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A shop keeper mixes 3 varieties of wheat costing ₹12, ₹14 and ₹17 per kg. Which of following represent ratio of mixing of varieties if the mixture is sold at ₹15 per kg and he gains 20% profit ? $ (1) 23:7:2 \quad (2)27:6:1 \quad (3)25:6:4 \quad (4)24:3:18 $

I am looking for a elegant way to solve this

I am stuck with my solution:
Let wheat1 = x kg
wheat2 = y kg
wheat3 = z kg are mixed.
Total cost price = 12x + 14y + 17z
Total selling price = (x+y+z)15

$profit = \frac{SP - CP}{CP}$
=> $\frac{20}{100} = \frac{[15(x+y+z)-(12x+14y+17z) ]}{ (12x+14y+17z)}$
=> $x = 3y + 9z$ ----- (1)

$15$ = total price of (x+y+z) / total wheat
=>$15 = \frac{(12x+14y+17z)}{(x+y+z)}$
=>$3x + y -2z = 0$ ----------------(2)

I am stuck after this. I don't know what mistake I am doing ?

Shailesh
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prashant singh
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    Welcome to Math. SE where you may certainly ask questions but you have to also show your own effort in solving them – Shailesh Sep 20 '15 at 00:14
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    Now I added my partial solution below the question. – prashant singh Sep 20 '15 at 00:37
  • If he sells at 15. @20% profit, then the cost must be 12.5. Instead of the second equation, concentrate on this fact and see what combination will make the average cost 12.5 – Shailesh Sep 20 '15 at 00:43
  • So from x = 3(y + 3z) we know that x must be a multiple of 3 so only 2 option we have for that 2 and 4 . But only 2 satisfies the equation. Means we have to get answer from by option. a bit tricky question for me. – prashant singh Sep 20 '15 at 01:38
  • Absolutely. I think you have understood the answer – Shailesh Sep 20 '15 at 02:10

1 Answers1

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I'm writing out this solution because just like the OP has, many students overly complicate solutions of such problems.

He sold at $15$ and made a profit of 20%. That means the cost was $12.5$. Now use $\frac{12x + 14y + 17z}{x + y + z} = 12.5$, which will give $x = 3y +9z = $ which means in a case like this, the first variety amount is divisible by $3$. A quick look at the choices and you have to choose among choice (2) and choice (4). Obviously choice (2) staisfies the equation. So the ratio is $27:6:1$

Shailesh
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