Logical Reasoning and Analytical Ability

Given: Principal (P) = Rs 50,000
Rate (R) = 10% per annum
Time (T) = 2 years

Simple Interest: $SI = \frac{P \times R \times T}{100}$ $SI = \frac{50000 \times 10 \times 2}{100} = Rs\ 10,000$

Compound Interest: $CI = P\left[\left(1+\frac{R}{100}\right)^T - 1\right]$ $CI = 50000\left[\left(1+\frac{10}{100}\right)^2 - 1\right]$ $CI = 50000[(1.1)^2 - 1] = 50000(1.21 - 1)$ $CI = 50000 \times 0.21 = Rs\ 10,500$

Difference: $CI - SI = 10500 - 10000 = Rs\ 500$

Therefore, the difference between compound interest and simple interest is Rs 500.

Let the present salaries of Dipesh and Sumit be:

[ \text{Dipesh}=2x,\qquad \text{Sumit}=3x ]

After each salary is increased by Rs. 4000, the new ratio is (40:57).

[ \frac{2x+4000}{3x+4000}=\frac{40}{57} ]

Cross-multiply:

[ 57(2x+4000)=40(3x+4000) ]

[ 114x+228000=120x+160000 ]

[ 228000-160000=120x-114x ]

[ 68000=6x ]

[ x=\frac{68000}{6}=11333.33 ]

Sumit’s present salary:

[ 3x=3\times 11333.33=34000 ]

[ \boxed{\text{Sumit’s present salary} = \text{Rs. } 34000} ]

(After the increment, Sumit’s salary would be (\text{Rs. } 38000).)

The LCM of 8, 12 and 4 is 24.

$\frac58+\frac7{12}-\frac34=\frac{15}{24}+\frac{14}{24}-\frac{18}{24}=\frac{15+14-18}{24}=\frac{11}{24}$

In decimal, $\frac{11}{24}=0.4583\ldots\approx 0.46$.

Total work $= 12 \times 15 = 180$ worker-days.

Work done in first 5 days $= 12 \times 5 = 60$ worker-days.

Remaining work $= 180 - 60 = 120$ worker-days.

After 4 more workers join, total workers $= 12 + 4 = 16$.

Time to finish remaining work $= \dfrac{120}{16} = 7.5$ days.

Total days from the start $= 5 + 7.5 = 12.5$ days.

Answer: 12.5 days (12 days 12 hours) in total; 7.5 days for the remaining work.