Percent Change and Growth | Varsity Tutors

SECTION 1

Historical Context & Motivation

The concept of expressing change as a fraction of a starting quantity is far older than the modern percent sign. Ancient merchants in Mesopotamia and Rome routinely computed proportional differences when pricing goods across trade routes, though they lacked a unified notation. The Latin phrase per centum—meaning "by the hundred"—did not crystallize into standard mathematical practice until European commerce demanded a universal way to compare profits, taxes, and interest across different currencies and quantities. Understanding this lineage reveals why percent change remains the dominant metric for comparing quantities on standardized tests such as the GRE: it normalizes raw differences against a common scale of 100.

c. 300 BCE

Proportional Reasoning in Antiquity

Greek mathematicians, including Euclid, formalized the theory of ratios and proportions in Elements, establishing the conceptual bedrock for comparing magnitudes.

15th c.

Per Centum in Italian Commerce

Italian merchants began writing interest calculations as fractions with a denominator of 100, eventually abbreviating 'per cento' into a symbol that evolved into the modern '%' sign.

1614

Napier's Logarithms

John Napier published logarithmic tables, enabling efficient computation of compound growth—a natural extension of repeated percent increases.

1935

Standardized Aptitude Testing

The first GRE administration required quantitative reasoning about data, making percent change a staple topic for graduate-school admission.

At its core, the question that percent change answers is deceptively simple: by what proportion of the original quantity did something increase or decrease? Yet the GRE exploits the subtleties hidden within this question—confusing the base of comparison, chaining successive changes, and distinguishing absolute from relative differences. The sections that follow will equip you to handle every variant with precision.

SECTION 2

Core Principles & Definitions

Before diving into formulas, it is essential to internalize the foundational ideas that govern percent change problems. Every GRE question in this domain ultimately rests on a small set of principles, and misunderstanding even one of them can lead to an incorrect answer on test day.

The Base Matters

Percent change is always computed relative to the original (initial) value. A $10 increase on a $50 base is 20%, but the same $10 increase on a $200 base is only 5%.

Direction: Increase vs. Decrease

A positive percent change signals growth; a negative percent change signals decline. The formula is identical in both cases—only the sign of the numerator differs.

Successive Changes Don't Simply Add

A 20% increase followed by a 20% decrease does not return you to the original value. Each change re-defines the base, so compounding effects arise.

Multiplier Representation

Every percent change can be expressed as a single multiplier: a 25% increase multiplies the original by 1.25, a 30% decrease multiplies by 0.70. This representation simplifies chained calculations.

Compound Growth

When the same percent change is applied repeatedly over n periods, the result is exponential: Final = Initial × (1 + r)ⁿ. This is the engine behind interest, population growth, and depreciation.

✦ KEY TAKEAWAY

Think of percent change like a recipe that scales to the size of your batch. If you double the sugar in a small cookie recipe, the taste changes dramatically; double the sugar in a commercial bakery batch, and the shift may barely register. The base quantity is the batch size—it determines how significant any absolute change actually is when expressed as a percentage.

SECTION 3

Visual Explanation

A powerful way to grasp percent change is to visualize it as a bar comparison. The diagram below contrasts an original value with a new value, making explicit both the absolute difference and the proportional relationship that defines the percent change.

The violet bar represents the original value (200). The cyan bar represents the new value (280). The pink-highlighted segment shows the absolute difference (80), and the formula below demonstrates how that difference, divided by the original, yields a 40% increase.

Notice that the pink region measures the change relative to the violet bar (the original), not the cyan bar (the new value). If you were instead asked for the percent decrease from 280 back to 200, the base would shift to 280, giving a different percentage—approximately 28.6%. This asymmetry is a favorite testing point on the GRE: the percent increase from A to B and the percent decrease from B back to A are not the same number, precisely because the base changes.

SECTION 4

Mathematical Framework

Percent change problems on the GRE distill into a small toolkit of equations. The essential skill is recognizing which formula applies and identifying the correct base. Below are the four equations you need.

BASIC PERCENT CHANGE

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

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

MULTIPLIER FORM

New = Original × (1 + r)

r = the rate of change expressed as a decimal (e.g., +0.25 for a 25% increase, −0.15 for a 15% decrease). The quantity (1 + r) is called the multiplier.

SUCCESSIVE PERCENT CHANGES

Final = Original × (1 + r₁) × (1 + r₂) × … × (1 + rₙ)

r₁, r₂, …, rₙ are the individual rates of change (positive or negative). Each multiplier is applied sequentially. The net percent change equals ((Final / Original) − 1) × 100.

COMPOUND GROWTH / DECAY

A = P × (1 + r)ⁿ

A = final amount, P = initial principal or quantity, r = rate per period (as a decimal), n = number of periods. This is the exponential extension of the multiplier form when the same rate applies repeatedly.

💡 GRE TIP

When the GRE asks for the "overall percent change" after two successive changes, resist the urge to add the percentages. Instead, multiply the corresponding multipliers and convert back. For instance, a 10% increase then a 10% decrease gives 1.10 × 0.90 = 0.99, a net 1% decrease, not 0%.

SECTION 5

Linear vs. Exponential Growth

The GRE occasionally presents data interpretation questions that require you to distinguish between linear growth (a fixed amount added each period) and exponential growth (a fixed percentage applied each period). The difference is subtle when periods are few, but it becomes dramatic over time. The diagram below contrasts the two trajectories for an initial value of 100 growing at either a flat +20 per period or a 20% compound rate.

The amber line shows linear growth: 100 → 120 → 140 → … (constant increments of 20). The cyan curve shows 20% compound growth: 100 → 120 → 144 → 172.8 → … By period 8, the exponential path reaches roughly 430, far surpassing the linear path at 260.

Numerical comparison of linear vs. exponential growth from an initial value of 100

Period	Linear Value (+20)	Exponential Value (×1.20)	Difference
0	100	100.00	0.00
2	140	144.00	4.00
5	200	248.83	48.83
8	260	429.98	169.98
10	300	619.17	319.17

The key insight is that in linear growth the absolute increase per period is constant, while in exponential (compound) growth the absolute increase itself grows because each period's change is a fixed percentage of an ever-larger base. GRE Data Interpretation sets frequently embed tables or graphs that require you to identify which model best describes the data.

SECTION 6

Worked Example

Let us walk through a GRE-style problem that combines successive percent changes with the multiplier approach.

📝 PROBLEM

A store raises the price of a jacket by 25% at the start of winter. At the end of winter, the store applies a 20% discount to the new price. What is the overall percent change from the original price?

Step-by-Step Solution

Step 1 — Identify the MultipliersA 25% increase corresponds to a multiplier of 1 + 0.25 = 1.25. A 20% decrease corresponds to a multiplier of 1 − 0.20 = 0.80.

Multipliers: 1.25 and 0.80

Step 2 — Compute the Combined MultiplierMultiply the two multipliers together: 1.25 × 0.80 = 1.00. At first glance this appears to be zero net change, but let us compute more carefully: 1.25 × 0.80 = 1.000. Indeed, in this specific case the combined multiplier is exactly 1.00.

Combined multiplier = 1.00

Step 3 — Convert to Percent ChangeNet percent change = (Combined Multiplier − 1) × 100 = (1.00 − 1) × 100 = 0%. The price returns exactly to its original level.

Net percent change = 0%

Step 4 — Verify with a Concrete ValueLet the original price be $100. After a 25% increase: $100 × 1.25 = $125. After a 20% discount: $125 × 0.80 = $100. The jacket is back at $100, confirming 0% net change.

Verification: $100 → $125 → $100 ✓

Step 5 — Reflect on the TrapMany test-takers would guess that 25% up and 20% down yields a 5% net increase, simply adding the percentages. The multiplier method reveals the truth: because the 20% discount was applied to the higher price of $125, the dollar amount removed ($25) exactly offset the dollar amount added ($25). This coincidence arises because 25% of 100 equals 20% of 125. In general, such exact cancellations are rare.

SECTION 7

Common Pitfalls & Comparisons

GRE percent change questions are designed to exploit predictable errors. The table below catalogues the most frequent mistakes alongside the correct reasoning.

Common GRE pitfalls with percent change

Pitfall	Why It's Wrong	Correct Approach
Adding successive percentages (e.g., +30% then −30% = 0%)	The base changes after the first operation, so the second percentage acts on a different value.	Multiply the multipliers: 1.30 × 0.70 = 0.91, yielding a 9% decrease.
Using the wrong base in a comparison (e.g., using New instead of Original)	Percent change is defined relative to the Original. Switching the base yields a different—and incorrect—percentage.	Always use the value before the change as the denominator.
Confusing percent change with percentage-point change	Going from 20% to 25% is a 5 percentage-point increase but a 25% percent increase (5/20 × 100).	Read the question carefully: 'percent change' and 'percentage points' are distinct quantities.
Ignoring the sign convention	A negative percent change indicates a decrease. Dropping the sign leads to selecting the wrong answer choice.	Keep track of signs throughout; label your final answer as an increase or decrease.

✦ KEY TAKEAWAY

Think of the multiplier method as a universal translator: no matter how many successive changes stack up, converting each to a multiplier and taking the product gives you a single number that encodes the entire history of changes. It is the same principle engineers use when cascading amplifier stages—each stage's gain multiplies, it never merely adds. On the GRE, this one technique eliminates the most common class of errors.

SECTION 8

Connection to Advanced Quantitative Topics

Percent change is not merely an isolated topic; it connects deeply to several other areas tested on the GRE and encountered in graduate-level coursework. The table below maps percent change concepts to their more advanced counterparts, illustrating how the foundational material in this lesson extends into broader quantitative reasoning.

From basic percent change to advanced quantitative reasoning

Percent Change Concept	Advanced Extension	Where You'll See It
Single percent change	Elasticity (economics): % change in quantity / % change in price	Graduate microeconomics, GRE Data Interpretation
Compound growth (1 + r)ⁿ	Continuous compounding: Peʳᵗ, where e is Euler's number	Finance, differential equations, GRE Quantitative Comparison
Successive percent changes	Geometric series and products in discrete mathematics	Computer science, actuarial science
Percentage-point vs. percent change	Marginal vs. relative effect sizes in statistics	Graduate research methods, meta-analysis

On the GRE itself, you may encounter Quantitative Comparison questions that test whether an expression involving percent changes is greater than, less than, or equal to a given benchmark. The multiplier framework transfers directly: convert each scenario into a product of multipliers, compare the products, and you have your answer without ever needing to choose a specific starting value. Looking beyond the exam, the ability to reason fluently about relative change is indispensable in data science, finance, and any field that relies on interpreting trends.

SECTION 9

Practice Problems

Work through the following five problems in order. Each builds on the principles discussed in this lesson, and the difficulty escalates from conceptual recall to critical analysis.

PROBLEM 1 — CONCEPTUAL

Explain why a 50% increase followed by a 50% decrease does not return a quantity to its original value. What is the net percent change?

PROBLEM 2 — BASIC CALCULATION

A town's population increased from 12,500 to 15,000 over five years. What is the percent increase?

PROBLEM 3 — INTERMEDIATE

A stock price rises by 10% in January, falls by 15% in February, and rises by 20% in March. What is the overall percent change from the beginning of January to the end of March?

PROBLEM 4 — APPLIED

A savings account earns 6% annual interest, compounded once per year. If $2,000 is deposited today, what is the account balance after 4 years, and what is the total percent increase over that period? Round to the nearest cent.

PROBLEM 5 — CRITICAL THINKING

Quantity A: The overall percent decrease when a value is reduced by 10% three times in succession. Quantity B: 30%. Compare the two quantities (Quantitative Comparison format). Justify your answer algebraically.

SUMMARY

Lesson Summary

Percent change measures how much a quantity has increased or decreased relative to its original value, expressed on a scale of 100. The fundamental formula— % Change = ((New − Original) / Original) × 100 —underpins every problem in this domain. For successive changes, convert each percent change into a multiplier (1 + r for increases, 1 − r for decreases) and take the product; never simply add or subtract percentages. Compound growth extends this idea over n periods via A = P × (1 + r)ⁿ, producing exponential behavior that diverges sharply from linear growth.

Key pitfalls to avoid on the GRE include confusing percentage-point change with percent change, using the wrong base value, and naively adding successive percent changes. The multiplier method is your single most reliable tool: it handles increases, decreases, and chains of changes in a unified framework. Master it, and you will approach any GRE percent change question with confidence.