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

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! • @saulspatz See the new edit. Jul 25 '18 at 15:40 • 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$1.2$Jul 25 '18 at 15:44 • @saulspatz Too right, let me rewrite those equations. Jul 25 '18 at 15:45 • 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. Jul 25 '18 at 15:56 • @saulspatz I'm truly out of my mind. Jul 25 '18 at 16:01 ## 2 Answers 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 • Can you explain this ratio$1:(\frac{20\%}{80\%}\cdot \frac{100\%}{30\%}) = 1:\frac{5}{6}$? Jul 25 '18 at 17:06 • 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$1 : \frac{5}{6} = 6 : 5$Jul 25 '18 at 17:19 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. • Can you explain it further? Jul 25 '18 at 16:31 • @Cargobob Have you decided that the correct answer is not$40$cents? I'm checking my work. Jul 25 '18 at 16:32 • The correct answer seems to be$40\$ cents. Jul 25 '18 at 16:32
• See my latest edit. Jul 25 '18 at 16:40
• However, I truly didn't how you found that equation. Jul 25 '18 at 16:40