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PREPITUS
Data AnalysisFoundational

Percentages

Move between percent of, percent change, and percent error without slipping — and undo a percent change correctly.

⏱ 6 min📍 Learn first: Ratios, Rates, and Units
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Free KC Learning Toolkit

Learn Percentages in the format that works for you

Start with a quick audio review, take the deeper podcast lesson, or move directly into the visual guide and knowledge check below.

13 free resources
00:00 / 01:34

DSAT Successive and Reverse Percentage Shortcuts

A short spoken refresher on this concept.

00:00 / 18:20

How shifting bases create percentage illusions

The full lesson, worked through in depth.

Concept Snapshot

Definition
A percent is a ratio out of 100. Percent change = (new − old)/old × 100.
What the SAT tests
Percent of a quantity, successive percent changes, reverse percent problems, and percent change from data.
Key rule
x% of N = (x/100)·N. Increase by r% → multiply by (1 + r/100). The base of a percent is ALWAYS the original value.
Most common mistake
Undoing a 20% increase by subtracting 20% — the base changed, so that overshoots.
Prerequisite
Ratios, Rates, and Units

Visual Explanation

What to picture: x% of N = (x/100)·N. Increase by r% → multiply by (1 + r/100). The base of a percent is ALWAYS the original value.

A diagram for this concept is in production. Until it is ready, the worked example below shows the same relationship step by step.

Concept Mind Map

DefinitionA percent is a ratio out of 100. Percent change = (new − old)/old × 100.
Key rulex% of N = (x/100)·N. Increase by r% → multiply by (1 + r/100). The base of a percent is ALWAYS the original value.

Percentages

Data Analysis

Percent OfStraight computation or find-the-whole-from-a-part.
Successive ChangesTwo percent changes compound — multiply factors, never add percents.
Reverse PercentGiven the after value, recover the original.

How the SAT Tests It

Pattern

Percent Of

Straight computation or find-the-whole-from-a-part.

Clue: "is what percent of"
Pattern

Successive Changes

Two percent changes compound — multiply factors, never add percents.

Clue: "increased by 10%, then decreased by 10%"
Pattern

Reverse Percent

Given the after value, recover the original.

Clue: "after a 25% discount, the price was…"

Worked Example

A jacket's price after a 30% discount is $84. What was the original price?

We have the AFTER value and must recover the original — a reverse percent problem.
30% off → pay 70% → after = 0.70 × original.
0.70p = 84 → p = 120.
30% of $120 = $36; $120 − $36 = $84 ✓. Note: adding 30% to $84 gives $109.20 — the classic wrong answer.

Try It Yourself

A stock rises 25% one year, then falls 20% the next. Overall, the stock is:

Common Traps

What students do

Adding successive percent changes.

Why it fails

Each change acts on a different base — the second percent is taken of an already-changed value.

Do this instead

Convert every change to a factor (1 ± r) and multiply the factors. Read the final factor as the total change.

Interactive Flashcards

Review deckCard 1 of 8

Four-Question Knowledge Check

4 quick questions drawn from this concept's own snapshot, patterns and traps.

Question 1 of 4. Which statement is the key rule for this concept?

Audio Lesson

How shifting bases create percentage illusions

Factors, reverse problems, and compounding changes.

00:00 / 18:20

18:20 · the full lesson

What to notice

x% of N = (x/100)·N. Increase by r% → multiply by (1 + r/100). The base of a percent is ALWAYS the original value.

What the lesson covers

  • 0:00 Percent as factor
  • 1:20 Reverse percent
  • 2:50 Successive changes
  • 4:00 Practice

Study Guide and Transcript

One-page study guide

The snapshot, the key rule, the patterns and the traps, on one page.

Source-Grounded AI Tutor

Ask the Percentages notebook

A tutor grounded in this concept's own sources — ask for a simpler explanation, another example, or why a particular answer is wrong. It is in development and will appear here once its sources pass academic review.

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