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Given:

Sunita scored 66% which is 50 marks more to secure pass marks. Gita score 38% and failed by 6 marks.

Calculation:

Let x be the total marks scored by Sunita.

According to the question,

Passing marks in the first case:

Passing marks = 66% of x - 50

Passing marks in the second case:

Passing marks = 38% of x + 6

Now, on equating the passing marks we get,

⇒ 66% of x - 50 = 38% of x + 6

⇒ 66% of x - 38% of x = 50 + 6

⇒ 28% of x = 56

⇒ x = 200

Again according to the question,

⇒ The marks scored by Vinay = 17.5% of 200 = 35

Therefore, '35' is the required answer.

Shortcut TrickLet full marks be 100%

⇒ 66% - 50 = 38% + 6

⇒ 28% = 56 marks

⇒ 1% = 2 marks

⇒ 17.5% = 35 marks

∴  The marks scored by Vinay = 35 marks

Given:

Diameter of spheres = 2 cm and 12 cm

External radius of hollow sphere = 9 cm

Formula Used:

Volume of sphere = (4/3) × π × r³

Volume of hollow sphere = (4/3) × π × (R₁³ – R₂³)

Thickness of hollow sphere = R₁ – R₂

Where,

r → Radius of sphere

R1 → External Radius of sphere

R2 → Internal Radius of sphere

Calculation:

Total volume of melted sphere = (4/3) × π × 1³ + (4/3) × π × 6³

⇒ (4/3) × π × (1 + 216)

⇒ (4/3) × π × (217)

Let the internal radius of hollow sphere be x

Volume of hollow sphere = (4/3) × π × (R13 – R23)

⇒ (4/3) × π × 217 = (4/3) × π × (9³ – x³)

⇒ 217 = 729 – x³

⇒ x³ = 729 – 217

⇒ x³ = 512

⇒ x = 8 cm

Thickness of hollow sphere = R1 – R2

⇒ 9 – 8

⇒ 1 cm

∴ The thickness of hollow sphere is 1 cm.

Shortcut Trick

For a 3-year period at 10%, the Effective Percentage Rate of Compound Interest is 33.1%.

Compound Interest = Principal × 33.1%

⇒ 12,000 × 0.331 = 3,972

∴ The correct answer is Rs. 3972.

Alternate Method

Given: Principal (P) = Rs. 12,000, Time (n) = 3 years, Rate (R) = 10% p.a.

Formula Used: Amount (A) = P(1 + R/100)ⁿ and Compound Interest (CI) = A − P

⇒ A = 12000 × (1 + 10/100)³

⇒ A = 12000 × (11/10) × (11/10) × (11/10)

⇒ A = 12 × 11 × 11 × 11

⇒ A = 12 × 1331 = 15,972

⇒ CI = 15,972 − 12,000

⇒ CI = 3,972

∴ The correct answer is Rs. 3972.

Additional Information

Effective Rate of Interest

For 2 years, effective rate = x + y + (xy/100). For 3 years at 10%, it is 33.1%.

Difference between CI and SI

For 3 years, Difference = P(R/100)² × (3 + R/100).

Compound Interest Formula

CI = P[(1 + R/100)ⁿ − 1] where P is principal, R is rate, and n is time.

Formula Used:

Average = Sum of observations / No. of observations

Calculation:

Let, the present age of Son = S

And, the present age of Father = F

We have, F = 3 × S     -----(1)

Three years ago, F - 3 = {(S - 3) × 4} - 3

⇒ F - 3 = 4 × S - 15     -----(2)

Put the value of eq(1) in eq(2)

⇒ 4 × S - 3 × S = 12

⇒ S = 12 & F = 36

Average = (12 + 36) / 2 = 24

Hence, average of Father's and Son's current age is 24 years.

Shortcut Trick

Current Ratio (diff 2) = 5 : 7 | 12 years ago Ratio (diff 1) = 1 : 2

Make differences equal: 12 years ago Ratio = 2 : 4 (multiplied by 2)

Unit change for Sudhir = 5 − 2 = 3 units → 12 years

1 unit = 4 years ∴ Sudhir current age = 5 × 4 = 20 years

Sudhir age after 5 years = 20 + 5 = 25 years

∴ The correct answer is 25 years.

Alternate Method

Given: Current age ratio of Sudhir and Ashish is 5 : 7. Twelve years ago, the ratio was 1 : 2.

Let the current ages of Sudhir and Ashish be 5x and 7x respectively.

According to the question, 12 years ago:

⇒ (5x − 12) / (7x − 12) = 1 / 2

⇒ 2 × (5x − 12) = 1 × (7x − 12)

⇒ 10x − 24 = 7x − 12

⇒ 10x − 7x = 24 − 12

⇒ 3x = 12

⇒ x = 12 ÷ 3 = 4

⇒ Current age of Sudhir = 5x = 5 × 4 = 20 years

⇒ Age of Sudhir after 5 years = 20 + 5 = 25 years

∴ The correct answer is 25 years.

Additional Information

Constant Age Difference

The difference between the ages of two individuals always remains the same, regardless of how many years pass.

Ratio Balancing Method

To solve age problems quickly, multiply ratio terms so that the numerical difference between the ratios of both people becomes identical.

Future and Past Ages

If current age is 'A', age 'n' years ago was (A − n) and age 'n' years later will be (A + n).

Given:

Speed of boy = 3 km/hr

Time to cover the distance = 3 h 40 minutes

Calculation:

Speed of walking = 3 km/hr

Time taken by walking = 3 hours 40 min = 3 + (40/60) = 3 + (2/3) = 11/3 hours

Distance covered by walking = speed × time = 3 × (11/3) = 11 km

Time taken by cycle = 11 hours

Speed of cycle = distance/time = 11/11 = 1 km/hr

∴ The correct answer is 1 km/hr.

Shortcut TrickConcept used:

If the distance is constant, then

Speed ∝ (1/time)

Calculation:

According to the question:

Total Time taken to cover the distance while walking = 3 hr 60 = 11/3 hrs

Now 

So, 

11 units = 3 km/hr

1 unit = 3 /11

11/3 units = 3/11 × 11/3 = 1 km/hr

∴ The correct answer is 1 km/hr.

Given:

The first pipe can fill the tank =12 hours

The second pipe can fill the pipe = 6 hours

The third pipe can fill the pipe = 4 hours

Formula used:

W = E × T     Where, W = Work, E = Efficiency, and T = Time

Calculation:

Let all three pipes together can fill the tank in X hours

The efficiency of the first pipe = \({1\over 12}\)

The efficiency of the second pipe = \({1\over 6}\)

The efficiency of the third pipe = \({1\over 4}\)

All three can complete the work in one hour = \({1\over 12}\ +\ {1\over 6}\ +\ {1\over 4}\ =\ {1\ +\ 2\ +\ 3\over 12}\ =\ {6\over 12}\ =\ {1\over 2}\)

They can complete the work all three together = 2 hours

∴ The required result will be 2 hours.

Shortcut TrickLet A, B ad C be the three pipes that fill the tank in 12, 6, and 4 hour respectively

LCM of 12, 6, and 4 = 12 units be the total quantity to be filled.

Combined Efficiency of A, B, and C = 6 units/hour

Total time taken by all three pipes to fill tank = 12/6 = 2 hours

Given:

The birth rate = 10%

The death rate = 20%

The current population = 40000

Calculation:

The population of the city after 2 years = 40000 × \(({110\over100})^2\) × \(({80\over100})^2\)

⇒ 40000 × (1.1)² × (0.8)²

⇒ 30976

∴ The population of the city after 2 years will be 30976.

Calculation:  

 \([6\frac{3}{4} + (10\frac{5}{6} - 8\frac{1}{3})] \div (7\frac{7}{8} \div \frac{27}{32})\)   

\(⇒ [\frac{27}{4} + (\frac{65}{6} - \frac{25}{3})] \div (\frac{63}{8} \div \frac{27}{32})\)

\(⇒ [\frac{27}{4} + (\frac{65 - 50}{6})] \div (\frac{63}{8} \div \frac{27}{32})\)

\(⇒ [\frac{27}{4} + \frac{15}{6}] \div (\frac{63}{8} \div \frac{27}{32})\)

\(⇒ [\frac{81 + 30}{12}] \div (\frac{63}{8} \times \frac{32}{27})\)

\(⇒ \frac{111}{12} \times \frac{3}{28}\)

\(⇒ \frac{111}{112}\)

\(∴ [6\frac{3}{4} + (10\frac{5}{6} - 8\frac{1}{3})] \div (7\frac{7}{8} \div \frac{27}{32}) =\frac{111}{112}\)

The correct option is 2 i.e. \(\frac{111}{112}\)

Shortcut Trick

Combine efficiencies based on LCM of 6, 12, and 24 which is 24.

Efficiency of A = 24 ÷ 6 = 4

Efficiency of B = 24 ÷ 12 = 2

Efficiency of C = 24 ÷ 24 = 1

Total efficiency = 4 + 2 + 1 = 7

Total time = Total Work ÷ Total Efficiency = 24 ÷ 7

∴ The correct answer is 24/7 days.

Alternate Method

Given: A = 6 days, B = 12 days, C = 24 days.

Formula Used: Total Work = LCM of individual days; Time = Total Work ÷ Efficiency.

⇒ Total efficiency of A + B + C = 4 + 2 + 1 = 7 units/day.

⇒ Total work to be completed = 24 units.

⇒ Time taken together = 24 ÷ 7 days.

∴ The correct answer is 24/7 days.

Additional Information

Total Work Concept

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

Reciprocal Method

1/T_total = 1/T₁ + 1/T₂ + 1/T₃. For this case: 1/6 + 1/12 + 1/24 = (4+2+1)/24 = 7/24.

Work and Wages

Wages are always distributed in the ratio of efficiencies if all persons work for the same duration of time.