Mixture Problem: Find Amount of Stronger Solution Needed

[bergausstein]

A chemist has 5 gallons of salt solution with
a concentration of 0.2 pound per gallon and another solution
with a concentration of 0.5 pound per gallon.
How many gallons of the stronger solution
must be added to the weaker solution to get
a solution that contains 0.3 pound per gallon?

this my attempt

let $x=$ amount of stronger solution needed(in gallons)
$5-x=$ amount of weaker solution(in gallons)

$0.5(x)+0.2(5-x)=0.3(5)$

$x=$ 1.67 gallons

Is this correct? if not can you tell me why my method didn't work. thanks!

 

[MarkFL]

We are going to add $x$ gallons of the stronger solution to the 5 gallons of weaker solution. So, we require, by equating two expressions for the total amount of salt:

$$5\cdot0.2+x\cdot0.5=(5+x)0.3$$

What do you get for $x$?

Do you see this is a weighted average?

 

[bergausstein]

We are going to add $x$ gallons of the stronger solution to the 5 gallons of weaker solution. So, we require, by equating two expressions for the total amount of salt:

$$5\cdot0.2+x\cdot0.5=(5+x)0.3$$

What do you get for $x$?

Do you see this is a weighted average?

that's stubborn weighted average again? :D:p

so there are two 5 gallons of mixture here? one with .2 lb/gal and the other .5lb/gal? Am I correct?

x = 2.5 gallons

 

[MarkFL]

Correct! :D

The problem you actually solved is how much of the weaker solution must be replaced by the stronger solution to get 5 gallons of a solution with 0.3 lb./gal concentration of salt. :D

...so there are two 5 gallons of mixture here? one with .2 lb/gal and the other .5lb/gal? Am I correct?...

You added this...we are not told how much of the stronger solution is available, but are left to assume that enough is available to get the desired solution.

 

[paulmdrdo1]

:D good

 

[soroban]

Hello, bergausstein!

Here is my approach to mixture problems.

A chemist has two solutions.
Solution A: concentration of 0.2 pound of salt per gallon .
Solution B: concentration of 0.5 pound per gallon.
How many gallons of solution B must be added to 5 gallons of solution A to get a solution that contains 0.3 pound per gallon?

He has 5 gallons of A which has 0.2 pound salt per gallon.
This contains: .$$(0.2)(5) = 1$$ pound of salt.

He adds $$x$$ gallons of B which has 0.5 pound salt per gallon.
This contains: .$$0.5x$$ pounds of salt.

Hence, the mixture will contain $$1 + 0.5x$$ pounds of salt.But we know that the mixture has $$x +5$$ gallons
. . which has 0.3 pounds of salt per gallon.
This contains $$0.3(x+5)$$ pounds of salt.We just described the final amount of salt in two ways.

There is our equation! . . . $$1 + 0.5x \;=\;0.3(x+5)$$