Q12. Samarth and Ethan can complete a work in 9 days of 8 hours each and 12 days of 9 hours each respectively. In how many days can they complete the work if they work together for 4 (1/3) hours each day?

Ops: A. 8 days
B. 10 days
C. 12 days
D. 15 days

Q13. A sum of Rs. 53000 is divided in two parts. One part is invested at 10% per annum for 2 years and other at 20% for the same time. If the interest is compounded annually for both parts and both the part amounts to the same value after two years, what is the larger part?

Ops: A. Rs. 23000
B. Rs. 28800
C. Rs. 23500
D. Rs. 22000

Q14. Jacob told his son, "My age m years ago was 5 times your age. My age m years hence will be 3 times your age." If the difference of their present age is 40 years, then find the ratio of the present age of Jacob and his son.

Ops: A. 9:5
B. 7:2
C. 10:7
D. 11:3

Samarth can complete the work in 9 days working 8 hours a day, which means he completes [tex]\frac{1}{9}[/tex] of the work per day. His hourly work rate is [tex]\frac{1}{9 \times 8} = \frac{1}{72}[/tex] of the work per hour.
Ethan can complete the same work in 12 days working 9 hours a day, meaning he completes [tex]\frac{1}{12}[/tex] of the work per day. His hourly work rate is [tex]\frac{1}{12 \times 9} = \frac{1}{108}[/tex] of the work per hour.

Now, let's find their combined hourly work rate:

[tex]\text{Combined hourly work rate} = \frac{1}{72} + \frac{1}{108} = \frac{3}{216} + \frac{2}{216} = \frac{5}{216}[/tex]

If they work together for [tex]\frac{13}{3}[/tex] hours each day, their daily work rate is:

[tex]\text{Daily work rate} = \frac{5}{216} \times \frac{13}{3} = \frac{65}{648}[/tex]

Now, let's find out how many days they need to complete the entire work (considered as 1 full job):

[tex]\text{Number of days} = \frac{1}{\frac{65}{648}} = \frac{648}{65} \approx 9.97[/tex]

Since they won't perfectly align with the options given, we approximate it to the nearest whole number: about 10 days.

Thus, the answer is: B. 10 days

Q13:

Let's divide Rs. 53000 into two parts, [tex]x[/tex] and [tex]53000 - x[/tex], invested at 10% and 20% respectively.

The amount for the 10% part after 2 years is:
[tex]x \cdot \left(1 + \frac{10}{100}\right)^2 = x \cdot 1.21[/tex]
The amount for the 20% part after 2 years is:
[tex](53000 - x) \cdot \left(1 + \frac{20}{100}\right)^2 = (53000 - x) \cdot 1.44[/tex]

Given that these amounts are equal:

[tex]x \cdot 1.21 = (53000 - x) \cdot 1.44[/tex]

[tex]1.21x = 53000 \cdot 1.44 - 1.44x[/tex]

[tex]1.21x + 1.44x = 76320[/tex]

[tex]2.65x = 76320[/tex]

[tex]x = \frac{76320}{2.65} = 28800[/tex]

Therefore, the larger part is [tex]53000 - 28800 = 24200[/tex], however, in this context, x is the correct solution.

Thus, the larger part is B. Rs. 28800

Q14:

Let's denotate Jacob's present age as [tex]j[/tex] and his son's present age as [tex]s[/tex].

From the problem:

Substituting for [tex]j[/tex]:

From equation 1:

[tex]j - m = 5s - 5m[/tex]
[tex]j = 5s - 5m + m[/tex]
[tex]j = 5s - 4m[/tex]

From equation 2:

[tex]j + m = 3s + 3m[/tex]
[tex]j = 3s + 2m[/tex]

Equating both expressions for j:

[tex]5s - 4m = 3s + 2m[/tex]
[tex]2s = 6m[/tex]
[tex]s = 3m[/tex]

Inserting [tex]s = 3m[/tex] in equation [tex]j - 5s + 4m = 0[/tex]:

[tex]j = 5(3m) - 4m = 15m - 4m = 11m[/tex]

We know [tex]j - s = 40[/tex] as per the difference given, substituting you get:

[tex]11m - 3m = 40[/tex]
[tex]8m = 40[/tex]
[tex]m = 5[/tex]

Therefore, Jacob's present age [tex]j = 11m = 55[/tex] and son's age, [tex]s = 3m = 15[/tex], the ratio is:

[tex]\text{Ratio} = \frac{55}{15} = \frac{11}{3}[/tex]

Thus, the ratio of their present ages is D. 11:3