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Calculation:

Let the number be N.

15% × (55% × N) = 2475

⇒ 15/100 × 55 N/100 = 2475 

⇒ N = (2475 × 100 × 100)/(15 × 55)

⇒ N = (2475 × 20 × 20)/(3 × 11)

⇒ N = 75 × 20 × 20 

⇒ N = 30,000

Now, 41% of that number is:

⇒ 30,000 × 41/100 

⇒ 300 × 41

⇒ 12,300

∴ The 41% of the number is 12,300.

Given:

Initial ratio of two numbers is 5 : 4

y is subtracted from each of the two numbers then the ratio becomes 2 : 1 

Calculation:

Let the two numbers be 5x and 4x.

According to the question,

(5x - y)/(4x - y) = 2/1

⇒ 5x - y = 8x - 2y

⇒ 3x = y     -----(1)

Now, The ratio when y is added to the two numbers,

⇒ (5x + y) : (4x + y)

Take the value of y from equation (1), we get

⇒ (5x + 3x) : (4x + 3x)

⇒ 8x : 7x

⇒ 8 : 7

∴ The ratio of the two numbers when y added to them is 8 : 7

Shortcut Trick

Total Work (LCM of 20, 30) = 60 units.

Efficiency: A = 3 units/day, B = 2 units/day.

Work done in 5 days: A = 3 × 5 = 15 units; B = 2 × 5 = 10 units.

Remaining work done by C = 60 − (15 + 10) = 35 units.

Payment Ratio (Work Done) = 15 : 10 : 35 = 3 : 2 : 7.

∴ The correct answer is 3 : 2 : 7.

Alternate Method

Given: Time taken by A = 20 days, B = 30 days. They work together for 5 days. C finishes the rest.

Formula: Payment is distributed in the ratio of the Total Work done by each individual.

⇒ Work done by A in 5 days = Efficiency × Time = 3 × 5 = 15 units.

⇒ Work done by B in 5 days = Efficiency × Time = 2 × 5 = 10 units.

⇒ Total work done by A and B together = 15 + 10 = 25 units.

⇒ Remaining work completed by C = Total Work − Work done by (A + B)

⇒ Work done by C = 60 − 25 = 35 units.

⇒ Ratio of Payment = Work done by A : Work done by B : Work done by C

⇒ Ratio = 15 : 10 : 35

⇒ Dividing by common factor 5: Ratio = 3 : 2 : 7.

∴ The correct answer is 3 : 2 : 7.

Additional Information

Wages and Work Done

Wages are always distributed in direct proportion to the actual work performed by each person, not necessarily their individual efficiencies unless they work for the same duration.

Efficiency and Time

Efficiency is inversely proportional to the time taken to complete a task. Efficiency = Total Work ÷ Time Taken.

Combined Work Rule

If multiple people work together, their combined efficiency is the sum of their individual efficiencies (Units/Day).

Given:

SI on ₹6000 in 4 years at R% = SI on ₹9000 in 2 years at 12%

Formula Used:

SI = PRT/100

Where, P = Principal, R = Rate, T = Time(in years)

Calculation:

According to the question

$\frac{6000\times R\times 4}{100}$ = $\frac{9000\times 12\times 2}{100}$ 

⇒ R = $\frac{9\times 12\times 2}{6\times4}$ = 9%

∴ The value of R is 9%.

Given:

(32)√x = 41√x - 40√x    

Concept used:

We will follow the options eliminated method.

Calculation:

According to the Option 3

If x² = 16

⇒ x = 4

Put the value in the equation.

⇒ 9^√4 = 41^√4 - 40^√4

⇒ 9² = 41² - 40²

⇒ 81 = 1681 - 1600

⇒ 81 = 81

Hence option 3 is fit to the equation.

According to the option 1

x² = 4

x = 2 

option 1 does not fit the equation because 2 is not a perfect square.

option 2 and 4 also do not fit the equation because they are also not a perfect square.

∴ The correct answer is 16

Shortcut Trick

Since the time of travel is constant for both trains, the distance ratio is equal to the speed ratio.

Speed ratio (Train 1 : Train 2) = 10 : 27

Difference in distance units = 27 − 10 = 17 units

17 units = 340 km → 1 unit = 20 km

Total distance = (10 + 27) units = 37 units = 37 × 20 km = 740 km

∴ The correct answer is 740 km.

Alternate Method

Given: Speed_A = 10 km/h, Speed_B = 27 km/h, Distance difference = 340 km

Formula: Distance = Speed × Time

Let the time after which they meet be 't' hours.

⇒ Distance covered by first train (d₁) = 10 × t

⇒ Distance covered by second train (d₂) = 27 × t

⇒ According to the question: d₂ − d₁ = 340

⇒ 27t − 10t = 340

⇒ 17t = 340

⇒ t = 20 hours

⇒ Total Distance = d₁ + d₂ = 10t + 27t = 37t

⇒ Total Distance = 37 × 20 = 740 km

∴ The correct answer is 740 km.

Additional Information

Relative Speed

When two objects move towards each other, their relative speed is the sum of their individual speeds (S₁ + S₂).

Meeting Time Formula

Time taken to meet = (Initial distance between them) ÷ (Relative Speed).

Constant Time Property

When time is constant, the ratio of distances covered is directly proportional to the ratio of their speeds (D₁/D₂ = S₁/S₂).

Given:

Average weight of X, Y, Z = 50 kg

Average weight of X, Y = 44.5 kg

Average weight of Y, Z = 53.5 kg

Weights of X, Y, Z are represented by p, q, r respectively.

Concept used:

Concept Used: Weighted Averages.

Calculation:

According to the question,

The Average weight of X, Y, and Z = 50 kg

⇒ p + q + r = 50 × 3

⇒ p + q + r = 150      .....(1)

The average weight of X and Y = 44.5 kg

⇒ p + q = 44.5 × 2

⇒ p + q = 89     .....(2)

The average weight of Y and Z = 53.5 kg

⇒ q + r = 53.5 × 2

⇒ q + r = 107     .....(3)

Solving these equations,

Subtract equation 2 from equation 1,

⇒ r = (p + q + r ) - (p + q)

⇒ r = 150 - 89 = 61     .....(4)

Subtract equation 4 from equation 3,

⇒ (q + r) - (r)

⇒ q = 107 - 61 = 46     .....(5)

Substitute q's value in equation 2,

⇒ p + q = 89

⇒ p + 46 = 89

⇒ p = 89 - 46  = 43     .....(6)

To get $\rm \frac{p+q}{r}$ ,

⇒ $\rm \frac{43 + 46 }{61}$ = 1.46 (to 2 decimal places)

∴ The value of  $\rm \frac{p+q}{r}$ correct to two decimal places, is 1.46

Given:

1 + 4 + 9 + 16 + 25 + ..... + 81 + 100 + 81 + ...... + 16 + 9 + 4 + 1 

Concept used:

Sum of Square of 1^st  'n' natural numbers is given by

12 + 22 + 32 + ..... + n2 = $\frac{n(n+1)(2n+1)}{6}$

Calculation:

According to the question,

1 + 4 + 9 + 16 + 25 + ..... + 81 + 100 + 81 + ..... + 16 + 9 + 4 + 1

⇒ 2 (1 + 4 + 9 + 16 + 25 + ..... + 81) + 100

⇒ 2 (1² + 2² + 3² + 4² + .... + 9²) + 100

By using the formula for sum of squares of 1^st  '9' natural numbers,

⇒ 2 $[\frac{9(9+1)(18 +1)}{6}]$ + 100

⇒ 2 $(\frac{9; \times ;10 ;\times ;19}{6} )$ + 100 

⇒ 570 + 100 = 670

∴ The sum of the given series is 670.

Given:

The distance between two trains A & B is 530 km

The speed of train A is 90 km/h

Speed of train B is 90 km/h

Formula Used:

Distance = Time × Speed

Calculation:

Train A, 1 hours distance = 1 × 90 = 90 km 

So, train A started time is 10:00 A.M.

Remaining distance = 530 - 90 = 440 km

Time = 440/(90 + 90)

Time = 440/180 = 22/9 hour

Train meet after 22/9 hours after 10:00 A.M.

⇒ Time they meet = 10:00 A.M. + 22/9 hours

22/9 hours = $2\frac{4}{9}$hours = 2 hours, 26 minutes and 40 seconds 

since,

4/9 hours × 60 minutes = 80/3 min = 26 $\frac{2}{3}$ min

2/3 min = 2/3 × 60 = 120/3 = 40 sec

⇒ So, they meet at 12:26:40 P.M.

∴ Train meet after 12:26:40 P.M. hours after 10:00 A.M.

Given : 

Principal = Rs. 64000

Rate = 5% per annum

Time = 3/2

Calculation : 

Since the interest is paid half yearly.

So, Time = 3 years, Rate = 2.5%

2.5% = 1/40

According to the question:

P :  A 

⇒ (40)³ : (41)³

⇒ 64000 : 68921

⇒ 64000 unit = 64000

⇒ 1 unit = 1

Interest  = 68,921 - 64,000 = 4,921

∴ The correct answer is Rs. 4,921.