Quantitative Aptitude PYQs

Shortcut Trick

Apply the successive multiplying factor for the discounts.

Final Price = Initial Price × (100 − d₁)% × (100 − d₂)%

Final Price = 1200 × 80% × 85%

Final Price = 1200 × 0.8 × 0.85 = 816

∴ The correct answer is 816.

Alternate Method

Given:

Initial price of the laptop = 1200

First discount percentage = 20%

Second discount percentage = 15%

Calculations:

⇒ Amount of first discount = 20% of 1200 = (20/100) × 1200 = 240

⇒ Price after first discount = 1200 − 240 = 960

⇒ Amount of second discount = 15% of 960 = (15/100) × 960 = 144

⇒ Final price of the laptop = 960 − 144 = 816

∴ The correct answer is 816.

Additional Information

Single Equivalent Discount Formula

For two successive discounts of a% and b%, the single effective discount is calculated as (a + b − (a × b)/100)%.

Relationship between MP and SP

Selling Price (SP) = Marked Price (MP) × (100 − Discount%)/100. Discounts are always applied to the Marked Price.

Successive Percentage Change

Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100), where r represents the percentage increase (+) or decrease (−).

Shortcut Trick

Total capacity (LCM of 20, 30, 60) = 60 units

Efficiency: A = 3, B = 2, C = −1 (outlet)

Work done in 10 minutes (A+B+C) = (3 + 2 − 1) × 10 = 40 units

Time taken by A+B to fill the remaining work (20 units) = 20 ÷ 5 = 4 minutes

Total time = 10 + 4 = 14 minutes

∴ The correct answer is 14 minutes.

Alternate Method

Given:

Time for pipe A = 20 minutes

Time for pipe B = 30 minutes

Time for outlet pipe C = 60 minutes

Formula used:

Total work = Efficiency × Time

Calculation:

⇒ Initially, all three pipes (A, B, and C) are opened together for 10 minutes.

⇒ Part filled in 10 minutes = (3 + 2 − 1) × 10 = 4 × 10 = 40 units

⇒ Remaining part of the tank = 60 − 40 = 20 units

⇒ Now Pipe C is closed, so only A and B will work.

⇒ Combined efficiency of A and B = 3 + 2 = 5 units/minute

⇒ Time taken to fill the remaining part = 20 ÷ 5 = 4 minutes

⇒ Total time taken to fill the tank = 10 minutes + 4 minutes = 14 minutes

∴ The correct answer is 14 minutes.

Additional Information

Efficiency of an outlet pipe (Negative Efficiency)

The efficiency of a pipe that empties a tank is always considered negative (−) because it reduces the total filled volume.

Combined Efficiency

When multiple pipes work together, their total efficiency is the algebraic sum of their individual efficiencies.

Relationship between time and efficiency

For the same amount of work, time and efficiency are inversely proportional to each other: Time ∝ 1/Efficiency.

Shortcut Trick

The age difference between two people always remains constant (10 years).

5 years ago, the ratio of Rahul to Brother was 2 : 1.

Difference in ratio = 2 − 1 = 1 unit → 10 years.

Rahul's age 5 years ago = 2 units = 2 × 10 = 20 years.

Rahul's present age = 20 + 5 = 25 years.

∴ The correct answer is 25 years.

Alternate Method

Given:

Age difference = 10 years

Condition 5 years ago: Rahul = 2 × Brother

Let Brother's present age be x years.

Then, Rahul's present age = x + 10 years.

⇒ 5 years ago, Brother's age = x − 5

⇒ 5 years ago, Rahul's age = (x + 10) − 5 = x + 5

According to the question:

⇒ x + 5 = 2 × (x − 5)

⇒ x + 5 = 2x − 10

⇒ 2x − x = 10 + 5

⇒ x = 15 years (Brother's present age)

⇒ Rahul's present age = 15 + 10 = 25 years.

∴ The correct answer is 25 years.

Additional Information

Age Difference Invariance

The numerical difference between the ages of two individuals remains the same at any point in time (past, present, or future).

Ratio Method for Ages

If the ratio of ages changes over time, calculate the value of 1 unit by equating the constant difference to the difference in ratio parts.

Time Transition

To find age 'n' years ago, subtract 'n' from the present age. To find age 'n' years hence, add 'n' to the present age.

Shortcut Trick

Ratio of investment for Amit and Raghav = 5 : 7

Total ratio units = 5 + 7 = 12 units

12 units → ₹48,000 (total profit)

1 unit → ₹4,000

Raghav's share = 7 units = 7 × ₹4,000 = ₹28,000

∴ The correct answer is ₹28,000.

Alternate Method

Given:

Investment ratio (Amit : Raghav) = 5 : 7

Total profit = ₹48,000

Formula Used:

Individual profit share = (Individual ratio part ÷ Total ratio parts) × Total profit

Calculations:

⇒ Total ratio parts = 5 + 7 = 12

⇒ Raghav's share = (7 ÷ 12) × 48,000

⇒ Raghav's share = 7 × 4,000

⇒ Raghav's share = ₹28,000

∴ The correct answer is ₹28,000.

Additional Information

Partnership Profit Rule

When the investment time is the same for all partners, the profit is distributed strictly in the ratio of their capital investments.

Compound Partnership

If partners invest for different time periods, the profit ratio is (C₁ × T₁) : (C₂ × T₂), where C is capital and T is time.

Ratio to Value Conversion

To find an individual share from a total value V and ratio a:b, the share of the first person is [a ÷ (a + b)] × V.

Shortcut Trick

The Mode is defined as the value that appears most frequently in a dataset.

Scan the data for repetitions: 53 appears 3 times (at positions 1, 4, and 7).

Other values like 55, 54, 47, and 43 appear only 2 times each.

Since 53 has the maximum frequency, it is the mode.

∴ The correct answer is 53.

Alternate Method

Given Data: 53, 46, 55, 53, 40, 48, 53, 47, 41, 54, 43, 55, 54, 51, 43, 42, 44, 47

To find the Mode, we construct a frequency table for the repeating values:

⇒ Frequency of 43: 2

⇒ Frequency of 47: 2

⇒ Frequency of 53: 3

⇒ Frequency of 54: 2

⇒ Frequency of 55: 2

⇒ Frequency of other values (40, 41, 42, 44, 46, 48, 51): 1 each

⇒ The value with the highest frequency (3) is 53.

∴ The correct answer is 53.

Additional Information

Mean

The arithmetic average calculated as the sum of all observations divided by the total number of observations (Sum ÷ n).

Median

The middle-most value when the data is arranged in ascending or descending order. For even n, it is the average of the two middle terms.

Empirical Relationship

For a moderately skewed distribution, the relationship between measures of central tendency is: Mode = 3 × Median − 2 × Mean.

Types of Mode

A dataset can be unimodal (one mode), bimodal (two modes), trimodal (three modes), or multimodal.

Given:

Equation: 2^x − 2^x−1 = 32

Formula used:

a^m − a^m−n = a^m−n(aⁿ − 1)

Calculations:

2^x − 2^x−1 = 32

⇒ 2^x−1(2¹ − 1) = 32

⇒ 2^x−1(2 − 1) = 32

⇒ 2^x−1(1) = 32

⇒ 2^x−1 = 2⁵

⇒ x − 1 = 5

⇒ x = 6

Now, finding the value of (2x − 1) / (2x + 3) by substituting x = 6:

⇒ (2 × 6 − 1) / (2 × 6 + 3)

⇒ (12 − 1) / (12 + 3)

⇒ 11 / 15

∴ The correct answer is option (1).

Shortcut Trick

Total Interest Percentage = Rate × Time = 8% × 3 = 24%.

Given, 24% of the Principal = ₹1,920.

Principal (100%) = (1,920 ÷ 24) × 100 = 80 × 100 = ₹8,000.

∴ The correct answer is ₹8,000.

Alternate Method

Given: Simple Interest (SI) = ₹1,920, Rate (R) = 8% p.a., Time (T) = 3 years.

Formula Used: SI = (P × R × T) ÷ 100

⇒ 1,920 = (P × 8 × 3) ÷ 100

⇒ 1,920 = (P × 24) ÷ 100

⇒ P × 24 = 1,920 × 100

⇒ P = (1,920 × 100) ÷ 24

⇒ P = 80 × 100

⇒ P = 8,000

∴ The correct answer is ₹8,000.

Additional Information

Simple Interest Principle

Simple Interest is always calculated on the original Principal amount for every time period, unlike compound interest where interest is added back.

Total Amount Formula

The total Amount (A) received at the end of the term is the sum of the Principal and the interest earned: A = P + SI.

Rate and Time Calculations

To find the Rate (R), use R = (SI × 100) ÷ (P × T). To find the Time (T), use T = (SI × 100) ÷ (P × R).

Shortcut Trick

First Number = 3 × 7 = 21

Second Number = (LCM × HCF) ÷ First Number

Second Number = (210 × 3) ÷ 21 = 10 × 3 = 30

∴ The correct answer is 30.

Alternate Method

Given:

L.C.M. = 210

H.C.F. = 3

For the first number: Divisor = 3, Quotient = 7, Remainder = 0

Formula Used:

1. Dividend = (Divisor × Quotient) + Remainder
2. Product of two numbers = L.C.M. × H.C.F.

Calculations:

⇒ First number = (3 × 7) + 0

⇒ First number = 21

⇒ Let the second number be x.

⇒ 21 × x = 210 × 3

⇒ 21 × x = 630

⇒ x = 630 ÷ 21

⇒ x = 30

∴ The correct answer is 30.

Additional Information

Fundamental Property of LCM & HCF

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. This property only applies to exactly two numbers.

HCF as a Factor of LCM

The HCF of a set of numbers is always a factor of the LCM of those same numbers.

HCF and LCM of Fractions

HCF of fractions = (HCF of Numerators) ÷ (LCM of Denominators) and LCM of fractions = (LCM of Numerators) ÷ (HCF of Denominators).

Shortcut Trick

Ratio of Speed = 40 : 30 = 4 : 3

Since Distance is constant, Time Ratio = Inverse of Speed Ratio = 3 : 4

Difference in time units = 4 − 3 = 1 unit → 20 minutes

Original time (3 units) = 3 × 20 = 60 minutes = 1 hour

Distance = Usual Speed × Usual Time = 40 km/hr × 1 hr = 40 km

∴ The correct answer is 40 km.

Alternate Method

Given:

Usual Speed (S₁) = 40 km/hr

New Speed (S₂) = 30 km/hr

Time Delay (Δt) = 20 minutes = 20/60 hour = 1/3 hour

Formula Used:

Distance (D) = [ (S₁ × S₂) ÷ (S₁ − S₂) ] × Δt

Calculations:

⇒ D = [ (40 × 30) ÷ (40 − 30) ] × (1/3)

⇒ D = [ 1200 ÷ 10 ] × (1/3)

⇒ D = 120 × (1/3)

⇒ D = 40 km

∴ The correct answer is 40 km.

Additional Information

Average Speed Formula

When a person covers two equal distances at speeds x and y, the Average Speed = (2 × x × y) ÷ (x + y).

Speed Conversion

To convert speed from km/hr to m/s, multiply by 5/18. To convert from m/s to km/hr, multiply by 18/5.

Relative Speed

If two objects move in the same direction at speeds u and v, relative speed = u − v. In opposite directions, relative speed = u + v.

Shortcut Trick

Number of smaller cylinders (n) = (Radius_large / Radius_small)² × (Height_large / Height_small)

n = (35 / 3.5)² × (50 / 15) = 10² × (10 / 3)

n = 100 × 3.333 ≈ 333.33

Since we need the number of complete cylinders, we take the integer part.

∴ The correct answer is 333 cylinders.

Alternate Method

Given:

Radius of large cylinder (R) = 35 cm

Height of large cylinder (H) = 50 cm

Radius of small cylinder (r) = 3.5 cm

Height of small cylinder (h) = 15 cm

Formula Used:

Volume of a cylinder = πr²h

Calculations:

⇒ Volume of large cylinder (V_L) = π × 35 × 35 × 50

⇒ Volume of one small cylinder (V_s) = π × 3.5 × 3.5 × 15

⇒ Let 'n' be the number of smaller cylinders formed.

⇒ n × V_s = V_L

⇒ n = (π × 35 × 35 × 50) ÷ (π × 3.5 × 3.5 × 15)

⇒ n = (35/3.5) × (35/3.5) × (50/15)

⇒ n = 10 × 10 × (10/3)

⇒ n = 1000 / 3 = 333.33...

⇒ Complete cylinders = 333

∴ The correct answer is 333 cylinders.

Additional Information

Volume of Cylinder

The volume of a solid cylinder is calculated as πr²h, where r is the radius of the circular base and h is the vertical height.

Conservation of Volume

When one solid object is melted and recast into other objects, the total volume remains constant: Initial Volume = Total Volume of New Objects.

Surface Area of Cylinder

Curved Surface Area = 2πrh and Total Surface Area = 2πr(r + h).

Complete Objects Calculation

When calculating the number of complete objects, always round down the decimal result to the nearest whole number (Floor value).

Shortcut Trick

Test the options for the larger number.

If the larger number is 20, then the smaller number is 20 − 8 = 12.

Condition: Twice the larger number = 2 × 20 = 40.

Condition: Three times the smaller number + 4 = (3 × 12) + 4 = 36 + 4 = 40.

Since both conditions match (40 = 40), the larger number is 20.

∴ The correct answer is 20.

Alternate Method

Given:

Difference between two numbers = 8

Let the larger number be x and the smaller number be y.

Formula Used:

Linear equations in two variables: x − y = d and ax + by = c.

Calculations:

⇒ x − y = 8

⇒ y = x − 8

⇒ 2x = 3y + 4

⇒ 2x = 3(x − 8) + 4

⇒ 2x = 3x − 24 + 4

⇒ 2x = 3x − 20

⇒ 3x − 2x = 20

⇒ x = 20

∴ The correct answer is 20.

Additional Information

Linear Equation Formulation

Word problems are solved by translating phrases like 'twice' as 2×, 'more than' as +, and 'difference' as subtraction.

Substitution Method

Expressing one variable in terms of another (e.g., y = x − 8) and substituting it into the second equation helps reduce variables.

Verification of Results

Always verify the calculated values in both original conditions to ensure accuracy in algebraic solutions.

Shortcut Trick

The standard Trigonometric Identity is cos²θ + sin²θ = 1.

Another fundamental identity is 1 + tan²θ = sec²θ, which implies 1 + tan²θ − sec²θ = 0.

By substituting this into the expression: (1 + tan²θ) + 1 − sec²θ ⇒ sec²θ + 1 − sec²θ = 1.

Since both values equal 1, the expression matches the given condition.

∴ The correct answer is 1 + tan²(θ) + (1− sec²(θ)).

Alternate Method

Given: Primary value = cos²θ + sin²θ

Formula Used:

1. Pythagorean Identity: sin²θ + cos²θ = 1
2. Secant Identity: sec²θ − tan²θ = 1 ⇒ tan²θ − sec²θ = −1

Calculations:

⇒ Target value = cos²θ + sin²θ = 1

⇒ Evaluating the expression: 1 + tan²θ + (1 − sec²θ)

⇒ Rearranging terms: (1 + tan²θ) + 1 − sec²θ

⇒ Substitute (1 + tan²θ) = sec²θ

⇒ Result = sec²θ + 1 − sec²θ

⇒ Result = 1

⇒ Since 1 = 1, the values are identical.

∴ The correct answer is 1 + tan²(θ) + (1− sec²(θ)).

Additional Information

Fundamental Pythagorean Identities

The three main identities are sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Reciprocal Relations

Key relations: sinθ = 1/cosecθ, cosθ = 1/secθ, and tanθ = 1/cotθ.

Ratio Identities

Tanθ is the ratio of sinθ to cosθ (tanθ = sinθ/cosθ), while cotθ is the inverse (cotθ = cosθ/sinθ).

Shortcut Trick

Ratio of Expenditure : Savings = 75 : 25 = 3 : 1 (Total = 4 units)

Apply the weighted average rule for percentage increases:

(Total Units × % Income Increase) = (Exp Units × % Exp Increase) + (Sav Units × % Sav Increase)

(4 × 20) = (3 × 10) + (1 × x)

80 = 30 + x ⇒ x = 50%

∴ The correct answer is 50% increase.

Alternate Method

Given: Initial spending = 75% of income. Income increases by 20%, Expenditure by 10%.

Let Initial Income = ₹100

⇒ Initial Expenditure = 75% of 100 = ₹75

⇒ Initial Savings = Income − Expenditure = 100 − 75 = ₹25

⇒ New Income = 100 + (20% of 100) = ₹120

⇒ New Expenditure = 75 + (10% of 75) = 75 + 7.5 = ₹82.5

⇒ New Savings = New Income − New Expenditure = 120 − 82.5 = ₹37.5

⇒ Change in Savings = 37.5 − 25 = ₹12.5

⇒ Percentage Change = (12.5 ÷ 25) × 100 = 50%

∴ The correct answer is 50% increase.

Additional Information

Income-Expenditure Formula

Income = Expenditure + Savings. Any increase in income must be balanced by the sum of increases in expenditure and savings.

Weighted Average Method

If components A and B form total T, then % change in T = (A/T × % change in A) + (B/T × % change in B).

Percentage Change

Percentage Increase/Decrease = [(New Value − Old Value) ÷ Old Value] × 100.

Shortcut Trick

Equate the common term Female students using LCM of 3 and 8, which is 24.

New Ratio Male : Female = 32 : 24 (multiplied by 8).

New Ratio Female : Staff = 24 : 3 (multiplied by 3).

Combined Ratio Male : Female : Staff = 32 : 24 : 3.

Total students : Staff = (32 + 24) : 3 = 56 : 3.

∴ The correct answer is 56 : 3.

Alternate Method

Given: Ratio of Male to Female students = 4 : 3

Ratio of Female students to Teaching staff = 8 : 1

⇒ Let Male students (M) = 4x and Female students (F) = 3x.

⇒ From the second ratio, F : T = 8 : 1, so Teaching staff (T) = F ÷ 8.

⇒ Total Students = M + F = 4x + 3x = 7x.

⇒ T = 3x ÷ 8 = 3x/8.

⇒ Ratio of Total Students to Teaching Staff = 7x : (3x/8).

⇒ Multiply both sides by 8 to clear the fraction: (7 × 8) : 3 = 56 : 3.

∴ The correct answer is 56 : 3.

Additional Information

Compound Ratio

When two or more ratios are multiplied together (a/b × c/d), the result (ac : bd) is called the compound ratio.

Balancing Ratios

To merge ratios A : B and B : C, find the LCM of the common term B and adjust both ratios accordingly to create a unified A : B : C sequence.

Invertendo Property

If a : b = c : d, then b : a = d : c. This is useful when the denominator or second term of the ratio needs to be compared or simplified.

Shortcut Trick

Ratio of students in three classes = 60 : 40 : 50 = 6 : 4 : 5.

Assume an average marks of 60.

Net deviation = (6 × (65 − 60)) + (4 × (72 − 60)) + (5 × (60 − 60)) = 30 + 48 + 0 = 78.

Average deviation = 78 ÷ (6 + 4 + 5) = 78 ÷ 15 = 5.2.

Combined average = Assumed Average + Average deviation = 60 + 5.2 = 65.2.

∴ The correct answer is 65.2.

Alternate Method

Given:

Class 1: 60 students, Average = 65

Class 2: 40 students, Average = 72

Class 3: 50 students, Average = 60

Formula Used:

Combined Average = (n₁a₁ + n₂a₂ + n₃a₃) ÷ (n₁ + n₂ + n₃)

Calculations:

⇒ Total marks for Class 1 = 60 × 65 = 3900

⇒ Total marks for Class 2 = 40 × 72 = 2880

⇒ Total marks for Class 3 = 50 × 60 = 3000

⇒ Sum of all marks = 3900 + 2880 + 3000 = 9780

⇒ Total number of students = 60 + 40 + 50 = 150

⇒ Combined Average = 9780 ÷ 150

⇒ Combined Average = 978 ÷ 15 = 65.2

∴ The correct answer is 65.2.

Additional Information

Weighted Average

It is an average where each quantity is given a specific weight according to its importance or size. Formula: Average = (Σ w_ix_i) ÷ Σ w_i.

Average Property

If each value in a data set is increased or decreased by a constant k, the new average also increases or decreases by k.

Net Deviation Rule

The sum of deviations of all observations from their arithmetic mean is always zero.

Shortcut Trick

A number divisible by 24 must be divisible by both 3 and 8.

For divisibility by 8, last three digits 44B must be divisible by 8 ⇒ B = 0 or 8.

For divisibility by 3, the sum of digits (26 + A + B) must be divisible by 3.

To maximize A + B, we take the largest possible value B = 8.

Now, 26 + A + 8 = 34 + A must be divisible by 3 ⇒ largest digit A = 8.

∴ The correct answer is 16.

Alternate Method

Given: The 7-digit number is 77A444B and it is divisible by 24.

Formula: If a number is divisible by 24, it must satisfy the divisibility rules for its co-prime factors 3 and 8.

⇒ Divisibility by 8: The last three digits 44B must be divisible by 8.

⇒ 440 + B ÷ 8 ⇒ Since 440 is divisible by 8 (55 × 8), B must be 0 or 8.

⇒ Divisibility by 3: The sum of digits 7 + 7 + A + 4 + 4 + 4 + B must be divisible by 3.

⇒ 26 + A + B must be a multiple of 3.

⇒ To find the maximum value of (A + B), let us test B = 8 (the maximum possible value for B).

⇒ Sum = 26 + A + 8 = 34 + A.

⇒ For 34 + A to be divisible by 3, possible values for A are 2, 5, 8.

⇒ To maximize the expression, we take A = 8.

⇒ Maximum (A + B) = 8 + 8 = 16.

∴ The correct answer is 16.

Additional Information

Divisibility Rule of 8

A number is divisible by 8 if its last three digits are divisible by 8 or if the last three digits are all zeros.

Divisibility Rule of 3

A number is divisible by 3 if the sum of all its individual digits is a multiple of 3.

Composite Divisibility

To check divisibility by a composite number like 24, check divisibility by its co-prime factors (3 and 8 in this case).

Shortcut Trick

Net Profit Ratio = (Weight actually received while buying / Standard weight) × (Standard weight / Weight actually given while selling)

Net Ratio = (1200 / 1000) × (1000 / 840) = 1200 : 840

Simplified Profit Ratio (SP : CP) = 10 : 7

Profit % = (Profit / CP) × 100 = (10 − 7) / 7 × 100 = 3/7 × 100

Profit % = 42.857...% ≈ 42.86%

∴ The correct answer is 42.86%.

Alternate Method

Given:

Weight obtained by dishonest shopkeeper while purchasing = 1200 g (instead of 1000 g)

Weight given by shopkeeper while selling = 840 g (instead of 1000 g)

Let the Cost Price (CP) of 1 g of rice be ₹1.

While purchasing, he pays for 1000 g but gets 1200 g.

⇒ Actual CP for 1200 g = ₹1000

⇒ CP of 1 g = 1000 / 1200 = ₹5/6

While selling, he charges for 1000 g (at CP) but gives only 840 g.

⇒ Selling Price (SP) of 840 g = ₹1000

⇒ SP of 1 g = 1000 / 840 = ₹25/21

Overall percentage profit:

⇒ Profit = SP − CP = (25/21) − (5/6)

⇒ Profit = (50 − 35) / 42 = 15/42 = 5/14

⇒ Profit % = (Profit / CP) × 100

⇒ Profit % = (5/14) / (5/6) × 100

⇒ Profit % = (5/14) × (6/5) × 100 = 6/14 × 100 = 3/7 × 100

⇒ Profit % = 42.857...%

∴ The correct answer is 42.86%.

Additional Information

Dishonest Dealer Formula

When a dealer sells at cost price but uses a false weight, Profit % = [Error / (True Value − Error)] × 100.

Combined Cheating (Buying and Selling)

If a dealer cheats by x% while buying and y% while selling, the net gain ratio is [(100+x)/100] × [100/(100−y)].

Successive Percentage Change

For successive profits of a% and b%, the effective profit is [a + b + (a × b)/100]%, though the ratio method is safer for weight errors.

Shortcut Trick

Volume of material in a pipe = π × h × (R + r) × (R − r)

Given: R = 8 cm, r = 7 cm, h = 20 cm, π = 3.14

Volume = 3.14 × 20 × (8 + 7) × (8 − 7)

Volume = 3.14 × 20 × 15 × 1 = 3.14 × 300 = 942

∴ The correct answer is 942 cubic cm.

Alternate Method

Given:

External radius (R) = 8 cm

Internal radius (r) = 7 cm

Length of the pipe (h) = 20 cm

Formula Used:

Volume of cylindrical pipe = πR²h − πr²h = πh(R² − r²)

Calculations:

⇒ Volume = 3.14 × 20 × (8² − 7²)

⇒ Volume = 3.14 × 20 × (64 − 49)

⇒ Volume = 3.14 × 20 × 15

⇒ Volume = 3.14 × 300

⇒ Volume = 942 cubic cm

∴ The correct answer is 942 cubic cm.

Additional Information

Total Surface Area (TSA) of Hollow Cylinder

TSA = External CSA + Internal CSA + 2 × Area of Base Rings = 2πRh + 2πrh + 2π(R² − r²).

Curved Surface Area (CSA)

For a solid cylinder, CSA = 2πrh. For a hollow pipe, the total curved area is the sum of the inner and outer curved surfaces.

Volume of a Solid Cylinder

The volume of a solid cylinder is calculated as V = πr²h, where r is the radius and h is the vertical height.

Shortcut Trick

The Effective Interest Rate for 4 years at 12% is: 12% × 4 = 48%.

Since 48% of the Principal equals the Simple Interest of ₹1,320.

Principal (100%) = (1,320 ÷ 48) × 100 = 27.5 × 100 = 2,750.

∴ The correct answer is ₹2,750.

Alternate Method

Given:

Simple Interest (SI) = ₹1,320

Rate of Interest (R) = 12% per annum

Time Period (T) = 4 years

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Calculations:

⇒ 1,320 = (P × 12 × 4) ÷ 100

⇒ 1,320 = (P × 48) ÷ 100

⇒ P × 48 = 1,320 × 100

⇒ P = 132,000 ÷ 48

⇒ P = 2,750

∴ The correct answer is ₹2,750.

Additional Information

Total Amount in Simple Interest

The total Amount (A) is the sum of the Principal and the Simple Interest: A = P + SI = P × [1 + (R × T) ÷ 100].

Constant Interest Property

In Simple Interest, the interest earned remains the same every year because it is always calculated on the original Principal only.

Rate and Time Calculation

To find the Rate (R): R = (SI × 100) ÷ (P × T). To find the Time (T): T = (SI × 100) ÷ (P × R).

Shortcut Trick

Total number of pens = 4 + 1 = 5

Total cost = (4 × 47) + (50% of 47)

Total cost = 188 + 23.5 = ₹211.5

Effective price = 211.5 ÷ 5 = ₹42.30

∴ The correct answer is ₹42.30.

Alternate Method

Given:

Price of one pen = ₹47

Scheme: Buy 4 pens and get a 50% discount on 1 extra pen.

Formula:

Effective price = Total payment / Total number of items

Calculation:

⇒ Cost of 4 pens = 4 × 47 = ₹188

⇒ Discount on the 5^th pen = 50% of 47 = 47 × (50/100) = ₹23.50

⇒ Discounted price of the 5^th pen = 47 − 23.50 = ₹23.50

⇒ Total amount paid for 5 pens = 188 + 23.50 = ₹211.50

⇒ Effective price per pen = 211.50 ÷ 5

⇒ Effective price per pen = ₹42.30

∴ The correct answer is ₹42.30.

Additional Information

Effective Discount %

To find the effective discount in such schemes: (Total Discount / Total Marked Price) × 100. In this question, Discount = ₹23.5 and Total Marked Price = 5 × 47 = ₹235. Effective Discount % = (23.5 / 235) × 100 = 10%.

Buy X Get Y Free

If the scheme were 'Buy 4 Get 1 Free', then the Effective Discount % would be [1 / (4 + 1)] × 100 = 20%.

Marked Price and Selling Price

Selling Price (SP) = Marked Price (MP) − Discount. Here, ₹42.30 is the effective selling price of each pen.