Up 10%, then down 10%. You end up with less.
The story so far Business at the bike shop is booming, so the owner gives you a 10% raise: your $2,400 a month becomes $2,640. A year later, sales slow down and everyone takes a 10% cut. You figure you're back at $2,400. But your new pay is $2,376, $24 less than where you started. Nobody cheated you. That's just how percents work. This lesson shows the math behind almost every money question: tips, raises, discounts, interest and investment returns.
A percent means "out of 100." Growth in percent is how money questions are asked: rates, returns, raises and prices.
Percents can fool you: a 50% loss needs a 100% gain to recover, and two discounts don't simply add up.
Turn every percent into a decimal, multiply, and always ask: percent of what?
Financial analysts measure growth in sales and profits. Actuaries use percents to price risk. Pricing analysts set discounts that still make money.
What you're looking at: your monthly pay, drawn to scale. The dashed line marks your starting $2,400. The raise added 10% of $2,400, which is $240. The cut took 10% of a bigger number, $2,640, which is $264. So you end $24 below the line.
Percent, piece by piece.
Six ideas cover nearly all the percent math you'll meet in money: finding a percent, percent change, points vs. percent, the comeback trap, stacked discounts and compounding.
A percent of a number
A percent means "out of 100." To use one, turn it into a decimal by moving the point two places left (15% = 0.15), then multiply.
- Your raise
- 10% = 0.10. $2,400 × 0.10 = $240. New pay: $2,400 + $240 = $2,640, or in one step $2,400 × 1.10 = $2,640.
- A tip
- 15% on a $60 meal: $60 × 0.15 = $9.
Percent change
Percent change compares the change to the starting number: new minus old, divided by old.
- The cut
- $2,376 − $2,640$2,640 = −$264$2,640 = −0.10, a 10% drop.
- Start to finish
- $2,376 − $2,400$2,400 = −$24$2,400 = −0.01, 1% less overall.
Percentage points vs. percent
When a rate itself changes, there are two ways to say it. Percentage points are the plain difference. Percent change compares that difference to the old rate. News headlines mix them up all the time.
- A savings rate from 4% to 5%
- Up 1 percentage point (5 − 4). But that's 5 − 44 = 0.25, a 25% increase in what you earn.
The comeback trap
A loss and the gain needed to undo it are not the same size, because the gain starts from a smaller number. That's why your 10% cut didn't cancel your 10% raise, and why big investment losses hurt so much.
- A stock falls 50%
- $100 × 0.50 = $50. A 50% gain only gets it to $50 × 1.50 = $75. To get back to $100 it needs $100 − $50$50 = 1.00, a 100% gain.
Stacked discounts
"20% off, plus an extra 10% off" is not 30% off. The second discount comes off the already-lower price. Multiply the parts you keep.
- A $100 jacket
- $100 × 0.80 = $80, then $80 × 0.90 = $72. That's 28% off, not 30%.
Simple vs. compound growth
Simple growth adds the same amount every year, figured on the starting number. Compound growth figures each year's gain on the new total, so the growth itself grows. That's the engine behind savings interest (Lesson 3.3), investing (Lesson 5.1) and, in reverse, credit card debt (Lesson 4.2).
- $1,000 at 10% for 3 years, simple
- $1,000 + 3 × $100 = $1,300.
- Same, compound
- $1,000 × 1.10 × 1.10 × 1.10 = $1,331. The extra $31 is growth earned on growth.
A quick shortcut for compounding is the rule of 72: divide 72 by the yearly percent to estimate how many years it takes to double.
- Growing 8% a year
- 728 = 9 years to double: $1,000 becomes about $2,000.
What you're looking at: $1,000 growing at the rate and for the years you pick. Each column is one year. The gray part is what simple growth would give you; the green on top is the extra from compounding. It's a math example, not a promise of any real return.
Two minutes. Just listen.
Tap any line to jump to it.
Percent sprint.
Eight quick money-math questions. Each one gives you the numbers. Work it out, then tap your answer. One try each.
What you're looking at: made-up money situations. Each lists the numbers, then asks one question.
The logic, step by step.
Three moves solve almost any percent problem.
Step 1: Turn the percent into a decimal
- Your raise
- 10% becomes 0.10.
Step 2: Use a multiplier
- Raise, then cut
- $2,400 × 1.10 × 0.90 = $2,400 × 0.99 = $2,376.
Step 3: Ask "percent of what?"
- The cut
- 10% of $2,640, not 10% of $2,400. That's why it's $264, not $240.
Worked example: getting back to $2,400
- Where you are. $2,376 a month after the raise and the cut.
- The gap. $2,400 − $2,376 = $24.
- Percent of what? Of the new, lower pay: $24$2,376 = 0.0101.
- The answer. You'd need a raise of about 1.01%, a bit more than 1%, to get back where you started.
- The lesson. Losses and cuts always need a slightly bigger gain to undo, and the bigger the loss, the bigger the gap.
Duckweed doubles.
Duckweed, a tiny floating plant, can double the space it covers in about two days when conditions are right. That's compounding in nature: each new plant starts making more plants. A pond can look nearly empty, and then be covered within days.
What you're looking at: an illustration, not data. If duckweed doubled every 2 days and covered a pond on day 60, it would cover half the pond on day 58, a quarter on day 56, and only an eighth on day 54.
Why this sticks. Compounding looks slow, then sudden. Most of the growth comes at the end. That's why starting to save early matters so much, and why ignored debt can swamp you.
The page you'd copy down.
- Percent = out of 100. 15% = 0.15. Percent of a number: multiply. $60 × 0.15 = $9.
- Percent change = new − oldold. Always measured from the start.
- Percentage points: 4% to 5% is +1 point, but a 25% increase.
- Comeback trap: −50% needs +100% to recover. +10% then −10% = × 0.99: $2,400 → $2,376.
- Stacked discounts multiply: 20% then 10% off = × 0.80 × 0.90 = × 0.72, so 28% off.
- Compound growth: start × (1 + rate)years. $1,000 at 10% for 3 years = $1,331 vs. $1,300 simple.
- Rule of 72: years to double ≈ 72yearly %. At 8%: 9 years.
Fill in the blanks
Grab a friend or a study group.
Headline check
Find a news headline with a percent in it. Is it percent or percentage points? Percent of what? Rewrite it to be clearer.
Store sale race
Find a real sale with stacked discounts. Everyone works out the final price on paper. First right answer wins.
Double or nothing
Each person picks a growth rate. Use the slider above to see how many years it takes $1,000 to double. Who guessed closest before looking?
Just you and your money.
These stay on this device. Nobody else sees them.
Exam-style questions.
Written the way a college exam would ask them. Each right answer on the first try is worth 10 xp.
On the job: you're a financial analyst
Your boss wants to know which of two companies is growing faster. Here are their yearly sales:
- Company A: $2,000,000 last year, $2,500,000 this year.
- Company B: $8,000,000 last year, $9,000,000 this year.
- Company B's manager says, "We grew by $1,000,000, double Company A's $500,000. We're growing faster."
Your task: work out each company's growth in percent. Is the manager right? If Company A keeps its growth rate, what will its sales be next year?
See a model answer
Company A: $2,500,000 − $2,000,000$2,000,000 = 0.25, so 25% growth.
Company B: $9,000,000 − $8,000,000$8,000,000 = 0.125, so 12.5% growth.
The manager's claim: B added more dollars, but from a much bigger start. In percent, A is growing twice as fast.
Next year for A: $2,500,000 × 1.25 = $3,125,000, if the 25% rate holds.
The math under every lesson.
What you're looking at: a map of where this lesson fits in the course. Read it left to right: the lessons it builds on, this lesson in the middle, and the lesson that comes next.
- Back · Unit 3Make your savings earn more
- Back · Unit 5Why invest at all?
- Back · Unit 4Credit cards
- You are hereThe math of percent and growth
- Next · Unit 10The time value of money