1. ## percent equation

Hi, everyone! I am so happy to find this friendly and supportive forum , glad to be part of it. I just started taking classes in college this fall, and math is one of the subject I have to work hard. I am doing OK so far but there are problems I am finding hard to solve. One of them is a word problem using the percentage. I had spent almost all day today, trying to solve it and finally I am giving up and asking for help. I will appreciate the help. A hospital needs to dilute a 60% boric acid solution to a 10% solution. If it needs 40 liters of the 10% solution, how much of the 60% solution and how much water should it use?

2. Hello, mirdita59!

Welcome aboard!

A hospital needs to dilute a 60% boric acid solution to a 10% solution.
If it needs 40 liters of the 10% solution, how much of the 60% solution
and how much water should it use?

Let $\displaystyle x$ = amount of 60% solutions to be used.
Then $\displaystyle 40-x$ = amount of water to be used.

Consider the amount of boric acid in each stage.

We used $\displaystyle x$ liters of the 60% solution.
. . This contains: $\displaystyle 0.6x$ liters of boric acid.

We add $\displaystyle 40-x$ liters of pure water.
. . This contains: $\displaystyle 0$ liters of boric acid.

The mixture contains: .$\displaystyle 0.6x + 0 \:=\:0.6x$ liters of boric acid.

But we are told that the final 40 liters should be 10% boric acid.
. . Hence, it will contain: .$\displaystyle (40)(0.10) \:=\:4$ liters of boric acid.

There is our equation: . $\displaystyle 0.6x \:=\:4\quad\Rightarrow\quad x \,=\,\frac{20}{3}$

Therefore, we should use $\displaystyle 6\frac{2}{3}$ liters of the 60% solution and $\displaystyle 33\frac{1}{3}$ liters of water.

3. A hospital needs to dilute a 60% boric acid solution to a 10% solution. If it needs 40 liters of the 10% solution, how much of the 60% solution and how much water should it use?

Water is 0% acid.

So final volume of liquid is 40 liters.

Let x = volume of 60% acid solution to be mixed, in liters.
So, (40-x) liters of water is to be added.

x(60%) +(40-x)(0%) = 40(10%)
Divide both sides by 1%,
x(60) +0 = 40(10)
x = 400/60 = 6.667liters, or (6 and 2/3) liters

Therefore,
6 and 2/3 liters of 60% boric acid solution
and (40 -6.667) = 33.333 liters, or 33 and 1/3 liters of water