Answer :
Let's solve each question step by step:
Q12:
To determine how many days Samarth and Ethan can complete the work together, we must first find their rates of work.
- 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:
- [tex]j - m = 5(s - m)[/tex]
- [tex]j + m = 3(s + m)[/tex]
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