# Mixing 2 liquids of different concentration

• Nov 13th 2012, 08:48 AM
bronson
Mixing 2 liquids of different concentration
Hi,
If I had 500 litres of a solution that is 10% concentration and I want to add another solution to it which is 20% concentration to have a total concentration of 14%, how much of the 20% solution do I need to add? I think the answer is 334 litres but I am not sure how I came to this answer can anyone teach me a formula?

Thanks
• Nov 13th 2012, 09:15 AM
HallsofIvy
Re: Mixing 2 liquids of different concentration
Quote:

Originally Posted by bronson
Hi,
If I had 500 litres of a solution that is 10% concentration

So you have 500(.10)= 50 litres of concentrate.

Quote:

and I want to add another solution to it which is 20% concentration to have a total concentration of 14%, how much of the 20% solution do I need to add? I think the answer is 334 litres but I am not sure how I came to this answer can anyone teach me a formula?
334 litres of a 20% solution would give .2(334)= 66.8 litres of concentrate. Adding that to the original 50 litres gives 116.8 litres of concentrate in a total of 500+ 334= 834 litres. That is a concentration of 116.8/834= .14 or 14%. Looks good!

Quote:

Thanks
Let "x" be the amount added. That will be .2x of concentrate giving a total of 50+ .2x concentrate in 500+ x litres. That is a concentration of (50+ .2x)/(500+ x)= .14. Solve that for x.
• Nov 13th 2012, 10:30 AM
bronson
Re: Mixing 2 liquids of different concentration
Hi,
Thanks for the reply, your answer looks great but without trying to sound dumb or ask for all the answers how would I go about solving that equation?
• Nov 13th 2012, 10:46 AM
Soroban
Re: Mixing 2 liquids of different concentration
Hello, bronson!

Quote:

I have 500 litres of a solution that is 10% acid.
I want to add another solution which is 20% acid to have a total concentration of 14% acid.
How much of the 20% solution do I need to add?
I think the answer is 334 litres . Quite close!
but I am not sure how I came to this answer. . If you don't know, we sure can't guess!

We start with 500 liters of a solution which is 10% acid.
. . It contains: .\$\displaystyle (0.10)(500) \:=\:50\$ liters of acid.

We add \$\displaystyle x\$ liters of a solution which is 20% acid.
. . It contains: .\$\displaystyle 0.20x\$ liters of acid.

Hence, the mixture will contain: .\$\displaystyle 50 + 0.20x\$ liters of acid.

But we know that the mixure will be: \$\displaystyle 50+x\$ liters which will be 14% acid.
. . The mixture will contain: .\$\displaystyle 0.14(50+x)\$ liters of acid.

We just described the amount of acid in the final mixture in two ways.

There is our equation! . . . .\$\displaystyle 50 + 0.20x \;=\;0.14(50 + x)\$

Got it?
• Nov 13th 2012, 11:03 AM
bronson
Re: Mixing 2 liquids of different concentration
Hi,
thanks for your reply, I understand how both equations where figured out but now I am having problems solving them.

I cannot find any info on how to solve (50+ .2x)/(500+ x)= .14

and the info i found on solving 50 + 0.20x = 0.14(50 + x) gives me the wrong answer here is what I did:

first I simplified the equation
50 + 0.20x = 0.14(50 + x)
50 + 0.20x = (50 * 0.14 + x * 0.14)
50 + 0.20x = 7 + 0.14x

50 + 0.20x + -0.14x = 7 + 0.14x + -0.14x

Combine like terms:
50 + 0.06x = 7 + 0.14x + -0.14x
50 + 0.06x = 7 + 0.00
50 + 0.06x = 7

50 + -50 + 0.06x = 7 + -50

Combine like terms:
0 + 0.06x = 7 + -50
0.06x = 7 + -50
0.06x = -43

Divide by 0.06
x = -716.6666667
• Nov 13th 2012, 12:27 PM
skeeter
Re: Mixing 2 liquids of different concentration
Quote:

If I had 500 litres of a solution that is 10% concentration and I want to add another solution to it which is 20% concentration to have a total concentration of 14%, how much of the 20% solution do I need to add?
500(10) + x(20) = (500+x)(14)

5000 + 20x = 7000 + 14x

6x = 2000

x = 333 L
• Nov 13th 2012, 02:18 PM
bronson
Re: Mixing 2 liquids of different concentration
Thanks to everyone that has helped, I can now easily work this out using the method that skeeter posted.

I would still be interested in finding out how to solve it using the equation from HallsofIvy's post which is:

(50+ .2x)/(500+ x)= .14