The 15% Up and 15% Down Problem
In math class, Priya and Dev are checking a store's pricing. A jacket originally costs $80. The store first raises the price by 15%, then advertises '15% off the raised price.' Dev says the two changes cancel, so the final price is $80. Priya says the final price will be lower than $80, and the tag shows $78.20. Their teacher asks them to explain who is right, to find the single percent change from the original price to the final price, and to write a sentence the store could use that would be accurate. Dev's argument is that adding 15% and subtracting 15% is a net change of 0. Priya's argument is that the 15% discount is taken from a larger amount. The class must also decide whether the store's claim 'same as the original price' is true.
1. Calculate the final price step by step and decide who is right.
Read the worked answer
Start with the original price: $80.
- A 15% increase: 15% of 80 = 0.15 × 80 = 12, so the raised price is 80 + 12 = $92.
- A 15% discount on the raised price: 15% of 92 = 0.15 × 92 = 13.80, so the final price is 92 − 13.80 = $78.20.
Priya is right. The final price is lower than $80 by $1.80. Dev's mistake is treating both 15% changes as if they were taken from the same base. The increase is 15% of $80, but the discount is 15% of $92, so the discount is larger in dollars.
2. Find the single percent change from the original price to the final price.
Read the worked answer
Use the final price divided by the original price:
78.20 ÷ 80 = 0.9775
A multiplier of 0.9775 means the final price is 97.75% of the original price. Therefore the single percent change is:
100% − 97.75% = 2.25% decrease
A second way is to multiply the two percent multipliers:
1.15 × 0.85 = 0.9775
This confirms a 2.25% decrease, not 0%. The order does not matter: a 15% increase followed by a 15% decrease gives the same result as the reverse order, because multiplication is commutative.
3. Write a mathematically accurate sentence the store could use, and explain why '15% off after a 15% markup is the same as the original price' is false.
Read the worked answer
An accurate sentence: 'After a 15% markup, the jacket is discounted 15%, for a final price 2.25% below the original price.'
The claim is false because the two percentages are calculated from different bases. The markup uses the original $80 as its base, but the discount uses the raised $92 as its base. The combined multiplier is:
1.15 × 0.85 = 0.9775
Since 0.9775 is less than 1, the result is a decrease, not a return to the original price. Saying the changes cancel would be true only if both percentages were taken from the same starting amount, which they are not.