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 Single, Double & Triple Scoops (Posted on 2019-05-03)
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.

 No Solution Yet Submitted by Danish Ahmed Khan No Rating

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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  x3171    1    031    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

11725*71 - 26800*31 = 1675

giving

7*71 + 38*31 = 1675

telling us there were 7 71-cent discounts and 38 31-cent discounts.

 Posted by Charlie on 2019-05-03 09:06:09

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