# A milkman has a basket of capacity 60 L, which is full of pure milk. He gives his 1^{st} customer 5 L from the basket, and then he adds 5 L water to the basket. Again, he does the same to his next customer and he keeps on doing this up to his 4^{th} customer. What will be the ratio of milk and water the 4^{th} customer will get?

## Answer

Correct Answer : a ) 1331 : 397

Explanation :**Quick Approach**

The amount of pure quantity A in a mixture, if initially x litres of mixture contains only pure quantity A and from that in each operation y litres of quantity A is taken out and replaced with the same amount of other quantity B will be

Pure quantity A in mixture after n operation=\(x * (1-{y \over x})^n \)

So, fourth customer will get his mixture after 3 operation

Pure quantity of milk in mixture after 3 operation=\(60 * (1-{5 \over 60})^3=60* ({11 \over 12})^3\)

Thus, Ratio of milk and water in mixture after 3 operation = Milk : (Total mixture – Milk)

\(60* ({11 \over 12})^3 :(60-60* ({11 \over 12})^3)\)

⇒\( ({11 \over 12})^3:(1-({11 \over 12})^3)\)

⇒\({1331 \over 1728}:(1-{1331 \over 1728})\)

⇒\(1331:397\)

The ratio of milk and water in the mixture will be equal to the ratio of milk and water 4th customer will get.

So, ratio of milk and water 4^{th} customer will get = 1331 : 397

Hence, (a) is the correct answer.

**Basic Approach**

From a basket of capacity 60 L full of milk when milkman gives 5L of pure milk

Remaining pure milk in the mixture after replacement of 5L milk with water = 60 – 5 = 55 L

Amount of pure milk customer will get when he buys 5 L of milk will be =\(5* ({55 \over 60})L\)

We can see the pattern here that after every replacement of milk with water in the mixture, the amount of pure milk in the mixture will become

⇒\({55 \over 60}\) times

So, Amount of pure milk third customer will get =\(5* ({55 \over 60})^2L\)

Similarly amount of pure milk fourth customer will get =\(5* ({55 \over 60})^3=5* ({11 \over 12})^3=5*{1331 \over 1728}\)

Now, on a purchase of 5 L mixture fourth customer will get a mixture of milk and water in the ratio

⇒ Milk : (Total mixture – Milk)

⇒\(5*{1331 \over 1728}:(5- 5*{1331 \over 1728})\)

⇒\(5*{1331 \over 1728}:5*(1- {1331 \over 1728})\)

⇒\(1331:397\)

Hence, (a) is the correct answer.

Such type of question is asked in various government exams like SSC CGL, SSC MTS, SSC CPO, SSC CHSL, RRB JE, RRB NTPC, RRB GROUP D, RRB OFFICER SCALE-I, IBPS PO, IBPS SO, RRB Office Assistant, IBPS Clerk, RBI Assistant, IBPS RRB OFFICER SCALE 2&3, UPSC CDS etc.

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