6.2 Percentages, Percent Change & Successive Percents

6.2 Percentages, Percent Change & Successive Percents

Percentages is one of the seven Problem-Solving and Data Analysis skill/knowledge testing points College Board lists for SAT Math. That whole domain is about 15% of the section (5 to 7 questions), and percent reasoning also surfaces inside Algebra and Advanced Math modeling questions, so the topic reaches further than its domain share suggests. While percentage arithmetic appears elementary, the SAT tests subtle conceptual distinctions: distinguishing between original bases, tracking successive compound changes, and algebraically recovering initial values.

Mastering the multiplier method simplifies these problems and eliminates the arithmetic traps associated with multi-step percentage calculations.

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1. Percentage Translation Framework

A percent literally means "per hundred" ($P% = \frac{P}{100}$). Word problems can be translated directly into algebraic equations using this translation guide:

+-----------------------------------------------------------------------------+
|                        PERCENT ALGEBRAIC TRANSLATOR                         |
|                                                                             |
|   English Keyword           Mathematical Symbol                             |
|   -----------------------   -------------------                             |
|   "is", "was", "equals"     -->   =                                         |
|   "of"                      -->   * (multiplication)                        |
|   "what number / value"     -->   x (unknown variable)                      |
|   "what percent"            -->   p / 100                                   |
|   "percent increase by r%"  -->   * (1 + r/100)                             |
|   "percent decrease by r%"  -->   * (1 - r/100)                             |
+-----------------------------------------------------------------------------+

The Three Core Percentage Equations

1. Finding the Part: "What is $35%$ of $80$?" $x = 0.35 \times 80 = 28$
2. Finding the Percentage: "$42$ is what percent of $120$?" $42 = \left(\frac{p}{100}\right) \times 120 \implies p = \frac{42}{120} \times 100 = 35\%$
3. Finding the Base (Whole): "$54$ is $60%$ of what number?" $54 = 0.60 \times x \implies x = \frac{54}{0.60} = 90$

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2. The Multiplier Method for Percent Change

Rather than calculating the amount of change and adding/subtracting it in two separate steps, apply a single decimal multiplier:

$\text{New Value} = \text{Original Value} \times (1 \pm r)$

where $r$ is the percentage change expressed as a decimal.

+-----------------------------------------------------------------------------+
|                     PERCENT MULTIPLIER REFERENCE TABLE                      |
|                                                                             |
|   Percent Change            Multiplier Formula         Decimal Multiplier   |
|   -----------------------   ------------------------   ------------------   |
|   5% Increase               1 + 0.05                   1.05                 |
|   25% Increase              1 + 0.25                   1.25                 |
|   80% Increase              1 + 0.80                   1.80                 |
|   150% Increase             1 + 1.50                   2.50                 |
|   10% Decrease (Discount)   1 - 0.10                   0.90                 |
|   35% Decrease              1 - 0.35                   0.65                 |
|   75% Decrease              1 - 0.75                   0.25                 |
+-----------------------------------------------------------------------------+

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3. Percent Change Formula & The Base Value Rule

The percent change formula calculates the relative change relative to the original base value:

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

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

Worked Example 1: Calculating Percent Increase

Problem: In 2020, a municipal solar farm generated $640\text{ megawatt-hours (MWh)}$ of electricity. In 2024, the solar farm generated $896\text{ MWh}$. By what percentage did electricity generation increase from 2020 to 2024?

Step-by-Step Solution:

1. Identify the original and new values: $V_{\text{original}} = 640, \quad V_{\text{new}} = 896$
2. Calculate the absolute difference: $\Delta V = 896 - 640 = 256\text{ MWh}$
3. Divide by the original base value (2020 value): $\text{Percent Increase} = \frac{256}{640} \times 100\% = 0.40 \times 100\% = 40\%$

[!WARNING] The Wrong Denominator Trap: A common mistake is dividing by the new value ($896$) instead of the original value ($640$). $\frac{256}{896} \approx 28.6%$ is a classic SAT distractor choice. Always place the starting point in the denominator.

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4. Successive (Compound) Percentage Changes

A critical SAT rule: Percentages cannot be added or subtracted directly when applied sequentially. Successive percentage changes apply to different base amounts at each step and must be multiplied.

$\text{Final Value} = \text{Initial Value} \times (1 \pm r_1) \times (1 \pm r_2) \times \dots \times (1 \pm r_n)$

+-----------------------------------------------------------------------------+
|                  SUCCESSIVE PERCENTS: MULTIPLY, DO NOT ADD                  |
|                                                                             |
|   Scenario: An item receives a 20% discount, followed by a 15% coupon.       |
|                                                                             |
|   [ INCORRECT ADDITIVE THINKING ]:                                          |
|   20% + 15% = 35% discount  --> Multiplier = 0.65   (WRONG!)                |
|                                                                             |
|   [ CORRECT MULTIPLICATIVE COMPOUNDING ]:                                   |
|   Step 1 Multiplier (20% off):  1 - 0.20 = 0.80                             |
|   Step 2 Multiplier (15% off):  1 - 0.15 = 0.85                             |
|   Combined Multiplier:          0.80 * 0.85 = 0.68                          |
|                                                                             |
|   Effective Total Discount:     (1 - 0.68) * 100% = 32% discount (NOT 35%!) |
+-----------------------------------------------------------------------------+

Symmetric Change Fallacy (Increase followed by Equal Decrease)

Consider an asset that increases by $25%$ and subsequently decreases by $20%$:

$\text{Combined Multiplier} = (1 + 0.25) \times (1 - 0.20) = 1.25 \times 0.80 = 1.00$

The asset returns exactly to $100%$ of its original value. (A $25%$ gain is fully erased by a $20%$ loss because the $20%$ loss is taken on a larger base).

Worked Example 2: Quarterly Stock Fluctuations

Problem: The share price of a technology company increases by $30%$ during the first quarter and then decreases by $20%$ during the second quarter. What is the net overall percentage change from the start of the year to the end of the second quarter?

Step-by-Step Solution:

1. Determine the respective multipliers:
   • First quarter increase of $30%$: $M_1 = 1 + 0.30 = 1.30$
   • Second quarter decrease of $20%$: $M_2 = 1 - 0.20 = 0.80$
2. Multiply the individual multipliers: $M_{\text{net}} = 1.30 \times 0.80 = 1.04$
3. Interpret the combined multiplier: $\text{Net Change} = (1.04 - 1.00) \times 100\% = +4\%\text{ (a } 4\%\text{ net increase)}$

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5. Working Backwards to the Original Base Value

When given the final price after a percentage change and asked for the initial price, set up the multiplier equation and solve for the unknown base by division:

$\text{Final} = \text{Original} \times (1 \pm r) \implies \text{Original} = \frac{\text{Final}}{1 \pm r}$

+-----------------------------------------------------------------------------+
|                        REVERSE PERCENTAGE CALCULATION                       |
|                                                                             |
|   Problem: After a 25% discount, a jacket costs \$108. What was original?    |
|                                                                             |
|   [ FATAL MISTAKE: Adding 25% of \$108 ]                                     |
|   \$108 + 0.25(\$108) = \$108 + \$27 = \$135  --> (WRONG!)                       |
|                                                                             |
|   [ CORRECT ALGEBRAIC EQUATION ]:                                           |
|   Original * (1 - 0.25) = \$108                                              |
|   Original * 0.75 = \$108                                                    |
|   Original = \$108 / 0.75 = \$144                                             |
+-----------------------------------------------------------------------------+

Worked Example 3: Sale Price with Sales Tax

Problem: A consumer purchases a laptop at a $20%$ discount off its original list price. A local sales tax of $8%$ is then added to the discounted price. If the consumer pays a final total of $$864.00$, what was the original list price of the laptop?

Step-by-Step Solution:

1. Let $P$ be the original list price.
2. Express the discounted price: $P \times (1 - 0.20) = 0.80 P$.
3. Apply the $8%$ sales tax multiplier ($1 + 0.08 = 1.08$): $0.80 P \times 1.08 = 864 \implies 0.864 P = 864$
4. Divide to solve for $P$: $P = \frac{864}{0.864} = \$1{,}000.00$

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6. Percentage Points vs. Percent Change

• Percentage Points: The simple arithmetic difference between two percentage values. $\text{Change in Percentage Points} = P_2 - P_1$
• Percent Change: The relative growth rate of the percentage itself. $\text{Percent Change} = \frac{P_2 - P_1}{P_1} \times 100\%$

Example: If an interest rate rises from $4%$ to $5%$, it has increased by $1$ percentage point, but has experienced a $25%$ percent increase ($\frac{5-4}{4} \times 100% = 25%$).

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7. Desmos Playbook for Percentage Problems

+-----------------------------------------------------------------------------+
|                     DESMOS PERCENTAGE SHORTCUT PLAYBOOK                     |
|                                                                             |
|   1. Reverse Base Calculation:                                              |
|      Type: 864 / (0.80 * 1.08)  --> Desmos outputs 1000 instantly.          |
|                                                                             |
|   2. Net Multiplier Verification:                                           |
|      Type: (1 + 0.30) * (1 - 0.20)  --> Desmos outputs 1.04 (+4% change).   |
|                                                                             |
|   3. Multi-Step Growth Comparisons:                                         |
|      Graph: y = 500 * (1.05)^x and y = 500 + 25x                            |
|      Compare linear vs. compounding growth side-by-side on coordinate canvas|
+-----------------------------------------------------------------------------+

[!TIP] Desmos Reverse Equation Solver: For complex multi-tiered percentage problems, you can type the full equation directly into Desmos as x * 0.80 * 1.08 = 864. Desmos displays a vertical line at $x = 1000$, allowing you to read the answer directly off the graph without manual arithmetic.