Percentages

Why percents exist

You scored $43$ out of $50$ on one quiz and $17$ out of $20$ on another. Which went better? It’s genuinely hard to say — the wholes are different sizes, so the tops can’t be compared directly. But rescale both to marks out of $100$ and the fog lifts: $86$ versus $85$. That rescaling is the whole idea of a percent: a common ruler for parts of different-sized wholes, so anything can be compared with anything at a glance. It’s why test scores, interest rates, discounts, tips, and battery indicators all speak percent.

A percent is a fraction you already know

Percent means “per hundred.” So $30\%$ is simply $30$ out of $100$ — and you already own two ways of writing that. From F3, it’s the fraction $\tfrac{30}{100}$, which simplifies to $\tfrac{3}{10}$. From F4, it’s the decimal $0.30$, because dividing by $100$ moves the point two places left. Every percent is those same three things wearing different clothes, and changing outfits is just two hops of the decimal point: percent → decimal hops left ($45\% = 0.45$), decimal → percent hops right ($0.07 = 7\%$).

Before you scroll on, make three predictions and check each in the grid: type $7$ — is that $0.7$ or $0.07$ as a decimal? Type $0.5$ — half a percent, which is much less than a half. And type $150$ — the grid runs out of cells, but the number doesn’t: a percent above $100$ just means more than one whole.

Taking a percent of a number

Dinner cost $$80$ and the service was great, so you want to tip $35\%$. No formula yet — build it from pieces you can hold in your head:

Now the shortcut. In F3 you saw that multiplying is taking a fraction of something. ”$35\%$ of $80$” is exactly that sentence: $\tfrac{35}{100} \times 80 = 0.35 \times 80 = 28$. Same answer, one multiplication — and that’s the general rule:

One habit worth naming: a percent is never an amount by itself — it’s always a percent of something. $25\%$ of a coffee and $25\%$ of a house are wildly different sums of money. Until you know the base $b$, "$25\%$" has no size.

Try $15\%$ of $60$ — but predict it first with the anchors ($10\%$ is $6$, half of that is $3$…), then let the widget confirm and show the breakdown.

Measuring change

Your favorite sneakers went from $$80$ to $$100$. The raw change is $$20$ — but is that a big jump? On an $$80$ base it’s a quarter of what you started with, so it’s a $25\%$ increase. That’s the rule: compare the change to the original value.

Here’s why the classic mistake — dividing by the new value — feels so natural: the new number is the one in front of you, and $\tfrac{20}{100} = 20\%$ comes out cleaner than $\tfrac{20}{80}$. But watch it break: run the same price in reverse, $$100$ down to $$80$, and the drop is $\tfrac{20}{100} = 20\%$. Same $$20$ gap, different percent — because what changed is the starting point you measure from. A percent change always answers “how big was the move compared to where I began?”

Type $80 \to 100$, then swap to $100 \to 80$, and watch the up-and-down percents refuse to match.

The multiplier shortcut

Here’s the idea that turns percent problems into one-step problems. After a $20\%$ increase you have all of what you started with plus $20\%$ of it — that’s $120\%$ of it, or $\times 1.20$ in a single multiplication. A $20\%$ decrease means you keep $80\%$: $\times 0.80$. Every percent change is secretly one multiplier.

Stacking changes

A store raises prices $20\%$, then announces a ”$20\%$ off” sale. Back to the original price? It certainly feels like it — $+20$ and $-20$ cancel, don’t they? Predict what the chain below will land on, then look:

$100 \to 120 \to 96$. The intuition "$20 - 20 = 0$" fails because the two percents stand on different bases: the $20\%$ cut acts on $120$, a bigger number, so it takes away more than the raise added. Percent changes never add — their multipliers multiply: $1.20 \times 0.80 = 0.96$, a $4\%$ loss. Try $+50$ then $-50$ (worse: $\times 0.75$), and $+10$ then $+10$ (more than $+20\%$: $\times 1.21$).

Working backwards

A jacket costs $$120$ after a $20\%$ markup. What did it cost before? The tempting move is to take $20\%$ off $120$ and answer $$96$ — but you can now say exactly why that fails: the $20\%$ was measured on the original price, not on $120$. Think forward first: $\text{original} \times 1.20 = 120$. Undoing a multiplication is division, so the original is $120 \div 1.20 = 100$ dollars. Check it: $100 \times 1.20 = 120$ ✓ — while $96 \times 1.20 = 115.2$ ✗.

Toggle between the right and wrong approaches in the widget and watch how far apart they land as the percent grows.

The one thing to remember

Percent means per hundred — a fraction and a decimal in different clothes. From there, every percent question is a multiplier question: $p\%$ of $b$ is $\tfrac{p}{100} \times b$, a $p\%$ change is one multiplication by $1 \pm \tfrac{p}{100}$, and undoing a change is dividing by that multiplier. And a percent change is always measured against where you started.

Conversions

Formulas

• Percent of a number: $\dfrac{p}{100}\times b$
• Percent change: $\dfrac{\text{new}-\text{old}}{\text{old}}\times100$
• Apply a change: increase by $p\% \Rightarrow \times\left(1+\tfrac{p}{100}\right)$; decrease $\Rightarrow \times\left(1-\tfrac{p}{100}\right)$
• Undo a change: divide the final value by that multiplier

Percent change

Stacked changes

Working backwards