5.2 Switching simultaneously and concentrations do not become equal

There is no shortcut/formula we recommend here. We apply the basics to solve this.

Example 18

Two alloys A and B weigh $10$ kg each. The percentage of copper in A and B are $80 \%$ and $60 \%$ respectively. Two kg is removed from each of the alloys A and B. The portion removed from A is added to B and vice versa. What is the percentage point difference in the copper concentration of the two alloys?

Solution

	Alloy A	Alloy B
Copper in $10$ kg of	$0.8 \times 10 = 8$ kg	$0.6 \times 10 = 6$ kg
Copper in $2$ kg of	$0.8 \times 2 = 1.6$ kg	$0.6 \times 2 = 1.2$ kg
*After Replacement*
Copper Quantity	$8 - 1.6 + 1.2 = 7.6$ kg	$6 - 1.2 + 1.6 = 6.4$ kg
Copper $\%$	$76 \%$	$64 \%$

Percentage point difference $= 76 \% - 64 \% = 12 \%$

Answer: $12 \%$

5.3 Non-Simultaneous Transfer

Example 19

Vessel A contains $32$ litres of milk solution where the ratio of milk to water is $3 : 1$. Vessel B contains $40$ litres of milk solution where the ratio of milk to water is $1 : 3$. $8$ litres of the solution is transferred from Vessel A to Vessel B. $12$ litres of the solution in Vessel B is then transferred to Vessel A. What is the ratio of milk to water in Vessel A after the two transfers?

Solution

Applying the given ratios, milk and water in Vessel A are $24$ litres and $8$ litres respectively, while in Vessel B are $10$ litres and $30$ litres respectively.

Initially one-fourth of Vessel A ($8$ out of $32$ litres) is transferred to B. Therefore, Milk and Water in Vessel A will reduce by one-fourth.

Now, the total quantity in Vessel B is $40 + 8 = 48$ litres. Now, one-fourth of Vessel B ($12$ out of $48$ litres) is transferred to A. Therefore, Milk and Water in Vessel B will reduce by one-fourth.

(in litres)	Vessel A			Vessel B
	Milk	Water	Total	Milk	Water	Total
Initial Quantity	$24$	$8$	$32$	$10$	$30$	$40$
Transferring $8$ litres to B	$24 - 6$ $= 18$	$8 - 2$ $= 6$	$32 - 8$ $= 24$	$10 + 6$ $= 16$	$30 + 2$ $= 32$	$40 + 8$ $= 48$
Transferring $12$ litres to A	$18 + 4$ $= 22$	$6 + 8$ $= 14$	$24 + 12$ $= 36$	$16 - 4$ $= 12$	$32 - 8$ $= 24$	$48 - 12$ $= 36$

Final ratio of milk and water in Vessel A $= 22 : 14 = 11 : 7$

Answer: $11 : 7$

6. Evaporation

These questions involve mixture of two items, where only one item is evaporating (or reducing), with the other remaining constant. Here, we first find the quantity of the item that does not reduce and then use this to calculate the total quantity of the mixture.

Example 20

Fresh grapes contain $90 \%$ water by weight while dry grapes contain $20 \%$ water by weight. What is the weight of dry grapes available from $20$ kg of fresh grapes?
[CAT 2001]

(1) $2$ kg (2) $2.4$ kg (3) $2.5$ kg (4) None of the above

Solution

When water evaporates from Fresh Grapes ($90 \%$ water and $10 \%$ pulp), we get Dry Grapes ($20 \%$ water and $80 \%$ pulp). This is because only the water quantity reduces, whereas the quantity of pulp remains the same.

Quantity of pulp in $20$ kg of Fresh Grapes $= 10 \%$ of $20$ kg $= 2$ kg

Let the quantity of Dry Grapes be $x$ kg. In these dry Grapes $2$ kg should constitute the $80 \%$ pulp.

$\dfrac{80}{100} \times x = 2$

⇒ $ x = 2.5$

Answer: (3) $2.5$ kg

7. Past Questions

Question 1

A manufacturer has $200$ litres of acid solution which has $15 \%$ acid content. How many litres of acid solution with $30 \%$ acid content may be added so that acid content in the resulting mixture will be more than $20 \%$ but less than $25 \%$?
[XAT 2010]

		More than $100$ litres but less than $300$ litres
		More than $120$ litres but less than $400$ litres
		More than $100$ litres but less than $400$ litres
		More than $120$ litres but less than $300$ litres
		None of the above

Question 2

Gopal sells fruit juice mixture using orange juice and pineapple juice. Gopal prepares this mixture by drawing out a jug of orange juice from a $10 litre container filled with orange juice, and replacing it with pineapple juice. If Gopal draws out another jug of the resultant mixture and replaces it with pineapple juice, the container will have equal volumes of orange juice and pineapple juice. The volume of the jug in litres, is
[XAT 2012]

		$2$
		$> 2$ and $< 2.5$
		$2.5$
		$> 2.5$ and $\le 3$
		$\ge 3$

Question 3

There are two alloys P and Q made up of silver, copper and aluminium. Alloy P contains $45 \%$ silver and rest aluminium. Alloy Q contains $30 \%$ silver, $35 \%$ copper and rest aluminium. Alloys P and Q are mixed in the ratio of $1 : 4.5$. The approximate percentages of silver and copper in the newly formed alloy is:
[IIFT 2015]

		$33 \%$ and $29 \%$
		$29 \%$ and $26 \%$
		$35 \%$ and $30 \%$
		None of the above