Percent Increase & Decrease — Compute percent increase and...

SECTION 1

Historical Context & Motivation

The concept of expressing change as a fraction of a whole — what we now call percent change — traces its roots to the earliest days of commerce and taxation. Whenever merchants needed to communicate how much a price had risen or fallen relative to its starting value, they required a standardized measure that transcended currency and unit systems. The Latin phrase per centum, meaning "for every hundred," provided exactly that universal reference point, enabling traders across the Roman Empire and medieval Europe to compare proportional changes on a common scale. Today, percent increase and decrease remain indispensable in fields ranging from economics and public health to standardized test preparation, where the ACCUPLACER Arithmetic section regularly asks you to compute these values quickly and accurately.

c. 300 BCE

Ancient Taxation Systems

Ancient civilizations in Mesopotamia and Egypt levied taxes as fixed fractions of harvests and goods, establishing the idea of proportional change long before formal percent notation existed.

1st c. CE

Roman Per Centum

Roman Emperor Augustus imposed a centesima rerum venalium — a 1/100 tax on goods sold at auction — embedding the "per hundred" concept into law and commerce.

15th c.

Italian Mercantile Notation

Italian merchants began abbreviating 'per cento' as 'p. cento' and eventually 'p. 100' or '%', giving rise to the modern percent sign and standardizing proportional calculations in trade ledgers.

19th–20th c.

Statistical & Financial Adoption

Percent change became the default language of economics, journalism, and science. Inflation rates, GDP growth, and population shifts were all reported as percent increases or decreases, making the concept a pillar of quantitative literacy.

1985–present

Standardized Testing Era

The ACCUPLACER, introduced by the College Board, features percent change prominently in its Arithmetic section, reflecting the skill's importance for college-level coursework in business, science, and social sciences.

The fundamental question this lesson addresses is straightforward but deceptively rich: given an original value and a new value, how do you express the magnitude and direction of the change as a percentage of the original? Mastering this question is not merely an exercise in arithmetic — it is the gateway to interpreting data, evaluating financial decisions, and succeeding on timed placement exams like the ACCUPLACER.

SECTION 2

Core Principles & Definitions

Before diving into formulas, it is essential to establish the foundational ideas that underpin every percent-change calculation. These principles ensure you always set up the problem correctly, regardless of context or complexity.

Original (Base) Value

The starting amount before any change occurs. This value always serves as the denominator in the percent-change formula, because percent change is always measured relative to where you began.

Amount of Change

The absolute difference between the new value and the original value: New − Original. A positive difference signals an increase; a negative difference signals a decrease.

Direction Matters

Percent increase and percent decrease are conceptually symmetric but computationally identical — both use the same formula. The sign of the numerator (positive or negative) determines the label you attach to the result.

Multiply by 100

The ratio (Amount of Change ÷ Original) yields a decimal. Multiplying by 100 converts it to a percent. Forgetting this step — or applying it twice — is one of the most common errors on timed tests.

Non-Symmetry of Successive Changes

A 20% increase followed by a 20% decrease does NOT return you to the original value. The base changes after the first operation, so the second percentage acts on a different number — a critical insight tested on the ACCUPLACER.

✦ KEY TAKEAWAY

Think of percent change like an odometer reading on a road trip. The original value is where you started, the new value is where you ended, and the percent change tells you what fraction of the starting distance the difference represents — always measured against the starting point, never the destination.

SECTION 3

Visual Explanation

The diagram below illustrates the anatomy of a percent-change calculation using a bar model. The original value is represented as a full bar, and the change — whether an increase or a decrease — is shown as an additional segment or a removed segment, with the resulting percentage labeled.

The violet bar represents the original value of 200. The green extension shows a 50-unit increase (25%), while the red-shaded region shows a 50-unit decrease (25%). Notice that both computations divide by the original value.

Observe that the green segment (increase) and the red segment (decrease) are the same absolute size — 50 units — yet they both correspond to 25% because the reference denominator remains the original value of 200 in both cases. This visual reinforces the principle that percent change is always relative to the starting point. A common exam trap is to divide by the new value instead; the bar model makes it visually clear why that would distort the comparison.

SECTION 4

Mathematical Framework

Two closely related formulas govern every percent-change problem you will encounter on the ACCUPLACER. Although they are often presented separately, they are really the same equation — the sign of the numerator simply determines whether the result is labeled an increase or a decrease.

PERCENT INCREASE

Percent Increase = ((New Value − Original Value) ÷ Original Value) × 100

Use this form when the new value is greater than the original value. The numerator (New − Original) is positive, yielding a positive percentage.

PERCENT DECREASE

Percent Decrease = ((Original Value − New Value) ÷ Original Value) × 100

Use this form when the new value is smaller than the original value. By subtracting in the order Original − New, you keep the result positive and label it a 'decrease.'

UNIFIED PERCENT CHANGE

Percent Change = ((New − Original) ÷ Original) × 100

If the result is positive, you have a percent increase. If it is negative, you have a percent decrease. This single formula eliminates guesswork about subtraction order.

⚠️ Common ACCUPLACER Trap

Some problems ask: "A price dropped from $80 to $60. What is the percent decrease?" Students frequently divide 20 by 60 (the new value) instead of 80 (the original value). The correct answer is (20 ÷ 80) × 100 = 25%, not 33.3%. Always verify which number is the starting value and place it in the denominator.

SECTION 5

Detailed Breakdown — The Four-Step Method

To solve any percent increase or decrease problem efficiently under timed conditions, follow a systematic four-step method. This approach prevents the most common computational and conceptual errors that cost points on the ACCUPLACER.

The four-step flowchart on the left shows the systematic method: identify, subtract, divide, convert. The right side demonstrates the same process applied to both an increase (80 → 100) and a decrease (80 → 60), each yielding 25%.

Step 1 — Identify Original & New: Read the problem carefully. The original value is always the value that existed earlier in time or before the change. On the ACCUPLACER, phrases like "was," "started at," or "originally" signal the denominator.
Step 2 — Compute the Difference: Subtract the original from the new value. A positive result means increase; negative means decrease.
Step 3 — Divide by the Original: This ratio expresses the change as a decimal fraction of the starting value.
Step 4 — Multiply by 100: Convert the decimal to a percent and label the result as an increase or decrease.

SECTION 6

Worked Example

Let's walk through a representative ACCUPLACER-style problem from start to finish, applying the four-step method developed in the previous section.

Percent Decrease — Rent Reduction

Step 1 — Read and Identify ValuesA tenant's monthly rent was $1,250. After negotiating a new lease, the rent dropped to $1,050. What is the percent decrease in rent? Here, the original value is $1,250 and the new value is $1,050.

Step 2 — Compute the DifferenceAmount of Change = New − Original = $1,050 − $1,250 = −$200. The negative sign confirms we are dealing with a decrease.

Change = −$200

Step 3 — Divide by the Original ValueDecimal Change = −200 ÷ 1,250 = −0.16. We divide by the original ($1,250), not the new rent.

Decimal = −0.16

Step 4 — Convert to PercentPercent Change = −0.16 × 100 = −16%. The rent decreased by 16%. On an ACCUPLACER answer sheet, you would select 16% (decrease) or simply 16%.

16% decrease

Percent Increase — Test Score Improvement

Step 1 — Identify ValuesA student scored 72 on a practice test and then scored 90 on the actual exam. What is the percent increase? The original score is 72 and the new score is 90.

Step 2 — Compute the DifferenceChange = 90 − 72 = 18. Positive, confirming an increase.

Change = +18

Step 3 — Divide by the Original18 ÷ 72 = 0.25.

Decimal = 0.25

Step 4 — Convert to Percent0.25 × 100 = 25%. The student's score improved by 25%.

25% increase

SECTION 7

Common Errors & How to Avoid Them

On a timed placement exam, speed can lead to habitual mistakes. The table below catalogs the most frequent percent-change errors and provides the corrective strategy for each.

Common Percent Change Errors on the ACCUPLACER

Error	Why It Happens	Correction
Dividing by the new value	Students confuse which number is the reference. In a decrease problem, the smaller number feels like the 'base.'	Always ask: 'What was the value BEFORE the change?' That value is the denominator.
Forgetting to multiply by 100	After dividing, students report the decimal (e.g., 0.25) as the final answer instead of 25%.	Make Step 4 a non-negotiable habit. If your answer is less than 1, you probably need to multiply by 100.
Subtracting in the wrong order	In decrease problems, subtracting New − Original gives a negative number, which confuses students who expect a positive percent.	Use the unified formula (New − Original). If the result is negative, label it a 'decrease' and report the absolute value.
Assuming symmetry of successive changes	A 50% increase followed by a 50% decrease seems like it should return to the original — but it doesn't.	After an increase, the new, larger base is used for the next calculation. Test with numbers: 100 → 150 → 75, not 100.

✦ KEY TAKEAWAY

Think of percent change like measuring how far you've traveled compared to where you started, not where you ended. If you start a hike at a 2,000-foot trailhead and climb to 2,500 feet, your elevation gain is measured against 2,000 — your starting elevation — yielding a 25% increase. The summit's altitude is irrelevant to the denominator.

SECTION 8

Connection to Advanced Percent Topics

Percent increase and decrease form the foundation for several more advanced topics that appear on higher-level placement and entrance exams. Understanding how the basic formula extends will deepen your conceptual mastery and prepare you for questions that layer additional complexity onto the core idea.

Basic vs. Advanced Percent Concepts

Basic Concept	Advanced Extension	Key Difference
Single percent increase/decrease	Successive percent changes	Each change uses the result of the previous change as its new base, so you cannot simply add or subtract the percentages.
Percent change between two values	Finding the original from a known percent change	Here the unknown is the denominator. You rearrange: Original = New Value ÷ (1 + r), where r is the decimal rate of change.
One-time change	Compound growth/decay	Repeated identical percent changes over n periods use the formula: Final = Original × (1 ± r)ⁿ. This extends percent change into exponential territory.
Percent of a single quantity	Percent-of-percent (e.g., tax on marked-up price)	Layered percentages require multiplying multipliers: e.g., a 10% markup followed by 8% tax means × 1.10 × 1.08, not × 1.18.

For the ACCUPLACER Arithmetic section, mastery of the single-step percent increase and decrease formulas is sufficient. However, being aware of successive and compound percent changes will help you recognize and avoid trap answers that assume percentages are additive. If you continue into college algebra or statistics, these extensions become essential tools in financial modeling, population dynamics, and inferential statistics.

SECTION 9

Practice Problems

The following five problems progress from conceptual understanding to critical analysis. Work through each one using the four-step method, and check your reasoning against the detailed answers provided.

PROBLEM 1 — CONCEPTUAL

A jacket originally costs $120. During a sale, its price is reduced to $90. A student claims the percent decrease is 33.3% because 30 ÷ 90 = 0.333. Identify the error in the student's reasoning and state the correct percent decrease.

PROBLEM 2 — BASIC CALCULATION

A city's population grew from 45,000 to 54,000 over five years. What is the percent increase in population?

PROBLEM 3 — INTERMEDIATE

An electronics store raises the price of a tablet from $320 to $400, then later discounts the new price by 15%. What is the final price of the tablet, and what is the overall percent change from the original $320 price?

PROBLEM 4 — APPLIED

A company's quarterly revenue fell from $2.4 million to $1.8 million. Management wants to return to the original revenue next quarter. By what percent must revenue increase from the current $1.8 million to reach $2.4 million?

PROBLEM 5 — CRITICAL THINKING

Prove algebraically that a p% increase followed by a p% decrease on any original value V always results in a net decrease. Express the overall percent change in terms of p.

SUMMARY

Lesson Summary

Percent increase and percent decrease both measure how much a quantity has changed relative to its original value. The unified formula — ((New − Original) ÷ Original) × 100 — produces a positive result for increases and a negative result for decreases. The critical rule is that the original value always serves as the denominator; dividing by the new value is the most common source of incorrect answers on the ACCUPLACER.

Apply the four-step method — Identify, Subtract, Divide, Convert — to every problem for consistent accuracy under timed conditions. Remember that successive percent changes are not additive because each subsequent change operates on a new base. A p% increase followed by a p% decrease always yields a net loss of (p²/100)%, a fact that can save you from trap answers. With these principles internalized and practiced, percent-change problems on the ACCUPLACER should become reliable points in your score.