Mixture

[Guest]

A certain medicine is a 5 per cent solution. An order is received for the medicine to be reduced to a 1 per cent solution. Determine amount of water added.

My partial solution:

Let W = water added.

The solution is 5 per cent medicine, 95 per cent water.

The resulting medicine is 1 per cent medicine, 99 per cent water.

0.05 + W = 0.01

How do I proceed from here ?

[Guest]

You don't really need algebra for this question. If we compare a 1% solution and 5% solution of the same volume then it is obvious that 5% solution has 5 times as many active ingredients (i.e. non-water elements) as the 1% solution. So for them to have the same amount of active ingredients you need 5 times the volume of the 1% solution, i.e. if the volume of the 5% solution is V, then the volume required for the 1% solution is 5V. So we need to dilute V to 5V to go from a 5% solution to a 1% solution, which means adding 4V of water, where V is the volume of the 5% solution.

If you want to use algebra, let
A = the amount of active ingredients in the 5% solution,
W = the amount of water in the 5% solution,
V = total volume of the 5% solution,
x = the amount of water added
(To make things simpler you can assume the 5% solution has a volume of 1 Litre and set V=1 then your answer will tell you how much water to add per litre of 5% solution.)

The equations we get from the question are:
A/(A+W) = 5/100
A/(A+W+x) = 1/100
A+W = V

The solution is x = 4V (as expected), A = 0.05V, W=0.95V.

Hope this helped,

R. Baber.

[Guest]

See also the dilution equation:
https://en.wikipedia.org/wiki/Dilution_%28equation%29

Which tells us that the final volume is 5 times the initial volume, meaning we have to add 4 times the initial volume.

Hope this helped,

R. Baber.

[Guest]

"A = the amount of active ingredients in the 5% solution,
W = the amount of water in the 5% solution,
V = total volume of the 5% solution,
x = the amount of water added"

I understand this.


"The equations we get from the question are:
A/(A+W) = 5/100
A/(A+W+x) = 1/100
A+W = V"

I am not clear on this part.

[Guest]

The percentage (i.e. 5) is just the amount of stuff you're interested in (i.e. A) over the total amount of stuff (i.e. A+W) times by 100.
So [tex]\frac{A}{(A+W)}\times 100= 5[/tex] or rearranged this is just [tex]\frac{A}{A+W}= \frac{5}{100}[/tex].

The amount of water you add is obviously going to depend on the amount of 5% solution you start with. The question doesn't state how much you start with, so you can either choose a convenient amount such as 1 Litre (in which case your solution will be "per litre") or label it as a variable V, and write the solution in terms of V. It's not hard to see that V = A+W.

Hope this helped,

R. Baber.

[Guest]

"A/(A+W+x) = 1/100
A+W = V"


A/(A + W + x) x 100 = 1
A + W = 1

That helped some. I still can't solve the equations. I'm sorry.

[Guest]

Are you saying you can't solve the equations or that you don't see where they come from?

If it is the former:

A+W = V, so W = V-A, substituting this into the first equation gives
A/(A+(V-A)) = 5/100
A/V = 0.05
which implies A = 0.05V
and since W = V-A this means W = V-0.05V which simplifies to W = 0.95V

The second equation can therefore be written as
(0.05V)/(0.05V+0.95V+x) = 1/100
which rearranges to
0.05V/(V+x) = 1/100
5V/(V+x) = 1
5V = V+x
x = 4V

Hope this helped,

R. Baber.

[Guest]

I know what you are saying but I don't understand.

[Guest]

If you go back to your original partial solution

My partial solution:

Let W = water added.

The solution is 5 per cent medicine, 95 per cent water.

The resulting medicine is 1 per cent medicine, 99 per cent water.
and continue from here-

For the 5% solution the ratio of medicine to water is 5/95
For the 1% solution the ratio of medicine to water is 1/99
We are not dealing with ratios here in this question

A "percent solution" implies what is the concentration?

For 5% solution the concentration is 5/100
For 1% solution the concentration is 1/100

Therefore considering the question as a problem of concentrations

Then 5/((5+95) + W) = 1/100

5/(100 + W) = 1/100
100 + W = 500
W = 400 = water added per 100 units of original solution

[Guest]

I understand now. The letters were throwing me off- they shouldn't - I understood what each represented.