Quantitative Aptitude PYQs

Shortcut Trick

Effective Discount % = 15 + 10 − (15 × 10) ÷ 100 = 23.5%.

Net Selling Price = 100% − 23.5% = 76.5% of Marked Price.

76.5% of MP = ₹3,672 ⇒ MP = 3,672 ÷ 0.765 = ₹4,800.

∴ The correct answer is ₹4,800.

Alternate Method

Given:

First Successive Discount (d₁) = 15%

Second Successive Discount (d₂) = 10%

Final Selling Price (SP) = ₹3,672

Formula Used:

SP = MP × (100 − d₁)/100 × (100 − d₂)/100

⇒ 3,672 = MP × (100 − 15)/100 × (100 − 10)/100

⇒ 3,672 = MP × 85/100 × 90/100

⇒ 3,672 = MP × 17/20 × 9/10

⇒ 3,672 = MP × 153/200

⇒ MP = (3,672 × 200) ÷ 153

⇒ MP = 24 × 200

⇒ MP = 4,800

∴ The correct answer is ₹4,800.

Additional Information

Successive Discount Formula

For two successive discounts of a% and b%, the single equivalent discount is (a + b − ab/100)%.

Selling Price Calculation

SP = MP − Discount. If multiple discounts are applied, the base for the second discount is the price after the first discount.

Marked Price (List Price)

The price indicated on the product before any discount is applied. Discounts are always calculated on the Marked Price.

Shortcut Trick

Net Profit = Sum of individual profits and losses

Net Profit = 300 − 50 + 200 = ₹450

Overall profit percentage = (Net Profit ÷ Total CP) × 100

Profit % = (450 ÷ 3,951) × 100 ≈ 11.39%

∴ The correct answer is 11.39% profit.

Alternate Method

Given:

Total Cost Price (CP) of 3 bed sheets = ₹3,951

Profit on 1^st sheet = ₹300

Loss on 2^nd sheet = ₹50

Profit on 3^rd sheet = ₹200

Formula Used:

Overall Profit/Loss = Sum of individual Profit/Loss

Profit Percentage = (Overall Profit ÷ Total CP) × 100

Calculations:

⇒ Total Overall Profit = 300 − 50 + 200

⇒ Total Overall Profit = ₹450

⇒ Overall Profit Percentage = (450 ÷ 3,951) × 100

⇒ Profit % = 45000 ÷ 3,951

⇒ Profit % ≈ 11.3895...

⇒ Rounding off to two decimal places, we get 11.39%.

∴ The correct answer is 11.39% profit.

Additional Information

Profit and Loss Percentage

Profit % is always calculated on the Cost Price (CP) unless stated otherwise. Formula: Profit % = (Profit ÷ CP) × 100.

Successive Profit/Loss

When multiple items are sold, the net profit is the algebraic sum of individual profits (positive) and losses (negative).

Rounding Rule

To round to two decimal places, look at the third decimal digit; if it is 5 or more, increase the second decimal digit by 1.

Shortcut Trick

Efficiency of A = 3 and B = 2 (taking Total Work as 36 units).

Combined work of A + B for 2 days = (3 + 2) × 2 = 10 units.

Remaining work to be done = 36 − 10 = 26 units.

Time taken by B alone = 26 ÷ 2 = 13 days.

∴ The correct answer is 13 days.

Alternate Method

Given: Time taken by A = 12 days, Time taken by B = 18 days, Duration of joint work = 2 days.

Formula Used: Efficiency = Total Work ÷ Time taken; Work Done = Efficiency × Time.

⇒ Efficiency of A = 3 units/day

⇒ Efficiency of B = 2 units/day

⇒ Total work done by A and B in 2 days = (3 + 2) × 2 = 10 units

⇒ Remaining work = Total Work − Work Done = 36 − 10 = 26 units

⇒ Time taken by B to complete the remaining work = 26 ÷ 2 = 13 days

∴ The correct answer is 13 days.

Additional Information

Relationship between Efficiency and Time

Efficiency is inversely proportional to the time taken to complete a work. If the ratio of time taken by two people is x : y, the ratio of their efficiencies will be y : x.

Work and Wages

Wages are generally distributed in the ratio of the work done by each individual, which is the product of their efficiency and the time for which they worked.

Combined Work Formula

If A can do a work in x days and B in y days, together they can do the work in (x × y) ÷ (x + y) days.

Shortcut Trick

Net percentage Increase after 15% and 20% = 15 + 20 + (15 × 20 ÷ 100) = 38%.

Let the initial Price be 100, then the current Price = 138.

Required Price for 5% Overall Effective Increment = 105.

Percentage Decrease needed = (138 − 105) ÷ 138 × 100 ≈ 23.91%.

∴ The correct answer is 23.91%.

Alternate Method

Given: First Increase = 15%, Second Increase = 20%, Target Effective Increment = 5%.

⇒ Let the initial Price of the Item = 100.

⇒ Price after first 15% Increase = 100 + 15 = 115.

⇒ Price after second 20% Increase = 115 + (20 ÷ 100 × 115) = 115 + 23 = 138.

⇒ Required Final Price with 5% Overall Effective Increment = 100 + 5 = 105.

⇒ Absolute Decrease needed from the current price = 138 − 105 = 33.

⇒ Approximate Percentage Decrease = (33 ÷ 138) × 100.

⇒ Percentage Decrease = 23.913% ≈ 23.91%.

∴ The correct answer is 23.91%.

Additional Information

Successive Percentage Formula

For two successive percentage changes a% and b%, the net effective change is given by the formula: (a + b + ab ÷ 100)%.

Multiplier Method

The final value after multiple changes can be calculated using multipliers: Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100).

Base for Percentage Decrease

When calculating a percentage decrease to return to a target value, the denominator must always be the current (larger) value.

Shortcut Trick

Ratio of Men to Women = 60% : 40% = 3 : 2

Using Weighted Average shortcut:

Average = (Ratio_Men × Avg_Men + Ratio_Women × Avg_Women) ÷ Total Ratio

Average = (3 × 51 + 2 × 55) ÷ (3 + 2) = (153 + 110) ÷ 5 = 263 ÷ 5 = 52.6

∴ The correct answer is 52.6 years.

Alternate Method

Given:

Percentage of Men = 60%

Percentage of Women = 40%

Average age of Men = 51 years

Average age of Women = 55 years

Formula Used:

Combined Average = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂)

Calculations:

Assume total members in the health club = 100

⇒ Number of Men = 60% of 100 = 60

⇒ Number of Women = 40% of 100 = 40

⇒ Total age of Men = 60 × 51 = 3060 years

⇒ Total age of Women = 40 × 55 = 2200 years

⇒ Total age of all members = 3060 + 2200 = 5260 years

⇒ Average age of all members = 5260 ÷ 100

⇒ Average age = 52.6 years

∴ The correct answer is 52.6 years.

Additional Information

Weighted Average Formula

It is used when two groups with different averages are combined. Formula: A_avg = (w₁a₁ + w₂a₂) ÷ (w₁ + w₂), where w represents weight/number and a represents average.

Net Deviation Method

Assume a baseline average and calculate the net increase or decrease relative to that baseline, then distribute the net change over the total number of items.

Alligation Rule

(Quantity 1) ÷ (Quantity 2) = (Average 2 − Combined Average) ÷ (Combined Average − Average 1). This is useful for finding ratios or combined means.

Shortcut Trick

Assume Total Capacity of the cistern = LCM(15, 20, 30) = 60 units.

Efficiencies: Pipe A = 4, Pipe B = 3, Pipe C = −2 (outlet).

Net efficiency for first 5 hours = 4 + 3 − 2 = 5 units/hr. Work done = 5 × 5 = 25 units.

Remaining work = 60 − 25 = 35 units. Time taken by (A + B) = 35 ÷ 7 = 5 hours.

Total time = 5 hr (initial) + 5 hr (remaining) = 10 hours.

∴ The correct answer is 10 hr.

Alternate Method

Given: Filling time for Pipe A = 15 hr, Pipe B = 20 hr; Emptying time for Pipe C = 30 hr.

⇒ Work done by A, B, and C together in 5 hours = 5 × (4 + 3 − 2)

⇒ Work done = 5 × 5 = 25 units

⇒ Remaining Work to be filled = Total Work − Completed Work

⇒ Remaining Work = 60 − 25 = 35 units

⇒ Since Pipe C is closed, only A and B will fill the remaining cistern.

⇒ Combined efficiency of (A + B) = 4 + 3 = 7 units/hr

⇒ Time taken to fill remaining part = 35 ÷ 7 = 5 hours

⇒ Total Time = Initial 5 hours + Remaining 5 hours = 10 hours

∴ The correct answer is 10 hr.

Additional Information

Inlet and Outlet Pipes

Inlet pipes add water (positive efficiency), while outlet pipes remove water (negative efficiency). Net work is the algebraic sum of their efficiencies.

Time and Efficiency Relationship

Efficiency is inversely proportional to the time taken to complete the work. Efficiency = Total Work ÷ Time.

Pipes and Cisterns Concept

If an inlet pipe fills a tank in x hours and an outlet pipe empties it in y hours, the net work done in 1 hour when both are open is (1/x − 1/y).

Given:

Expression: 0.5² × 5⁵ × 2⁻⁴ × 4⁴ × 10⁻³ ÷ a = 1

Formula used:

1. x^m × xⁿ = x^m+n
2. (x^m)ⁿ = x^m×n
3. x⁻ⁿ = 1/xⁿ

Calculation:

0.5² × 5⁵ × 2⁻⁴ × 4⁴ × 10⁻³ ÷ a = 1

⇒ a = (1/2)² × 5⁵ × 2⁻⁴ × (2²)⁴ × (2 × 5)⁻³

⇒ a = 2⁻² × 5⁵ × 2⁻⁴ × 2⁸ × 2⁻³ × 5⁻³

⇒ a = 2^{(−2 − 4 + 8 − 3)} × 5^{(5 − 3)}

⇒ a = 2⁻¹ × 5²

⇒ a = (1/2) × 25

⇒ a = 12.5

The simplified value of the variable a is 12.5.

∴ The correct answer is option (4).

Shortcut Trick

Let the HCF of the three numbers be x.

Since the ratio is 11 : 3 : 2, and 11, 3, and 2 are co-prime numbers, their LCM will be 11 × 3 × 2 × x.

⇒ 66x = 2706

⇒ x = 2706 ÷ 66

⇒ x = 41

∴ The correct answer is 41.

Alternate Method

Given:

⇒ Ratio of three numbers = 11 : 3 : 2

⇒ LCM of the numbers = 2706

Formula Used:

⇒ Numbers = (Ratio Part) × HCF

⇒ LCM of (ax, bx, cx) = x × LCM(a, b, c), where x is the HCF.

Calculations:

Let the common multiplier (which is the HCF) be x.

⇒ First number = 11x

⇒ Second number = 3x

⇒ Third number = 2x

⇒ LCM (11x, 3x, 2x) = x × LCM (11, 3, 2)

⇒ Since 11, 3, and 2 have no common factors other than 1, their LCM is 11 × 3 × 2 = 66.

⇒ 66x = 2706

⇒ x = 2706 ÷ 66

⇒ x = 41

∴ The correct answer is 41.

Additional Information

HCF (Highest Common Factor)

The greatest number that divides two or more numbers without leaving a remainder. For numbers in a ratio, the HCF is the common multiplier.

LCM (Least Common Multiple)

The smallest positive integer that is divisible by each of the given numbers. LCM = HCF × product of remaining co-prime factors.

Property of Two Numbers

For any two numbers A and B: A × B = HCF(A, B) × LCM(A, B). This property does not directly apply to more than two numbers.

Shortcut Trick

To find the total cost of 9 items when the cost of 5 items is given, find the price per item and multiply by the required quantity.

Cost per dress = 4955 ÷ 5 = 991

Cost of 9 dresses = 991 × 9 = 8919

∴ The correct answer is ₹8,919.

Alternate Method

Given:

Price of 5 identical dresses = ₹4,955

Number of dresses Meena buys = 9

Formula Used:

Unitary Method: Value of 1 unit = Total Value ÷ Total Units

Total Required Value = Value of 1 unit × Required Units

Calculations:

⇒ Cost of 5 dresses = ₹4,955

⇒ Cost of 1 dress = 4955 ÷ 5

⇒ Cost of 1 dress = ₹991

⇒ Cost of 9 dresses = 991 × 9

⇒ Cost of 9 dresses = ₹8,919

∴ The correct answer is ₹8,919.

Additional Information

Unitary Method

A technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.

Direct Proportion

When two quantities increase or decrease at the same rate. In this case, more dresses lead to a higher cost: x₁/y₁ = x₂/y₂.

Inverse Proportion

When an increase in one quantity results in a decrease in the other. Product remains constant: x₁ × y₁ = x₂ × y₂.

Shortcut Trick

Relative Speed in opposite directions = S₁ + S₂ = 36 + 54 = 90 km/h.

Convert Relative Speed to m/s: 90 × (5/18) = 25 m/s.

Total distance to be covered = Sum of lengths of trains = 100 + 150 = 250 m.

Required Time = Total Distance ÷ Relative Speed = 250 ÷ 25 = 10 sec.

∴ The correct answer is 10 sec.

Alternate Method

Given:

Length of Train 1 (L₁) = 100 m

Length of Train 2 (L₂) = 150 m

Speed of Train 1 (S₁) = 36 km/h

Speed of Train 2 (S₂) = 54 km/h

Formula Used:

Time = Total Distance ÷ Relative Speed

Relative Speed (Opposite Direction) = S₁ + S₂

Calculations:

⇒ Total Distance = L₁ + L₂ = 100 + 150 = 250 m

⇒ Relative Speed = S₁ + S₂ = 36 + 54 = 90 km/h

⇒ Convert km/h to m/s: 90 × (5/18) m/s

⇒ Relative Speed = 5 × 5 = 25 m/s

⇒ Time taken = 250 ÷ 25

⇒ Time taken = 10 sec

∴ The correct answer is 10 sec.

Additional Information

Relative Speed Concepts

When two objects move in opposite directions, their relative speed is the sum of their individual speeds (S₁ + S₂). If they move in the same direction, the relative speed is the difference between their speeds (|S₁ − S₂|).

Unit Conversion

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

Crossing Different Objects

When a train crosses a stationary point/pole, distance = length of train. When it crosses a platform or bridge, distance = length of train + length of platform.

Shortcut Trick

Present Ratio (Father : Son) = 6 : 1 (Difference = 5 units)

Ratio after 5 years = 7 : 2 (Difference = 5 units)

Increase in units for both = 7 − 6 = 1 unit (or 2 − 1 = 1 unit).

This 1 unit increase corresponds to the 5-year time gap.

⇒ 1 unit = 5 years.

Son's present age = 1 unit = 5 years.

∴ The correct answer is 5 years.

Alternate Method

Given:

Present ratio of Father and Son = 6 : 1

Ratio after 5 years = 7 : 2

Calculations:

Let the present age of the Father be 6x and the Son be x.

⇒ Age of Father after 5 years = 6x + 5

⇒ Age of Son after 5 years = x + 5

According to the given ratio after 5 years:

⇒ (6x + 5) ÷ (x + 5) = 7 ÷ 2

⇒ 2 × (6x + 5) = 7 × (x + 5)

⇒ 12x + 10 = 7x + 35

⇒ 12x − 7x = 35 − 10

⇒ 5x = 25

⇒ x = 5

Present age of the Son = x = 5 years.

∴ The correct answer is 5 years.

Additional Information

Constant Age Difference

The difference between the ages of two individuals remains the same throughout their lives. In ratio problems, ensure the unit difference between the two terms is balanced across time periods.

Ratio Balancing Method

If the differences between ratio terms (e.g., 6−1=5 and 7−2=5) are already equal, the change in units directly represents the time interval. If not equal, multiply ratios by the other ratio's difference to balance them.

Linear Equation in Ages

For 'n' years ago, subtract n from the present age; for 'n' years hence, add n to the present age to form the required equation.

Given:

Expression: {1 / 0.01} × {[(5 / 8) + 0.25] × (4 / 5)}

Formula used:

BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Calculations:

{1 / 0.01} × {[(5 / 8) + 0.25] × (4 / 5)}

⇒ 100 × {[(5 / 8) + (2 / 8)] × (4 / 5)}

⇒ 100 × {7 / 8 × 4 / 5}

⇒ 100 × 28 / 40

⇒ 100 × 7 / 10

⇒ 10 × 7

⇒ 70

∴ The correct answer is option (2).

Shortcut Trick

The sum of cubes property states that if a + b + c = 0, then a³ + b³ + c³ = 3abc.

Here, the sum of the base values is 25 + (−20) + (−5) = 0.

Applying the trick: 3 × 25 × (−20) × (−5) = 75 × 100 = 7500.

∴ The correct answer is 7500.

Alternate Method

Given:

Expression: 25³ + (−20)³ + (−5)³

Let a = 25, b = −20, and c = −5.

Formula Used:

The Algebraic Identity used is: a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca).

Calculations:

⇒ First, calculate the sum of the variables: a + b + c = 25 + (−20) + (−5)

⇒ a + b + c = 25 − 20 − 5 = 0

⇒ Substitute (a + b + c) = 0 into the identity: a³ + b³ + c³ − 3abc = 0 × (a² + b² + c² − ab − bc − ca)

⇒ a³ + b³ + c³ = 3abc

⇒ Value = 3 × 25 × (−20) × (−5)

⇒ Value = 75 × 100

⇒ Value = 7500

∴ The correct answer is 7500.

Additional Information

Conditional Identity

When the sum of three variables is zero, their sum of cubes is always equal to three times their product (3abc).

Sum of Cubes Formula

The factorized form is a³ + b³ = (a + b)(a² − ab + b²), useful for binomial cubic expressions.

Trinomial Square Identity

The square of a sum is given by (a + b + c)² = a² + b² + c² + 2(ab + bc + ca).

Shortcut Trick

Scheme A: Net Discount = 45 + 44 − (45 × 44)/100 = 69.2%

Scheme B: Net Discount = 100 − (64% × 59% × 55%) ≈ 79.2%

Scheme C: Total Discount = [(4 × 49) + (6 × 19)] / 10 = 31%

Scheme D: Discount = 7 ÷ (10 + 7) × 100 ≈ 41.2%

Comparing all, Scheme C offers the minimum discount (31%).

∴ The correct answer is Scheme C.

Alternate Method

Assume the Marked Price (MP) of each toy is ₹100.

For a customer buying 10 toys, the total MP = 10 × 100 = ₹1000.

Formula Used: Effective Discount % = (Total Discount ÷ Total MP) × 100

⇒ Scheme A: Net discount = 45 + 44 − (45 × 44)/100 = 89 − 19.8 = 69.2%

⇒ Scheme B: Final SP factor = 0.64 × 0.59 × 0.55 = 0.20768 → Discount = (1 − 0.20768) × 100 = 79.232%

⇒ Scheme C: Discount on first 4 = 4 × 49 = ₹196; Discount on next 6 = 6 × 19 = ₹114. Total = ₹310. Discount % = (310/1000) × 100 = 31%

⇒ Scheme D: Discount % = [Free items ÷ Total items] × 100 = [7 ÷ 17] × 100 ≈ 41.18%

⇒ The least beneficial scheme for the customer is the one with the lowest discount percentage.

⇒ 31% < 41.18% < 69.2% < 79.23%

∴ The correct answer is Scheme C.

Additional Information

Successive Discounts Formula

For two successive discounts of x% and y%, the single equivalent discount is [x + y − (xy/100)]%.

Buy X Get Y Free

In such schemes, the effective discount percentage is given by [Y ÷ (X + Y)] × 100.

Customer Benefit Rule

A scheme is most beneficial to a customer when the total discount is maximum and least beneficial when the total discount is minimum.

Shortcut Trick

When multiple identical cuboids are stacked, the total Volume is simply the sum of their individual volumes.

Volume of one cuboid = 10.5 × 6.4 × 2 = 134.4 cm³

Total Volume = 3 × 134.4 = 403.2 cm³

∴ The correct answer is 403.2 cm³.

Alternate Method

Given:

Length of each cuboid (l) = 10.5 cm

Breadth of each cuboid (b) = 6.4 cm

Height of each cuboid (h) = 2 cm

Number of cuboids stacked = 3

Formula Used:

Volume of a cuboid = Length × Breadth × Height

When three cuboids are stacked one on top of the other:

⇒ New Length (L) = 10.5 cm

⇒ New Breadth (B) = 6.4 cm

⇒ New Height (H) = 2 + 2 + 2 = 6 cm

⇒ Volume of new cuboid = L × B × H

⇒ Volume = 10.5 × 6.4 × 6

⇒ Volume = 67.2 × 6

⇒ Volume = 403.2 cm³

∴ The correct answer is 403.2 cm³.

Additional Information

Total Surface Area of Cuboid

The total area of all six faces is given by 2 × (lb + bh + hl).

Diagonal of a Cuboid

The longest rod that can be placed inside a cuboid is the diagonal, calculated as √(l² + b² + h²).

Lateral Surface Area

The area of the four side walls is given by 2 × h × (l + b).

Shortcut Trick

In an average calculation, the sum of deviations from the average must be zero.

Deviations from 18 are: (6.5 − 18) = −11.5, (10.8 − 18) = −7.2, and (19.9 − 18) = +1.9.

Sum of known deviations = −11.5 − 7.2 + 1.9 = −16.8.

To make the net deviation zero, the deviation of a must be +16.8.

Value of a = 18 + 16.8 = 34.8.

∴ The correct answer is 34.8.

Alternate Method

Given:

Numbers: 6.5, 10.8, 19.9, and a

Total count (n) = 4

Average = 18

Formula Used:

Average = (Sum of all observations) ÷ (Total number of observations)

Calculations:

⇒ (6.5 + 10.8 + 19.9 + a) ÷ 4 = 18

⇒ 6.5 + 10.8 + 19.9 + a = 18 × 4

⇒ 37.2 + a = 72

⇒ a = 72 − 37.2

⇒ a = 34.8

∴ The correct answer is 34.8.

Additional Information

Arithmetic Mean

The Arithmetic Mean is the central value of a set of numbers, calculated by adding them together and dividing by the count.

Sum Property

The total sum of any dataset can be found using the formula: Sum = Average × Number of terms.

Deviation Rule

The sum of deviations of all observations from their arithmetic mean is always zero (Σ(x − Mean) = 0).

Shortcut Trick

Effective rate of Compound Interest for 2 years at 10% = x + y + (xy/100)

⇒ Effective Rate = 10 + 10 + (10 × 10)/100 = 21%

⇒ Compound Interest = 21% of ₹10,000

⇒ Interest = 0.21 × 10,000 = ₹2,100

∴ The correct answer is ₹2,100.

Alternate Method

Given:

Principal (P) = ₹10,000

Rate (R) = 10% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Compound Interest (CI) = A − P

Calculations:

⇒ A = 10,000 × (1 + 10/100)²

⇒ A = 10,000 × (11/10) × (11/10)

⇒ A = 10,000 × (121 / 100)

⇒ A = 100 × 121 = ₹12,100

⇒ Compound Interest (CI) = 12,100 − 10,000

⇒ CI = ₹2,100

∴ The correct answer is ₹2,100.

Additional Information

Simple Interest vs Compound Interest

Simple interest is calculated only on the Principal, while compound interest is calculated on the Principal plus the accumulated interest of previous periods.

Difference for 2 Years

The difference between CI and SI for 2 years is given by the formula: Difference = P × (R/100)².

Rule of 72

To find the approximate time required to double your money at a given interest rate, divide 72 by the annual rate of return.

Shortcut Trick

The Surface Area of a sphere is given by 4πr².

Since π contains the factor 22 (2 × 11), the result must be a multiple of 11.

For r = 7 cm, Surface Area = 4 × (22/7) × 7 × 7 = 88 × 7.

88 × 7 = 616 sq cm.

∴ The correct answer is 616 sq cm.

Alternate Method

Given:

Radius of the sphere (r) = 7 cm

Value of π = 22/7

Formula Used:

Surface Area of a Sphere = 4πr²

Calculations:

⇒ Surface Area = 4 × (22/7) × (7)²

⇒ Surface Area = 4 × (22/7) × 7 × 7

⇒ Surface Area = 4 × 22 × 7

⇒ Surface Area = 88 × 7

⇒ Surface Area = 616 sq cm

∴ The correct answer is 616 sq cm.

Additional Information

Volume of a Sphere

The total space occupied by a sphere is given by the formula V = (4/3)πr³.

Curved Surface Area of a Hemisphere

A hemisphere is half of a sphere; its curved surface area is half that of a sphere: CSA = 2πr².

Total Surface Area of a Hemisphere

For a solid hemisphere, the total surface area includes the curved part and the flat circular base: TSA = 3πr².

Shortcut Trick

Net difference in percentage = Gain% − (Loss%) = 5% − (−10%) = 15%

According to the question, 15% of Cost Price = ₹60

1% of Cost Price = 60 ÷ 15 = ₹4

100% (Cost Price) = 4 × 100 = ₹400

∴ The correct answer is ₹400.

Alternate Method

Given:

Initial Loss = 10%

New Gain = 5%

Difference in Selling Price = ₹60

Formula Used:

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

Let the Cost Price (CP) of the article be ₹x.

⇒ Initial Selling Price (SP₁) = x − 10% of x = 0.90x

⇒ New Selling Price (SP₂) = x + 5% of x = 1.05x

⇒ Difference: SP₂ − SP₁ = 60

⇒ 1.05x − 0.90x = 60

⇒ 0.15x = 60

⇒ x = 60 ÷ 0.15

⇒ x = 400

∴ The correct answer is ₹400.

Additional Information

Direct CP Formula

When there is a shift from loss to gain, Cost Price = [Difference in SP ÷ (Loss% + Gain%)] × 100.

Profit and Loss Percentage

Always calculate Profit% and Loss% on the Cost Price unless stated otherwise. Formula: (Profit/Loss ÷ CP) × 100.

Marked Price (MP)

The price at which a Discount is offered is the Marked Price. SP = MP × (100 − Discount%)/100.

Shortcut Trick

Simplified ratio X : Y : Z = 1/2 : 1/4 : 5/16 → Multiply by LCM (16) = 8 : 4 : 5.

Total parts = 8 + 4 + 5 = 17 units.

17 units = ₹9,520 ⇒ 1 unit = 9520 ÷ 17 = ₹560.

Share of Y = 4 units = 4 × 560 = ₹2,240.

This corresponds to Option 3.

∴ The correct answer is ₹2,240.

Alternate Method

Given:

Total sum = ₹9,520

Ratio of X : Y : Z = 1/2 : 1/4 : 5/16

Formula Used:

Share of an individual = (Individual ratio part ÷ Sum of ratio parts) × Total Amount

Calculations:

⇒ To simplify the Ratio, find the LCM of denominators 2, 4, and 16, which is 16.

⇒ Ratio X : Y : Z = (1/2 × 16) : (1/4 × 16) : (5/16 × 16)

⇒ Ratio X : Y : Z = 8 : 4 : 5

⇒ Sum of ratios = 8 + 4 + 5 = 17 units

⇒ 17 units = ₹9,520

⇒ 1 unit = 9520 ÷ 17 = ₹560

⇒ Share of Y = 4 units = 4 × 560 = ₹2,240

⇒ Therefore, Option 3 is the correct result.

∴ The correct answer is ₹2,240.

Additional Information

Ratio Simplification

To convert a fractional ratio into a whole number ratio, multiply each term by the Least Common Multiple (LCM) of all the denominators.

Unitary Method in Ratios

Divide the total quantity by the sum of the ratio parts to find the value of a single unit (1 part = Total ÷ Sum of Ratios).

Direct Proportion

The share of any individual is directly proportional to their respective ratio part when a fixed sum is divided among multiple entities.