How much does the fruit concentrate cost in $1$ liter of fruit juice?

Question

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?

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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!

Answer 1

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.

Answer 2

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