**1. Find L.C.M. of [latex]\frac {2}{3}[/latex],[latex]\frac {8}{9}[/latex],[latex]\frac {64}{81}[/latex],[latex]\frac {10}{27}[/latex]**
A. [latex]\frac {250}{9}[/latex]
B. [latex]\frac {160}{3}[/latex]
C. [latex]\frac {128}{9}[/latex]
D. [latex]\frac {320}{3}[/latex]

**Answer**: Option D

**Explanation**:
L.C.M. = [latex]\frac {L.C.M. of Numerator}{H.C.F. of Denominator}[/latex]
L.C.M. of numerators = 2, 8, 64, 10
2 = 2[latex]^{1}[/latex]
8 = 2[latex]^{3}[/latex]
64 = 2[latex]^{6}[/latex]
10 = 2 × 5
L.C.M of 2, 8, 64, 10 = 2[latex]^{6}[/latex]6 × 5 = 320
H.C.F. of denominators = 3, 9, 81, 27
3 = 3[latex]^{1}[/latex]
9 = 3[latex]^{2}[/latex]
81 = 3[latex]^{4}[/latex]
27 = 3[latex]^{3}[/latex]
H.C.F. of 3, 9, 81, 27 = 3
L.C.M. of [latex]\frac {2}{3}[/latex],[latex]\frac {8}{9}[/latex], [latex]\frac {64}{81}[/latex] [latex]\frac {10}{27}[/latex], = [latex]\frac {320}{3}[/latex]

**2. In a mixture of 13 liters, the ratio of milk and water is 3: 2. If 3 liters of this mixture is replaced by 3 liters of milk, then what will be the ratio of milk and water in the newly formed mixture?**
A. 10 : 3
B. 8: 5
C. 9: 4
D. 1: 1

**Answer**: Option C

**Explanation**:
Given: Total quantity of mixture = 13 liters
3 liters of the mixture is removed from the container – So, let's forget this altogether!
Now, you have left with only 10 liters of the mixture in 3:2 ratio.
Milk in 10 litres mix = 10 x [latex]\frac {3}{(2 + 3)}[/latex] = 6 litres
Water in 10 litres mix = 10 x [latex]\frac {2}{(2 + 3)}[/latex] = 4 litres
We add 3 liters milk to this.
So, milk in new mix is = 6 liters + 3 litres = 9 litres
Water= 4 litres
Ratio of milk : water = 9 : 4

**3. Smith and Kate started a business investing Rs. 84,000 and Rs. 28,000 respectively. In what ratio the profit earned after 2 years be divided between Smith and Kate respectively? **
A. 2 : 3
B. 3 : 1
C. 13 : 3
D. None of these

**Answer**: Option B

**Explanation**:
P’s share of profit = [latex]\frac {x}{y}[/latex] - - - - - - (x and y are investments)
Q’s share of profit
x : y = P’s share of profit : Q’s share of profit
Therefore,
[latex]\frac {Smith’s share of profit}{Smita’s share of profit}[/latex] = [latex]\frac {84000}{28000}[/latex] =[latex]\frac {3}{1}[/latex]
The profit earned after 2 years will be divided between Smith and Kate in the ratio of 3: 1.

**4. The remainder is 29, when a number is divided 56. If the same number is divided by 8, then what is the remainder? **
**Answer**: Option D

**Explanation**:
We know that,
Dividend = [(Divisor × Quotient)] + Remainder
It is given that, the remainder is 29, when a number (dividend) is divided 56(divisor).
Dividend and quotient are unknown, hence assume dividend as X and quotient as Y.
Therefore,
X = 56 × Y + 29
56 is completely divisible by 8, but 29 is not completely divisible and we get the remainder as 5, which is the required answer.
OR
X = 56 × Y + 29
= (8 × 7Y) + (8 × 3) + 5
5 is the required remainder.

**5. There is a road besides a river. Two friends Ram & Shyam started their journey from place P, moved to the garden located at another place Q & then returned to place P. Ram moves by swimming at a speed of 15 km/hr while Shyam sails on a boat at a speed of about 12 km/hr. If the flow of water current is at the speed of 6 km/hr, what will be the average speed of boat sailor?**
A. 6 km/hr
B. 9 km/hr
C. 12 km/hr
D. 18 km/hr

**Answer**: Option B

**Explanation**:
Average Speed =[latex]\frac {Downstream Speed \times Upstream Speed}{Speed in still water}[/latex]
=[latex]\frac {(x + y) (x – y)}{x}[/latex]km/hr
Speed of boat in still water = y [latex]\frac {(t2 + t1)}{(t2 – t1)}[/latex]km/hr
As Ram swims both ways at the speed of 15 km/hr, the average speed of swimming is 15 km/hr.
Being a boat sailor, Shyam moves downstream at speed = 12 + 6 = 18 km/hr & upstream at speed = 12 – 6 = 6 km/hr
Therefore, average speed of boat sailor = Downstream speed x Upstream speed / speed in still water
=[latex]\frac {[Downstream Speed \times Upstream Speed]}{[(1/2) \times ([Downstream Speed + Upstream Speed])]}[/latex]
=[latex]\frac {(18 \times 6)}{[(1/2) \times (18 + 6)]}[/latex]
=[latex]\frac {2 \times 18 \times 6}{18 + 6}[/latex]
= 9 km/hr