Application Problems on Allegations and Mixtures: CAT Quantitative Aptitude

Let us jump straight to the problems asked in CAT Quantitative Aptitude Examination. We have avoided the formulae driven questions as you can find plenty of them in other text books.

Example:

River water consists of 0.5% salt in it; lake water consists of 1% salt in it, and sea water consists of 5% of salt in it. If these three are mixed in the ratio of 4:2:1 and boiled such that the concentration of salt in the solution gets doubled, (i.e. by evaporating water from the solution) approximately what percentage of water initially present in the solution has evaporated?

[1] 55%

[2] 45%

[3] 75%

[4] 50%

Solution:

Situation Before Boiling

We have to mix river water, lake water, and sea water in the ratio of 4:2:1.

Let us assume we are mixing 4x, 2x and x unit of each.

This will give us 7x units of the total mixture.

In this 7x units, amount of salt from riveris 0.5% of 4x, i.e. units of salt

Similarly, in 7x units, amount of salt from lake is 1% of 2x, i.e. units of salt

And, in 7x units, amount of salt from sea is 5% of x, i.e. units of salt

Adding all the units of salt in 7x units of water, we get overall salt in mixture, which is:

Concentration of salt in 7x units of water

This means, in 7x units of mixture we have unit of salt, which also means the amount of water in this mixture is

After Boiling

When the above mixture is boiled, the concentration of salt doubles (i.e., the salt does not change in amount, but the water evaporates in the mixture and salt remains such that it has double concentration in the mixture now).

That is,

Concentration of salt in the (remaining) mixture becomes

Since, the concentration of salt is doubling without any change in quantity of salt in the mixture, it is imperative that the total volume of the mixture is getting halved…this is possible only when the units of water in the mixture are evaporating in such a way that the volume of mixture remaining after boiling is exactly half the volume of the original mixture…in other words, as water evaporates on boiling 7x units of the mixture, the remaining volume of the mixture is now 3.5x only.

Therefore, water remaining in the mixture is units

Therefore, the amount of water evaporated =

Therefore, amount of water evaporated =

Therefore, the correct option is [4].

Example:

From a barrel containing 500ml of alcohol, 3 cups of alcohol are poured into a barrel containing 500 ml of water. After mixing the contents well, 3 cups of the mixture are poured into the barrel of alcohol. The percentage of water in the barrel of alcohol and the percentage of alcohol in the barrel of water are then compared. Which one of the following is true?
   
[1] The former is greater than the latter

[2] The two are equal

[3] The latter is greater than the former

[4] Cannot be determined

Solution:

This problem came in CAT 1998.

The problem is very intuitive and the correct option is [2].

A check may be carried out as follows:

Say, we have 500 ml of alcohol in beaker 1and 500 ml of water in beaker 2.
Now, assume 3 cups hold 300 ml of any liquid — alcohol or water, so if we take 3 cups of alcohol from beaker 1 (the cup will have 300 ml of alcohol) and pour it into beaker 2…

In Beaker 1:

200 ml of alcohol is left

In Beaker 2:

…we will have 500 ml of water and 300 ml of alcohol as a mixture, 800 ml mixture.
In this mixture, alcohol is parts and water is parts.

Now, if we fill 3 cups from beaker 2…

In Beaker 2 :

500 ml mixture remains, the constituents of the mixture are:

and ml water

In Beaker 1:

We were holding 200 ml of alcohol to which 300 ml of mixture is added from beaker 2.

The constituents of 300 ml of mixture that is added are:

and ml water

Therefore, final situation of constituents in Beaker 1 = 200 + 112.5 = 312.5 ml alcohol

…and187.5 ml water

Comparison:
Percentage of water in beaker 1 =

Percentage of alcohol in beaker 2 =