Solve percent increase and decrease problems. - ISEE Upper Level Quantitative Reasoning

SECTION 1

Historical Context & Motivation

The concept of percent comes from the Latin phrase per centum, meaning "out of one hundred." Long before modern algebra existed, merchants and tax collectors needed a standardized way to express how much a quantity had grown or shrunk relative to its starting value. Whether it was the price of grain rising after a poor harvest or a kingdom's treasury declining after a costly war, people needed a common language for relative change. That language became the percent.

~300 BCE • Ancient Fractions

Egyptian and Babylonian traders used unit fractions to calculate markups on goods. These early ratio methods planted the seeds for proportional reasoning.

~100 CE • Roman Taxation by Hundredths

Emperor Augustus levied a tax of 1/100 on auction sales, called the centesima rerum venalium. This is one of the earliest recorded uses of a rate expressed per hundred.

1400s • Italian Merchant Math

Italian merchants popularized the abbreviation 'per cento' in commercial arithmetic texts, standardizing percent notation across European trade.

1800s • Modern Statistical Use

Economists and scientists adopted percent change as the standard way to compare growth rates, inflation, and experimental results across different scales.

Today • Standardized Testing

Percent increase and decrease questions appear on virtually every standardized admissions test, including the ISEE Upper Level, because they test proportional reasoning — a skill essential for advanced math and real life.

The central question these problems address is deceptively simple: By what fraction of the original amount did something change? On the ISEE, this question takes many forms — population growth, sale discounts, successive changes, and quantitative comparisons. Mastering the underlying logic will make every variation feel familiar.

SECTION 2

Core Principles & Definitions

Before diving into formulas, you need a rock-solid understanding of four foundational ideas. Every percent change problem on the ISEE connects back to these principles, so take a moment to lock them in.

The Original (Base) Value

Percent change is always calculated relative to the original amount — the value before the change occurred. Misidentifying the base is the single most common error on these problems.

Amount of Change

The amount of change equals the new value minus the original value. A positive result means an increase; a negative result means a decrease.

Percent as a Ratio

A percent is simply a ratio with a denominator of 100. Writing 25% is the same as writing 25/100, or 0.25, or 1/4. Fluency in converting among these forms will save you time.

The Multiplier Shortcut

A 30% increase means the new value is 130% of the original, so you can multiply by 1.30. A 30% decrease means the new value is 70% of the original, so multiply by 0.70. This one-step approach is the fastest method on test day.

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KEY TAKEAWAY

Think of percent change like a speedometer in a car. The speedometer doesn't tell you where you are — it tells you how fast you're moving relative to standing still. Similarly, a percent change doesn't tell you the new value directly — it tells you how much the value shifted relative to where it started. Always anchor your calculation to the starting point.

SECTION 3

Visual Explanation

The bar model below shows how percent increase and percent decrease relate to an original value. The full blue bar represents the original quantity (100%). Notice how an increase adds a segment beyond 100%, while a decrease removes a segment, leaving less than 100%.

The green segment represents the amount added during a 25% increase, while the red-dashed segment represents the amount removed during a 25% decrease. Both changes are calculated from the same original value of 200.

Notice something important: even though both changes are 25%, the amount of change is the same (50) only because the original value is the same. If you increased to 250 and then decreased 250 by 25%, you would lose 62.5 — not 50 — because the base changed. This asymmetry is a favorite trap on the ISEE, so keep it in mind whenever you see successive percent changes.

SECTION 4

Mathematical Framework

There are two main formulas you need for percent change problems. The first finds the percent change when you know the original and new values. The second finds the new value when you know the original value and the percent change.

PERCENT CHANGE FORMULA

Percent Change = ((New − Original) / Original) × 100%

New = the value after the change. Original = the value before the change. A positive result means an increase; a negative result means a decrease.

NEW VALUE FORMULA (INCREASE)

New Value = Original × (1 + r)

r = the percent increase expressed as a decimal. For example, a 40% increase means r = 0.40, so the multiplier is 1.40.

NEW VALUE FORMULA (DECREASE)

New Value = Original × (1 − r)

r = the percent decrease expressed as a decimal. For example, a 15% decrease means r = 0.15, so the multiplier is 0.85.

SUCCESSIVE PERCENT CHANGES

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

For successive changes, multiply the individual multipliers together. You cannot simply add or subtract the percentages — each change applies to a different base.

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ISEE STRATEGY

When answer choices are numbers ordered from least to greatest, you can often estimate to eliminate two or three options before calculating. If a $300 item increases by about 30%, the new price is roughly $390 — any choice far from $390 can be crossed out immediately. This process of elimination saves valuable seconds and protects against careless errors.

SECTION 5

Common Problem Types on the ISEE

Percent change questions on the ISEE Upper Level fall into a handful of recognizable categories. The diagram below maps out five common types and the key strategy for each. Recognizing the type quickly helps you choose the right formula without hesitation.

The five most common percent change problem types on the ISEE Upper Level. Each type requires a slightly different algebraic setup, but all share the same core formula. Type 5 (Quantitative Comparison) is unique to the ISEE format.

Problem Type	What You're Given	What You Solve For	Key Tip
Find % Change	Original and new values	The percent increase or decrease	Divide by the original, not the new value
Find New Value	Original value and percent change	The value after the change	Use the multiplier (1 ± r) for speed
Find Original	New value and percent change	The starting value	Divide the new value by the multiplier
Successive Changes	Two or more percent changes in sequence	The final value or net percent change	Multiply multipliers — do NOT add the percents
QC Comparison	Two quantities involving percent change	Which column is greater (or can't be determined)	Test specific values if variables are present

Quick reference for the five common ISEE percent change problem types

SECTION 6

Worked Example

Let's walk through a multi-step ISEE-style problem to see the formulas in action. Pay close attention to how the base value changes between successive percent changes.

Successive Percent Change Problem

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Step 1 — Read the Problem

A store originally prices a jacket at $80. During a holiday sale, the price is reduced by 25%. One week later, the sale price is increased by 20%. What is the final price of the jacket?
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Step 2 — Apply the First Change (25% Decrease)

The multiplier for a 25% decrease is 1 − 0.25 = 0.75. Multiply: $80 × 0.75 = $60.

Sale price = $60
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Step 3 — Apply the Second Change (20% Increase)

Now the base is $60 (not $80!). The multiplier for a 20% increase is 1 + 0.20 = 1.20. Multiply: $60 × 1.20 = $72.

Final price = $72
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Step 4 — Verify with the Combined Multiplier

Alternatively, multiply the two multipliers together: 0.75 × 1.20 = 0.90. Then $80 × 0.90 = $72. This confirms the final price and shows that the net effect is a 10% decrease — not a 5% decrease, as you might mistakenly get by subtracting 25% − 20%.

Net change = −10% (not −5%)
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Step 5 — Choose the Answer

The final price of the jacket is $72. If this were a multiple-choice question with options like (A) $60, (B) $72, (C) $76, (D) $80, you would confidently select (B). Notice that $76 is the trap answer for students who incorrectly compute a 5% net decrease from $80.

Answer: $72

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TRAP ALERT

On the ISEE, one of the most popular wrong answers in successive percent change problems is the result of adding or subtracting the percentages directly. A 25% decrease followed by a 20% increase is NOT a 5% decrease. Always multiply the multipliers instead.

SECTION 7

Common Errors & How to Avoid Them

Understanding the formulas is only half the battle. On the actual ISEE, time pressure and tricky wording cause even well-prepared students to stumble. The table below catalogues the most frequent mistakes and offers a concrete fix for each one.

Common Error	Why It Happens	How to Fix It
Using the wrong base	The student divides by the new value instead of the original, or uses a changed value as the base for a successive change.	Circle the word 'original' or 'was' in the problem. The base is always the value BEFORE the change.
Adding successive percents	It feels intuitive to add +20% and −10% to get +10%, but this ignores the shifting base.	Always multiply multipliers: 1.20 × 0.90 = 1.08, which is an 8% increase — not 10%.
Forgetting to convert to/from decimal	The student writes 25 instead of 0.25, leading to wildly inflated answers.	Before computing, convert every percent to a decimal by dividing by 100. Check: does your answer make sense in context?
Confusing % of vs. % off	'30% off' means you pay 70% of the price, but '30% of' means you pay 30%. A subtle wording difference changes the answer completely.	Read the phrasing twice. '% off' → subtract from 1. '% of' → use the percent directly.
Choosing answer (D) too quickly on QC	Students assume 'cannot be determined' whenever they see variables, even when the relationship is fixed.	Test at least two different values. If both give the same relationship, (D) is likely wrong. Only choose (D) if different values yield different outcomes.

Five most frequent errors on ISEE percent change problems

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KEY TAKEAWAY

Most wrong answers on percent change problems aren't random — they're predictable traps designed around common mistakes. If you know the traps in advance, you can avoid them on test day and even use the wrong answers to double-check your work. If you got one of the trap answers, pause and re-examine your base value.

SECTION 8

Connection to Advanced Topics

Percent increase and decrease are stepping stones to more advanced mathematical ideas you will encounter in high school and beyond. Understanding these connections can deepen your grasp of the concept and occasionally help you solve ISEE problems more efficiently.

ISEE Concept	Advanced Connection	Why It Matters
Percent change formula	Rate of change (slope)	In algebra and calculus, the idea of 'change relative to a starting point' generalizes to slope and derivatives.
Successive percent changes	Compound interest / Exponential growth	Applying a percent change repeatedly leads to the compound interest formula A = P(1 + r)ⁿ.
Multiplier method (1 ± r)	Geometric sequences	Each successive term in a geometric sequence is the previous term multiplied by a constant ratio — the same structure as repeated percent change.
Finding the original from the new value	Inverse operations / Solving equations	Dividing by the multiplier is an inverse operation — a core algebraic skill used throughout math.

How ISEE percent change concepts connect to advanced math

For the ISEE Upper Level specifically, you should be comfortable with the idea that a percent increase followed by the same percent decrease does not return you to the original value. A 20% increase followed by a 20% decrease yields a net multiplier of 1.20 × 0.80 = 0.96, meaning a 4% net decrease. This pattern generalizes: an x% increase followed by an x% decrease always produces a net decrease equal to (x/100)² × 100%. Knowing this shortcut can save time on test day.

SECTION 9

Practice Problems

Work through these five problems in order. They increase in difficulty, and the last two use the quantitative comparison format you'll see on the actual ISEE. Remember: there is no penalty for guessing, so always select an answer. Use process of elimination to improve your odds.

Question 1: CONCEPTUAL

A town's population was 4,000 in 2020 and 5,000 in 2021. What was the percent increase in population? (A) 20% (B) 25% (C) 50% (D) 80%

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The correct answer is (B) 25%. The amount of change is 5,000 − 4,000 = 1,000. Percent change = (1,000 / 4,000) × 100% = 25%. Choice (A) 20% results from dividing the change by the new value (1,000/5,000) — a common base error. Choice (C) 50% and (D) 80% do not correspond to any correct calculation and can be eliminated because a jump from 4,000 to 5,000 is clearly less than doubling.

Question 2: BASIC CALCULATION

A laptop originally costs $1,200. It goes on sale for 15% off. What is the sale price? (A) $180 (B) $1,020 (C) $1,050 (D) $1,380

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The correct answer is (B) $1,020. A 15% decrease means you pay 85% of the original price. Using the multiplier: $1,200 × 0.85 = $1,020. Choice (A) $180 is just 15% of $1,200 — the discount amount, not the sale price. Choice (C) $1,050 might come from subtracting $150 (a miscalculation of 15% of $1,000). Choice (D) $1,380 is the result of a 15% increase instead of a decrease.

Question 3: INTERMEDIATE

After a 30% increase, the enrollment at a school is 650 students. What was the enrollment before the increase? (A) 195 (B) 455 (C) 500 (D) 845

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The correct answer is (C) 500. Let the original enrollment be x. After a 30% increase: x × 1.30 = 650. Solving: x = 650 / 1.30 = 500. Choice (A) 195 is 30% of 650 — the amount of the increase, not the original. Choice (B) 455 comes from subtracting 30% of 650 from 650 (650 − 195 = 455), which incorrectly uses the new value as the base. Choice (D) 845 is 650 × 1.30, which would be a 30% increase from 650 rather than finding the original.

Question 4: APPLIED

Column A: The value of a $500 investment after a 10% increase followed by a 10% decrease Column B: $500 (A) Column A is greater (B) Column B is greater (C) The two quantities are equal (D) Cannot be determined

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The correct answer is (B) Column B is greater. Column A: $500 × 1.10 × 0.90 = $500 × 0.99 = $495. Column B: $500. Since $495 < $500, Column B is greater. The trap here is choosing (C), which assumes a 10% increase and 10% decrease cancel out. They don't, because each change applies to a different base. This is a pure-number comparison, so (D) cannot be correct.

Question 5: CRITICAL THINKING

x is a positive number. Column A: The percent increase from x to 3x Column B: The percent decrease from 3x to x (A) Column A is greater. (B) Column B is greater. (C) The two quantities are equal. (D) Cannot be determined.

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The correct answer is (A) Column A is greater. Column A: Percent increase from x to 3x = ((3x − x) / x) × 100% = (2x / x) × 100% = 200%. Column B: Percent decrease from 3x to x = ((3x − x) / 3x) × 100% = (2x / 3x) × 100% = 66.67%. Since 200% > 66.67%, Column A is always greater regardless of the value of x. The variable cancels out, so (D) is incorrect. Choice (C) is the trap for students who assume matching amounts of change (2x in both cases) mean matching percents — but percent change depends on the base. Testing values confirms: if x = 10, Column A = 200%, Column B ≈ 66.7%. If x = 100, Column A = 200%, Column B ≈ 66.7%. The relationship is always the same.

SUMMARY

Lesson Summary

Summary

Percent change problems revolve around one core idea: the amount of change divided by the original value. To find the percent change, use (New − Original) / Original × 100%. To find a new value after a percent change, multiply the original by the multiplier (1 ± r). To find the original from a new value, divide the new value by the multiplier. For successive percent changes, multiply the individual multipliers — never add or subtract the percentages.

On ISEE quantitative comparison problems, watch for traps involving the asymmetry of percent increase and decrease — a percent increase followed by the same percent decrease does NOT return to the original value. Always identify the correct base value before computing, and use estimation and process of elimination to work efficiently under time pressure. With these strategies, percent change problems become some of the most reliable points you can earn on test day.