1. In two vessels A and B, the ratio of milk and water is $4:5$ and $5:1$ respectively. In what ratio should the liquids in both vessels be mixed so that the resulting mixture contains milk and water in the ratio $5:4$?

A) $2:5$

B) $5:2$

C) $3:4$

D) $4:3$

Solution:
Concentration of milk in A = $\frac{4}{9}$.
Concentration of milk in B = $\frac{5}{6}$.
Target concentration = $\frac{5}{9}$.
By Alligation:
Ratio = $(\frac{5}{6} - \frac{5}{9}) : (\frac{5}{9} - \frac{4}{9})$
Ratio = $(\frac{15-10}{18}) : (\frac{1}{9}) = \frac{5}{18} : \frac{1}{9} = 5:2$.
Correct Option: B

2. A shopkeeper has 100 kg of sugar. He sells a part of it at $7\%$ profit and the rest at $17\%$ profit. He gains $10\%$ on the whole. How much did he sell at $17\%$ profit?

A) 30 kg

B) 70 kg

C) 40 kg

D) 60 kg

Solution:
Profit 1: $7\%$, Profit 2: $17\%$, Mean Profit: $10\%$.
Ratio of quantities = $(17 - 10) : (10 - 7) = 7 : 3$.
Total ratio sum = $7 + 3 = 10$ units.
10 units = 100 kg $\implies$ 1 unit = 10 kg.
Quantity at $17\%$ profit (3 units) = $3 \times 10 = 30$ kg.
Correct Option: A

3. The average age of students in a class is $15.8$ years. The average age of boys in the class is $16.4$ years and that of girls is $15.4$ years. Find the ratio of the number of boys to the number of girls in the class.

A) $1:2$

B) $3:4$

C) $2:3$

D) $5:3$

Solution:
Using Alligation:
Boys avg: $16.4$, Girls avg: $15.4$, Combined avg: $15.8$.
Ratio (Boys : Girls) = $(15.8 - 15.4) : (16.4 - 15.8) = 0.4 : 0.6 = 2:3$.
Correct Option: C

4. A man borrowed $₹10,000$ in two parts, one at $8\%$ per annum simple interest and the other at $10\%$ per annum. If the total interest paid in a year was $₹880$, how much was borrowed at $8\%$?

A) $₹4,000$

B) $₹5,000$

C) $₹5,500$

D) $₹6,000$

Solution:
Overall interest rate = $(\frac{880}{10000}) \times 100 = 8.8\%$.
Rate 1: $8\%$, Rate 2: $10\%$, Mean Rate: $8.8\%$.
Ratio of Principal = $(10 - 8.8) : (8.8 - 8) = 1.2 : 0.8 = 3:2$.
Total units = $3 + 2 = 5$.
Amount at $8\%$ = $(\frac{3}{5}) \times 10000 = ₹6,000$.
Correct Option: D

5. In what ratio must rice at $₹72$ per kg be mixed with rice at $₹62$ per kg so that the mixture is worth $₹64.50$ per kg?

A) $1:2$

B) $1:3$

C) $2:5$

D) $3:5$

Solution:
Price 1: $72$, Price 2: $62$, Mean Price: $64.5$.
Ratio (Rice 1 : Rice 2) = $(64.5 - 62) : (72 - 64.5) = 2.5 : 7.5 = 1:3$.
Correct Option: B

6. A vessel contains $40$ litres of milk. From this vessel $4$ litres of milk were taken out and replaced by water. This process was repeated further one more time. How much milk is now contained by the vessel?

A) $30$ litres

B) $36$ litres

C) $32.4$ litres

D) $34.2$ litres

Solution:
Formula: Final Milk = Initial Milk $\times (1 - \frac{x}{C})^n$, where $x$ is amount replaced, $C$ is capacity, and $n$ is number of times.
Final Milk = $40 \times (1 - \frac{4}{40})^2 = 40 \times (0.9)^2$
Final Milk = $40 \times 0.81 = 32.4$ litres.
Correct Option: C

7. A person travels $150$ km in $6$ hours. He travelled some distance at $20$ km/h and the rest at $35$ km/h. How much distance did he cover at the speed of $20$ km/h?

A) $80$ km

B) $70$ km

C) $60$ km

D) $90$ km

Solution:
Average speed = $\frac{150}{6} = 25$ km/h.
Speed 1: $20$, Speed 2: $35$, Average: $25$.
Ratio of Time = $(35 - 25) : (25 - 20) = 10 : 5 = 2:1$.
Total time = $6$ hours. Time at $20$ km/h = $(\frac{2}{3}) \times 6 = 4$ hours.
Distance at $20$ km/h = $20 \times 4 = 80$ km.
Correct Option: A

8. Alloy A contains copper and tin in the ratio $1:2$ and alloy B contains copper and tin in the ratio $2:3$. If these two alloys are mixed in the ratio $3:4$ to form a new alloy, find the ratio of copper and tin in the new alloy.

A) $3:5$

B) $5:8$

C) $11:17$

D) $13:22$

Solution:
In alloy A, Copper = $\frac{1}{3}$. In alloy B, Copper = $\frac{2}{5}$.
Mixed in ratio $3:4$.
Total Copper = $(\frac{1}{3} \times 3) + (\frac{2}{5} \times 4) = 1 + \frac{8}{5} = \frac{13}{5}$.
Total Tin = $(\frac{2}{3} \times 3) + (\frac{3}{5} \times 4) = 2 + \frac{12}{5} = \frac{22}{5}$.
Ratio (Cu : Tin) = $\frac{13}{5} : \frac{22}{5} = 13:22$.
Correct Option: D

9. In what ratio must a $50\%$ alcohol solution be mixed with a $30\%$ alcohol solution to get a $45\%$ alcohol solution?

A) $3:1$

B) $1:3$

C) $2:1$

D) $1:2$

Solution:
Using Alligation:
Conc 1: $50\%$, Conc 2: $30\%$, Mean Conc: $45\%$.
Ratio = $(45 - 30) : (50 - 45) = 15 : 5 = 3:1$.
Correct Option: A

10. The average salary of $12$ officers in an office is $₹400$ and that of the rest is $₹56$. If the average salary of all the employees is $₹60$, what is the total number of employees in the office?

A) $1020$

B) $1008$

C) $1032$

D) $1044$

Solution:
Officers salary: $400$, Rest salary: $56$, Total average: $60$.
Ratio (Officers : Rest) = $(60 - 56) : (400 - 60) = 4 : 340 = 1 : 85$.
Given Officers = $12$ (which is 1 unit).
Rest employees = $85 \times 12 = 1020$.
Total employees = $12 + 1020 = 1032$.
Correct Option: C

Quantitative Aptitude: Alligation