Search Previous Year Questions

Given:

Total surface area of sphere = 324π

Formula:

Total surface area of sphere = 4πr²

Volume of sphere = (4/3)πr³

Calculation:

According to the question

4πr² = 324π

⇒ r² = 324/4

⇒ r² = 81

⇒ r = 9

Volume of sphere = (4/3) πr³

⇒ (4/3) × π × 9 × 9 × 9

⇒ 972π cm³

∴ 2/3^rd volume of sphere = 972π × (2/3) = 648π cm³

Shortcut Trick

When the Cost Price (CP) of two articles is the same, the net profit or loss percentage is the arithmetic mean of the individual profit/loss percentages.

Net Profit/Loss % = (Percentage 1 + Percentage 2) ÷ 2

Net % = (−10% + 20%) ÷ 2 = 10% ÷ 2 = 5%

Since the result is positive, it is a profit.

∴ The correct answer is 5% profit.

Alternate Method

Given:

CP of Article 1 = ₹3050

CP of Article 2 = ₹3050

Loss on Article 1 = 10%

Profit on Article 2 = 20%

Formula Used:

Selling Price (SP) = CP × (100 ± Profit/Loss %) ÷ 100

Profit % = (Total SP − Total CP) ÷ Total CP × 100

Calculations:

⇒ Total CP = 3050 + 3050 = ₹6100

⇒ SP₁ = 3050 × (90 ÷ 100) = ₹2745

⇒ SP₂ = 3050 × (120 ÷ 100) = ₹3660

⇒ Total SP = 2745 + 3660 = ₹6405

⇒ Net Profit = 6405 − 6100 = ₹305

⇒ Net Profit % = (305 ÷ 6100) × 100

⇒ Net Profit % = 5%

∴ The correct answer is 5% profit.

Additional Information

Same Selling Price Case

If two articles are sold at the same Selling Price (SP), one at x% profit and another at x% loss, there is always a net loss of (x/10)² %.

Profit and Loss Basics

Profit % and Loss % are always calculated on the Cost Price (CP) unless stated otherwise in the question.

Successive Percentage Change

For two successive changes of a% and b%, the net effective change is given by [a + b + (a × b ÷ 100)]%.

Given:

Number of days taken to complete the work = 18 days

Absent men = 6

Total number of days taken to complete the work = 20 days

Concept Used:

\(M_1D_1=M_2D_2\)

Calculation:

Let x number of men can do the work in 18 days.

Number of men completed the work in 20 days = x – 6

Equating both the equations for the work,

⇒ (x) × 18 = 20 × (x – 6)

⇒ 18x = 20x – 120

⇒ 20x – 18x = 120

⇒ 2x = 120

⇒ x = 60

Therefore, the original number of men is 60.

Given:

Teena, Reena and Sheena start a business with investment of respectively ₹ 24000, ₹ 28000 and ₹ 20000

Teena invests for 8 months, Reena invest for 10 months and Sheena invests for one year.

Total Profit = 25810

Concept:

Ratio in which profit divides = Sum of money invested by individual × Duration for which money is invested.

Calculation:

Ratio in which profit divides = 24000 × 8 : 28000 × 10 : 20000 × 12

⇒ Ratio = 24 × 8 : 28 × 10 : 20 × 12

⇒ Ratio = 24 : 35 : 30

Share of Teena 

⇒ 25810 × 24/(24 + 35 + 30)

⇒ 25810 × 24/89

⇒ 290 × 24

⇒ Rs. 6960

∴ The share of teena is Rs. 6960.

Given

Three speeds = 5 km/h, 8 km/h and 10 km/h

Formula used

Time = Distance/Speed; 1 hour = 60 minutes

Calculation

let the distance travelled in each speed  is x km

Time taken in each speed by a person to travel same disatnce = x/5; x/8 and x/10

According to question

(x/5) + (x/8) + (x/10) = 2 + (33/60)

⇒ (8x + 5x + 4x)/40 = 2 + (11/20)

⇒ 17x/40 = (40 + 11)/20

⇒ 17x/40 = 51/20

⇒ x = (51 × 40)/(17 × 20)

⇒ x = 6

Total distance travelled in three different speed = 6 + 6 + 6

⇒ 18 km

∴ Total distance travelled by a person is 18 km

Shortcut Trick

Initial Milk : Water = 3 : 2

Final Milk : Water = 1 : 2 ⇒ 3 : 6 (Multiplying by 3 to equate Milk)

Milk is constant at 3 units, while Water increases from 2 to 6 units (+4 units).

Initial mixture = 3 + 2 = 5 units = 75 liters ⇒ 1 unit = 15 liters.

Water added = 4 units × 15 = 60 liters.

∴ The correct answer is 60 liters.

Alternate Method

Initial 75 L

M : W = 3 : 2 Milk = 45 L Water = 30 L

Water x Liters

→

Final 75 + x L

M : W = 1 : 2 Milk = 45 L Water = 30 + x L

Given:

Total volume of the initial mixture = 75 liters

Ratio of milk to water = 3 : 2

Calculation:

⇒ Quantity of milk = (3 / 5) × 75 = 45 liters

⇒ Quantity of water = (2 / 5) × 75 = 30 liters

Let x liters of water be added to the mixture.

Since only water is added, the absolute quantity of milk remains unchanged.

The new ratio of milk to water becomes 1 : 2.

⇒ Milk / Water = 1 / 2

⇒ 45 / (30 + x) = 1 / 2

⇒ 45 × 2 = 30 + x

⇒ 90 = 30 + x

⇒ x = 90 − 30

⇒ x = 60

∴ The correct answer is 60 liters.

Additional Information

Constant Component Concept

When a specific component (e.g., water) is added or removed from a mixture, the quantity of the other component (e.g., milk) remains unchanged. Equating the constant element's ratio parts is the fastest shortcut to solving these problems.

Replacement Method

When a part of a mixture is withdrawn, the remaining mixture retains the original ratio. If an equal amount of one component is then added, only the new ratio changes based on the added component.

Rule of Alligation

Used to find the ratio in which two mixtures of varying concentrations should be mixed: (Quantity of Cheaper) ÷ (Quantity of Dearer) = (Price of Dearer − Mean Price) ÷ (Mean Price − Price of Cheaper).

Given:

First number : Second number = 3 : 4

Concept used:

New Number = Old Number × (100 + Increase%)/100

Calculation:

Let the first and second number be 3a and 4a respectively

According to the question,

3a × 60/100 = 144

⇒ a = 80

⇒ Second number = 4a = 4 × 80 = 320

30% of the second number = 320 x 30/100 = 96

∴  30% of the second number is 96.

Given:

Naveen got 21% marks and failed by 25 marks. If he gets 46% marks, he fails by 5 marks.

Solution:

Let the maximum marks of the exam be x.

Naveen’s required passing marks = Marks he got + marks he failed by.

At 21%:

⇒ 21% of x + 25 = Passing marks

At 46%:

⇒ 46% of x + 5 = Passing marks

Since the passing marks are the same:

⇒ 21% of x + 25 = 46% of x + 5

Simplify the equation:

⇒ 0.21x + 25 = 0.46x + 5

⇒ 0.25x = 20

⇒ x = 80

∴ The maximum marks of the exam are 80.

Shortcut Trick

Ratio of Expenditure to Savings = 300 : 100 = 3 : 1

Total Income parts = 3 + 1 = 4

Using Weighted Average: (Income % × Total) = (Exp % × Exp parts) + (Sav % × Sav parts)

18 × 4 = (x × 3) + (64 × 1)

72 = 3x + 64 ⇒ 3x = 8 ⇒ x = 8/3 = 2 2/3 %

∴ The correct answer is Increases by 2 2/3 %.

Alternate Method

Given:

Expenditure = 200% more than Savings

Income Increase = 18%, Savings Increase = 64%

Formula:

Income = Expenditure + Savings

⇒ Let Savings = 100

⇒ Expenditure = 100 + 200 = 300

⇒ Total Income = 300 + 100 = 400

⇒ New Income = 400 + (18% of 400) = 400 + 72 = 472

⇒ New Savings = 100 + (64% of 100) = 164

⇒ New Expenditure = New Income − New Savings = 472 − 164 = 308

⇒ Increase in Expenditure = 308 − 300 = 8

⇒ Percentage Increase = (8 ÷ 300) × 100 = 8/3 = 2 2/3 %

∴ The correct answer is Increases by 2 2/3 %.

Additional Information

Income, Expenditure, and Savings

Income is always the sum of Expenditure and Savings. Percentage changes in components follow the rule of weighted averages based on their initial ratios.

Percentage Change Shortcut

Total Change % = (v₁ × p₁ + v₂ × p₂) ÷ (p₁ + p₂), where v is value change and p is the weight or part ratio.

Base Value Consideration

Always ensure the base (initial value) for percentage calculation remains consistent. For increases 'X% more than Y', the value is Y × (1 + X/100).

Given:

Sum of diagonals of the rhombus = 98 cm

The perimeter of the rhombus = 164 cm

Formula used:

Area of a rhombus = xy/2

x and y are diagonals of the rhombus

The perimeter of the rhombus = 4a

a = \(\sqrt{(x/2)^2 + (y/2)^2}\)

a = edge of the rhombus

Calculation:

According to the question

x + y = 98 cm    ----(i)

4a = 164

⇒ a = 41 cm

⇒ 41² = (x/2)² + (y/2)²

⇒ 1681 = \(\dfrac{x^2 + y^2}{4}\)

⇒ x² + y² = 6724    ----(ii)

We know that (x + y)² = x2 + y2 + 2xy

From (i) and (ii), we have 

⇒ 98² = 6724 + 2xy

⇒ xy = 1440

Area of the rhombus = 1440/2

∴ The area of the rhombus is 720 cm².