17. On the last day of class, Professor Jansen took his entire class to Bafflin

Question

17. On the last day of class, Professor Jansen took his entire class to Bafflin and Loggins for ice cream. He had $125 to spend, money he made from the sale of his new book, Problems with No Solutions. The students had the choice of 1, 2, or 3 scoops of ice cream. If they decided to have more than one scoop, then each scoop had to be a different flavor. The flavors were chocolate, vanilla, or strawberry. After all the orders were made, it was found that 3/4 of the number of people who had 2 scoops had 3 scoops. One-fourth of the people who had strawberry had just strawberry. For two scoop combinations, chocolate and strawberry was twice as popular as either chocolate and vanilla or strawberry and vanilla. A dozen students had all three flavors and got stomach aches. Two times as many people had just vanilla as had just strawberry. Twice as many people had just strawberry as had just chocolate One-seventh of the people who had multiple scoops was equal to the number of people who had just chocolate. Two students didn't order any ice cream at all. After paying the bill, Professor Jansen received less than one dollar change from the $125. How many people are in the class, how many scoops were ordered, and how much does a scoop cost? 0

Explanation / Answer

Total amount present with professor Jansen=$125

the students had choice of 1,2, or 3 scoops of ice cram

let total number of students be n

n=x + y+ z +2

x-number of students who took 1 scoop

y-number of students who took 2 scoops

z-number of students who took 3 scoops

2-who didnot order any thing

the 3 flavours are chocolate (c), vanilla(v), strawberry (s)

if student takes more than one scoop it should be of different flavour.

c=just chocolate(jc)+chocolate combined with other(cc)+chocolate combined with other 2 flavours

similarly for v and s

v= jv+cv+vanilla combined with 2 other flavours

s=js+cs+strawberry combined with 2 other flavours

given 3/4(no.of people who had 2 scoops)=no.of people who had 3 scoops

3/4(y)=z------------eq1

1/4(who had strawberry)=just strawberry

1/4(s)=js-------------------eq2

a dozen(12) students had all 3 flavours

z=12

from eq1

3/4(y)=12

y=16

for 2 scoops combination chocolate & strawberry was twice as popular as either chocolate &vanilla or strawberry & vanilla

generally,

y=(c&s)+(c&v)+(s&v)-----------eq3

(c&s)=2(c&v)

or

(c&s)=2(s&v)

so,consider 2:1:1 ratio

assume (c&v)=(s&v)

then from eq3

2(c&v)+(c&v)+(c&v)=16

c&v=4

s&v=4

c&s=2*4=8=>c&s=8

two times as many people had just vanilla had just strawberry

2(jv)=js-------------eq4

twice as many people had just strawberry as had just chocolate

2(js)=jc-------------eq5

one seventh of people who had multiple scoops(2 or 3 scoops) was equal to number of people who had just chocolate

1/7(y+z)=jc

1/7(16+12)=jc

=>jc=4

from eq5

2(js)=4

=>js=2

from eq 4

2(jv)=2

=>jv=1

x-->number of students who eat only one scoop

x=jc+jv+js

x=4+1+2=7

x=7

chocolate scoops=jc+cc(c&s,c&v)+all 3 flavours=4+4+8+12=28

strawberry scoops=js+cs(c&s,s&v)+all 3 flavours=2+4+8+12=26

vanilla scoops=jv+cv(c&v,s&v)+all 3 flavours=1+4+4+12=21

therefore total number of students=x+y+z+2=7+16+12+2=37

noof students=37

total number of scoops=c+s+v=28+26+21=75

number of scoops=75

given that after paying bill professor Jansen was left with less than a dollar change from $125

assume $1 is left as change

then cost of each scoop =124/75=1.653333

suppose if we round it to nearest value ie.;$1.66

then total cost of all scoops will be 1.66*75=$124.5

so the change left will be less than a dollar

so,the cost of each scoop can be $1.66

  

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