} \times (1 + t) Total with Tax = Subtotal × ( 1 + t ) (Example: 8.5 % 8.5\% 8.5% tax means you pay 1 + 0.085 = 1.085 1 + 0.085 = 1.085 1 + 0.085 = 1.085 of subtotal)
Tip / Gratuity of g % g\% g % : Total with Tip = Meal Cost × ( 1 + g ) \text{Total with Tip} = \text{Meal Cost} \times (1 + g) Total with Tip = Meal Cost × ( 1 + g ) (Example: 18 % 18\% 18% tip means total paid is 1.18 1.18 1.18 of meal cost)
Successive (Chained) Discounts
A pervasive trap on math exams involves successive percentage changes. When multiple discounts are applied in sequence, they are never additive because each subsequent percentage applies to the newly reduced intermediate price. Final Multiplier = ( 1 − d 1 ) × ( 1 − d 2 ) × ⋯ × ( 1 − d n ) \text{Final Multiplier} = (1 - d_1) \times (1 - d_2) \times \dots \times (1 - d_n) Final Multiplier = ( 1 − d 1 ) × ( 1 − d 2 ) × ⋯ × ( 1 − d n ) Effective Total Discount = 1 − Final Multiplier \text{Effective Total Discount} = 1 - \text{Final Multiplier} Effective Total Discount = 1 − Final Multiplier
Worked Example 2: Successive Discounts vs. Direct Sum
A designer winter jacket with an original retail price of $ 240 \$240 $240 is placed on a 30 % 30\% 30% off clearance rack. During a weekend promotional sale, the store offers an additional 20 % 20\% 20% discount off the clearance price. What is the final sale price of the jacket, and what is the effective total percentage discount?
Calculate intermediate clearance price (Multiplier 1 − 0.30 = 0.70 1 - 0.30 = 0.70 1 − 0.30 = 0.70 ): Clearance Price = $ 240 × 0.70 = $ 168.00 \text{Clearance Price} = \$240 \times 0.70 = \$168.00 Clearance Price = $240 × 0.70 = $168.00
Apply the second 20 % 20\% 20% discount (Multiplier 1 − 0.20 = 0.80 1 - 0.20 = 0.80 1 − 0.20 = 0.80 ): Final Sale Price = $ 168.00 × 0.80 = $ 134.40 \text{Final Sale Price} = \$168.00 \times 0.80 = \$134.40 Final Sale Price = $168.00 × 0.80 = $134.40
Combine using overall chained multiplier: Net Multiplier = 0.70 × 0.80 = 0.56 \text{Net Multiplier} = 0.70 \times 0.80 = 0.56 Net Multiplier = 0.70 × 0.80 = 0.56 Final Price = $ 240 × 0.56 = $ 134.40 \text{Final Price} = \$240 \times 0.56 = \$134.40 Final Price = $240 × 0.56 = $134.40
Determine effective single discount rate: Effective Discount = ( 1 − 0.56 ) × 100 % = 44 % \text{Effective Discount} = (1 - 0.56) \times 100\% = 44\% Effective Discount = ( 1 − 0.56 ) × 100% = 44%
(Exam Trap Alert: The effective discount is 44 % 44\% 44% , NOT 30 % + 20 % = 50 % 30\% + 20\% = 50\% 30% + 20% = 50% . If a 50 % 50\% 50% discount were applied, the price would be $ 120.00 \$120.00 $120.00 , undercharging by $ 14.40 \$14.40 $14.40 .)
Retail Markup vs. Gross Profit Margin
In retail business arithmetic, students must distinguish between markup on cost and gross profit margin .
Metric Mathematical Formula Base Value (Denominator) Business Meaning
Markup Rate ( m m m ) Selling Price − Cost Cost × 100 % \frac{\text{Selling Price} - \text{Cost}}{\text{Cost}} \times 100\% Cost Selling Price − Cost × 100% Wholesale Cost Percentage added onto cost to set retail price
Gross Profit Margin Selling Price − Cost Selling Price × 100 % \frac{\text{Selling Price} - \text{Cost}}{\text{Selling Price}} \times 100\% Selling Price Selling Price − Cost × 100% Selling Price (Revenue) Percentage of each revenue dollar retained as profit
Worked Example 3: Markup vs. Margin
A boutique electronics retailer purchases wireless headphones from a wholesale distributor for $ 60.00 \$60.00 $60.00 per unit and applies a 65 % 65\% 65% markup on cost. Calculate the retail selling price and the retailer's gross profit margin.
Calculate selling price with 65 % 65\% 65% markup: Selling Price = Cost × ( 1 + 0.65 ) = $ 60.00 × 1.65 = $ 99.00 \text{Selling Price} = \text{Cost} \times (1 + 0.65) = \$60.00 \times 1.65 = \$99.00 Selling Price = Cost × ( 1 + 0.65 ) = $60.00 × 1.65 = $99.00
Determine gross dollar profit: Profit = $ 99.00 − $ 60.00 = $ 39.00 \text{Profit} = \$99.00 - \$60.00 = \$39.00 Profit = $99.00 − $60.00 = $39.00
Compute gross profit margin: Gross Margin = $ 39.00 $ 99.00 × 100 % = 0.3939 ⋯ × 100 % ≈ 39.39 % \text{Gross Margin} = \frac{\$39.00}{\$99.00} \times 100\% = 0.3939\dots \times 100\% \approx 39.39\% Gross Margin = $99.00 $39.00 × 100% = 0.3939 ⋯ × 100% ≈ 39.39%
Simple Interest ( I = P r t I = Prt I = P r t )
Simple interest is interest calculated solely on the original principal amount deposited or borrowed, ignoring any interest compounding over time.
The Simple Interest Formula I = P ⋅ r ⋅ t I = P \cdot r \cdot t I = P ⋅ r ⋅ t
I I I (Interest) : The dollar amount of interest earned by an investor or paid by a borrower.
P P P (Principal) : The original starting capital deposited or borrowed.
r r r (Annual Interest Rate) : The stated annual interest rate expressed in decimal form (e.g., 5.25 % = 0.0525 5.25\% = 0.0525 5.25% = 0.0525 ).
t t t (Time in Years) : The duration of the loan or investment expressed in years .
Converting Time Intervals to Years
If time is given in months ( m m m ) : t = m 12 t = \frac{m}{12} t = 12 m (Example: 18 months = 18 12 = 1.5 years 18\text{ months} = \frac{18}{12} = 1.5\text{ years} 18 months = 12 18 = 1.5 years ; 9 months = 9 12 = 0.75 years \; 9\text{ months} = \frac{9}{12} = 0.75\text{ years} 9 months = 12 9 = 0.75 years )
If time is given in days ( d d d ) : t = d 365 t = \frac{d}{365} t = 365 d (or d 360 \frac{d}{360} 360 d if using ordinary interest conventions)
Total Accumulated Balance ( A A A )
The total future value or repayment balance A A A combines the original principal with the total interest earned: A = P + I = P + P r t = P ( 1 + r t ) A = P + I = P + Prt = P(1 + rt) A = P + I = P + P r t = P ( 1 + r t )
Worked Example 4: Calculating Simple Interest Loan Repayment
A small business owner secures a $ 12 , 000 \$12,000 $12 , 000 equipment loan with a simple annual interest rate of 6.5 % 6.5\% 6.5% for a term of 42 months 42\text{ months} 42 months . Calculate the total interest accrued over the life of the loan and the total amount required to settle the balance.
Convert duration 42 months 42\text{ months} 42 months to years: t = 42 12 = 3.5 years t = \frac{42}{12} = 3.5\text{ years} t = 12 42 = 3.5 years
Calculate total interest accrued ( I = P r t I = Prt I = P r t ): I = $ 12 , 000 × 0.065 × 3.5 = $ 2 , 730.00 I = \$12,000 \times 0.065 \times 3.5 = \$2,730.00 I = $12 , 000 × 0.065 × 3.5 = $2 , 730.00
Calculate total future repayment balance ( A = P + I A = P + I A = P + I ): A = $ 12 , 000 + $ 2 , 730 = $ 14 , 730.00 A = \$12,000 + \$2,730 = \$14,730.00 A = $12 , 000 + $2 , 730 = $14 , 730.00
Worked Example 5: Solving for the Unknown Interest Rate
An investor deposits $ 5 , 000 \$5,000 $5 , 000 into a guaranteed fixed-rate certificate of deposit. After 4 years 4\text{ years} 4 years , the total balance in the account grows to $ 5 , 900 \$5,900 $5 , 900 . What was the annual simple interest rate paid by the certificate?
Find total interest earned ( I = A − P I = A - P I = A − P ): I = $ 5 , 900 − $ 5 , 000 = $ 900 I = \$5,900 - \$5,000 = \$900 I = $5 , 900 − $5 , 000 = $900
Set up the simple interest formula to isolate r r r : I = P ⋅ r ⋅ t I = P \cdot r \cdot t I = P ⋅ r ⋅ t $ 900 = $ 5 , 000 ⋅ r ⋅ 4 \$900 = \$5,000 \cdot r \cdot 4 $900 = $5 , 000 ⋅ r ⋅ 4 900 = 20 , 000 ⋅ r 900 = 20,000 \cdot r 900 = 20 , 000 ⋅ r
Solve for r r r and convert to percent: r = 900 20 , 000 = 0.045 = 4.5 % per year r = \frac{900}{20,000} = 0.045 = 4.5\%\text{ per year} r = 20 , 000 900 = 0.045 = 4.5% per year
Common Pitfalls & ACCUPLACER Exam Traps
Base Confusion in Percent Increase/Decrease : Always divide the difference Δ \Delta Δ by the original starting value. If an item drops from $ 100 \$100 $100 to $ 80 \$80 $80 , the decrease is 20 100 = 20 % \frac{20}{100} = 20\% 100 20 = 20% . If it rises back from $ 80 \$80 $80 to $ 100 \$100 $100 , the increase is 20 80 = 25 % \frac{20}{80} = 25\% 80 20 = 25% , NOT 20 % 20\% 20% .
Linear Addition of Successive Percentages : Remember that a 20 % 20\% 20% increase followed by a 20 % 20\% 20% decrease does NOT return you to the original value: ( 1 + 0.20 ) ( 1 − 0.20 ) = 1.20 × 0.80 = 0.96 (1 + 0.20)(1 - 0.20) = 1.20 \times 0.80 = 0.96 ( 1 + 0.20 ) ( 1 − 0.20 ) = 1.20 × 0.80 = 0.96 (a net 4 % 4\% 4% loss).
Using Months Directly for t t t in I = P r t I = Prt I = P r t : If a loan lasts 18 months 18\text{ months} 18 months , plugging in t = 18 t = 18 t = 18 calculates interest for 18 years 18\text{ years} 18 years ! Always convert months to years ( t = 18 12 = 1.5 t = \frac{18}{12} = 1.5 t = 12 18 = 1.5 ).
Applying Sales Tax to Discounted vs. Original Price : In retail problems, sales tax is almost universally assessed on the discounted subtotal , not the pre-sale original sticker price.