7.1 Percentage Calculations, Percent Change, Markups & Discounts

7.1 Percentage Calculations, Percent Change, Markups & Discounts

Percentage arithmetic constitutes one of the most heavily emphasized topics on the Wonderlic Basic Skills Test Quantitative section (WBST-QS). Whether evaluating retail markdowns, employee wage adjustments, invoice trade discounts, or financial performance metrics, candidates must solve percentage items accurately without a calculator under an unforgiving time budget of 26.67 seconds per question.

To achieve elite performance, you cannot rely on slow manual long division or trial-and-error arithmetic. You must master the algebraic relationships of the Percentage Triangle, leverage mental decomposition shortcuts, avoid baseline-value traps in percent change calculations, and navigate successive percentage adjustments using multiplicative factors.

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The Anatomy of Percentages: "Per Hundred" Mechanics

The term percent originates from the Latin per centum, meaning "by the hundred." A percentage is simply a ratio whose denominator is fixed at 100. Converting smoothly between percentages, decimals, and fractions is the prerequisite for all advanced commercial arithmetic.

                     THE THREE-WAY CONVERSION HIGHWAY
                     
              Shift decimal 2 places LEFT (÷ 100)
           [ Percent ] ----------------------------> [ Decimal ]
               45%     <----------------------------    0.45
                       Shift decimal 2 places RIGHT (× 100)
                 \                                  /
     Write over   \                                /  Read place value
     100 & reduce  \                              /   and simplify
                    v                            v
                              [ Fraction ]
                            45/100 = 9/20

Instant Conversion Rules

1. Percent to Decimal: Remove the % symbol and move the decimal point two positions to the left ($7.5% = 0.075$; $65% = 0.65$; $120% = 1.20$).
2. Decimal to Percent: Move the decimal point two positions to the right and append the % symbol ($0.04 = 4%$; $0.375 = 37.5%$; $1.85 = 185%$).
3. Fraction to Percent: Scale the denominator to 100 if possible, or convert the fraction to a decimal benchmark and shift the decimal point two places right ($\frac{7}{20} = \frac{35}{100} = 35%$; $\frac{3}{8} = 0.375 = 37.5%$).

Essential Benchmark Conversions for the WBST

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The Percentage Triangle: Three Core Formulas

Every percentage problem involves three variables: the Part (the portion or quantity being measured), the Percent / Rate (the ratio expressed per 100), and the Whole / Base (the total or original quantity upon which the rate operates).

                              THE PERCENTAGE TRIANGLE
                              
                                      / \ 
                                     /   \ 
                                    /  P  \ 
                                   /-------\ 
                                  / R  |  W \ 
                                 /_____|_____\ 
                                 
                     Cover P:  Part    = Rate (Decimal) × Whole
                     Cover R:  Rate    = Part ÷ Whole
                     Cover W:  Whole   = Part ÷ Rate (Decimal)

1. Solving for the Part ($P = R \times W$)

Problem Statement: A medical laboratory processes 450 blood samples in a day. If $18%$ of the samples require manual re-testing, how many samples must be re-tested?

• Identify variables: $W = 450$, $R = 18% = 0.18$.
• Mental calculation: $10% \text{ of } 450 = 45$. $20% \text{ of } 450 = 90$. $2% \text{ of } 450 = 4.5 \times 2 = 9$.
• $18% = 20% - 2% = 90 - 9 = 81$ samples.

2. Solving for the Percent / Rate ($R = \frac{P}{W}$)

Problem Statement: Out of 80 applicants interviewed for a clerical role, 28 possess advanced certification in electronic medical records. What percentage of applicants hold this certification?

• Identify variables: $P = 28$, $W = 80$.
• Setup: $R = \frac{28}{80}$. Divide numerator and denominator by 4: $\frac{7}{20}$.
• Scale to denominator of 100: $\frac{7 \times 5}{20 \times 5} = \frac{35}{100} = 35%$.

3. Solving for the Whole / Base ($W = \frac{P}{R}$)

Problem Statement: A regional distribution warehouse ships 42 damaged cartons during an audit week. If this scrap count represents exactly $3.5%$ of total inventory handled, how many total cartons were handled?

• Identify variables: $P = 42$, $R = 3.5% = 0.035$.
• Setup: $W = \frac{42}{0.035} = \frac{42{,}000}{35}$.
• Divide 42,000 by 35: $42 \div 7 = 6$; $35 \div 7 = 5$. Thus $\frac{6{,}000}{5} = 1{,}200$ cartons.

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Rapid Mental Percentage Decomposition

Under strict time constraints, never set up long-hand multiplication algorithms for standard percentages. Decompose the requested percentage into modular building blocks: 50%, 25%, 10%, 5%, and 1%.

                       MENTAL DECOMPOSITION TOOLKIT
                       
  To find 50%:   Divide by 2
  To find 25%:   Divide by 4 (or half of 50%)
  To find 10%:   Shift decimal 1 place LEFT
  To find 5%:    Take half of 10%
  To find 1%:    Shift decimal 2 places LEFT
  To find 15%:   10% + 5% (take 10%, then add half of that)
  To find 20%:   10% × 2
  To find 35%:   25% + 10%  (or 3 × 10% + 5%)

Practical Mental Math Exercises

• Find 15% of $68.00 (Restaurant Tip or Service Charge):
  • $10% = $6.80$
  • $5% = $3.40$ (half of $$6.80$)
  • $15% = $6.80 + $3.40 = $10.20$
• Find 35% of $160.00 (Retail Wholesale Markup):
  • $25% \text{ of } 160 = 160 \div 4 = $40.00$
  • $10% \text{ of } 160 = $16.00$
  • $35% = $40.00 + $16.00 = $56.00$
• Find 8% of $250.00 (Municipal Sales Tax):
  • $1% \text{ of } 250 = $2.50$
  • $8% = 8 \times $2.50 = $20.00$

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Commercial Retail Math: Wholesale, Markups, Discounts & Tax

Workplace arithmetic on the Wonderlic frequently tests the full commercial pricing cycle: wholesale cost, retail markup, promotional discount, and state sales tax.

Key Commercial Formulas

$\text{Retail Selling Price} = \text{Wholesale Cost} \times (1 + \text{Markup Rate})$ $\text{Discounted Sale Price} = \text{Original Price} \times (1 - \text{Discount Rate})$ $\text{Final Invoice Total} = \text{Discounted Subtotal} \times (1 + \text{Sales Tax Rate})$

Critical Commercial Rule: Sales tax is always assessed on the net discounted price, never on the pre-discount original retail price.

Full Commercial Workflow Walkthrough

Scenario: An appliance dealer purchases an industrial microwave oven at wholesale for $320.00. The dealer marks up the wholesale cost by 40% to determine the showroom retail price. During an annual promotional event, the dealer offers a 15% discount off the retail price. If local sales tax is 6%, what does a customer pay at the register?

1. Calculate Retail Showroom Price (40% Markup):
   • Wholesale = $$320.00$
   • $10% = $32.00 \implies 40% = 4 \times $32.00 = $128.00$
   • Retail Price = $$320.00 + $128.00 = $448.00$
2. Calculate Discounted Sale Price (15% Markdown):
   • Retail Price = $$448.00$
   • $10% = $44.80$
   • $5% = $22.40$
   • Discount = $$44.80 + $22.40 = $67.20$
   • Discounted Price = $$448.00 - $67.20 = $380.80$
3. Calculate Sales Tax and Final Amount (6% Tax):
   • Taxable Subtotal = $$380.80$
   • $1% = $3.808 \implies 6% = 6 \times 3.808 = $22.848 \approx $22.85$
   • Final Register Total = $$380.80 + $22.85 = $403.65$

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Percent of Change: The Original Baseline Rule

Percent of change quantifies relative growth or decline over time. On the WBST, this concept appears in production rate shifts, budget variances, and staffing alterations.

The Fundamental Formula

$\text{Percent of Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\% = \frac{\Delta}{\text{Original Value}} \times 100\%$

• If $\text{New} > \text{Original}$, the result is positive, representing a Percent Increase.
• If $\text{New} < \text{Original}$, the result is negative, representing a Percent Decrease.

                           THE BASELINE VALUE TRAP
                           
         CORRECT BASELINE                     FATAL TEST TRAP
      Divide by ORIGINAL Value             Divide by NEW Value
      
          New - Original                       New - Original
      ----------------------- × 100%       --------------------- × 100%
             ORIGINAL                               NEW
               [✓]                                  [✗]

Demonstrating the Baseline Trap

Scenario: A medical billing specialist improves daily claim processing from 40 claims per day to 50 claims per day. What is the percentage increase?

• Absolute Change: $50 - 40 = +10$ claims.
• Correct Method (Divide by Baseline 40): $\text{Percent Increase} = \frac{10}{40} \times 100\% = \frac{1}{4} \times 100\% = 25.0\%$
• The Trap Answer (Divide by New 50): $\frac{10}{50} = 20.0\% \quad \text{[WRONG — This distractor is ALWAYS present!]}$

Now reverse the scenario: If daily processing drops from 50 claims down to 40 claims:

• Absolute Change: $40 - 50 = -10$ claims.
• Correct Method (Divide by Baseline 50): $\text{Percent Decrease} = \frac{-10}{50} \times 100\% = -20.0\%$

Key Principle: A change of 10 units on a base of 40 is a $25%$ increase; the exact same change of 10 units on a base of 50 is a $20%$ decrease. The original value dictates the denominator.

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Sequential and Successive Percentage Changes

Standardized tests frequently test whether candidates understand that successive percentage changes cannot be added together directly.

The Additive Fallacy

A store offers a winter clearance sale with 20% off all jackets. An additional coupon at checkout provides 10% off the clearance price. Many test-takers fall for the trap of adding $20% + 10% = 30%$. This is completely wrong.

Why? The second discount (10%) is applied only to the reduced price (80% of original), not the original 100% price!

The Multiplier Method for Successive Changes

To compute successive percentage changes swiftly, multiply the respective decimal decay or growth factors together:

$\text{Net Multiplier} = (1 \pm r_1) \times (1 \pm r_2) \times \dots \times (1 \pm r_n)$

• First markdown: $20% \text{ discount} \implies \text{Multiplier} = 1 - 0.20 = 0.80$
• Second markdown: $10% \text{ discount} \implies \text{Multiplier} = 1 - 0.10 = 0.90$
• Combined Multiplier: $0.80 \times 0.90 = 0.72$
• Effective Price: Customer pays $72%$ of original price.
• Effective Total Discount: $100% - 72% = 28%$ total discount (not $30%$)!

Successive Increase and Decrease (Asymmetric Reversal)

Consider an asset worth $100 that increases by $25%$, then decreases by $20%$:

• After $25%$ increase: $$100 \times 1.25 = $125.00$.
• After $20%$ decrease: $$125 \times (1 - 0.20) = $125 \times 0.80 = $100.00$.
• Net Multiplier: $1.25 \times 0.80 = 1.00$ ($0%$ net change).

If the asset first decreases by $20%$ and then increases by $20%$:

• Net Multiplier: $0.80 \times 1.20 = 0.96$.
• Net Result: A $4%$ net loss ($100% - 96% = 4%$), not a return to even.