An ice cream shop sells 3 flavored scoops: lime, vanilla, and strawberry. Each customer may choose to buy single, double, or triple scoops, and no one orders repeated flavor on the same cone.
For the single scoop, the lime flavor costs 1 dollar each, vanilla 1.5 dollars each, and strawberry 2 dollars each. For double scoops, each order will get a discount of 31 cents off for any combination. For example, the double scoops of lime and strawberry flavors will cost 1+2-0.31=2.69 dollars. Finally, for the triple scoops of 3 flavors, it will be discounted to 3.79 dollars.
At the end of the day, 63 lime, 61 vanilla, and 56 strawberry scoops are sold, and the shopkeeper collects 249.75 dollars in total from customers for these sales.
How many customers bought the ice cream? Assume each ice cream is sold to a different person.
Without volume discounts the scoops would have individually totaled 63 + 61 * 1.5 + 56 * 2 = 266.50.
Doubles get .31 off, and triples get .71 off.
The total discounts were 266.50 - 249.75 = 16.75.
Using the table function of a graphing calculator with
y = (16.75 - .31*x) / .71
finds that if x, representing the number of doubles, is 38, then y, representing the number of triples, is 7. Other pairs are either non-integral or negative values.
There were a total of 63+61+56 = 180 scoops sold, from which we must subtract 38, for the second scoops of pairs, and 2*7 for the second and third scoops of triples. This leaves 128 as the number of customers.
0 23.59154929577465
1 23.1549295774648
2 22.71830985915493
3 22.28169014084507
4 21.84507042253521
5 21.40845070422535
6 20.9718309859155
7 20.53521126760564
8 20.09859154929578
9 19.66197183098592
10 19.22535211267606
11 18.7887323943662
12 18.35211267605634
13 17.91549295774648
14 17.47887323943662
15 17.04225352112676
16 16.6056338028169
17 16.16901408450704
18 15.73239436619718
19 15.29577464788732
20 14.85915492957747
21 14.42253521126761
22 13.98591549295775
23 13.5492957746479
24 13.11267605633803
25 12.67605633802817
26 12.23943661971831
27 11.80281690140845
28 11.3661971830986
29 10.92957746478873
30 10.49295774647887
31 10.05633802816902
32 9.619718309859156
33 9.183098591549296
34 8.746478873239438
35 8.309859154929578
36 7.87323943661972
37 7.43661971830986
38 7.000000000000001
39 6.563380281690141
40 6.126760563380281
41 5.690140845070424
42 5.253521126760564
43 4.816901408450704
44 4.380281690140844
45 3.943661971830987
46 3.507042253521127
47 3.070422535211268
48 2.63380281690141
49 2.19718309859155
50 1.76056338028169
51 1.32394366197183
52 .8873239436619704
53 .4507042253521131
54 .01408450704225572
55 -.4225352112676067
56 -.859154929577464
57 -1.295774647887321
58 -1.732394366197184
59 -2.169014084507041
...
105 -22.25352112676056
106 -22.69014084507042
107 -23.12676056338028
108 -23.56338028169014
109 -24
110 -24.43661971830986
111 -24.87323943661972
112 -25.30985915492958
113 -25.74647887323944
114 -26.1830985915493
Alternative, more analytic, method:
Above we needed to account for 1675 cents of discounts with numbers of 31-cent and 71-cent discounts. We can use an annotated Euclidean Algorithm for finding GCD to find out how this can be accomplished and to get the value to be positive for both numbers of discounts.
x71 x31
71 1 0
31 0 1
9 1 -2
4 -3 7
1 7 -16
0 -31 71
Aside from telling us that GCD(71,31) = 1, the penultimate line tells us that 7*71 - 16*31 = 1. The last line tells us the obvious: that 71*31 - 31*71 = 0.
So how many of each do we need to make 1675? Just multiply
7*71 - 16*31 = 1
by 1675:
11725*71 - 26800*31 = 1675
But we need to bring this up to a positive multiple of 31. How many times 71 do we need to raise -26000 to make it positive? ... It's 378, so we multiply
-31*71 + 71*31 = 0
by 378
giving -11718*71 + 26838*31 = 0
and add that to
11725*71 - 26800*31 = 1675
giving
7*71 + 38*31 = 1675
telling us there were 7 71-cent discounts and 38 31-cent discounts.
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Posted by Charlie
on 2019-05-03 09:06:09 |
solution two ways