Everybody loves a sale. Your bank balance might not. This week you work out exactly how many extra customers a discount has to bring in before it pays for itself.
Learning objectives. After this week you can (1) compute a percent change; (2) work out a discounted price; and (3) work out how much your units have to change to keep revenue the same after a price change.
Remember this: a price cut has to win back more customers than the percent it took off.
1) Percent change
Percent Change = (New − Old) ÷ Old × 100
$8.00 → $9.00 is (9 − 8) ÷ 8 × 100 = +12.5%. Always divide by the old value, the number you started from.
A drop comes out negative: $9.00 → $8.00 is (8 − 9) ÷ 9 × 100 ≈ −11.1%. Notice it is not −12.5%. The two moves start from different numbers, so they give different percents.
Try it
You dropped the hero from $8.00 to $7.00.
Percent change = (New − Old) ÷ Old × 100
Fill in the boxes.
2) Discounts
Discounted Price = Price × (1 − Discount Rate)
20% off $8.50 = $8.50 × 0.80 = $6.80. Multiplying by 0.80 is the same as taking 20% off, in one step.
3) The discount trap
A 10% price cut grows revenue only if units rise by more than 10%.
A lower price usually sells more, but every sale brings in less money. This formula tells you how much your units have to change just to bring in the same revenue as before:
Units Change to Keep Revenue the Same = Old Price ÷ New Price − 1
Try it
You were selling 120 a day at $8.00. At the new $7.00, how many must you sell to bring in the same money?
Units needed = Old units × Old price ÷ New price
Fill in the boxes.
Worked example: a price cut
You drop the burger from $8.50 to $7.65, a 10% cut. Yesterday you sold 100 burgers for $850.
8.50 ÷ 7.65 − 1 ≈ +11.1%. You need about 11% more burgers, so 112 instead of 100, just to bring in the same $850. Check it: 112 × $7.65 = $856.80.
The deeper the cut, the faster that number climbs:
| Price cut | Units needed to keep revenue the same |
|---|---|
| 5% | +5.3% |
| 10% | +11.1% |
| 20% | +25% |
| 30% | +42.9% |
And that only keeps revenue the same. Every discounted burger also earns you less profit, so a busy discount day can still be a worse day.
Worked example: a price rise
The same formula works the other way. Raise the burger from $8.50 to $9.35, a 10% rise:
8.50 ÷ 9.35 − 1 ≈ −9.1%. You could sell about 9% fewer burgers and still bring in the same money. A negative answer here is correct.
Quick self-check
(a) A price moves from $12.00 to $10.20. What is the percent change? (b) What change in units keeps revenue the same? (c) What is 25% off $6.00? (Answers: −15%; 12.00 ÷ 10.20 − 1 ≈ +17.6%; $4.50.)
Common mistakes
- Dividing by the new value. Percent change always divides by the old one.
- Thinking a 10% cut needs only 10% more sales. It needs more than that: 11.1%.
- Changing two things at once. Change a price and your district on the same day, and you cannot tell which one moved your sales. Change one thing.
This week in the arena
Run one clean experiment. Change one price, once, and keep your district and crew the same. Before the close, write down how many units you expect to sell. Afterwards, take that item's units sold from the Daily Report before the change and the one after it, and work out the actual change: (New Units − Old Units) ÷ Old Units × 100. Compare it with the number the formula gave you. Did you sell enough to keep your revenue the same?