Foundations — Percentages — finding %, % of a quantity, % increase - decrease

This page assumes you have seen nothing. We build each symbol the parent note (the topic) uses, one at a time, each resting on the one before.

1. A whole, a part, and cutting into pieces

Before any symbols, a picture. Take a bar. Call the entire bar the whole. Colour some of it — that coloured piece is a part.

Why the topic needs this: every percentage question is secretly "how big is the part compared to the whole?" If you cannot point to which thing is the whole, you cannot even start.

2. The fraction symbol $\frac{a}{b}$

How do we write "a part out of a whole"? With a fraction.

The bar $\to$ pieces $\to$ shaded-count idea is exactly Fractions and Decimals. We need it because a percent is a fraction — just a special one.

3. The decimal form — the same thing without a bar

Sometimes we don't want a stacked fraction; we want a single number. That number is the decimal.

Why we need it: decimals let us multiply easily. "$0.375\times 240$" is a one-step calculator move; "$\frac{3}{8}$ of $240$" needs mental gymnastics. Same value, friendlier shape. (More in Fractions and Decimals.)

4. The percent sign $\%$ — a fraction with a fixed floor

Now the star of the show.

Why lock the denominator at 100? Because two fractions with different bottoms are hard to compare by eye — is $\frac{17}{40}$ bigger than $\frac{9}{20}$? Force both onto a $100$-slice grid and the answer is instant: $42.5$ slices vs $45$ slices. That shared grid is the entire reason percentages exist.

5. The word "of" — the hidden multiplication

The parent note leans hard on "of means multiply." Here is why, from zero.

6. The base — the "whole" that 100% points at

This is the single idea students trip on, so we give it its own name and picture.

Why it matters so much: the very same physical change looks like a different percent depending on which bar you call "the whole." A jump of $50$ is $20\%$ of a $250$-bar but only $16.7\%$ of a $300$-bar. Get the base wrong and every later step is wrong. This is exactly the Ratio and Proportion idea of "compared to what?"

7. Old, new, and change — the ingredients of increase/decrease

The multiplier form $V_n=V_o\left(1+\frac{r}{100}\right)$ powers Simple and Compound Interest and Exponential Growth and Decay; the profit%/loss% idea in Profit and Loss is just this change formula with cost price as the base.

Prerequisite map

Equipment checklist

Read the checklist; hide the right side and see if you can answer each before revealing.