Quantitative Aptitude PYQs

Shortcut Trick

Let the total enrolled voters be x.

Successful candidates votes = x × (Cast%) × (Valid%) × (Candidate's share%)

⇒ 13572 = x × 0.78 × 0.96 × 0.58

⇒ x = 13572 ÷ (0.78 × 0.96 × 0.58) = 31,250

∴ The correct answer is 31,250.

Alternate Method

Given:

Voters cast = 78%

Invalid votes = 4% (So, Valid votes = 96% of cast votes)

Candidate's share = 58% of valid votes = 13,572

Formula:

Total Voters × (Cast/100) × (Valid/100) × (Share/100) = Candidate Votes

⇒ Let total enrolled voters = x

⇒ x × (78/100) × (96/100) × (58/100) = 13572

⇒ x × (78 × 96 × 58) / 1000000 = 13572

⇒ x = (13572 × 1000000) / (78 × 96 × 58)

⇒ Dividing 13572 by 58: 13572 ÷ 58 = 234

⇒ x = (234 × 1000000) / (78 × 96)

⇒ Dividing 234 by 78: 234 ÷ 78 = 3

⇒ x = (3 × 1000000) / 96

⇒ x = 1000000 / 32

⇒ x = 31250

∴ The correct answer is 31,250.

Additional Information

Election-based Percentages

Valid Votes = Total Cast Votes − Invalid Votes. The winner and loser share is always calculated from the valid votes unless stated otherwise.

Successive Percentage Application

When multiple reductions or shares are applied in sequence, use the chain rule: Final Value = Initial Value × a% × b% × c%.

Majority Margin

Margin of victory = (Votes of Winner − Votes of Loser). Usually expressed as a percentage of the total valid votes.

Shortcut Trick

Cost Price for the collector = ₹1,440

Profit = 25% of 1,440 = 1/4 × 1,440 = ₹360

Auction Selling Price = 1,440 + 360 = ₹1,800

Alternatively: SP = 1,440 × 125% = 1,440 × 5/4 = ₹1,800

∴ The correct answer is ₹1,800.

Alternate Method

Given:

Price at which dealer sells to collector (CP for collector) = ₹1,440

Profit percentage at auction = 25%

Formula used: Selling Price (SP) = CP × (1 + Profit%/100)

⇒ Profit = 25% of ₹1,440

⇒ Profit = (25/100) × 1440

⇒ Profit = 1/4 × 1440 = 360

⇒ Final Price = Cost Price + Profit

⇒ Final Price = 1440 + 360

⇒ Final Price = 1800

∴ The correct answer is ₹1,800.

Additional Information

Profit and Loss Basics

Profit = Selling Price − Cost Price. Profit percentage is always calculated on the CP: (Profit/CP) × 100.

Successive Transactions

When an item passes through multiple owners, the SP for one becomes the CP for the next. The final price is CP × (1 + p₁/100) × (1 + p₂/100)...

Fractional Equivalents

Common profit percentages: 20% = 1/5, 25% = 1/4, 33.33% = 1/3, 50% = 1/2. Use (1 + fraction) as a multiplier for SP.

Shortcut Trick

Sum of exterior angles is always 360°.

Given ratio (Exterior : Interior) = 2 : 43

2 units = 360° ⇒ 1 unit = 180°

Sum of interior angles = 43 × 180°

Since Sum = (n − 2) × 180°, we have n − 2 = 43 ⇒ n = 45

Measure of interior angle = (43 × 180°) / 45 = 43 × 4 = 172°

∴ The correct answer is 172°.

Alternate Method

Given:

Ratio of sum of exterior angles to sum of interior angles = 2 : 43

Formula Used:

Sum of exterior angles of a polygon = 360°

Sum of interior angles of a polygon = (n − 2) × 180°

Each interior angle of a regular polygon = [(n − 2) × 180°] / n

According to the question:

⇒ 360 / [(n − 2) × 180] = 2 / 43

⇒ 2 / (n − 2) = 2 / 43

⇒ n − 2 = 43

⇒ n = 45

Now, calculate the measure of each interior angle:

⇒ Interior Angle = [(45 − 2) × 180°] / 45

⇒ Interior Angle = (43 × 180°) / 45

⇒ Interior Angle = 43 × 4°

⇒ Interior Angle = 172°

∴ The correct answer is 172°.

Additional Information

Sum of Angles Property

For any convex polygon, the sum of all exterior angles (one at each vertex) is always 360°, regardless of the number of sides.

Interior and Exterior Relation

The sum of an interior angle and its adjacent exterior angle at any vertex of a polygon is always 180° (Linear Pair).

Number of Diagonals

The number of diagonals in a polygon with n sides is given by the formula: [n × (n − 3)] / 2.

Shortcut Trick

Net effective increase = x + y + (xy/100)

⇒ Net % = 15 + 30 + (15 × 30)/100 = 45 + 4.5 = 49.5%

Population after 2 years = 100% + 49.5% = 149.5% of 1,60,000

⇒ 1.495 × 1,60,000 = 2,39,200

∴ The correct answer is 2,39,200.

Alternate Method

Given: Initial Population (P) = 1,60,000

Increase in Year 1 (R₁) = 15%

Increase in Year 2 (R₂) = 30%

Formula: Population after n years = P × (1 + R₁/100) × (1 + R₂/100)

⇒ Population after 1^st year = 1,60,000 × (115/100)

⇒ Population after 1^st year = 1,600 × 115 = 1,84,000

⇒ Population after 2^nd year = 1,84,000 × (130/100)

⇒ Population after 2^nd year = 1,840 × 130 = 2,39,200

∴ The correct answer is 2,39,200.

Additional Information

Successive Percentage Change

When a value changes by x% and then y% successively, the net percentage change is given by [x + y + (xy/100)]%.

Population Decrease (Depreciation)

If the population decreases, the value of R becomes negative in the formula: Final = P × (1 − R/100)ⁿ.

Constant Growth Rate

If the growth rate remains constant for n years, the formula simplifies to: Final = P × (1 + R/100)ⁿ, which is identical to the Compound Interest formula.

Shortcut Trick

Sum of Length (L) and Breadth (B) = Perimeter ÷ 2 = 40 ÷ 2 = 20 cm.

Difference between Length and Breadth (L − B) = 5 cm.

Length (L) = (Sum + Difference) ÷ 2 = (20 + 5) ÷ 2 = 12.5 cm.

Breadth (B) = (Sum − Difference) ÷ 2 = (20 − 5) ÷ 2 = 7.5 cm.

∴ The correct answer is 12.5 cm, 7.5 cm.

Alternate Method

Given:

Perimeter (P) = 40 cm

Length (L) = Breadth (B) + 5 cm

Formula Used:

Perimeter of a rectangle = 2 × (Length + Breadth)

Calculations:

⇒ 2 × (L + B) = 40

⇒ L + B = 20

⇒ (B + 5) + B = 20

⇒ 2B + 5 = 20

⇒ 2B = 20 − 5

⇒ 2B = 15

⇒ B = 7.5 cm

⇒ L = B + 5 = 7.5 + 5 = 12.5 cm

∴ The correct answer is 12.5 cm, 7.5 cm.

Additional Information

Area of Rectangle

The total space occupied by the rectangle is given by Area = Length × Breadth.

Diagonal of Rectangle

The distance between opposite corners is d = √(L² + B²).

Square Property

A rectangle becomes a square when Length = Breadth, and its perimeter is 4 × Side.

Shortcut Trick

Maximum Selling Price is obtained when the total effective discount is minimum.

Check the sum of discounts for each scheme:

(1) 38%, (2) 58%, (3) 61%, (4) 20%.

Since Scheme 4 has the lowest sum and lowest individual values, its effective discount will be the minimum.

Minimum Discount ⇒ Maximum Selling Price.

∴ The correct answer is Two successive discounts of 10%.

Alternate Method

Successive Discounts: d₁% and d₂%

Effective Discount % = d₁ + d₂ − (d₁ × d₂)/100

Calculating effective discount for each scheme:

⇒ Scheme 1: 23 + 15 − (23 × 15)/100 = 38 − 3.45 = 34.55%

⇒ Scheme 2: 12 + 46 − (12 × 46)/100 = 58 − 5.52 = 52.48%

⇒ Scheme 3: 25 + 36 − (25 × 36)/100 = 61 − 9 = 52.00%

⇒ Scheme 4: 10 + 10 − (10 × 10)/100 = 20 − 1 = 19.00%

⇒ To maximize Selling Price, we need the Minimum Effective Discount.

⇒ Minimum discount is 19% from Scheme (4).

∴ The correct answer is Two successive discounts of 10%.

Additional Information

Effective Discount Formula

For three successive discounts a%, b%, and c%, the effective discount is calculated by applying the two-discount formula twice or using the multiplier method: 1 − [(1−a/100)(1−b/100)(1−c/100)].

Relationship between MP and CP

MP/CP = (100 + Profit%)/(100 − Discount%). This ratio is useful for finding the marked price when profit and discount are known.

Maximum vs Minimum Discount

For a fixed sum of two discounts (a+b = constant), the effective discount is maximum when the difference between a and b is maximum, and minimum when the difference is minimum.

Shortcut Trick

Total Work (LCM of 8, 12, 16) = 48 units.

Efficiencies: A = 6, B = 4, C = 3. Total efficiency (A+B+C) = 13 units/hr.

Work done in 2 hours = 13 × 2 = 26 units.

Remaining work = 48 − 26 = 22 units.

Time for A and B to finish = 22 ÷ (6 + 4) = 2.2 hours.

Total time = 2 + 2.2 = 4.2 hours.

∴ The correct answer is 4.2 hours.

Alternate Method

Given:

Time taken by A = 8 hrs, B = 12 hrs, C = 16 hrs.

C leaves after 2 hours.

Calculations:

⇒ Total units of work = LCM(8, 12, 16) = 48 units

⇒ Efficiency of A = 48 ÷ 8 = 6 units/hr

⇒ Efficiency of B = 48 ÷ 12 = 4 units/hr

⇒ Efficiency of C = 48 ÷ 16 = 3 units/hr

⇒ Combined efficiency of A + B + C = 6 + 4 + 3 = 13 units/hr

⇒ Work done in the first 2 hours = 13 × 2 = 26 units

⇒ Remaining work = 48 − 26 = 22 units

⇒ Workers remaining = A + B

⇒ Efficiency of A + B = 6 + 4 = 10 units/hr

⇒ Time taken to finish remaining work = 22 ÷ 10 = 2.2 hours

⇒ Total time = 2 + 2.2 = 4.2 hours

∴ The correct answer is 4.2 hours.

Additional Information

Work and Efficiency Relation

Total Work = Efficiency × Time. Efficiency is inversely proportional to the time taken to complete a task.

Pipes and Cisterns

The same logic applies where filling pipes have positive efficiency and outlet pipes (leaks) have negative efficiency.

Work and Wages

Wages are distributed based on the ratio of work done by each individual, which depends on their efficiency and time worked.

Shortcut Trick

Profit Ratio of Suman : Tariq : Neha = 6 : 11 : 12

Total units = 6 + 11 + 12 = 29 units

Difference between Neha and Suman = 12 − 6 = 6 units

Value of 6 units = (79721 ÷ 29) × 6 = 2749 × 6

Value of 6 units = ₹16,494

∴ The correct answer is ₹16,494.

Alternate Method

Given:

Investment ratio of Suman, Tariq and Neha = 6 : 11 : 12

Total annual profit = ₹79,721

Formula Used:

Profit Share = (Individual Ratio Part ÷ Sum of Ratio Parts) × Total Profit

Sum of ratio parts = 6 + 11 + 12 = 29

⇒ 29 units = ₹79,721

⇒ 1 unit = 79721 ÷ 29

⇒ 1 unit = ₹2,749

⇒ Suman share = 6 × 2749 = ₹16,494

⇒ Neha share = 12 × 2749 = ₹32,988

⇒ Difference = 32988 − 16494

⇒ Difference = ₹16,494

∴ The correct answer is ₹16,494.

Additional Information

Partnership Profit Rule

Profit is distributed based on the product of Capital (C) and Time (T). Profit Ratio P₁ : P₂ = C₁ × T₁ : C₂ × T₂.

Total Ratio Unitary Method

To find the value of any share or difference, divide the total amount by the sum of all ratio parts to find the value of 1 unit.

Working vs Sleeping Partner

A working partner manages the business and may receive a salary before the remaining profit is divided according to the investment ratio.

Shortcut Trick

Recognize x = 3 + 2√2 as a perfect square: (√2 + 1)².

∴ √x = √2 + 1.

Reciprocal 1/√x = 1/(√2 + 1) = √2 − 1.

Substitute in expression: (√2 + 1) − (√2 − 1).

Value = 2.

∴ The correct answer is 2.

Alternate Method

Given: x = 3 + √8

Formula: (√x − 1/√x)² = x + 1/x − 2

⇒ 1/x = 1 / (3 + √8) × (3 − √8) / (3 − √8) ⇒ 1/x = (3 − √8) / (9 − 8) = 3 − √8 ⇒ x + 1/x = (3 + √8) + (3 − √8) = 6 ⇒ (√x − 1/√x)² = 6 − 2 ⇒ (√x − 1/√x)² = 4 ⇒ √x − 1/√x = √4 = 2

∴ The correct answer is 2.

Additional Information

Conjugate Surds

For any surd of the form a + √b, its conjugate is a − √b. Their product results in a rational number a² − b.

Square Root of a Surd

If a surd is in the form (a + b) + 2√(ab), its square root is simply √a + √b.

Algebraic Identity

The expansion of (a − b)² = a² + b² − 2ab is frequently used to find values of √x − 1/√x by setting a = √x and b = 1/√x.

Shortcut Trick

Smallest number leaving same remainder k = LCM(divisors) + k

LCM(18, 35, 140):

LCM(35, 140) = 140

LCM(140, 18) = 1260

Required number = 1260 + 7 = 1267

∴ The correct answer is 1267.

Alternate Method

Given values: Divisors = 18, 35, 140 and Remainder = 7.

Formula: Smallest number = LCM(x, y, z) + r

⇒ Prime factors of 18 = 2 × 3²

⇒ Prime factors of 35 = 5 × 7

⇒ Prime factors of 140 = 2² × 5 × 7

⇒ LCM = 2² × 3² × 5 × 7

⇒ LCM = 4 × 9 × 35 = 1260

⇒ Number = 1260 + 7 = 1267

∴ The correct answer is 1267.

Additional Information

LCM (Least Common Multiple)

The smallest positive integer that is perfectly divisible by each of the given numbers.

Case of Different Remainders

If a number divided by x, y, z leaves remainders a, b, c such that (x−a) = (y−b) = (z−c) = k, the number is LCM(x, y, z) − k.

Relationship between HCF and LCM

For two numbers a and b, HCF(a, b) × LCM(a, b) = a × b.

Shortcut Trick

Ratio of articles P : Q : R = 4 : 2 : 5

Total cost per ratio unit = (4 × 280) + (2 × 185) + (5 × 105)

Total cost per ratio unit = 1120 + 370 + 525 = 2015

Common multiplier = Total budget ÷ Unit cost = 8060 ÷ 2015 = 4

Number of articles of type Q = 2 × 4 = 8

∴ The correct answer is 8.

Alternate Method

Given prices: P = ₹280, Q = ₹185, R = ₹105

Given ratio of quantity P : Q : R = 4 : 2 : 5

Let the quantities of P, Q, and R be 4x, 2x, and 5x respectively.

Total Expenditure = (Price_P × Quantity_P) + (Price_Q × Quantity_Q) + (Price_R × Quantity_R)

⇒ 8060 = (280 × 4x) + (185 × 2x) + (105 × 5x)

⇒ 8060 = 1120x + 370x + 525x

⇒ 8060 = 2015x

⇒ x = 8060 ÷ 2015

⇒ x = 4

⇒ Number of articles of type Q = 2x

⇒ Number of articles of type Q = 2 × 4 = 8

∴ The correct answer is 8.

Additional Information

Ratio of Values

The ratio of the total values of different items is the product of the ratio of their individual prices and the ratio of their quantities (Value Ratio = Price Ratio × Quantity Ratio).

Unitary Method in Ratios

When the total sum and the ratio are known, the value of one ratio unit is calculated as: Total Sum ÷ Sum of ratio parts (weighted by their respective values).

Compound Ratio

For two ratios a:b and c:d, the compound ratio is (a×c):(b×d), which is often used in problems involving price and quantity to find the total expenditure ratio.

Shortcut Trick

Efficiency Ratio (Amit : Gopal) = 100 : 130 = 10 : 13

Time Ratio is inverse of Efficiency Ratio ⇒ Amit : Gopal = 13 : 10

Given Amit's time = 13 days (13 units = 13 days)

⇒ Gopal's time = 10 units = 10 days

∴ The correct answer is 10.

Alternate Method

Given:

Time taken by Amit = 13 days

Efficiency of Gopal = Efficiency of Amit + 30% of Efficiency of Amit

Formula Used:

Total Work = Efficiency × Time

⇒ Let Amit's efficiency = 10 units/day

⇒ Gopal's efficiency = 10 + 30% of 10 = 13 units/day

⇒ Total Work = Amit's time × Amit's efficiency

⇒ Total Work = 13 × 10 = 130 units

⇒ Time taken by Gopal = Total Work ÷ Gopal's Efficiency

⇒ Time taken by Gopal = 130 ÷ 13 = 10 days

∴ The correct answer is 10.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete a task. If efficiency increases by x%, the time decreases proportionally.

Work Equation

Total Work = (Efficiency₁ × Time₁) = (Efficiency₂ × Time₂). This constant work principle allows solving for any missing variable.

Ratio Conversion

If the ratio of efficiency of A and B is a : b, then the ratio of time taken by them to complete the same work will be b : a.

Shortcut Trick

The average of the first n odd natural numbers is always equal to n.

Here, n = 12.

∴ Average = 12.

∴ The correct answer is 12.

Alternate Method

Given:

First 12 odd natural numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23.

Formula Used:

Sum of the first n odd natural numbers = n²

Average = Total Sum ÷ Number of terms

Calculations:

⇒ Total terms (n) = 12

⇒ Sum of 12 terms = 12² = 144

⇒ Average = 144 ÷ 12

⇒ Average = 12

Alternatively, for an Arithmetic Progression (AP):

⇒ Average = (First term + Last term) ÷ 2

⇒ Average = (1 + 23) ÷ 2 = 24 ÷ 2 = 12

∴ The correct answer is 12.

Additional Information

Average of first n even numbers

The average of the first n even natural numbers (2, 4, 6... 2n) is always (n + 1).

Average of first n natural numbers

The average of the first n natural numbers (1, 2, 3... n) is given by (n + 1) ÷ 2.

Sum of first n natural numbers

The sum of the first n natural numbers is calculated using the formula [n × (n + 1)] ÷ 2.

Shortcut Trick

Surface Area (SA) = 4πr² = 616 ⇒ r² = 49 ⇒ r = 7 cm.

Radius decreases by 25% ⇒ Ratio of radii = 1 : 0.75 = 4 : 3.

Ratio of volumes = (4)³ : (3)³ = 64 : 27.

Original Volume = (4/3) × (22/7) × 7³ = 4312/3 cm³.

New Volume = (4312/3) × (27/64) = 135.375 × 4.48 ≈ 606.375 cm³.

∴ The correct answer is 606.375 cm³.

Alternate Method

Given:

Surface area of the original sphere = 616 cm²

Reduction in radius = 25%

Formula Used:

Surface area of sphere = 4πr²

Volume of sphere = (4/3)πr³

Step 1: Finding the radius of the original sphere (r).

⇒ 4 × (22/7) × r² = 616

⇒ r² = (616 × 7) / (4 × 22)

⇒ r² = (616 × 7) / 88

⇒ r² = 7 × 7 = 49

⇒ r = 7 cm

Step 2: Finding the radius of the new sphere (r_new).

⇒ r_new = r − 25% of r

⇒ r_new = 7 − (25/100 × 7)

⇒ r_new = 7 − 1.75 = 5.25 cm

Step 3: Finding the volume of the new sphere.

⇒ Volume = (4/3) × π × (r_new)³

⇒ Volume = (4/3) × (22/7) × (5.25) × (5.25) × (5.25)

⇒ Volume = (4/3) × (22/7) × (21/4) × (21/4) × (21/4)

⇒ Volume = (22 × 21 × 21) / (4 × 4)

⇒ Volume = (11 × 441) / 8 = 4851 / 8

⇒ Volume = 606.375 cm³

∴ The correct answer is 606.375 cm³.

Additional Information

Properties of a Sphere

For a sphere of radius r, the Surface Area is 4πr² and the Volume is (4/3)πr³.

Ratio Relationship

If the radius of a sphere is changed by a factor of k, the surface area changes by k² and the volume changes by k³.

Hemisphere Calculations

For a hemisphere, the Curved Surface Area is 2πr² and the Total Surface Area is 3πr².

Shortcut Trick

Apply the algebraic identity: (a + b)² − (a − b)² = 4ab.

In this expression, a = 5pq and b = 16q.

Required value = 4 × 5pq × 16q = 320pq².

∴ The correct answer is 320pq².

Alternate Method

Given: (5pq + 16q)² − (5pq − 16q)²

Formula used: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²

⇒ [(5pq)² + 2 × 5pq × 16q + (16q)²] − [(5pq)² − 2 × 5pq × 16q + (16q)²]

⇒ [25p²q² + 160pq² + 256q²] − [25p²q² − 160pq² + 256q²]

⇒ 25p²q² + 160pq² + 256q² − 25p²q² + 160pq² − 256q²

⇒ 160pq² + 160pq²

⇒ 320pq²

∴ The correct answer is 320pq².

Additional Information

Algebraic Identity for Difference of Squares

x² − y² = (x + y)(x − y). This can also be used to solve the problem by letting x = (5pq + 16q) and y = (5pq − 16q).

Algebraic Identity for Sum of Squares

(a + b)² + (a − b)² = 2(a² + b²). This identity is used when the middle sign is positive.

Simplification Rule

While simplifying algebraic expressions, always check for common factors or standard identities to reduce calculation time.

Shortcut Trick

Expenditure % = 100% − Savings % = 100% − 88% = 12%

Monthly Expenditure = 12% of ₹9475

⇒ 12/100 × 9475 = 3/25 × 9475

⇒ 3 × 379 = ₹1137

∴ The correct answer is ₹1137.

Alternate Method

Given:Monthly Salary = ₹9475, Savings Percentage = 88%

Formula Used:Expenditure = Salary − Savings

⇒ Savings amount = 88% of 9475 = (88 × 9475) ÷ 100

⇒ Savings amount = 0.88 × 9475 = ₹8338

⇒ Expenditure = 9475 − 8338

⇒ Expenditure = ₹1137

∴ The correct answer is ₹1137.

Additional Information

Percentage Change

Percentage change is calculated as (New Value − Old Value) / Old Value × 100. In income problems, if income increases by r%, new income = Original × (1 + r/100).

Income-Expenditure Relation

Total Income is always the sum of Total Expenditure and Total Savings. If any two are known, the third can be calculated directly.

Base Value in Percentage

In salary-based questions, the Total Salary is always considered as the base (100%) for calculating percentage savings or expenses.

Shortcut Trick

Using the trigonometric identity: If A + B = 90°, then tan A × tan B = 1.

Here, (10° + 80° = 90°) ⇒ tan 10° × tan 80° = 1.

And, (18° + 72° = 90°) ⇒ tan 18° × tan 72° = 1.

Therefore, the product is 1 × 1 = 1.

∴ The correct answer is 1.

Alternate Method

Given: tan 10° tan 18° tan 72° tan 80°

Formula Used: tan(90° − θ) = cot θ and tan θ × cot θ = 1

⇒ tan 80° = tan(90° − 10°) = cot 10°

⇒ tan 72° = tan(90° − 18°) = cot 18°

Substitute these values back into the expression:

⇒ tan 10° × tan 18° × cot 18° × cot 10°

⇒ (tan 10° × cot 10°) × (tan 18° × cot 18°)

⇒ 1 × 1 = 1

∴ The correct answer is 1.

Additional Information

Complementary Angle Identities

For any angle θ, sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, and tan(90° − θ) = cot θ.

Reciprocal Identities

The product of trigonometric terms that are reciprocals of each other is 1, such as tan θ × cot θ = 1 and sin θ × cosec θ = 1.

Shortcut Trick

Effective Rate of Interest for 2 years at 20% = 20 + 20 + (20 × 20)/100 = 44%.

Amount = 100% + 44% = 144% of Principal.

Amount = 1.44 × 9400 = 13536.

∴ The correct answer is ₹13,536.

Alternate Method

Given:

Principal (P) = ₹9,400

Rate (R) = 20% p.a.

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 9400(1 + 20/100)²

⇒ A = 9400(1 + 1/5)²

⇒ A = 9400(6/5)²

⇒ A = 9400 × (36/25)

⇒ A = 376 × 36

⇒ A = 13536

∴ The correct answer is ₹13,536.

Additional Information

Effective Rate for Compound Interest

For 2 years, the effective rate is x + y + (xy/100)%, where x and y are the rates for the first and second years.

Compound Interest vs Simple Interest

The difference for 2 years is given by the formula D = P(R/100)².

Compounding Half-Yearly

If interest is compounded half-yearly, the rate is halved (R/2) and the time is doubled (2n).

Shortcut Trick

Identify the smallest two-digit prime numbers: 11 and 13.

Check if they are twin primes: 13 − 11 = 2 (Yes).

Calculate the sum: 11 + 13 = 24.

∴ The correct answer is 24.

Alternate Method

Given: We need to find the sum of the first pair of two-digit twin prime numbers.

Definition: Twin primes are pairs of prime numbers that have a difference of 2.

⇒ Prime numbers starting from 10 are: 11, 13, 17, 19, 23, 29, ...

⇒ The first two-digit prime number is 11.

⇒ The next prime number is 13.

⇒ Difference = 13 − 11 = 2.

⇒ Since the difference is 2, (11, 13) is the first pair of two-digit twin primes.

⇒ Sum = 11 + 13 = 24.

∴ The correct answer is 24.

Additional Information

Twin Prime Pairs

Common twin prime pairs include (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), and (41, 43).

Prime Number Properties

Except for (3, 5), all twin prime pairs are of the form (6n − 1, 6n + 1) for some integer n.

Smallest Two-Digit Prime

The smallest two-digit prime number is 11, and the largest is 97.

Shortcut Trick

Difference in costs after doubling the second set:

Cost of 1 cookie = (2 × 23) − 36 = 46 − 36 = ₹ 10.

Cost of 1 chocolate = 23 − (2 × 10) = ₹ 3.

Required cost of 4 chocolates and 5 cookies = (4 × 3) + (5 × 10) = 12 + 50 = ₹ 62.

∴ The correct answer is ₹ 62.

Alternate Method

Given:

Case 1: 2 Chocolates + 3 Cookies = ₹ 36

Case 2: 1 Chocolate + 2 Cookies = ₹ 23

Let the cost of one chocolate be ₹ x and one cookie be ₹ y.

⇒ 2x + 3y = 36 ....(i)

⇒ x + 2y = 23 ....(ii)

Multiplying equation (ii) by 2:

⇒ 2x + 4y = 46 ....(iii)

Subtracting equation (i) from equation (iii):

⇒ (2x + 4y) − (2x + 3y) = 46 − 36

⇒ y = 10

Substitute y = 10 in equation (ii):

⇒ x + 2(10) = 23

⇒ x + 20 = 23

⇒ x = 3

Cost of 4 chocolates and 5 cookies:

⇒ 4x + 5y = 4(3) + 5(10)

⇒ 12 + 50 = 62

∴ The correct answer is ₹ 62.

Additional Information

Linear Equations in Two Variables

Equations of the form ax + by = c represent a straight line. Solving two such equations finds the intersection point (x, y).

Elimination Method

A technique to solve a system of equations by making the coefficients of one variable equal and then adding or subtracting the equations.

Substitution Method

Solve one equation for one variable and substitute that expression into the second equation to find the value of the other variable.