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Given:

Volume of the tank = 256 litres

Depth = half of the length

Breath = 1/4 of the difference between length and depth

Formula used:

Volume of a cuboid = Length × Breadth × Depth

Calculation:

Let the length of the tank be 'L'.

Then, depth = L / 2

And, breadth = 1/4 × (L - L / 2) = 1/4 × (L / 2) = L / 8

Now, we use the volume formula:

Volume = L × (L / 8) × (L / 2)

256 = (L³) / 16

⇒ L³ = 256 × 16

⇒ L³ = 4096

⇒ L = ∛4096

⇒ L = 16 cm

∴ The length of the tank is 16 cm.

Given:

Ramesh's time = 2 × (Naresh's time + Mahesh's time)

Naresh's time = 3 × (Ramesh's time + Mahesh's time)

All three together complete the job in 30 days.

Formula used:

Work done in 1 day = 1 / (Total days taken)

Efficiency is inversely proportional to time.

Calculation:

Let R, N, and M be the efficiencies of Ramesh, Naresh, and Mahesh respectively.

The efficiency of all three together is 1/30.

⇒ R + N + M = 1/30

From the given conditions, we can write the efficiencies as follows:

R = (N + M) / 2

N = (R + M) / 3

From the first equation, we can substitute (N + M) with (1/30 - R) in the combined efficiency equation:

⇒ R = (1/30 - R) / 2

⇒ 2R = 1/30 - R

⇒ 3R = 1/30

⇒ R = 1 / (3 × 30)

⇒ R = 1/90

Ramesh's efficiency is 1/90. This means Ramesh does 1/90^th of the job in one day.

The time taken by Ramesh to finish the job alone is the reciprocal of his efficiency.

Time = 1 / R

⇒ Time = 1 / (1/90)

⇒ Time = 90 days

∴ Ramesh alone will take 90 days to finish the job.

Given:

Principal (P) = ₹6,400

Amount (A) = ₹12,500

Rate (r) = 25% per annum

Formula used:

A = P\((1+\frac{r}{100})^t\)

Where, t = time

Calculations:

12500 = 6400\((1+\frac{25}{100})^t\)

⇒ \(\frac{12500}{6400}= (1+\frac{25}{100})^t\)

⇒ \(\frac{125}{64}= (\frac{125}{100})^t\)

⇒ \(\frac{125}{64}= (\frac{5}{4})^t\)

Comparing powers:

⇒ \((\frac{5}{4})^3 = (\frac{5}{4})^t\)

⇒ t = 3 years

∴ The correct answer is option (4).

Given:

Marked price of the geyser = ₹48,000

First discount = 15%

Second discount = 20%

Additional discount = 4% (if net payment after discounts is above ₹30,000)

Reward points = 1680

Value of each reward point = ₹0.25

Formula used:

Net price after discount = Marked price × (1 - First discount) × (1 - Second discount)

Net price after additional discount = Net price after discount × (1 - Additional discount)

Value of reward points = Reward points × Value per reward point

Net selling price = Net price after additional discount - Value of reward points

Calculation:

First, we calculate the price after the two successive discounts:

Net price after first discount = ₹48,000 × (1 - 0.15) = ₹48,000 × 0.85 = ₹40,800

Net price after second discount = ₹40,800 × (1 - 0.20) = ₹40,800 × 0.80 = ₹32,640

Since the net payment (₹32,640) is above ₹30,000, the customer qualifies for an additional discount of 4%:

Net price after additional discount = ₹32,640 × (1 - 0.04) = ₹32,640 × 0.96 = ₹31,334.40

Next, we calculate the value of the reward points:

Value of reward points = 1680 × ₹0.25 = ₹420

Finally, we subtract the value of the reward points from the net price after the additional discount:

Net selling price = ₹31,334.40 - ₹420 = ₹30,914.40

∴ The net selling price of the geyser for the customer is ₹30,914.40.

Given:

A number is increased by 30% and then decreased by 30%. Finally, the number changes.

Formula used:

Final value = Initial value × (1 + Increase%) × (1 - Decrease%)

Calculation:

Let the initial number = 100

After 30% increase:

⇒ 100 × (1 + 30/100) = 100 × 1.3 = 130

After 30% decrease:

⇒ 130 × (1 - 30/100) = 130 × 0.7 = 91

Change in number:

⇒ 100 - 91 = 9

Percentage decrease:

⇒ (9/100) × 100 = 9%

∴ The correct answer is option (4).

Given:

Rate of milk = Rs 29 per litre

Rate of mixture = Rs 25 per litre

Formula Used:

Cost of mixture = Total Cost / Total Quantity

Solution:

We know that price of water is zero.

Cost of milk = Rs 29 per litre

Cost of water = Rs 25 per litre

Let the ratio of milk and water be x : 1

Then,

(1 × 0) + (x × 29) = (1 + x) × 25

0 + 29x = 25 + 25x

29x - 25x = 25

4x = 25

x = 25 / 4

The ratio of milk and water will be 25 / 4 : 1 = 25 : 4

Alternate Method By Alligation method:- 

The ratio of milk and water will be 25 : 4

Given:

We need to find the smallest value among the following options:

Option 1: (99 ÷ 6 of 3) + (58 ÷ 116) + (132 ÷ 11)

Option 2: ((2 ÷ 5) + (4 ÷ 3)) × 30

Option 3: (32 ÷ 8) + (3 ÷ 5 of 85)

Option 4: (21 ÷ 63 of 33) + (6.5 × 4)

Formula used:

Basic arithmetic operations: Addition, Division, Multiplication, and Simplification

Calculations:

Option 1:

⇒ (99 ÷ 6 × 3) + (58 ÷ 116) + (132 ÷ 11)

⇒ (99 ÷ 18) + (1 ÷ 2) + 12

⇒ 5.5 + 0.5 + 12

⇒ 18

Option 2:

⇒ ((2 ÷ 5) + (4 ÷ 3)) × 30

⇒ (0.4 + 1.333) × 30

⇒ 1.733 × 30

⇒ 52

Option 3:

⇒ (32 ÷ 8) + (3 ÷ 5 × 85)

⇒ 4 + (3 × 17)

⇒ 4 + 51

⇒ 55

Option 4:

⇒ (21 ÷ 63 × 33) + (6.5 × 4)

⇒ (1 ÷ 3 × 33) + 26

⇒ 11 + 26

⇒ 37

Comparison:

Option 1 = 18, Option 2 = 52, Option 3 = 55, Option 4 = 37

∴ The smallest value is from Option 1, and the correct answer is Option 1.

Given:

Cost Price (CP) = ₹800

Selling Price (SP) = ₹950

Formula used:

Profit Percentage = \(\frac{Profit}{CP} \times 100\)

Profit = SP - CP

Calculation:

Profit = ₹950 - ₹800

⇒ Profit = ₹150

Profit Percentage = \(\frac{150}{800} \times 100\)

⇒ Profit Percentage = \(\frac{15000}{800}\)

⇒ Profit Percentage = 18.75%

∴ The correct answer is option (2).

Given:

Mean proportional between two numbers = 18

First number is 15 greater than the second number.

Formula used:

If 'x' is the mean proportional between 'a' and 'b', then x² = ab.

Calculation:

Let the first number be 'a' and the second number be 'b'.

According to the problem:

a = b + 15

18² = a × b

⇒ 324 = a × b

Substitute the value of 'a' in the second equation:

⇒ 324 = (b + 15) × b

⇒ 324 = b² + 15b

⇒ b² + 15b - 324 = 0

By factoring the quadratic equation, we find two numbers that multiply to -324 and add up to 15, which are 27 and -12.

⇒ (b + 27)(b - 12) = 0

The possible values for 'b' are -27 and 12. Since the numbers are generally positive in this context, we take b = 12.

Now, we find the first number 'a':

a = b + 15

⇒ a = 12 + 15

⇒ a = 27

∴ The first number is 27.

Given:

Race distance = 800 m

Kamal is beaten by 150 m or 30 sec.

Formula used:

Speed = Distance / Time

Conversion from m/s to km/h = (Speed in m/s) × (18/5)

Calculation:

The phrase "beats by 150m or 30sec" implies that Kamal takes 30 seconds to cover a distance of 150m.

We first find the speed of Kamal:

Speed of Kamal = 150 m / 30 s

⇒ Speed of Kamal = 5 m/s

Now, we find the time taken by Kamal to complete the 800m race:

Time taken by Kamal = 800 m / 5 m/s

⇒ Time taken by Kamal = 160 seconds

Since Surya beats Kamal by 30 seconds, Surya's time to complete the race is:

Time taken by Surya = 160 s - 30 s

⇒ Time taken by Surya = 130 seconds

Now, we can find the speed of Surya in m/s:

Speed of Surya = 800 m / 130 s 

Finally, we convert the speed of Surya from m/s to km/h:

Speed of Surya = 800 m / 130 s × (18/5)

⇒ Speed of Surya = \(22\frac{2}{13}\text{km/h}\)

∴ The speed of Surya is  \(22\frac{2}{13}\text{km/h}\)