True Discount
A friend is selling a $100 gift card they got for Christmas but do not want. You have a planned purchase for the store and so they sell it to you for 5% off the value.
You go into store and there is 5% off the product you wish purchase, happily with the discount removed it costs you exactly $100, so you can use the gift card up in full.
Given the price you paid for the gift card and the original price of the product, what is the effective discount?
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1 solution
Sorry not Latex solution from me someone can make a prettier answer :)
The effective discount is given by:
Original pre discount price of item := P
price paid for giftcard = G
Effective Discount := E
P-(PE/100) = G
So we need to work out some of these figures.
Firstly let start with the easy part.
G, price paid for giftcard:
this is given by the price of the giftcard less 5% of that original value
100 - 100*(5/100) = 100 - 5 = $95
so G=95
Now we need to work out P, the original pre-discount price of if the item
we know 95% of the item is $100, so we need to work out what 100% of the item is
(100/95) * 100 = 2000/19 to be precise or $105.26 to the nearest cent (although this number can vary slightly given that the original price would have been subjected to rounding and we are working backwards with the rounding information. As the rounding could have happened in either direction I'll how working with the rounded answer and the most precise fraction.)
substituting the rounded numbers into the original equation we get
P-(PE/100) = G
105.26- 105.26E/100 = 95
105.26(1-E/100) = 95
1-E/100=95/105.26
1-E/100=0.902
E/100=1-0.902
E/100=0.098
E=9.8%
closest solution to choose from is 9.75% If you keep a couple more decimal places whilst calculating knowing you'll be multiply by 100, you would get close to 9.75.
substituting in the fraction answer we get.
2000/19-(2000/19)*E/100 =95
2000/19 (1 - E/100) = 95
1-E/100=95*19/2000
1-E/100=0.9025
E/100 = 0.0975
E=9.75
closest answer is
9.75%
Due to the lost of data when calculating the original price I gave the solution as multiple choice and not and exact decimal.
It is easy for many people to assume that 5% off gift card and 5% off item is a simple 5% + 5% = 10% when that is not the case.