Math & numbers Ideas & discovery

How to understand percentages once and for all

Understand percentages once and stop getting caught by them.

A percentage describes a share of a stated base. Name that base before calculating, especially when comparing a rise, a fall or two rates. The examples below show the arithmetic and give you a way to check the result.

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Start with three things you can do today

A starting goal

Percentages confuse me: percent off, percent increase, percentage points, percent of a percent. Explain them once, properly, with examples from money and the news, so I stop getting them wrong.

Change any word so it fits your situation. The course is built from what you type and a few setup questions.

Start today

Three things you can do before the course exists.

Useful on their own. The course is for when you want the whole skill, not just the first move.

Distinguish the discount from the final price
Original example: 20% of $50 is 0.20 x 50 = $10. A 20% discount therefore leaves a $40 price, before any other charges. Check: $40 + $10 returns to the original $50. The discount amount is not the amount you pay.
Use the new base for the second change
Starting at 100, a 20% rise gives 120. A subsequent 20% fall is 24, because 0.20 x 120 = 24, leaving 96. Try reversing the order: 100 to 80 to 96. These examples end 4% below the start; equal opposite percentages do not cancel.
Separate percentage points from relative change
A rate moving from 10% to 15% rises 5 percentage points. Relative to the original 10%, the increase is 5 divided by 10 =50%. Both are correct measures of different things. Check that a headline states the base and measure before judging its meaning.

Best for: Math & numbers
Prerequisites: None. Bring the situation you have.
Course length: 7 to 30 lessons
Pace: Yours. One lesson at a time, as often as you like.
Format: Read it, or play the audio
Access: Available on the App Store and Google Play, with an optional early testing list

Before the course is built

Setup asks about your situation first.

Lesson 1 opens on the moment you describe, and the examples use your role. For this request, expect questions like these.

Where do percentages trip you up: shopping, pay rises, interest, or the news?
Can you find ten percent of a number quickly now?
Is this for daily life, work, or exams?

Right for now

If any of this sounds familiar, start here.

Percent off never comes out right

You are not sure whether the deal is good.

Rises and falls confuse you

Up twenty, down twenty, and you are somehow worse off.

Percentage points sound like a trick

You want to know which measure a headline is using.

Course arc

Where the course goes, in order.

Cadence writes the lessons for you, so the exact path follows your answers. This is the ground a course built from this sentence covers.

01 / 06

Parts of a hundred

The idea, and the base it always depends on.

Percent of and percent off

Fast methods, in your head.

Increase and decrease

Why they are not symmetric.

Percentage points

The difference, and the headlines that blur it.

Reverse percentages

Finding the original price from the sale price.

Compounding

Repeated percentages, from interest to inflation.

What changes

What you will be able to do, and what you will stop believing.

You will be able to

Understand what a percent is

Parts of a hundred, and why the base matters most.

Calculate the common cases

Of, off, increase, decrease, and reverse.

Tell percent from percentage points

And spot when a headline is using the wrong one.

Handle growth and interest

Compounding, and why repeated percentages multiply.

Ideas this course corrects

A percentage is meaningful without its base

Name the amount or group to which it applies. A different base can change the result.

Percent and percentage points are interchangeable

Subtract rates for percentage points; divide the difference by the original rate for relative percentage change.

Course questions

What to know before you start.

How to understand percentages once and for all: what can I do today?

Distinguish the discount from the final price. Original example: 20% of $50 is 0.20 x 50 = $10. A 20% discount therefore leaves a $40 price, before any other charges. Check: $40 + $10 returns to the original $50. The discount amount is not the amount you pay. Use the new base for the second change. Starting at 100, a 20% rise gives 120. A subsequent 20% fall is 24, because 0.20 x 120 = 24, leaving 96. Try reversing the order: 100 to 80 to 96. These examples end 4% below the start; equal opposite percentages do not cancel. Separate percentage points from relative change. A rate moving from 10% to 15% rises 5 percentage points. Relative to the original 10%, the increase is 5 divided by 10 =50%. Both are correct measures of different things. Check that a headline states the base and measure before judging its meaning.

Is this a fixed course?

No. Cadence builds the course when you type the sentence and answer a few setup questions, so the lessons follow your situation. This page describes what a course built from this request covers. You see the whole path before Lesson 1.

Who is this course for?

Do I need any prior experience?

No. Setup asks what you already know and where you want to use it, and the course starts from there. Bring a real situation if you have one; the examples are built around it.

How long does it take?

Most courses run 7 to 30 lessons. The course length is sized to what you asked for. You set the pace: one lesson at a time, as often as you like. Lessons can be read or played as audio.

Does this course award a certificate or degree?

No. Cadence is for practical self-learning and does not award accredited degrees or professional certification.