Every problem in this lesson is the same picture: start with an original, then multiply by a factor. Going up 20% means ×1.20; going down 20% means ×0.80.
In Lesson 4.4 you learned that a percent is just a fraction out of 100 — a way to say "how much of the whole." Now we put percents to work in the place you will meet them most: money. A 20%-off sale, a 15% tip, a raise, the interest a bank pays — every one of these is the same single move. You take a starting amount, you decide whether it should grow or shrink, and you multiply by one number that does the whole job at once.
By the end of this lesson you will be able to: raise or cut a quantity by a percent using the factor (1 ± r); find a percent change from an old value to a new one; compute a discount and see why "20% off" means you pay 80%; work out profit and profit margin, plus sales tax and tips; calculate simple interest with I = P·r·t; and take a first look at compound interest, where interest earns interest. We keep one color habit throughout: the original / starting amount / the "part" is amber, the whole, or the "out of 100," is blue, a third quantity is purple, anything that grows is green, and anything that shrinks is warm red.
4.5.1 Percent increase and decrease — the master move
Suppose a price of $250 goes up 20%. The slow way is two steps: find 20% of 250 (that is 50), then add it on (250 + 50 = 300). That always works — but watch what happens when we write it in one line. The new amount is the whole original plus one fifth more of it:
new = 250 + 20% of 250 = 250 × (1 + 0.20) = 250 × 1.20 = 300
That little factor (1 + r) is the master move. The 1 keeps the whole original; the r adds the extra percent. A cut works the same way, except you take away the percent: a 20% decrease means multiply by (1 − 0.20) = 0.80. So 250 cut by 20% is 250 × 0.80 = 200.
Both start from the same original 250. The factor decides everything: 1.20 grows it to 300, 0.80 shrinks it to 200.
Run the movie backward and you get percent change: given an old value and a new one, what percent did it move? You compare the change to where you started:
percent change = new − oldold × 100%
From 250 to 300 the change is +50, and 50250 = 0.20 = +20%. From 250 down to 200 the change is −50, giving −20%. Notice the denominator is always the old value — the place you measured the move from.
Worked example — a raise
A worker earns $18 an hour and gets a 5% raise. Factor: 1 + 0.05 = 1.05. New wage = 18 × 1.05 = $18.90. Check by parts: 5% of 18 is 0.90, and 18 + 0.90 = 18.90. ✓
The percent is always OF the original (the base)
"20%" is meaningless until you ask 20% of what. In an increase or decrease, it is 20% of the starting amount — that starting amount is the base, the whole 100%. This is why the denominator in percent change is the old value, and it is exactly why an up-then-down trip does not return to the start (Section 4.5.1's idea returns in the exercises).
🎮 Try itThe increase / decrease machine
Pick an original amount, slide the rate, and flip between up and down. Watch the factor (1 ± r) and the before/after bars.
Original $250
Rate20%
Direction
4.5.2 Discounts and markdowns — "20% off" means you pay 80%
A discount is just a decrease in disguise, and it is the single most useful version of the master move. When a sign says 25% off, the store is taking away 25% of the list price. So the part you actually pay is the rest:
you pay (100% − 25%) = 75%, so sale price = list × (1 − 0.25) = list × 0.75
There are two equivalent routes to the same answer, and it is worth seeing both so you can pick whichever is quicker:
① Subtract route: find the discount, then take it off. 25% of $80 is $20, and 80 − 20 = $60.
② Pay-fraction route: multiply once by the pay-fraction. 80 × 0.75 = $60.
The list splits into the part you pay (75%) and the part you save (25%). Multiplying $80 by the pay-fraction 0.75 lands directly on $60.
Worked example — a jacket on sale
A jacket lists at $80 at 25% off. Pay-fraction = 1 − 0.25 = 0.75. Sale price = 80 × 0.75 = $60. You save 80 − 60 = $20. ✓
Key idea
A discount and a "pay" are two names for the same split: discount + pay = 100%. "30% off" is the same instruction as "pay 70%." The fast move is always to multiply the list by the pay-fraction(1 − d) in one shot.
🎮 Try itDiscount calculator
Set a list price and slide the % off. See the pay-fraction, the price you pay, and the dollars saved split across one bar.
List $80
% off25%
4.5.3 Profit, profit margin, and tax
A shop buys a thing for some cost and sells it for some price. The plain dollar gain is the profit:
profit = selling price − cost
But $5 of profit means very different things on a $10 item and a $1000 item. To compare fairly we measure the profit against the cost, as a percent. That is the profit margin:
profit margin = profitcost × 100%
Buy for $50, sell for $65: profit = 65 − 50 = $15, and margin = 1550 × 100% = 30%. If you sell for less than you paid, the profit is negative — that is a loss.
Profit is the green sliver the selling price rises above the cost. The margin rates that sliver against the cost bar: $15 out of $50 is 30%.
The other everyday percent on a receipt is tax (and its twin, a tip). Tax does not replace the price — it is added on top:
A meal costs $40. Sales tax is 8% and you leave a 15% tip, both figured on the $40.
Tax = 40 × 0.08 = $3.20. Tip = 40 × 0.15 = $6.00. Total = 40 + 3.20 + 6.00 = $49.20.
In one factor: 40 × (1 + 0.08 + 0.15) = 40 × 1.23 = $49.20. ✓
Margin is measured against COST, not selling price
Profit margin here uses cost as the base (this is often called markup on cost): margin = profit ÷ cost. Dividing by the selling price instead gives a different number (that one is called "margin on sales"). Always ask "a percent of what?" In this lesson, profit margin is always the percent of the cost.
🎮 Try itProfit & margin
Set the cost and the selling price. The widget finds the profit and the margin against cost — green for profit, red for a loss.
Cost $50
Sell $65
4.5.4 Principal, interest rate, and simple interest
When you put money in a bank, the bank pays you a percent of it every year for letting them use it. The money you start with is the principal, the yearly percent is the rate, and the money you earn is the interest. Simple interest pays the same amount each year, always figured on the original principal:
I = P · r · t (interest = principal × rate × time)
Here P is the principal, r is the rate written as a decimal, and t is the number of years. When you take your money out, you get back the principal plus the interest: total = P + I.
Simple interest stacks a fixed slice of interest on top of the principal each year. Two years at 3% on $500 adds $30, for a total of $530.
Worked example — $500 for two years
P = $500, r = 3% = 0.03, t = 2 years.
I = P·r·t = 500 × 0.03 × 2 = $30. Total = 500 + 30 = $530.
Check year by year: each year pays 500 × 0.03 = $15, and two years is $30. ✓
🎮 Try itSimple-interest calculator
Set the principal, rate, and years. The formula fills itself in and a stacked bar shows principal plus interest.
Principal $500
Rate %3
Years2
4.5.5 A first look at compound interest
Simple interest always pays on the original principal. But most real bank accounts do something cleverer: each year they add the interest back in, and the next year you earn interest on that larger amount too. Interest earning interest is called compound interest, and it is the same master move from Section 4.5.1 — applied again and again.
Each year your balance is multiplied by (1 + r). Do that for t years and the multipliers stack up:
after t years: balance = P × (1 + r)t
Compare that to simple interest over a few years on $1000 at 5%. Simple interest adds a flat $50 every year. Compound interest adds a little more each year, because each year's 5% is taken on a bigger balance:
Year 1 they tie. After that, compound pulls ahead a little more each year — the right column shows the growing lead. Over long stretches that gap becomes enormous.
Worked example — $1000 at 5%, year 3
Simple: 1000 × (1 + 0.05 × 3) = 1000 × 1.15 = $1150.00. Compound: 1000 × (1.05)3 = 1000 × 1.157625 = $1157.63.
Compound wins by $7.63 — and the lead keeps widening every year after.
Key idea
Simple interest adds the same slice each year: total = P(1 + r·t). Compound interest multiplies by (1 + r) each year: total = P(1 + r)t. They start equal after one year, but compound always pulls ahead from year two on, because it earns interest on interest.
🎮 Try itSimple vs compound
Set a principal and rate, then slide the years. Bars show the simple balance beside the compound balance, with the growing gap called out.
Principal $1000
Rate %5
Years5
★ The big ideas, in one breath
Almost every money percent is the same move: new = original × (1 ± r). Going up r% multiplies by (1 + r); going down r% multiplies by (1 − r). A discount of d% means you pay (1 − d) of the list; percent change compares the change to the old value, (new − old)/old × 100%. Profit is selling − cost and profit margin is profit/cost × 100%; tax and tips add on top. Simple interest is I = P·r·t on the original principal, while compound interest multiplies by (1 + r) every year to give P(1 + r)t — which always pulls ahead of simple after the first year.
Coming up next — 4.6
You will share a quantity by a ratio (split $60 in the ratio 2:3), and meet rates with units — miles per hour, dollars per pound, the price-per-unit that tells you which deal is really cheaper.
✎ Exercises 4.5
Work each one out first, then open the answer to check your thinking.
A bike costs $200 and its price goes up 15%. Use the factor (1 + r) to find the new price.
Show answer
Factor = 1 + 0.15 = 1.15. New price = 200 × 1.15 = $230. (Check: 15% of 200 is $30, and 200 + 30 = 230.)
A $48 game is marked down 25%. Find the new price using (1 − r).
Show answer
Factor = 1 − 0.25 = 0.75. New price = 48 × 0.75 = $36. (Check: 25% of 48 is $12, and 48 − 12 = 36.)
A plant grew from 40 cm to 50 cm. What was the percent change?
Show answer
Change = 50 − 40 = +10. Percent change = 10/40 × 100% = 0.25 × 100% = +25%. The denominator is the old value, 40.
A sweater lists at $60 with 30% off. What do you pay, and how much do you save?
Show answer
Pay-fraction = 1 − 0.30 = 0.70. You pay 60 × 0.70 = $42, and you save 60 − 42 = $18.
Double discount (not additive!) A $100 coat is "20% off, then an extra 10% off the reduced price." What is the final price? Is it the same as 30% off?
Show answer
First: 100 × 0.80 = $80. Then: 80 × 0.90 = $72. A flat 30% off would be 100 × 0.70 = $70. So the double discount ($72) is not the same as 30% off ($70) — the second cut is taken on the smaller $80, not the original $100. The combined factor is 0.80 × 0.90 = 0.72, i.e. only 28% off.
A store buys mugs for $8 each and sells them for $10. Find the profit per mug and the profit margin (on cost).
A restaurant bill is $50. You add 8% sales tax and a 20% tip, both on the $50. What is the total?
Show answer
Tax = 50 × 0.08 = $4. Tip = 50 × 0.20 = $10. Total = 50 + 4 + 10 = $64. (Or in one factor: 50 × (1 + 0.08 + 0.20) = 50 × 1.28 = $64.)
You deposit $800 at 4% simple interest for 3 years. Find the interest and the total.
Show answer
I = P·r·t = 800 × 0.04 × 3 = $96. Total = 800 + 96 = $896. (Each year pays 800 × 0.04 = $32, and 3 × 32 = $96.)
$1000 at 10%. Compare the balance after 2 years under simple interest versus compound interest. How much more does compound earn?
Show answer
Simple: 1000 × (1 + 0.10 × 2) = 1000 × 1.20 = $1200. Compound: 1000 × (1.10)2 = 1000 × 1.21 = $1210. Compound earns $10 more — that extra $10 is the 10% interest paid on year 1's $100 of interest.
Challenge — up then down. A $100 price goes up 10%, then the new price goes down 10%. Is it back to $100? Find the final price and explain.
Show answer
No — it ends below $100. Up 10%: 100 × 1.10 = $110. Down 10%: 110 × 0.90 = $99. The two factors multiply to 1.10 × 0.90 = 0.99, a net 1% drop. The reason: the 10% increase was 10% of 100 (= $10), but the 10% decrease was 10% of the larger 110 (= $11). Different bases, so they don't cancel.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
This lesson develops multistep percent reasoning and is aligned to 7.RP.A.3 — solve multistep ratio and percent problems, including percent increase and decrease, markups and markdowns, simple interest, tax, tips, and commissions — and to 7.EE.B.3, which asks students to solve multistep real-life problems with positive and negative rational numbers and to assess the reasonableness of answers. The organizing idea is the single multiplicative factor new = original × (1 ± r), which unifies raises, discounts, tax/tip totals, and the per-period growth of interest; students who internalize it stop computing percents in fragile two-step pieces. Two classic misconceptions are confronted head-on: that a discount followed by a discount is additive (it is multiplicative — Exercise 5), and that an x% increase undone by an x% decrease returns to the start (it does not, because the two percents are taken on different bases — Exercise 10 and the quiz). The simple-vs-compound comparison previews exponential growth (P(1 + r)ᵗ) without yet requiring formal exponent rules.
eastmath.com · Stage 4 · 4.5 Percentages in Action · Intuition before notation
eastmath.com · 4.5 Percentages in Action: Increase, Discount, and Interest · 4.5.3 Profit, profit margin, and tax