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Single, Double & Triple Scoops (Posted on 2019-05-03) Difficulty: 3 of 5
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    
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Solution solution two ways Comment 1 of 1
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.


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