Apply the first percentage to the base, then apply the second percentage to the new intermediate value.
⚡ SAT Speed Trick & Desmos Hack
Use the '100' method: Start with 100, apply the first multiplier, then apply the second multiplier to the result.
💡 Worked SAT Archetype Example
Problem: A value increases by 20% then decreases by 10%. What is the net percentage change?
📘 Step-by-Step Textbook Solution:
Step 1
Assume initial value: $100$
Step 2
Apply 20% increase: $100 \times 1.20 = 120$
Step 3
Apply 10% decrease to 120: $120 \times 0.90 = 108$
Step 4
Calculate net change: $108 - 100 = 8\%$
⚡ Speed / Desmos Tactic:
Step 1
Type into Desmos: 1.2 * 0.9
Step 2
Result: 1.08
Step 3
Interpret: 1.08 - 1 = 0.08 or 8%
Concept 3
Concept 3: Reverse Percentages
Finding the original value when the final value and the percentage change are known.
Original Value: $Original = \frac{Final}{1 \pm r}$
Common Error: Do not multiply the final value by $(1 \mp r)$
Verification: $Original \times (1 \pm r) = Final$
📘 Traditional Algebraic Method
Set up an algebraic equation where x is the original value and solve for x by dividing the final value by the multiplier.
⚡ SAT Speed Trick & Desmos Hack
Define a function $f(x) = x * (1 \pm r)$ in Desmos and use the slider or intersection to find x when $f(x) = Final$.
💡 Worked SAT Archetype Example
Problem: After a 20% discount, a shirt costs $40. What was the original price?
📘 Step-by-Step Textbook Solution:
Step 1
Let x be the original price
Step 2
Equation: $x \times (1 - 0.20) = 40$
Step 3
Simplify: $0.8x = 40$
Step 4
Solve: $x = \frac{40}{0.8} = 50$
⚡ Speed / Desmos Tactic:
Step 1
Type into Desmos: 40 / 0.8
Step 2
Result: 50
Practice Questions (20)
Question 1Percentage Increase and Decrease
Easy
A shirt originally costs $40. During a sale, the price is reduced by 20%. What is the sale price of the shirt?
Hint: Calculate 20% of 40 and subtract it from the original price.
📘 Step-by-Step Algebraic Solution
Step 1
Calculate the discount amount: $40 \times 0.20 = 8$
Step 2
Subtract the discount from the original price: $40 - 8 = 32$
⚡ Desmos Shortcut / Speed Hack
Step 1
Recognize that a 20% reduction means paying 80% of the price.
Step 2
Calculate $40 \times 0.8 = 32$
Question 2Percentage Increase and Decrease
Easy
The population of a town increased from 500 to 600. What was the percentage increase in the population?
Hint: Use the formula: (New - Old) / Old * 100.
📘 Step-by-Step Algebraic Solution
Step 1
Find the increase: $600 - 500 = 100$
Step 2
Divide by the original: $100 / 500 = 0.2$
Step 3
Convert to percentage: $0.2 \times 100 = 20\%$
⚡ Desmos Shortcut / Speed Hack
Step 1
Type (600-500)/500 into Desmos.
Step 2
Result is 0.2, which is 20%.
Question 3Percentage Increase and Decrease
Medium
An item's price is increased by 25% and then decreased by 20%. What is the net percentage change in the price?
Hint: Assume the initial price is 100.
📘 Step-by-Step Algebraic Solution
Step 1
Let initial price = 100.
Step 2
After 25% increase: $100 \times 1.25 = 125$.
Step 3
After 20% decrease: $125 \times 0.80 = 100$.
Step 4
Net change: $(100 - 100) / 100 = 0\%$
⚡ Desmos Shortcut / Speed Hack
Step 1
Use the multiplier method: $1.25 \times 0.80 = 1.00$.
Step 2
Since the result is 1, there is no change.
Question 4Percentage Increase and Decrease
Medium
The value of a car depreciates by 15% each year. If the car is worth $20,000 today, what will it be worth in 2 years?
Hint: Multiply the value by 0.85 twice.
📘 Step-by-Step Algebraic Solution
Step 1
Year 1 value: $20,000 \times 0.85 = 17,000$.
Step 2
Year 2 value: $17,000 \times 0.85 = 14,450$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Enter $20000 \times 0.85^2$ into Desmos.
Step 2
Result is 14450.
Question 5Percentage Increase and Decrease
Hard
A store increases the price of an item by $x$% and then decreases the new price by $x$%. If the final price is 96% of the original price, what is the value of $x$?
Hint: Set up the equation: $(1 + x/100)(1 - x/100) = 0.96$.
📘 Step-by-Step Algebraic Solution
Step 1
$(1 + r)(1 - r) = 0.96$ where $r = x/100$.
Step 2
$1 - r^2 = 0.96$.
Step 3
$r^2 = 0.04$.
Step 4
$r = 0.2$, so $x = 20$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Test the options in the equation $(1 + x/100)(1 - x/100)$.
Step 2
For $x=20$, $(1.2)(0.8) = 0.96$.
Question 6Percentage of a Whole
Easy
In a class of 40 students, 15% are absent. How many students are present?
Hint: If 15% are absent, 85% are present.
📘 Step-by-Step Algebraic Solution
Step 1
Calculate absent: $40 \times 0.15 = 6$.
Step 2
Subtract from total: $40 - 6 = 34$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Calculate $40 \times 0.85$ directly.
Step 2
Result is 34.
Question 7Percentage of a Whole
Easy
What is 25% of 40% of 200?
Hint: Convert percentages to decimals and multiply.
📘 Step-by-Step Algebraic Solution
Step 1
$0.25 \times 0.40 \times 200$.
Step 2
$0.10 \times 200 = 20$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Enter $0.25 \times 0.4 \times 200$ into Desmos.
Step 2
Result is 20.
Question 8Percentage of a Whole
Medium
A survey of 500 people found that 60% prefer brand A, 25% prefer brand B, and the rest prefer brand C. How many people prefer brand C?
Hint: Find the percentage for brand C first.
📘 Step-by-Step Algebraic Solution
Step 1
Percentage for C: $100\% - 60\% - 25\% = 15\%$.
Step 2
Calculate: $500 \times 0.15 = 75$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Calculate $500 \times (1 - 0.6 - 0.25)$ in Desmos.
Step 2
Result is 75.
Question 9Percentage of a Whole
Medium
If $x$ is 20% of $y$, and $y$ is 50% of $z$, what percentage of $z$ is $x$?
Hint: Pick a value for z, e.g., 100.
📘 Step-by-Step Algebraic Solution
Step 1
Let $z = 100$.
Step 2
$y = 0.5 \times 100 = 50$.
Step 3
$x = 0.2 \times 50 = 10$.
Step 4
$10$ is $10\%$ of $100$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Multiply the percentages: $0.2 \times 0.5 = 0.1$.
Step 2
Convert to percentage: 10%.
Question 10Percentage of a Whole
Hard
A solution is 20% acid. If 5 liters of water are added to 20 liters of this solution, what is the new percentage of acid?
Hint: Find the amount of pure acid, then divide by the new total volume.
📘 Step-by-Step Algebraic Solution
Step 1
Amount of acid: $20 \times 0.20 = 4$ liters.
Step 2
New total volume: $20 + 5 = 25$ liters.
Step 3
New percentage: $4 / 25 = 0.16$ or $16\%$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Enter $(20 \times 0.2) / (20 + 5)$ into Desmos.
Step 2
Result is 0.16.
Question 11Reverse Percentages
Easy
After a 10% discount, a jacket costs $90. What was the original price?
Hint: Let original price be $x$. $0.9x = 90$.
📘 Step-by-Step Algebraic Solution
Step 1
$0.9x = 90$.
Step 2
$x = 90 / 0.9 = 100$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Divide 90 by 0.9 in Desmos.
Step 2
Result is 100.
Question 12Reverse Percentages
Easy
A number increased by 25% is 125. What is the number?
Hint: 1.25 * x = 125.
📘 Step-by-Step Algebraic Solution
Step 1
$1.25x = 125$.
Step 2
$x = 125 / 1.25 = 100$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Divide 125 by 1.25 in Desmos.
Step 2
Result is 100.
Question 13Reverse Percentages
Medium
A company's revenue increased by 20% in 2022 and then decreased by 20% in 2023. If the 2023 revenue was $480,000, what was the 2021 revenue?
Hint: Let R be 2021 revenue. $R * 1.2 * 0.8 = 480,000$.
📘 Step-by-Step Algebraic Solution
Step 1
$R \times 1.2 \times 0.8 = 480,000$.
Step 2
$R \times 0.96 = 480,000$.
Step 3
$R = 480,000 / 0.96 = 500,000$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Enter $480000 / (1.2 \times 0.8)$ into Desmos.
Step 2
Result is 500000.
Question 14Reverse Percentages
Medium
A shirt is on sale for 30% off. If the discount amount is $15, what is the original price?
Hint: 0.30 * x = 15.
📘 Step-by-Step Algebraic Solution
Step 1
$0.30x = 15$.
Step 2
$x = 15 / 0.30 = 50$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Divide 15 by 0.3 in Desmos.
Step 2
Result is 50.
Question 15Reverse Percentages
Hard
The price of a stock rose by 25% and then fell by $x$%. If the final price is the same as the original price, what is $x$?
Hint: 1.25 * (1 - x/100) = 1.
📘 Step-by-Step Algebraic Solution
Step 1
$1.25(1 - r) = 1$.
Step 2
$1 - r = 1 / 1.25 = 0.8$.
Step 3
$r = 0.2$, so $x = 20$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Calculate $1 - (1 / 1.25)$ in Desmos.
Step 2
Result is 0.2, which is 20%.
Question 16Comparative Percentages
Easy
Company A has 200 employees and Company B has 300 employees. What percentage of the total employees does Company A have?
Hint: Part / Total * 100.
📘 Step-by-Step Algebraic Solution
Step 1
Total employees = $200 + 300 = 500$.
Step 2
Percentage = $200 / 500 = 0.4 = 40\%$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Enter $200 / (200 + 300)$ into Desmos.
Step 2
Result is 0.4.
Question 17Comparative Percentages
Easy
If 40 is what percent of 80?
Hint: 40 / 80 * 100.
📘 Step-by-Step Algebraic Solution
Step 1
$40 / 80 = 0.5$.
Step 2
$0.5 \times 100 = 50\%$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Enter $40 / 80$ into Desmos.
Step 2
Result is 0.5.
Question 18Comparative Percentages
Medium
The number of students in Club X is 25% more than the number of students in Club Y. If Club Y has 80 students, how many students are in Club X?
Hint: 80 * 1.25.
📘 Step-by-Step Algebraic Solution
Step 1
$80 \times (1 + 0.25) = 80 \times 1.25$.
Step 2
$80 \times 1.25 = 100$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Enter $80 \times 1.25$ into Desmos.
Step 2
Result is 100.
Question 19Comparative Percentages
Medium
If $a$ is 50% more than $b$, then $b$ is what percentage less than $a$?
Hint: Let b = 100, then a = 150.
📘 Step-by-Step Algebraic Solution
Step 1
Let $b = 100$, so $a = 150$.
Step 2
Difference = $150 - 100 = 50$.
Step 3
Percentage less = $50 / 150 = 1/3 \approx 33.3\%$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Use formula $(r / (1+r)) \times 100$ where $r=0.5$.
Step 2
$0.5 / 1.5 = 1/3$.
Question 20Comparative Percentages
Hard
In a town, 60% of the population are adults. Of the adults, 40% are employed. What percentage of the total population are employed adults?
Hint: Multiply the percentages.
📘 Step-by-Step Algebraic Solution
Step 1
Let total population = 100.
Step 2
Adults = $100 \times 0.60 = 60$.
Step 3
Employed adults = $60 \times 0.40 = 24$.
Step 4
$24 / 100 = 24\%$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Multiply $0.6 \times 0.4$ in Desmos.
Step 2
Result is 0.24.
Official SAT Exam Blueprint & Weightage
Metric
Exam Pattern & Weightage
Question Frequency
1-3 questions per test module
Module 1 Appearance
Medium Frequency (Foundational tests)
Module 2 Appearance
High Frequency (Hard module score differentiator)
Pattern Analysis
College Board frequently tests reverse percentages and multi-step percentage changes in word problems involving population growth or price adjustments.
🏛️
Official SAT PYQ Drill Bank (2023–2026)
We are compiling real verified College Board exam patterns with step-by-step Desmos shortcuts for Percentages.
Updated for 2026 Testing Season
Revision, Traps & Desmos Syntax
Percentage Change
$\frac{New - Old}{Old} \times 100$
Used to find the percent increase or decrease between two values.
Multiplier Method
$Value \times (1 \pm r)^t$
Used for repeated percentage changes over time (compound interest/growth).
🚨 Top SAT Traps & Misconceptions
⚠️ SAT Trap: The 'Add/Subtract' Trap
Adding or subtracting percentages directly (e.g., 20% increase then 10% decrease is NOT a 10% increase). Always use multipliers.
⚠️ SAT Trap: The 'Reverse' Trap
Applying the percentage change to the final value instead of dividing by the multiplier to find the original.
⚡ Essential Desmos Cheatsheet
🎯 The 100-Base Method
100 * (1 + r1) * (1 + r2)
Always use 100 as your starting value to visualize percentage changes as simple numbers.
3-Level Mock Test (30 Questions)
🟢 Level 1: Foundation
10 Qs · Sub-600 Score
🟡 Level 2: Target 700+
10 Qs · 600–740 Score
🔴 Level 3: 800-Mastery
10 Qs · 750–800 Score
Question 1Level 1: Foundation
What is $20\%$ of $80$?
Explanation:
Step 1
Convert the percentage to a decimal.
Step 2
$0.20 \times 80 = 16$
Step 3
The correct answer is $16$.
Question 2Level 1: Foundation
A shirt costs $40$. If there is a $10\%$ discount, how much is the discount amount?
Explanation:
Step 1
Calculate $10\%$ of the original price.
Step 2
$0.10 \times 40 = 4$
Step 3
The discount is $4$.
Question 3Level 1: Foundation
If $50$ is $25\%$ of a number $x$, what is $x$?
Explanation:
Step 1
Set up the equation $0.25x = 50$.
Step 2
$x = \frac{50}{0.25}$
Step 3
$x = 200$
Question 4Level 1: Foundation
A student answered $18$ out of $20$ questions correctly. What percentage is this?
Explanation:
Step 1
Write the fraction of correct answers.
Step 2
$\frac{18}{20} = 0.9$
Step 3
$0.9 \times 100 = 90\%$
Question 5Level 1: Foundation
Increase $60$ by $50\%$.
Explanation:
Step 1
Find $50\%$ of $60$.
Step 2
$0.50 \times 60 = 30$
Step 3
Add the increase to the original: $60 + 30 = 90$.
Question 6Level 1: Foundation
What is $15\%$ of $200$?
Explanation:
Step 1
Multiply $200$ by $0.15$.
Step 2
$200 \times 0.15 = 30$
Step 3
The result is $30$.
Question 7Level 1: Foundation
A price decreases from $80$ to $60$. What is the percentage decrease?
Explanation:
Step 1
Find the difference: $80 - 60 = 20$.
Step 2
Divide by the original: $\frac{20}{80} = 0.25$.
Step 3
Convert to percentage: $25\%$.
Question 8Level 1: Foundation
If $x$ is $10\%$ of $500$, what is $2x$?
Explanation:
Step 1
Calculate $x = 0.10 \times 500 = 50$.
Step 2
Calculate $2x = 2 \times 50$.
Step 3
$2x = 100$.
Question 9Level 1: Foundation
What is $200\%$ of $15$?
Explanation:
Step 1
Convert $200\%$ to $2.0$.
Step 2
$2.0 \times 15 = 30$.
Step 3
The result is $30$.
Question 10Level 1: Foundation
A jacket costs $120$. If the tax is $5\%$, what is the total cost?
Explanation:
Step 1
Calculate tax: $0.05 \times 120 = 6$.
Step 2
Add tax to price: $120 + 6 = 126$.
Step 3
Total cost is $126$.
Question 11Level 2: Target 700+
The population of a town increased from $12,000$ to $15,000$. What was the percentage increase?
Explanation:
Step 1
Find the increase: $15,000 - 12,000 = 3,000$.
Step 2
Divide by original: $\frac{3,000}{12,000} = 0.25$.
Step 3
$0.25 = 25\%$.
Question 12Level 2: Target 700+
An item is marked down by $20\%$, then the new price is marked down by another $10\%$. What is the total percentage discount?
Explanation:
Step 1
Let original price be $100$.
Step 2
After $20\%$ off: $100 \times 0.8 = 80$. After $10\%$ off: $80 \times 0.9 = 72$.
Step 3
Total discount is $100 - 72 = 28\%$.
Question 13Level 2: Target 700+
If $x$ is $20\%$ more than $y$, and $y$ is $50$, what is $x$?
Explanation:
Step 1
$x = y + 0.20y = 1.2y$.
Step 2
$x = 1.2 \times 50$.
Step 3
$x = 60$.
Question 14Level 2: Target 700+
A store sells a product for $P$. If the price is increased by $25\%$, what is the new price in terms of $P$?
Explanation:
Step 1
Increase $P$ by $25\%$ means $P + 0.25P$.
Step 2
$1P + 0.25P = 1.25P$.
Step 3
The new price is $1.25P$.
Question 15Level 2: Target 700+
A solution contains $20\%$ salt. If there are $40$ grams of salt, how many grams of total solution are there?
Explanation:
Step 1
$0.20 \times \text{Total} = 40$.
Step 2
$\text{Total} = \frac{40}{0.20}$.
Step 3
$\text{Total} = 200$.
Question 16Level 2: Target 700+
If $40\%$ of $x$ is equal to $60\%$ of $y$, what is the ratio $x:y$?
If $a$ is $20\%$ of $b$, and $b$ is $25\%$ of $c$, then $a$ is what percentage of $c$?
Explanation:
Step 1
$a = 0.2b$ and $b = 0.25c$.
Step 2
$a = 0.2(0.25c) = 0.05c$.
Step 3
$0.05 = 5\%$.
Question 23Level 3: 800 Mastery
A stock price drops $50\%$ and then rises $50\%$. What is the total percentage change?
Explanation:
Step 1
Start at $100$.
Step 2
$100 \times 0.5 = 50$. Then $50 \times 1.5 = 75$.
Step 3
$100 - 75 = 25$ decrease, which is $25\%$.
Question 24Level 3: 800 Mastery
In a class, $60\%$ of students are girls. If $20\%$ of the girls and $40\%$ of the boys wear glasses, what percentage of the total class wears glasses?
Explanation:
Step 1
Let class size be $100$. Girls = $60$, Boys = $40$.
Step 2
Girls with glasses: $0.20 \times 60 = 12$. Boys with glasses: $0.40 \times 40 = 16$.
Step 3
Total with glasses: $12 + 16 = 28$ out of $100$, so $28\%$.
Question 25Level 3: 800 Mastery
If the length of a rectangle is increased by $20\%$ and the width is decreased by $20\%$, what is the change in area?
Explanation:
Step 1
Area $A = L \times W$. New Area $A' = (1.2L)(0.8W)$.
Step 2
$A' = 0.96LW = 0.96A$.
Step 3
$1 - 0.96 = 0.04$, which is a $4\%$ decrease.
Question 26Level 3: 800 Mastery
A candidate receives $45\%$ of the vote in an election and loses by $2,000$ votes. How many total votes were cast?
Explanation:
Step 1
Winner received $55\%$ of the vote.
Step 2
Difference is $55\% - 45\% = 10\%$. $0.10x = 2,000$.
Step 3
$x = 20,000$.
Question 27Level 3: 800 Mastery
A solution is $30\%$ acid. How much water must be added to $10$ liters of this solution to make it $20\%$ acid?
Explanation:
Step 1
Acid amount: $0.30 \times 10 = 3$. Let $x$ be water added.
Step 2
$\frac{3}{10+x} = 0.20$.
Step 3
$3 = 2 + 0.2x \implies 1 = 0.2x \implies x = 5$.
Question 28Level 3: 800 Mastery
If $x$ is $25\%$ of $y$, and $y$ is $400\%$ of $z$, then $x$ is what percentage of $z$?
Explanation:
Step 1
$x = 0.25y$ and $y = 4z$.
Step 2
$x = 0.25(4z) = 1z$.
Step 3
$1z$ is $100\%$ of $z$.
Question 29Level 3: 800 Mastery
A price is increased by $x\%$ and then decreased by $x\%$. The final price is $96\%$ of the original. What is $x$?