Calculate Percent Change — Calculate percent increase, decrease, and percent change. - TEAS Mathematics

SECTION 1

Historical Context & Motivation

The concept of expressing change as a fraction of an original quantity has deep roots in the history of commerce, taxation, and finance. The Latin phrase per centum, meaning "for every hundred," gave rise to the modern notion of percentage, a convention that standardizes comparison across quantities of vastly different magnitudes. Before percentages became widespread, merchants in ancient Mesopotamia and Rome used fractional markups and discounts expressed as ratios, but the lack of a universal denominator made cross-context comparison cumbersome. The emergence of a base-100 framework allowed traders, scientists, and governments to communicate proportional shifts—whether in price, population, or concentration—with immediate clarity.

c. 300 BCE • Proportional Taxation in Rome

Roman tax collectors computed the centesima rerum venalium, a 1/100 tax on goods sold at auction—one of the earliest systematic uses of a per-hundred framework.

1400s • Italian Merchant Manuscripts

Italian merchants began writing "per cento" in accounting ledgers. The modern percent sign (%) evolved from abbreviations of this phrase over the following centuries.

1700s • Standardization in Science

Scientists such as Lavoisier adopted percent composition to describe chemical mixtures, cementing percentage as a universal tool for expressing proportional relationships.

1900s • Percent Change in Economics & Health Sciences

Government agencies and public-health researchers standardized percent change as the primary metric for reporting shifts in GDP, inflation rates, morbidity statistics, and medication dosage adjustments.

Despite its ubiquity, percent change is a source of frequent error on standardized examinations such as the TEAS. The central question this lesson addresses is: given two values representing a before and an after state, how do we rigorously quantify the magnitude and direction of the shift on a per-hundred scale, and how do we avoid the conceptual pitfalls—such as confusing the reference value or conflating absolute with relative change—that lead to incorrect answers under time pressure?

SECTION 2

Core Principles & Definitions

Before deriving formulas, it is essential to internalize four foundational ideas that govern every percent-change calculation. These principles clarify the logic behind the arithmetic and prevent the most common test-day mistakes.

Original Value as the Reference

Percent change is always computed relative to the original (initial) value. Swapping the numerator's reference point will produce an entirely different—and incorrect—result.

Sign Indicates Direction

A positive result denotes a percent increase, while a negative result denotes a percent decrease. The general formula unifies both cases under one expression.

Absolute vs. Relative Change

The absolute change (new − original) reports magnitude in the same units as the data. Percent change normalizes this difference against the original, making quantities on different scales directly comparable.

Symmetry Does Not Hold

A 50% increase followed by a 50% decrease does not return to the original value. Because the reference base shifts after the first change, percent changes are non-symmetric.

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KEY TAKEAWAY

Think of percent change like measuring how far a runner has deviated from a starting line. The starting line is your original value, and the deviation is always expressed as a fraction of that starting distance—not the finish line. If a researcher's grant budget changes from $200,000 to $250,000, the $50,000 increase is measured against the $200,000 origin (a 25% increase), not against the $250,000 endpoint (which would yield a different ratio entirely). This asymmetry is the single most common source of error on standardized quantitative sections.

SECTION 3

Visual Explanation

The following diagram illustrates the anatomy of a percent-change calculation by mapping each component—original value, new value, absolute change, and the resulting percentage—onto a visual bar representation. Observing these relationships spatially reinforces why the denominator must always be the original value.

The upper two bars represent the original and new values. The highlighted amber segment shows the absolute change, which is then divided by the original value (blue bar) and multiplied by 100 to yield the percent change.

Notice that the amber segment (the absolute change of 50) is measured only against the length of the blue bar (the original value of 200), not against the full green/violet bar (the new value of 250). This spatial relationship is the geometric essence of percent change: the ratio of the deviation to the starting point, scaled to a per-hundred basis. If the new value were smaller than the original—say 150 instead of 250—the amber segment would extend to the left, and the computed percentage would carry a negative sign, indicating a percent decrease.

SECTION 4

Mathematical Framework

All percent-change problems reduce to one general formula, from which the specific cases of percent increase and percent decrease are derived. Understanding the algebraic structure makes it possible to rearrange the formula for any unknown—original value, new value, or percent change—depending on what a problem provides.

GENERAL PERCENT CHANGE

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

Where New Value is the final quantity, and Original Value is the initial quantity. A positive result indicates an increase; a negative result indicates a decrease.

PERCENT INCREASE (SPECIFIC CASE)

Percent Increase = ((New − Original) / Original) × 100, where New > Original

When the new value exceeds the original, the numerator is positive, and the result is expressed as a percent increase. For example, a dosage rising from 40 mg to 52 mg yields ((52 − 40) / 40) × 100 = 30%.

PERCENT DECREASE (SPECIFIC CASE)

Percent Decrease = ((Original − New) / Original) × 100, where New < Original

When the new value is less than the original, some texts use this rearranged form so the result is a positive number explicitly labeled as a decrease. Equivalently, the general formula returns a negative value.

FINDING THE NEW VALUE

New Value = Original Value × (1 + Percent Change / 100)

This rearrangement is useful when a problem provides the original value and the percent change and asks for the resulting quantity. For a decrease, the percent change is negative, so the multiplier falls below 1.

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Common TEAS Pitfall

Problems sometimes phrase the question as "by what percent did X change?" and provide answer choices that include both the absolute change and the percent change. Always confirm you are reporting the relative (percentage) change, not the raw numerical difference. Additionally, watch for the word "of" in the stem—"20% of 150" (= 30) is a different operation than "a 20% change from 150" (= new value of 180).

SECTION 5

Detailed Breakdown — Increase vs. Decrease vs. Successive Changes

While the general percent-change formula handles both directions, it is instructive to examine the three most common scenarios—simple increase, simple decrease, and successive (compound) changes—side by side. The diagram below provides a flowchart that guides you from a raw data pair to the correct classification and answer.

Start at the top by identifying original and new values. Compute the difference, determine its sign to classify the change, then divide by the original and multiply by 100. For successive changes, multiply the individual change factors before computing the net percentage.

Scenario	Formula Variant	Example
Simple Increase	((New − Orig) / Orig) × 100	80 → 100: ((100 − 80) / 80) × 100 = 25%
Simple Decrease	((Orig − New) / Orig) × 100	100 → 80: ((100 − 80) / 100) × 100 = 20%
Successive Changes	Net factor = (1 + p₁/100)(1 + p₂/100); Net % = (factor − 1) × 100	+20% then −10%: (1.20)(0.90) = 1.08 → net +8%

Summary of the three main percent-change scenarios

The successive-changes row deserves special attention because it reveals the non-additive nature of percentages. A 20% increase followed by a 10% decrease does not net to a 10% increase; the net change is only 8%, because the 10% decrease applies to the already-enlarged value. This multiplicative compounding is a frequent TEAS distractor and a principle that extends directly into pharmacokinetic half-life calculations and financial interest problems you will encounter in graduate-level coursework.

SECTION 6

Worked Example

A hospital pharmacy records that its monthly usage of a particular antibiotic dropped from 1,250 doses to 925 doses after implementing a new antimicrobial stewardship protocol. Determine the percent change in monthly usage.

Antibiotic Usage — Percent Decrease

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Step 1 — Identify the Original and New Values

The original (initial) value is the usage before the protocol: 1,250 doses. The new (final) value is the usage after: 925 doses.
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Step 2 — Compute the Absolute Change

Absolute Change = New − Original = 925 − 1,250 = −325 doses. The negative sign confirms the quantity decreased.

Absolute Change = −325 doses
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Step 3 — Divide by the Original Value

Relative Change = −325 / 1,250 = −0.26. This decimal represents the fractional change.

Relative Change = −0.26
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Step 4 — Convert to a Percentage

Percent Change = −0.26 × 100 = −26%. Because the result is negative, this is a 26% decrease in antibiotic usage.

Percent Change = −26% (a 26% decrease)
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Step 5 — Verify with the Reverse Calculation

Check: 1,250 × (1 + (−26/100)) = 1,250 × 0.74 = 925 ✓. The new value matches, confirming the answer.

SECTION 7

Common Pitfalls & Conceptual Comparisons

Understanding percent change also means knowing where it can mislead. The table below contrasts percent change with related but distinct measures, highlighting the errors that arise when test-takers conflate them.

Concept	What It Measures	Common Error
Percent Change	Relative shift from the original value, expressed per 100.	Using the new value as the denominator instead of the original.
Absolute Change	Raw numerical difference (new − original), in the same units.	Reporting the absolute change when the question asks for a percentage.
Percentage Points	Arithmetic difference between two percentages (e.g., 40% → 55% = 15 pp).	Saying "15% increase" when the correct statement is a 15 percentage-point increase (which is actually a 37.5% relative increase).
Percent of Total	A part's share of a whole, not a change at all.	Confusing "X is 30% of Y" with "X increased by 30%."

Percent change vs. related measures

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KEY TAKEAWAY

The distinction between percent change and percentage-point change is analogous to the distinction between relative and absolute error in experimental science. When a clinical trial reports that a drug reduced infection rates from 8% to 5%, the absolute reduction is 3 percentage points, but the relative (percent) reduction is (3/8) × 100 = 37.5%. TEAS questions may test whether you can distinguish these two framings; always check whether the problem asks for a relative change or a simple difference in rates.

SECTION 8

Connection to Advanced Quantitative Reasoning

Percent change is not merely an arithmetic exercise; it serves as the conceptual gateway to several advanced topics you will encounter in graduate coursework and professional practice. Understanding how the basic formula scales prepares you for compound growth, pharmacokinetics, and inferential statistics.

Basic Concept	Advanced Extension	Why It Matters
Single percent change	Compound growth / decay	Bacterial colony growth, drug half-life calculations, and financial interest all iterate percent change over multiple periods via A = P(1 + r)ⁿ.
Percent increase / decrease	Elasticity measures (economics)	Price elasticity of demand = (% change in quantity demanded) / (% change in price). The percent-change formula is applied to both numerator and denominator.
Relative change as a ratio	Percent error in laboratory science	Percent error = \|experimental − theoretical\| / theoretical × 100. Structurally identical to percent change, reinforcing the universal applicability of the formula.
Successive percent changes	Geometric mean return	In biostatistics and finance, the geometric mean of successive growth factors yields the true average rate of change over time, avoiding the distortion of arithmetic averaging.

From basic percent change to advanced applications

Recognizing that percent change is a special case of a ratio-of-differences framework allows you to transfer the same reasoning to unfamiliar contexts on the TEAS—and beyond. Whether you are evaluating a change in patient blood pressure, a shift in laboratory reagent concentration, or the efficacy of a public-health intervention, the underlying logic remains: quantify the deviation, normalize it against the baseline, and express it per hundred.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A student claims that a 25% increase followed by a 25% decrease returns any quantity to its original value. Explain, with conceptual reasoning, why this claim is incorrect.

Show answer

The claim is incorrect because each percentage change is computed relative to a different base. After a 25% increase, the base becomes 1.25 times the original. The subsequent 25% decrease removes 25% of this larger base—not 25% of the original. Numerically, if the original is 100, a 25% increase yields 125; a 25% decrease of 125 gives 125 × 0.75 = 93.75, which is 6.25% below the original. The non-symmetric behavior arises from the multiplicative nature of percent changes: (1.25)(0.75) = 0.9375 ≠ 1.00.

Question 2: BASIC CALCULATION

A patient's resting heart rate decreased from 88 bpm to 72 bpm after beginning a cardiovascular exercise program. Calculate the percent decrease.

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Using the general formula: Percent Change = ((72 − 88) / 88) × 100 = (−16 / 88) × 100 ≈ −18.18%. Therefore, the patient's resting heart rate decreased by approximately 18.2%. The absolute change is 16 bpm, but the question asks for the relative change, which requires dividing by the original value of 88 bpm.

Question 3: INTERMEDIATE

The price of a medical textbook was $135. After a 12% discount, a 7% sales tax was applied to the discounted price. What is the final price, and what is the overall percent change from the original list price?

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First, apply the 12% discount: $135 × (1 − 0.12) = $135 × 0.88 = $118.80. Next, apply the 7% sales tax: $118.80 × (1 + 0.07) = $118.80 × 1.07 = $127.116, which rounds to $127.12. The net percent change is ((127.12 − 135) / 135) × 100 = (−7.88 / 135) × 100 ≈ −5.84%. Alternatively, multiply the factors: 0.88 × 1.07 = 0.9416, and (0.9416 − 1) × 100 = −5.84%. The overall effect is a net decrease of about 5.84% from the original list price.

Question 4: APPLIED

A hospital's emergency department saw 4,200 visits in January and 5,460 visits in February. The hospital administrator states that visits increased by 1,260. A reporter writes that visits "rose by 30%." Confirm or correct the reporter's claim, and explain the distinction between absolute and relative change.

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The absolute change is indeed 5,460 − 4,200 = 1,260 visits; the administrator's figure is correct. To verify the reporter's claim, compute the percent change: (1,260 / 4,200) × 100 = 30.0%. The reporter's figure is also correct. This problem illustrates an important distinction: the absolute change (1,260 visits) describes the raw magnitude, while the percent change (30%) normalizes that magnitude against the baseline. If another hospital with 10,000 January visits also saw a 1,260-visit increase, its percent change would be only 12.6%, demonstrating why relative change is more informative when comparing entities of different scales.

Question 5: CRITICAL THINKING

A laboratory standard solution is prepared at 0.500 mol/L. A student measures it three times and obtains 0.485, 0.510, and 0.492 mol/L. (a) Calculate the percent error for each measurement relative to the standard. (b) Argue whether reporting percent change or percentage-point difference is more appropriate when communicating the accuracy of these measurements, and explain your reasoning.

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(a) Percent error uses the same structure as percent change with the standard as the original value. For 0.485: |0.485 − 0.500| / 0.500 × 100 = 3.0%. For 0.510: |0.510 − 0.500| / 0.500 × 100 = 2.0%. For 0.492: |0.492 − 0.500| / 0.500 × 100 = 1.6%. (b) Percent error (a relative measure) is more appropriate than a percentage-point difference because the concentrations are continuous measurements, not themselves percentages. A percentage-point difference applies when comparing two values that are already expressed as percentages (e.g., two exam scores). Reporting relative percent error also enables comparison across experiments with different standard concentrations—an absolute deviation of 0.015 mol/L means very different things when the standard is 0.500 mol/L versus 5.000 mol/L. This parallels the broader principle that percent change normalizes differences against a reference baseline, making it the correct tool for scale-independent comparison.

SUMMARY

Lesson Summary

Summary

Percent change quantifies a shift between two values by computing the absolute difference (New − Original), dividing by the original value, and multiplying by 100. A positive result indicates a percent increase, while a negative result indicates a percent decrease. The denominator must always be the original (baseline) value—never the new value—because percent change measures deviation from a fixed reference point.

For successive percent changes, multiply the individual factors (1 ± p/100) rather than adding the percentages, because each subsequent change applies to a new base. Be careful to distinguish percent change (a relative measure) from percentage-point difference (an absolute difference between two rates) and from simple absolute change (the raw numerical difference). This formula extends naturally into compound growth, percent error, and elasticity calculations encountered across the health sciences and quantitative disciplines.