anonymous
  • anonymous
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
Does each one need to be used?
anonymous
  • anonymous
@bluepig148: Of-course.
anonymous
  • anonymous
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

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dumbcow
  • dumbcow
wait now i got it: proportion is 1:2:10
.Sam.
  • .Sam.
I'm not sure but is it ------------------------------------------ Total= $5+$6 +$6.80=17.8 Ratio= \[\frac{5}{6.8}*6.5~,~\frac{6}{6.8}*6.5~,~\frac{6.8}{6.8}*6.5\] \[Ratio=~~4.779~,~5.735~,~8.84\] -------------------------------------------- \[4.779+5.735+8.84=17.014\]
Hero
  • Hero
None of the above makes any sense without adequate explanation
anonymous
  • anonymous
@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
dumbcow
  • dumbcow
messed up again... --> 2:3:15 2*$5 + 3*$6 +15*$6.8 = 130 130/20 = 6.5
dumbcow
  • dumbcow
@bluepig148, you are right, thanks
anonymous
  • anonymous
@Hero What about my solution? Looking at the numbers, I see a clear pattern. I'll work on making some equations now.
Hero
  • Hero
@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
anonymous
  • anonymous
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?
dumbcow
  • dumbcow
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
anonymous
  • anonymous
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).
.Sam.
  • .Sam.
My second thought: \[5x~,~6y~,~6.8z~=~6.5~,~6.5~,~6.5\] 5x=6.5 , 6y=6.5 , 6.8z=6.5 \[x=\frac{13}{10}~~,~~y=\frac{13}{12}~~,~~z=\frac{65}{68}\]
anonymous
  • anonymous
If have a feeling that there are infinite solutions to this problem ;)
anonymous
  • anonymous
For the first time, Fool's problem is solvable.
anonymous
  • anonymous
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)?
anonymous
  • anonymous
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
anonymous
  • anonymous
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.
anonymous
  • anonymous
Congratz Ishaan :)

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