Grade 10 Contest Problems.... ASAP?
Can you help me answer this contest question....
A shopper plans to purchase an item that has a listed price greater than $100 and can use any one of three coupons. Coupon A gives 15% off the listed price, Coupon B gives $30 off the listed price, and Coupon C gives 25% off the amount by which the listed price exceeds $100.
Let x and y be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is y−x?
I actually do not know how to get the answer... PLease explain in great detail :DDDDD
THANKSSSS
Comments
Let p be the listed price
Coupon A saves 0.15*p
Coupon B saves 30
Coupon C saves 0.25(p-100)
Coupon A saves at least as much as B, so
0.15*p >= 30
which rearranges to p >= 30*100/15 = 200
Coupon A saves at least as much as C, so
0.15*p >= 0.25(p-100)
which rearranges to
0.15*p >= 0.25*p - 25
25 >= 0.1*p
250 >= p
So the price p lies between $200 and $250, which is x and y respectively
y - x = $250 - $200 = $50