Question

Fool's problem of the day,

Today's problem of the day is based on arithmetic.

If the cost price of three kinds of sugar are \( \$5, \$6 \) and \( \$6.80 \) per bag respectively. In what proportion should they be mixed so that the price of the mixture may be \( \$6.50 \) per bag?

PS:The fastest solution to this problem is of just three lines.

Answer 1

Does each one need to be used?

Answer 2

@bluepig148: Of-course.

Answer 3

Wrote a program to brute force it, lol.
$5.00 $6.00 $6.80
1 3 10
1 6 15
1 9 20
1 12 25
1 15 30
1 18 35
1 21 40
1 24 45
1 27 50
1 30 55
1 33 60
1 36 65
1 39 70
1 42 75
1 45 80
1 48 85
1 51 90
1 54 95
2 3 15
2 9 25
2 15 35
2 21 45
2 27 55
2 33 65
2 39 75
2 45 85
2 51 95
3 3 20
3 6 25
3 12 35
3 15 40
3 21 50
3 24 55
3 30 65
3 33 70
3 39 80
3 42 85
3 48 95
4 3 25
4 9 35
4 15 45
4 21 55
4 27 65
4 33 75
4 39 85
4 45 95
5 3 30
5 6 35
5 9 40
5 12 45
5 18 55
5 21 60
5 24 65
5 27 70
5 33 80
5 36 85
5 39 90
5 42 95
6 3 35
6 15 55
6 21 65
6 33 85
6 39 95
7 3 40
7 6 45
7 9 50
7 12 55
7 15 60
7 18 65
7 24 75
7 27 80
7 30 85
7 33 90
7 36 95

Were some solutions...
http://ideone.com/B9hYq

Answer 4

wait now i got it:
proportion is 1:2:10

Answer 5

I'm not sure but is it
------------------------------------------
Total= $5+$6 +$6.80=17.8

Ratio=
\[\frac{5}{6.8}*6.5~,~\frac{6}{6.8}*6.5~,~\frac{6.8}{6.8}*6.5\]
\[Ratio=~~4.779~,~5.735~,~8.84\]
--------------------------------------------
\[4.779+5.735+8.84=17.014\]

Answer 6

None of the above makes any sense without adequate explanation

Answer 7

@dumbcow I may be misunderstanding the problem, but:
1:2:10
5:6:6.8
1 5 brings total cash spent to 5.
2 6 means +12 brings total to 17.
10 6.8 means +68 brings total to 85.

1 + 2 + 10 = 13

85/13 doesn't = 6.5

Answer 8

messed up again...

--> 2:3:15

2*$5 + 3*$6 +15*$6.8 = 130
130/20 = 6.5

Answer 9

@bluepig148, you are right, thanks

Answer 10

@Hero What about my solution? Looking at the numbers, I see a clear pattern. I'll work on making some equations now.

Answer 11

@dumbcow provided a solution. I suspect he used one of the proportion sets @bluepig148 came up with using his program. He will probably deny it though

Answer 12

For every 3 the second column goes up, the third goes up 5.
For every 1 the first column goes up, the third goes up 5.

Anyone wanna generalize?

Answer 13

since there are many solutions, i think the algabraic way of solving this is to fix 2 amounts and find the weight of the 3rd so as to obtain an avg of 6.5

Lets say you use 1 bag of each the $5 and $6, then you can find how much of the $6.80 sugar you need to get a $6.50 mixture

5+6+6.8N = 6.5(N+2)
0.3N = 2
N = 20/3 = 6.667

so another solution is 1:1:20/3 or 3:3:20

Answer 14

I believe there are exactly 119 solutions that are not simply scalings of another. Tried all possibilities of combinations from 1- 100 and it stopped when the $5 bin hit 18. Unless there's some disconnected set of solutions, I don't think there are any answers past that (whole numbers).

Answer 15

My second thought:
\[5x~,~6y~,~6.8z~=~6.5~,~6.5~,~6.5\]
5x=6.5 , 6y=6.5 , 6.8z=6.5
\[x=\frac{13}{10}~~,~~y=\frac{13}{12}~~,~~z=\frac{65}{68}\]

Answer 16

If have a feeling that there are infinite solutions to this problem ;)

Answer 17

For the first time, Fool's problem is solvable.

Answer 18

Can you give a whole numbered and fully simplified example that's not on my list (the one the program generates, not the one I copy pasted)?

Answer 19

Oh doh. Silly me. Just came up with a ton more.
192 21 995
193 3 970
193 6 975
193 9 980
193 12 985
193 15 990
193 18 995
194 3 975
194 9 985
194 15 995
195 3 980
195 6 985
195 12 995
196 3 985
196 9 995
197 3 990
197 6 995
198 3 995
To name a few. XD

Answer 20

Let x, y and z be the amount of sugar in the mixture for the bags worth $5, $6 and $6.80.

\[\implies \frac{5 x +6y + 6.80z}{x+y+z} = 6.50\]\[\implies 5x + 6y + 6.80z = 6.50x+ 6.50y + 6.50z\]\[\implies 1.5x + .5y - .3z = 0\]\[\implies 15x + 5y - 3z = 0\]
Every point on the plane \(15x + 5y - 3z = 0\) must be the solution for the problem.

Answer 21

Congratz Ishaan :)