Successive Discounts and a Gift Card: A Reasoning and Communication Problem
Maya is comparing two bike shops for a bicycle with a sticker price of $360. Shop A advertises '20% off every bike, then take an extra 15% off the reduced price.' Shop B advertises '30% off every bike' and gives a $25 gift card for the helmet Maya plans to buy anyway. Maya has saved $300 and wants the lowest actual out-of-pocket cost for the bike and helmet. A friend tells her the two shops are equivalent because 20% + 15% = 35%, which is more than 30%. Maya is not sure, because the second discount at Shop A is applied after the first. The helmet costs $40 at both shops, and Maya will buy it regardless. Sales tax is ignored. Maya also wants to know what single percentage discount would be equivalent to Shop A's two successive discounts.
1. Calculate the final price of the bicycle at each shop before the gift card. Show why successive discounts are not additive.
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Shop A: the first discount of 20% leaves 80%, so 360×0.80=288. The extra 15% is taken off 288, not off 360: 288×0.85=244.80. The overall multiplier is 0.80×0.85=0.68, so the equivalent single discount is 32% off, not 35% off. Shop B: 360×0.70=252. Successive percent discounts multiply because each is applied to a new base; they are not additive.
2. Compare Maya's total out-of-pocket cost for the bike and helmet at each shop. What should she choose if the gift card can be used immediately and she definitely buys the helmet?
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Shop A total for bike and helmet: 244.80+40=284.80. Shop B: bike 252 plus helmet 40 minus 25 gift card =267.00, assuming the gift card can be used immediately and Maya definitely buys the helmet. Shop B is cheaper by 284.80−267.00=17.80. If the gift card has restrictions or cannot be combined with the helmet purchase, the comparison changes: without subtracting the gift card, Shop B costs 292.00, and Shop A would be cheaper. The judgement depends on whether the gift card is as good as cash for Maya; since she needs the helmet, Shop B is the better choice.
3. Generalize: if a price is increased by 20% and then decreased by 20%, is the final price the same? Use algebra and give the single percentage change. Then explain how this connects to Maya's discount question.
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No. Start with P. A 20% increase gives 1.20P. A 20% decrease gives 0.80(1.20P)=0.96P, which is 4% below P. The single equivalent change is a 4% decrease. This is the same multiplication principle as Maya's discounts: 0.80×0.85=0.68, a 32% discount, not 20%+15%=35%. Percent changes compose multiplicatively, not additively.