Lesson: Mixture Questions

Comment on Mixture Questions

Hello Brent,

Could you please explain this question?

A mixture of 12 ounces of vinegar and oil is 40%n vinegar by weight. How many ounces of oil must be added to the mixture to produce a new mixture that is only 25% vinegar? Also is there an easier and faster approach than sketching the figures of solutions? There are many in other sources, but I do not want to confuse. I want a method from you.
greenlight-admin's picture

INITIAL mixture: 12 ounces, which is 40% vinegar
40% of 12 = 4.8
So, the INITIAL mixture contains 4.8 ounces of vinegar

Let x = number of ounces of oil that we must ADD to the initial mixture.

IMPORTANT: Once we add x ounces of oil to the initial mixture, the RESULTING mixture will have a total volume of 12+x ounces

Also, the RESULTING mixture will contain 4.8 ounces of vinegar

GOAL: We want the resulting mixture to be 25% vinegar.
In other words, we want the resulting mixture to be 1/4 vinegar

So, we get: 4.8/(x+12) = 1/4
Cross multiply to get: (1)(x+12) = (4)(4.8)
Expand and simplify: x + 12 = 19.2
Solve: x = 7.2

So, we must add 7.2 ounces of oil

Hello Sir,

For a mixture problem, what I normally do is, draw 3 boxes, and try to set up an algebraic equation.

for this problem, my equation was,
.40(12) + x = .25(12 + x)

but I see the, the correct equation is, .40(12) = .25(12 + x)

I don't know why, why the [ +x ] portion from the left side of the equation i wrote wasn't included into the equation.

is it because, the question is asking for a final mixture that is "only vinegar" instead of the phrase "resulting solution", that we normally see on other questions!

eagerly waiting for your reply.

Thanks :)
greenlight-admin's picture

Notice that (0.40)(12) represents the volume of VINEGAR in the initial mixture.
Likewise, (0.25)(12 + x) represents the volume of VINEGAR in the resulting mixture.
So, our equation is keeping track of the volume of VINEGAR throughout the process.

In the equation, x = number of ounces of OIL that we must add to the initial mixture.
Since there's no vinegar in the oil, we don't add x.

Alternatively, we COULD say that the oil we're adding is 0% vinegar.
So, we could write: 0.40(12) + (0.0)(x) = 0.25(12 + x)

Does that help?

100 percent clear Sir,

Thank you so much, you’re amazing :)

It is helpful

Thank you! The catch in this question is resulting mixture change to 12+x.

Another way to approach this question is to solve for the concentration of the combined solution know the concentration of the contributing parts:
300(30%) + 200(70%) = 500(c) --> 300 * .3 + 200 * .7 = 500*c
90 + 140 =230 = 500 * c
c = (230/500) *100 = 46%
greenlight-admin's picture


A cup full of water when added to 100 cc of a jar containing 100 cc of 30% grape solution results in a 10%
grape solution.
Column A : volume of the cup
Column B: 400 cc
sir how to solve this question?
greenlight-admin's picture

ASIDE: I'll use milliliters (ml) instead of cubic centimeters (cc)

Start with 100 ml of 30% grape solution.
So, this initial solution contains 30 ml of grape solution, and 70 ml of water.

Now add x ml of pure water (i.e., x = the volume of the cup).
So, this part contains 0 ml of grape solution, and x ml of water.

When we combine the two amounts we get: 30 ml of grape solution and (70 + x) mls of water.

Also, note that the TOTAL volume of the resulting solution = (100 + x) mls

We want the resulting solution to be 10% grape solution.
In other words, we want: (volume of grape solution in resulting solution)/(TOTAL volume of resulting solution) = 10%
In other words: (30)/(100 + x) = 10/100
Simplify: (30)/(100 + x) = 1/10
Cross multiply to get: (1)(100 + x) = (30)(10)
Expand: 100 + x = 300
Solve: x = 200

So, the volume of the cup = 200 ml

We get:

Answer: B


During a college fun-fair, the entrance fee for students was $12 and for teachers (and their family members) it was $30. 75 people came to attend that event and a total of $1800 was gathered. How many students came to this fun-fair?
Pls solve
greenlight-admin's picture

Yikes! That's a poorly-worded question!

The addition of "(and their family members)" makes this question unanswerable.
However, if we IGNORE that proviso, and assume that all 75 attendees are either students or teachers, (no family members), then we can answer the question as follows:

Let t = # of teachers who went
Let x = # of students who went

So, we can write:
t + x = 75
30t + 12x = 1800

When we solve the system, we get: t = 50 and x = 25

Answer: 25 students.


I got the same answer but doubted myself due to the exclusion of the family members. Thanks Brent


Can you provide an explanation to the question posted above, I'm not quite understanding how the answer came to be.

Thank you!
greenlight-admin's picture


When 6l pure acid is added isnt the resulting mixture total 26? since there is 8l of some other liquid?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/a-container-holds-10-liters-of-a-solution-...

We start with 10 liters of ORIGINAL solution
Although this original solution consists of 2 liters of acid and 8 liters of some other liquid, the TOTAL volume is 10 liters.

So, if we add 6 more liters to the ORIGINAL 10 liters, the resulting volume = 10 + 6 = 16

Does that help?


This technique took some practice, but once I got the hang of it, I thought it's a really good way of solving these problems. It really helps you keep track of what's going on and what you are looking for. Kahn and PurpleMath both used tables, but I think the method here of drawing out the containers is far more intuitive.
greenlight-admin's picture

Thanks, Kevin!!!

Hello Sir,

Is there an algebraic solution to this question, I couldn't solve it the way I normally do, like drawing boxes and breaking it into parts!

"Suppose you have a 200-liter mixture that is 90% water and 10% bleach and ask how much water you would need to add to make it 5% bleach"

greenlight-admin's picture

We start with 200 liters of solution

10% of 200 = 20
So, the original solution contains 20 liters of bleach

Let x = the volume of water (in liters) we must add to get a solution that is 5% bleach
This means the NEW volume of the solution = 200 + x
The volume of bleach is still 20 liters (since we didn't add any bleach the original solution)

We want the new solution to be 5% bleach.
In other words, we want: (volume of bleach)/(volume of solution) = 5/100 (aka 5%)
Substitute to get: 20/(200 + x) = 5/100
Simplify to get: 20/(200 + x) = 1/20
Cross multiply: (1)(200 + x) = (20)(20)
Expand: 200 + x = 400
Solve: x = 200

So we must add 200 liters of water.

Got it, Sir, Thank you so much :)

How many liters of 20% vinegar solution should be added to 4 liters of 50% vinegar solution to make a 30% vinegar solution?
greenlight-admin's picture

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