SOLUTION: A store owner wants to provide a discount of 40%. They discover an item is already marked 25% off. Rather than change the sign the owner decides to stack another discount on top

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Question 1209201: A store owner wants to provide a discount of 40%. They discover an item is already marked 25% off.
Rather than change the sign the owner decides to stack another discount on top of this current one.
What additional discount is necessary for the two combined discounts to be equivalent to 40% off? Hint: the answer is not 15%
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: 20%


Explanation
1-0.25 = 0.75 is the first discount multiplier
1-x is the second discount multiplier where 0 < x < 1
(1-0.25)*(1-x) = 0.75(1-x) = proportion of the final price that the customer ends up paying
1-thatProportion = 1-0.75*(1-x) = combined discount multiplier = 0.40
1-0.75*(1-x) = 0.40 solves to x = 1/5 = 0.20 = 20%
The second discount must be 20%
I'll let the student handle the scratch work when it comes to solving that equation.

Check:
Let's say there's an item worth $100 before any discounts are used.
100*(1-0.25)*(1-0.20) = 60 is the final price after both discounts of 25% and 20% are applied in either order.
The customer ultimately saves 100-60 = 40 dollars, which leads to the 40/100 = 0.40 = 40% discount. This example helps confirm the answer.
Notice that 100*(1-0.40) = 60.

As the hint strongly implies, many students might easily mistake the answer to be 15% since 25+15 = 40. But unfortunately the percentage discounts don't add like that. The percentages do not add because the 1st discount applies to the sticker price while the 2nd discount applies to an amount smaller than the sticker price.

Another way to see why we cant add the percentages:
Let's say the store owner offered the discounts 30% and 80%.
If an item starts off at $100 then its final price would be
100*(1-0.30)*(1-0.80) = 14 dollars.
But if we follow the practice of naively adding the percentages then we'd get a 30% + 80% = 110% off discount. Which is completely silly because we can't go over 100%.