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In a factory, $1$ liter of fruit juice contains water cost, fruit concentrate cost and packing cost.

  • There's fruit concentrate $\%25$ of water in the fruit concentrate and water mixture.

  • Water cost equals $\%30$ of fruit concentrate cost.

  • Packing cost is $0.32\$$

If you sell $1$ liter of fruit juice with the price of $2.4\$$ which gives $\%100$ of profit, How much does the fruit concentrate cost in $1$ liter of fruit juice?


Let's recall $F =$ fruit concentrate cost, $W = $ Water cost and $P= $ packing cost,

$$W + F + \underbrace{P}_{0.32} = 1.2$$

Which yields

$$W + F = 0.88$$

Water cost equals $\%30$ of fruit concentrate cost.

$$W = \dfrac{30}{100}F$$

Then we have that

$$F + \dfrac{30}{100}F = 0.88 \implies F = 0.67692$$

I can't think of any way to proceed right now.

Regards!

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  • $\begingroup$ @saulspatz See the new edit. $\endgroup$
    – Cargobob
    Jul 25 '18 at 15:40
  • $\begingroup$ Your first equation is incorrect. On the left-hand side you have the cost of production, and on the right-hand side you have the selling price. We are told that at that price, there is a $100\%$ profit, so the right-hand side should be $1.2$ $\endgroup$
    – saulspatz
    Jul 25 '18 at 15:44
  • $\begingroup$ @saulspatz Too right, let me rewrite those equations. $\endgroup$
    – Cargobob
    Jul 25 '18 at 15:45
  • $\begingroup$ First, there's a variable $S$ that appears out of nowhere. That must be a typo for $W$. Second, you aren't using the given that the fruit concentrate mixture is $25\%$ water. The statement that water cost is $30\%$ of fruit concentrate cost must apply to the unit cost. Try it again. $\endgroup$
    – saulspatz
    Jul 25 '18 at 15:56
  • $\begingroup$ @saulspatz I'm truly out of my mind. $\endgroup$
    – Cargobob
    Jul 25 '18 at 16:01
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If one liter of juice sells for $\$2.40$ at $100\%$ profit then the cost is $\$1.20$. Subtracting the $\$0.32$ cost of packaging, the concentrate and water costs $\$0.88$.

The water to concentrate cost ratio in the juice is $1:(\frac{20\%}{80\%}\cdot \frac{100\%}{30\%}) = 1:\frac{5}{6}$ or $6:5$

The water in one liter of juice costs $\frac{6}{11}\cdot 0.88 = 48$ cents

The concentrate in one liter of juice costs $\frac{5}{11}\cdot 0.88 = 40$ cents

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  • $\begingroup$ Can you explain this ratio $1:(\frac{20\%}{80\%}\cdot \frac{100\%}{30\%}) = 1:\frac{5}{6}$? $\endgroup$
    – Cargobob
    Jul 25 '18 at 17:06
  • $\begingroup$ For every 1 part cost of water in a mixture, the concentrate cost will be the volume ratio $\frac{20}{80}$ of concentrate to water times the cost ratio $\frac{100}{30}$ of concentrate to water. Hence $1 : \frac{5}{6} = 6 : 5$ $\endgroup$
    – Phil H
    Jul 25 '18 at 17:19
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You are correct up to the point where you compute that the cost of the water plus the cost of the fruit concentrate is $88$ cents in a liter of the product.

The sentence "There's fruit concentrate %25 of water in the fruit concentrate and water mixture," is rather awkward. I guess it means that the amount of fruit concentrate in the mixture is $25\%$ of the amount of fruit concentrate, or that a liter of the product contains $200$ ml of fruit concentrate and $800$ ml of water.

Let $x$ be the cost of a liter of fruit concentrate. A liter of water costs $30\%$ of this or $.3x.$ Since we only have $20\%$ of a liter of fruit concentrate in a liter of mixture, the cost of the concentrate is $.2x.$ Also, we have $80\%$ of a liter of water, so the cost of the water is $.8(.3x)=.24x$ Then the cost of the concentrate and water in a liter of product is$$ .2x +.24x=.44x$$ Since we've already determined that the cost of the of the water and concentrate is $88$ cents we have $$ .44x=88\implies x=200\text{ cents}$$

EDIT

I misread the question. I thought we were looking for the cost of a liter of fruit concentrate, but it's actually the cost of the amount of fruit concentrate in a liter of the mixture. That's $20\%$ of $200$ cents or $40$ cents.

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  • $\begingroup$ Can you explain it further? $\endgroup$
    – Cargobob
    Jul 25 '18 at 16:31
  • $\begingroup$ @Cargobob Have you decided that the correct answer is not $40$ cents? I'm checking my work. $\endgroup$
    – saulspatz
    Jul 25 '18 at 16:32
  • $\begingroup$ The correct answer seems to be $40$ cents. $\endgroup$
    – Cargobob
    Jul 25 '18 at 16:32
  • $\begingroup$ See my latest edit. $\endgroup$
    – saulspatz
    Jul 25 '18 at 16:40
  • $\begingroup$ However, I truly didn't how you found that equation. $\endgroup$
    – Cargobob
    Jul 25 '18 at 16:40

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