Quantitative Aptitude PYQs

Shortcut Trick

Directly calculate the perfect squares: √196 = 14 and √81 = 9.

Apply the square of a square root property: (√16)² = 16.

Perform the operations: 14 − 9 + 16 = 5 + 16 = 21.

∴ The correct answer is 21.

Alternate Method

Given: √196 − √81 + (√16)²

Evaluate the expression using the BODMAS rule and properties of square roots:

⇒ 14 − 9 + (4)²

⇒ 14 − 9 + 16

⇒ 5 + 16

⇒ 21

∴ The correct answer is 21.

Additional Information

Square Root Properties

The square root of a number is a value that gives the original number when squared. Specifically, (√x)² = x for any non-negative number x.

BODMAS / PEMDAS

This rule dictates the order of operations: Brackets, Orders (Powers and Roots), Division, Multiplication, Addition, and Subtraction.

Common Perfect Squares

Memorizing perfect squares up to 25 (e.g., 13² = 169, 14² = 196, 15² = 225) is essential for fast simplification calculation.

Shortcut Trick

The effective interest rate difference is calculated by multiplying the rate difference with the time period.

Difference in Rate per year = 7% − 5% = 2%

Total difference for 2 years = 2% × 2 = 4%

According to the question, 4% of Principal = ₹40

⇒ 1% of Principal = 40 ÷ 4 = ₹10

⇒ 100% (Principal) = 10 × 100 = ₹1,000

∴ The correct answer is ₹1,000.

Alternate Method

Given:

Time period (T) = 2 years

Rate 1 (R₁) = 5% per annum

Rate 2 (R₂) = 7% per annum

Difference in Simple Interest = ₹40

Formula Used:

Simple Interest (SI) = (Principal × Rate × Time) ÷ 100

Calculations:

Let the Principal be ₹P.

⇒ SI₁ = (P × 5 × 2) ÷ 100 = 10P/100

⇒ SI₂ = (P × 7 × 2) ÷ 100 = 14P/100

⇒ SI₂ − SI₁ = 40

⇒ (14P − 10P) ÷ 100 = 40

⇒ 4P ÷ 100 = 40

⇒ 4P = 40 × 100

⇒ P = 4000 ÷ 4

⇒ P = 1,000

Thus, the Principal of each sum is ₹1,000, which corresponds to Option (2).

∴ The correct answer is ₹1,000.

Additional Information

Simple Interest (SI) Properties

Simple interest is calculated only on the Principal amount and remains constant for every year if the rate and principal are unchanged.

Principal Formula

When the difference in interest (D) is given for two different rates (R₁ and R₂) over the same time (T), then P = (D × 100) ÷ [T × (R₂ − R₁)].

Total Amount

The total Amount after interest is calculated as A = Principal + Simple Interest = P × [1 + (R × T ÷ 100)].

Shortcut Trick

Assume Total Work = 24 units (LCM of 12 and 24).

Efficiency of A = 2 units/day and B = 1 unit/day.

Combined work in 4 days = 4 × (2 + 1) = 12 units.

Remaining work = 24 − 12 = 12 units.

Time taken by B = 12 ÷ 1 = 12 days.

∴ The correct answer is 12 days.

Alternate Method

Given:

Time taken by A = 12 days

Time taken by B = 24 days

Formula Used:

Total Work = Time × Efficiency

Calculations:

⇒ Total Work = LCM of 12 and 24 = 24 units

⇒ Efficiency of A = 24 ÷ 12 = 2 units/day

⇒ Efficiency of B = 24 ÷ 24 = 1 unit/day

⇒ Combined efficiency of A and B = 2 + 1 = 3 units/day

⇒ Work completed by both in 4 days = 4 × 3 = 12 units

⇒ Remaining work = 24 − 12 = 12 units

⇒ Days taken by B to finish remaining work = 12 ÷ 1 = 12 days

∴ The correct answer is 12 days.

Additional Information

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken. If the ratio of time taken by two people is x:y, then the ratio of their efficiencies is y:x.

Work and Wages

Wages are distributed based on the amount of work done. If people work for the same duration, wages are divided in the ratio of their efficiencies.

Pipes and Cisterns Analogy

This concept is identical to Pipes and Cisterns, where filling a tank is positive work and a leak represents negative work (negative efficiency).

Shortcut Trick

When two equal halves of a distance are covered at speeds x and y, the Average Speed = (2 × x × y) ÷ (x + y).

Average Speed = (2 × 30 × 60) ÷ (30 + 60) = 3600 ÷ 90 = 40 km/h.

Total Time = Total Distance ÷ Average Speed = 120 ÷ 40 = 3 hr.

∴ The correct answer is 3 hr.

Alternate Method

Given:

Total Distance = 120 km

First half distance (D₁) = 120 ÷ 2 = 60 km

Second half distance (D₂) = 120 ÷ 2 = 60 km

Speed during first half (S₁) = 30 km/h

Speed during second half (S₂) = 60 km/h

Formula used: Time = Distance ÷ Speed

Calculations:

⇒ Time taken for the first half (T₁) = 60 ÷ 30 = 2 hr

⇒ Time taken for the second half (T₂) = 60 ÷ 60 = 1 hr

⇒ Total Time taken = T₁ + T₂ = 2 + 1 = 3 hr

Comparing the result with the options, it corresponds to Option 3.

∴ The correct answer is 3 hr.

Additional Information

Average Speed Formula

Average Speed = Total Distance ÷ Total Time. It is not the simple arithmetic mean of speeds unless the time intervals are equal.

Equal Distance Case

If a journey is divided into n equal distance parts with speeds v₁, v₂... v_n, the average speed is the Harmonic Mean of the speeds.

Speed-Time-Distance Relationship

Distance = Speed × Time. If distance is constant, speed is inversely proportional to time (S ∝ 1/T).

Shortcut Trick

A number is divisible by 68 if and only if it is divisible by both 4 and 17 (since 68 = 4 × 17).

Check divisibility by 4: The last two digits must be divisible by 4.

For ( i ) 17350 and ( iv ) 35350, the last two digits are 50. Since 50 is not divisible by 4 (50 ÷ 4 = 12.5), these numbers cannot be divisible by 68.

Numbers ( ii ) 23800 and ( iii ) 28560 end in 00 and 60, both of which are divisible by 4; checking with 17 confirms their divisibility by 68.

∴ The correct answer is ( i ) and ( iv ).

Alternate Method

Given: Numbers (i) 17350, (ii) 23800, (iii) 28560, (iv) 35350. Divisor = 68.

⇒ Divisibility of ( i ) 17350: 17350 ÷ 68 = 255.147 (Remainder ≠ 0)

⇒ Divisibility of ( ii ) 23800: 23800 ÷ 68 = 350 (Remainder = 0)

⇒ Divisibility of ( iii ) 28560: 28560 ÷ 68 = 420 (Remainder = 0)

⇒ Divisibility of ( iv ) 35350: 35350 ÷ 68 = 519.852 (Remainder ≠ 0)

∴ The correct answer is ( i ) and ( iv ).

Additional Information

Divisibility Rule for 4

A number is divisible by 4 if its last two digits are 00 or form a number that is exactly divisible by 4.

Divisibility by Composite Numbers

To check divisibility by a composite number L, the number must be divisible by all coprime factors of L. For 68, the coprime factors are 4 and 17.

Divisibility Rule for 17

Multiply the last digit of the number by 5 and subtract it from the rest of the number. If the result is divisible by 17, the whole number is divisible by 17.

Shortcut Trick

Compare the effective discount percentage of all four schemes for 4 toys:

(A) 6 + 39 − (6 × 39 / 100) = 45 − 2.34 = 42.66%

(B) Successive 48%, 16%, 11% ≈ 61.12%

(C) Average of (37 + 37 + 29 + 29) / 4 = 132 / 4 = 33%

(D) 2 free out of (4 + 2) = 2 / 6 × 100 = 33.33%

The lowest discount (least beneficial) is 33% from Scheme C.

∴ The correct answer is C.

Alternate Method

Assume the Marked Price (MP) of each toy is ₹100. A customer buys 4 toys, so total Marked Price = ₹400.

Calculation for Scheme A:

⇒ Net Discount = x + y − (xy / 100)

⇒ 6 + 39 − (6 × 39 / 100) = 45 − 2.34 = 42.66%

Calculation for Scheme B:

⇒ Successive discount of 48% and 16% = 48 + 16 − (48 × 16 / 100) = 64 − 7.68 = 56.32%

⇒ Now combine 56.32% and 11% = 56.32 + 11 − (56.32 × 11 / 100) = 67.32 − 6.1952 = 61.1248%

Calculation for Scheme C:

⇒ Total MP for 4 toys = 4 × 100 = ₹400

⇒ Discount on first 2 toys = 2 × 37% of 100 = ₹74

⇒ Discount on next 2 toys = 2 × 29% of 100 = ₹58

⇒ Total Discount = 74 + 58 = ₹132

⇒ Discount % = (132 / 400) × 100 = 33%

Calculation for Scheme D:

⇒ Customer gets 2 toys free on buying 4 toys (Total = 6 toys)

⇒ Discount % = (Free Items / Total Items) × 100 = (2 / 6) × 100 = 33.33%

⇒ Comparing all: 42.66%, 61.12%, 33%, and 33.33%.

∴ The correct answer is C.

Additional Information

Successive Discount Formula

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

Buy X Get Y Free

In a 'Buy X Get Y Free' scheme, the effective discount percentage is given by [Y / (X + Y)] × 100.

Marked Price and Selling Price

Discount is always calculated on the Marked Price (MP). SP = MP × (100 − Discount%)/100.

Shortcut Trick

Ratio of employees in Branch A : B : C = 40 : 30 : 30 = 4 : 3 : 3.

Weighted Average = [(1,20,000 × 4) + (1,80,000 × 3) + (1,50,000 × 3)] ÷ (4 + 3 + 3)

Average = (4,80,000 + 5,40,000 + 4,50,000) ÷ 10

Average = 14,70,000 ÷ 10 = 1,47,000.

∴ The correct answer is ₹1,47,000.

Alternate Method

Given:

Branch A: 40 employees, Avg Sales = ₹1,20,000

Branch B: 30 employees, Avg Sales = ₹1,80,000

Branch C: 30 employees, Avg Sales = ₹1,50,000

Formula Used: Average = Total Sales ÷ Total Employees

⇒ Total Employees = 40 + 30 + 30 = 100

⇒ Total Sales = (40 × 1,20,000) + (30 × 1,80,000) + (30 × 1,50,000)

⇒ Total Sales = 48,00,000 + 54,00,000 + 45,00,000

⇒ Total Sales = ₹1,47,00,000

⇒ Average per employee = 1,47,00,000 ÷ 100 = ₹1,47,000

∴ The correct answer is ₹1,47,000.

Additional Information

Weighted Average Formula

It is used when data sets have different levels of importance or quantity. Formula: X_avg = (w₁x₁ + w₂x₂ + w₃x₃) ÷ (w₁ + w₂ + w₃).

Combined Mean

The combined mean of two or more groups is the sum of all observations of all groups divided by the total number of observations in all groups.

Average Property

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

Shortcut Trick

Profit Ratio P : Q : R = (20 × 8) : (15 × 9) : (12 × t) = 160 : 135 : 12t

Given: Q < R < P ⇒ 135 < 12t < 160

For complete months within a year (t ≤ 12), only t = 12 satisfies the condition.

R profit (144 units) = ₹72,000 ⇒ 1 unit = 72,000 ÷ 144 = ₹500

P profit (160 units) = 160 × 500 = ₹80,000

∴ The correct answer is ₹80,000.

Alternate Method

Given:

Investment of P = ₹2,00,000 for 8 months

Investment of Q = ₹1,50,000 for 9 months

Investment of R = ₹1,20,000 for t months

Profit of R = ₹72,000

Formula: Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂) : (Capital₃ × Time₃)

⇒ P's Profit Units = 2,00,000 × 8 = 16,00,000

⇒ Q's Profit Units = 1,50,000 × 9 = 13,50,000

⇒ R's Profit Units = 1,20,000 × t

⇒ Since Profit Q < Profit R < Profit P

⇒ 13,50,000 < 1,20,000 × t < 16,00,000

⇒ 11.25 < t < 13.33

⇒ For complete months in a business year, t = 12 months.

⇒ R's actual units = 1,20,000 × 12 = 14,40,000

⇒ 14,40,000 units corresponds to ₹72,000 profit.

⇒ 1 unit value = 72,000 ÷ 14,40,000 = 1 ÷ 20

⇒ Profit of P = 16,00,000 × (1 ÷ 20) = ₹80,000

∴ The correct answer is ₹80,000.

Additional Information

Partnership Profit Sharing

Profit is distributed in the ratio of Effective Capital, where Effective Capital = Investment × Time period.

Compound Partnership

When partners invest their capitals for different time periods, it is known as a compound partnership.

Working vs. Sleeping Partner

A working partner manages the business and may receive an additional salary, while a sleeping partner only invests capital.

Shortcut Trick

Ratio of goods: 50 : (30% of 50) : (remaining) ⇒ 50 : 15 : 35 = 10 : 3 : 7.

Apply Weighted Average Method for overall profit percentage x.

x = [(10 × 21) + (3 × 43) + (7 × −17)] ÷ (10 + 3 + 7)

x = [210 + 129 − 119] ÷ 20 = 220 ÷ 20 = 11.

∴ The correct answer is 11.

Alternate Method

Given:

Let Total Goods = 100 units at ₹1/unit. Total CP = ₹100.

Part 1: 50% of the goods = 50 units sold at 21% profit.

Part 2: 30% of the remaining goods = 30% of 50 = 15 units sold at 43% profit.

Part 3: Still remaining goods = 100 − (50 + 15) = 35 units sold at 17% loss.

Formula Used:

Overall Profit/Loss % (x) = [Total Profit or Loss ÷ Total CP] × 100

Calculations:

⇒ Profit from Part 1 = 50 × (21 ÷ 100) = ₹10.5

⇒ Profit from Part 2 = 15 × (43 ÷ 100) = ₹6.45

⇒ Loss from Part 3 = 35 × (17 ÷ 100) = ₹5.95

⇒ Total Profit = 10.5 + 6.45 − 5.95

⇒ Total Profit = 16.95 − 5.95 = ₹11

⇒ Overall profit % (x) = (11 ÷ 100) × 100 = 11%

⇒ x = 11

∴ The correct answer is 11.

Additional Information

Weighted Average Formula

When a whole is divided into parts n₁, n₂... with individual changes p₁, p₂..., the net change is (Σ n_i × p_i) ÷ Σ n_i.

Profit and Loss Percentage

Profit % = (Profit ÷ CP) × 100 and Loss % = (Loss ÷ CP) × 100. Always calculate on Cost Price unless stated otherwise.

Successive Percentage Change

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

Shortcut Trick

A regular hexagon is composed of 6 equilateral triangles.

Area of 1 equilateral triangle = (√3 / 4) × (side)²

Total Area = 6 × (1.732 / 4) × 10²

Total Area = 1.5 × 1.732 × 100 = 259.8 m².

∴ The correct answer is 259.8 m².

Alternate Method

Given:

Side of the regular hexagon (s) = 10 m

Value of √3 ≈ 1.732

Formula Used:

Area of a regular hexagon = [3√3 / 2] × s²

Calculations:

⇒ Area = [3 × 1.732 / 2] × (10)²

⇒ Area = [5.196 / 2] × 100

⇒ Area = 2.598 × 100

⇒ Area = 259.8 m²

∴ The correct answer is 259.8 m².

Additional Information

Perimeter of a Regular Hexagon

The perimeter is the total length of the boundary, calculated as P = 6 × side.

Interior Angles

Each interior angle of a regular hexagon is 120°, and the sum of all interior angles is 720°.

Longest Diagonal

The longest diagonal connects two opposite vertices and its length is equal to twice the side (D = 2 × s).

Shortcut Trick

Total interest percentage for the entire duration = Rate × Time

Total Interest % = 12% × 5 = 60%

Interest earned = 60% of the Principal (₹3200)

Interest = 0.60 × 3200 = 1920

∴ The correct answer is ₹1920.

Alternate Method

Given:

Principal (P) = ₹3200

Rate of Interest (R) = 12% per annum

Time (T) = 5 years

Formula Used:

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

Calculations:

⇒ SI = (3200 × 12 × 5) ÷ 100

⇒ SI = 32 × 12 × 5

⇒ SI = 32 × 60

⇒ SI = 1920

∴ The correct answer is ₹1920.

Additional Information

Total Amount in Simple Interest

The total amount (A) payable after the time period is the sum of the Principal and the Simple Interest: A = P + SI.

Compound Interest (CI)

If the interest was compounded annually, the amount would be calculated as A = P(1 + R/100)ⁿ, where n is the number of years.

Rate and Time Consistency

Always ensure the Rate (R) and Time (T) are in the same unit (e.g., if R is per annum, T must be in years).

Shortcut Trick

The surface area of a sphere is proportional to the square of its radius (r²).

For an increase in radius by x%, the net increase in surface area is calculated using the successive percentage formula: x + x + (x × x)/100.

Given x = 20%, the calculation is 20 + 20 + (20 × 20)/100.

Net increase = 40 + 4 = 44%.

∴ The correct answer is 44%.

Alternate Method

Given:

Initial radius increase = 20%

Formula Used:

Surface Area of a spherical balloon (S) = 4πr²

Calculations:

⇒ Let the initial radius (r₁) be 5 units.

⇒ New radius (r₂) = 5 + (20% of 5) = 5 + 1 = 6 units.

⇒ Initial Surface Area (S₁) = 4π(5)² = 100π (or simply consider ratio r² = 25).

⇒ New Surface Area (S₂) = 4π(6)² = 144π (or ratio r² = 36).

⇒ Increase in Surface Area = S₂ − S₁ = 144π − 100π = 44π.

⇒ Percentage Increase = (Increase / Original) × 100

⇒ Percentage Increase = (44π / 100π) × 100 = 44%.

∴ The correct answer is 44%.

Additional Information

Volume Change of a Sphere

The volume of a sphere is 4/3πr³. If the radius increases by 20%, the volume increases by (1.2)³ − 1 = 1.728 − 1 = 72.8%.

Successive Percentage Rule

For any product based relation Z = A × B, if A changes by a% and B by b%, the net change in Z is [a + b + (ab/100)]%.

Hemisphere Properties

For a hemisphere, Curved Surface Area = 2πr² and Total Surface Area = 3πr². The percentage change logic for radius remains the same as a sphere.

Shortcut Trick

Identify the bases: a = 22, b = −6, and c = −16.

Check the sum of bases: 22 + (−6) + (−16) = 22 − 22 = 0.

Apply the Algebraic Identity: If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Calculate the result: 3 × 22 × (−6) × (−16) = 66 × 96 = 6336.

∴ The correct answer is 6336.

Alternate Method

Given: 22³ + (−6)³ + (−16)³

Formula Used: a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

⇒ Let a = 22, b = −6, and c = −16.

⇒ First, evaluate the sum (a + b + c) = 22 + (−6) + (−16) = 0.

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

⇒ This simplifies to a³ + b³ + c³ = 3abc.

⇒ Required Value = 3 × 22 × (−6) × (−16).

⇒ Required Value = 66 × 96 = 6336.

∴ The correct answer is 6336.

Additional Information

Special Case of Cubic Identity

When the sum of three terms is zero (a + b + c = 0), the sum of their cubes is always equal to three times their product (3abc). This is frequently used in competitive exams to avoid long cubic calculations.

Sum of Squares Transformation

The term (a² + b² + c² − ab − bc − ca) can also be written as 1/2[(a − b)² + (b − c)² + (c − a)²], which is always non-negative for real numbers.

Common Cubic Values

Memorizing cubes up to 25 is helpful: 22³ = 10648, 16³ = 4096, and 6³ = 216.

Shortcut Trick

To find the greatest number leaving the same remainder, calculate the HCF of the differences.

Difference 1 = 321 − 186 = 135

Difference 2 = 402 − 321 = 81

The HCF of 135 and 81 is 27 since 135 = 27 × 5 and 81 = 27 × 3.

∴ The correct answer is 27.

Alternate Method

Given: The numbers are 186, 321, and 402.

Formula Used: The greatest number that divides x, y, and z leaving the same remainder in each case is HCF(|x − y|, |y − z|, |z − x|).

⇒ Difference (d₁) = 321 − 186 = 135

⇒ Difference (d₂) = 402 − 321 = 81

⇒ Difference (d₃) = 402 − 186 = 216

⇒ Prime factorization of 135 = 3 × 3 × 3 × 5 = 3³ × 5

⇒ Prime factorization of 81 = 3 × 3 × 3 × 3 = 3⁴

⇒ Prime factorization of 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2³ × 3³

⇒ Highest Common Factor (HCF) = 3³ = 27

∴ The correct answer is 27.

Additional Information

HCF with Given Remainders

To find the greatest number that divides x, y, and z leaving remainders a, b, and c respectively, find the HCF of (x−a), (y−b), and (z−c).

LCM with Same Remainder

The smallest number which when divided by x, y, and z leaves the same remainder r in each case is given by [LCM(x, y, z) + r].

Property of HCF

The HCF of any two or more numbers must also divide the difference between those numbers.

Shortcut Trick

When a single discount (D) equals the sum of two successive discounts (x + y), the difference in discount amount is always (x × y / 100)% of the Marked Price.

Here, D = 30%, x = 25%, and y = 5%. Note that 30 = 25 + 5.

Difference % = (25 × 5) / 100 = 1.25%.

Difference amount = 1.25% of ₹800 = 1.25 × 8 = ₹10.

∴ The correct answer is ₹10.

Alternate Method

Given:

Marked Price (MP) = ₹800

Offer 1: Single Discount = 30%

Offer 2: Successive Discounts = 25% and 5%

Formula Used:

Total Discount = MP × Discount%

Effective Discount for x% and y% successive = x + y − (xy / 100)

Calculations:

⇒ Offer 1 Discount amount = 30% of 800 = (30/100) × 800 = ₹240

⇒ Offer 2 Effective Discount% = 25 + 5 − (25 × 5 / 100)

⇒ Offer 2 Effective Discount% = 30 − 1.25 = 28.75%

⇒ Offer 2 Discount amount = 28.75% of 800 = (28.75 / 100) × 800 = 28.75 × 8 = ₹230

⇒ Difference in Discount amount = 240 − 230 = ₹10

∴ The correct answer is ₹10.

Additional Information

Successive Discount Formula

For two successive discounts of a% and b%, the net discount is (a + b − ab/100)%. For three discounts a, b, and c, apply the formula twice.

Marked Price and Selling Price

Selling Price = Marked Price × (100 − Discount%)/100. The discount is always calculated on the Marked Price (MP) unless specified otherwise.

Single vs Successive Discount

A single discount of (x + y)% is always greater than two successive discounts of x% and y% because in the successive case, the second discount is applied to a reduced amount.

Shortcut Trick

Simplify the ratio of the numbers: 6 : 4 → 3 : 2 (co-prime terms).

For co-prime ratio parts 3 and 2, and HCF = 49:

LCM = Product of ratio terms × HCF

LCM = 3 × 2 × 49 = 6 × 49 = 294.

∴ The correct answer is 294.

Alternate Method

Given:

Ratio of the two numbers = 6 : 4

HCF of the numbers = 49

Formula used:

If two numbers are in the ratio a : b, then the numbers can be written as ax and bx, where x is the HCF of the numbers.

LCM of ax and bx = a × b × x

Shortcut Trick

Total cost of Notebooks and Pencils = (20 × 50) + (50 × 10) = 1000 + 500 = ₹1500.

Remaining amount for Pens = 3500 − 1500 = ₹2000.

Average cost of one Pen = Total Pen cost ÷ Number of Pens = 2000 ÷ 100 = ₹20.

∴ The correct answer is ₹20.

Alternate Method

Given:

Total quantity: 20 Notebooks, 100 Pens, and 50 Pencils.

Total Expenditure = ₹3,500.

Average cost of Notebook = ₹50 | Average cost of Pencil = ₹10.

Calculations:

⇒ Total cost of 20 Notebooks = 20 × 50 = ₹1,000

⇒ Total cost of 50 Pencils = 50 × 10 = ₹500

⇒ Combined cost of Notebooks and Pencils = 1000 + 500 = ₹1,500

⇒ Total cost for 100 Pens = Total Expenditure − Combined cost

⇒ Total cost for 100 Pens = 3500 − 1500 = ₹2,000

⇒ Average cost of a Pen = Total cost of Pens ÷ Total number of Pens

⇒ Average cost of a Pen = 2000 ÷ 100 = ₹20

∴ The correct answer is ₹20.

Additional Information

Fundamental Average Formula

The Average of a set of values is the sum of the values divided by the number of values: Average = Total Sum ÷ Number of Items.

Total Value Derivation

To find the Total Value when the average is known, use the formula: Total Value = Average × Number of Items.

Weighted Average Concept

When multiple groups are combined, the Overall Average is calculated as: (n₁×a₁ + n₂×a₂ + ...) ÷ (n₁ + n₂ + ...).

Shortcut Trick

Identify the triangle type: Since 5² + 12² = 13² (25 + 144 = 169), it is a right-angled triangle.

In a right-angled triangle, the perpendicular from the right-angle vertex to the hypotenuse (h) is given by:

h = (Product of legs) ÷ Hypotenuse

h = (5 × 12) ÷ 13 = 60 ÷ 13 ≈ 4.615...

∴ The correct answer is 4.62 cm.

Alternate Method

Given: Sides of the triangle are 5 cm, 12 cm, and 13 cm.

Formula: Area of triangle = 1/2 × base × height

⇒ First, calculate Area using the two legs: Area = 1/2 × 5 × 12

⇒ Area = 30 cm²

⇒ Now, consider the longest side (13 cm) as the base and 'h' as the perpendicular height.

⇒ Area = 1/2 × 13 × h

⇒ 30 = 1/2 × 13 × h

⇒ 60 = 13 × h

⇒ h = 60 ÷ 13 ≈ 4.6153... cm

⇒ Rounding to two decimal places, h = 4.62 cm.

∴ The correct answer is 4.62 cm.

Additional Information

Pythagorean Triplet

A set of three positive integers a, b, and c such that a² + b² = c². Common triplets include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Altitude of a Right Triangle

The altitude (h) to the hypotenuse (c) of a right triangle with legs a and b is given by h = (a × b) ÷ c.

Right Triangle Area

Area can be calculated either as (1/2 × leg₁ × leg₂) or (1/2 × hypotenuse × altitude to hypotenuse).

Shortcut Trick

Change in students = 40 − 25 = 15; Change in bill = 25,000 − 17,500 = ₹7,500.

Variable charge per student = 7,500 ÷ 15 = ₹500.

To find the bill for 60 students, add the cost of 20 more students to the 40-student bill.

Total bill for 60 students = 25,000 + (20 × 500) = 25,000 + 10,000 = ₹35,000.

∴ The correct answer is ₹35,000.

Alternate Method

Given: Total bill for 25 students = ₹17,500 and for 40 students = ₹25,000.

Formula Used: Total Charge = Fixed Part (F) + (Number of Students (n) × Variable Part per student (v))

⇒ Let Fixed Charge be F and Variable Charge per student be v.

⇒ F + 25v = 17,500 ... (i)

⇒ F + 40v = 25,000 ... (ii)

⇒ Subtracting (i) from (ii): (40v − 25v) = 25,000 − 17,500

⇒ 15v = 7,500 ⇒ v = 500

⇒ Substituting v in (i): F + 25(500) = 17,500 ⇒ F + 12,500 = 17,500

⇒ Fixed Charge (F) = 17,500 − 12,500 = ₹5,000

⇒ Total bill for 60 students = F + 60v = 5,000 + (60 × 500)

⇒ Total bill = 5,000 + 30,000 = ₹35,000

∴ The correct answer is ₹35,000.

Additional Information

Linear Variation

This problem follows a linear equation pattern y = mx + c, where 'c' is the fixed cost and 'm' is the variable rate per unit.

Fixed and Variable Costs

Fixed costs remain constant regardless of student count, while variable costs increase proportionally with the number of students.

System of Linear Equations

Problems involving two unknowns (Fixed and Variable) are typically solved using the Elimination method or Substitution method.

Shortcut Trick

Profit = 575 − 500 = 75

Profit % = (Profit ÷ CP) × 100

Profit % = (75 ÷ 500) × 100 = 15%

∴ The correct answer is 15%.

Alternate Method

Given: Cost Price (CP) of the article = ₹500 and Selling Price (SP) = ₹575.

Formula used: Profit Percentage = [(Selling Price − Cost Price) ÷ Cost Price] × 100

⇒ Profit = Selling Price − Cost Price

⇒ Profit = 575 − 500 = ₹75

⇒ Profit Percentage = (75 ÷ 500) × 100

⇒ Profit Percentage = 15%

∴ The correct answer is 15%.

Additional Information

Loss Percentage

If Selling Price is less than Cost Price, then Loss % = [(CP − SP) ÷ CP] × 100.

Selling Price Calculation

To find Selling Price when Profit % is given: SP = CP × (100 + Profit %) ÷ 100.

Cost Price Calculation

To find Cost Price when Profit % is given: CP = [SP × 100] ÷ (100 + Profit %).