Quantitative Aptitude PYQs

Shortcut Trick

Total days from 7 February to 21 April 2023 is 73 days.

73 days is exactly 1/5 of a year (73 ÷ 365 = 1/5).

Simple Interest = Principal × Rate × Time = 2000 × 7% × 1/5

SI = 20 × 7 ÷ 5 = 4 × 7 = 28

∴ The correct answer is ₹28.

Alternate Method

Given:

Principal (P) = ₹2,000

Rate (R) = 7% per annum

Time Period: 7 February 2023 to 21 April 2023

Calculation of Time (T):

⇒ Number of days in February (28 − 7) = 21 days

⇒ Number of days in March = 31 days

⇒ Number of days in April = 21 days

⇒ Total Days = 21 + 31 + 21 = 73 days

⇒ Time in years (T) = 73 ÷ 365 = 1/5 years

Calculation of Simple Interest (SI):

⇒ Formula: SI = (P × R × T) ÷ 100

⇒ SI = (2000 × 7 × 1/5) ÷ 100

⇒ SI = (2000 × 7) ÷ (5 × 100)

⇒ SI = 14000 ÷ 500

⇒ SI = 28

∴ The correct answer is ₹28.

Additional Information

Simple Interest Formula

SI = (P × R × T) ÷ 100, where P is Principal, R is Rate, and T is Time in years.

Day Counting Rule

While calculating interest for a period, the day on which money is deposited is usually excluded, and the day on which money is withdrawn is included.

Time Conversion

If time is given in days, divide by 365 (or 366 for a leap year) to convert it into years. Note that 73 days = 1/5 year, 146 days = 2/5 year, and 219 days = 3/5 year.

Shortcut Trick

Since Area of a Square is side², the percentage error is calculated using the Successive Percentage formula.

Successive Change = x + y + (x × y) / 100

Here, x = 4% and y = 4% (as both sides increase by 4%).

Percentage Error in Area = 4 + 4 + (4 × 4) / 100 = 8 + 0.16 = 8.16%.

∴ The correct answer is 8.16%.

Alternate Method

Given:

Percentage error in side = 4%

Formula Used:

Area of a Square = side × side = (side)²

Percentage Error = [(Measured Value − Actual Value) / Actual Value] × 100

Calculations:

⇒ Let the actual side of the square be s = 100 units.

⇒ Actual Area = s² = 100 × 100 = 10,000 sq. units.

⇒ Measured side (4% more) = 100 + 4% of 100 = 104 units.

⇒ Measured Area = (104)² = 10,816 sq. units.

⇒ Error in Area = 10,816 − 10,000 = 816 sq. units.

⇒ Percentage Error in Area = (816 / 10,000) × 100

⇒ Percentage Error in Area = 8.16%.

∴ The correct answer is 8.16%.

Additional Information

Successive Percentage Change

For two successive changes of a% and b%, the effective change is (a + b + ab/100)%. This applies to area calculations where dimensions are multiplied.

Error in Volume

For a cube, if the side error is x%, the volume error involves three successive changes: [(1 + x/100)³ − 1] × 100%.

Relative Error Rule

If Z = Aⁿ, then the relative error in Z is approximately n × (relative error in A), provided the error is very small (usually < 5%). For 4%, 2 × 4% = 8%, but exact calculation gives 8.16%.

Shortcut Trick

To find the maximum possible side length of a square tile with minimal wastage, calculate the Highest Common Factor (HCF) of the wall dimensions.

Dimensions: 36 m and 64 m

36 = 4 × 9

64 = 4 × 16

Common factor = 4

∴ The correct answer is 4 m.

Alternate Method

Given:

Length of the rectangular wall (L) = 64 m

Width of the rectangular wall (W) = 36 m

Formula Used:

Side of the largest square tile = HCF(Length, Width)

Calculations:

⇒ Prime factorization of 36 = 2 × 2 × 3 × 3 = 2² × 3²

⇒ Prime factorization of 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2⁶

⇒ HCF (36, 64) = 2² = 4

⇒ Therefore, the maximum side length of each tile is 4 m.

∴ The correct answer is 4 m.

Additional Information

Highest Common Factor (HCF)

The largest number that divides two or more numbers without leaving a remainder. It is used to find maximum size or equal distribution.

Number of Tiles Formula

Total number of tiles = (Area of wall) ÷ (Area of one tile) = (L × W) ÷ (Side²).

Least Common Multiple (LCM)

In contrast to HCF, LCM is used to find the minimum dimensions of a larger square that can be formed using smaller rectangular tiles.

Shortcut Trick

Ratio of Profit Share = Ratio of Investments.

Suman : Tariq : Neha = 1,71,000 : 24,000 : 1,05,000 = 171 : 24 : 105.

Simplify by dividing by 3: 57 : 8 : 35.

Total units = 57 + 8 + 35 = 100 units.

100 units = &rupee;7,16,200 ⇒ 1 unit = &rupee;7,162.

Neha share (35 units) = 35 × 7,162 = &rupee;2,50,670.

∴ The correct answer is &rupee;2,50,670.

Alternate Method

Given: Investment of Suman (I_s) = &rupee;1,71,000, Tariq (I_t) = &rupee;24,000, Neha (I_n) = &rupee;1,05,000. Total Profit = &rupee;7,16,200.

Formulas: Profit Share Ratio = (I₁ × T₁) : (I₂ × T₂) : (I₃ × T₃). Since time (T) is same, Profit ∝ Investment.

⇒ Ratio of profits = 171,000 : 24,000 : 1,05,000

⇒ Dividing by 1000 = 171 : 24 : 105

⇒ Dividing by 3 = 57 : 8 : 35

⇒ Neha profit share = [35 ÷ (57 + 8 + 35)] × 7,16,200

⇒ Neha profit share = (35 ÷ 100) × 7,16,200

⇒ Neha profit share = 35 × 7,162 = 2,50,670

∴ The correct answer is &rupee;2,50,670.

Additional Information

Partnership Principle

In a business partnership, if the time periods for investments are different, the profit is distributed in the ratio of the product of Capital and Time (C₁T₁ : C₂T₂).

Simple vs Compound Partnership

A Simple Partnership is where all partners invest capital for the same duration. A Compound Partnership involves different capital amounts for different time intervals.

Shortcut Trick

In any quadrilateral, the angle formed by the bisectors of two adjacent angles is equal to half the sum of the other two interior angles.

⇒ ∠LZK = (∠I + ∠J) ÷ 2

⇒ ∠LZK = (17° + 51°) ÷ 2

⇒ ∠LZK = 68° ÷ 2 = 34°

∴ The correct answer is 34°.

Alternate Method

Given:

In quadrilateral IJKL, ∠I = 17° and ∠J = 51°.

KZ and LZ are the angle bisectors of ∠K and ∠L respectively.

Formula Used:

1. Sum of interior angles of a quadrilateral = 360°
2. Sum of interior angles of a triangle = 180°

Calculations:

⇒ In quadrilateral IJKL: ∠I + ∠J + ∠K + ∠L = 360°

⇒ 17° + 51° + ∠K + ∠L = 360°

⇒ 68° + ∠K + ∠L = 360°

⇒ ∠K + ∠L = 360° − 68° = 292°

⇒ Since LZ and KZ are bisectors, in ΔLZK:

⇒ ∠LZK + ∠ZLK + ∠ZKL = 180°

⇒ ∠LZK + (∠L ÷ 2) + (∠K ÷ 2) = 180°

⇒ ∠LZK + (∠L + ∠K) ÷ 2 = 180°

⇒ ∠LZK + (292° ÷ 2) = 180°

⇒ ∠LZK + 146° = 180°

⇒ ∠LZK = 180° − 146° = 34°

∴ The correct answer is 34°.

Additional Information

Angle Bisector Property in Quadrilaterals

The angle between the bisectors of two adjacent angles of a quadrilateral is equal to half the sum of the remaining two angles.

Sum of Interior Angles

For any n-sided polygon, the sum of interior angles is given by (n − 2) × 180°. For a quadrilateral (n=4), it is 360°.

Angle Bisector

An angle bisector divides an angle into two equal parts. If Z is the meeting point of bisectors of ∠K and ∠L, then ∠ZKL = ∠K/2 and ∠ZLK = ∠L/2.

Shortcut Trick

Convert percentages and fractions into a single product: 0.4 × 0.2 × (4/7) × Number = 336.

Number = (336 × 100 × 100 × 7) ÷ (40 × 20 × 4)

Number = (336 × 7) ÷ (0.4 × 0.2 × 4) = 336 ÷ 0.0457... → 7350.

∴ The correct answer is 7350.

Alternate Method

Given: 40% of 20% of 4/7 of a number = 336.

Formula Used: Percentage value = (Value/100) × Total.

Let the number be x.

⇒ (40/100) × (20/100) × (4/7) × x = 336

⇒ (2/5) × (1/5) × (4/7) × x = 336

⇒ (8/175) × x = 336

⇒ x = (336 × 175) ÷ 8

⇒ x = 42 × 175

⇒ x = 7350

∴ The correct answer is 7350.

Additional Information

Percentage to Fraction Conversion

To simplify calculations, convert percentages to fractions: 40% = 2/5, 20% = 1/5, 25% = 1/4, etc.

Successive Percentages

When percentages are applied successively ('of' operator), they are always multiplied together.

Base Change

If a number is increased by x%, the new value is (100+x)/100 times the original value.

Shortcut Trick

Mode is defined as the observation that occurs the maximum number of times in a data set.

In the given series, the number 21 appears 3 times.

All other numbers appear fewer than 3 times (26, 22, 28 appear twice; others appear once).

∴ The correct answer is 21.

Alternate Method

Given: Observations = 26, 26, 22, 21, 31, 24, 21, 28, 28, 30, 33, 22, 25, 23, 21

Step 1: Count the frequency (repetition) of each unique number.

⇒ Frequency of 21 = 3

⇒ Frequency of 22 = 2

⇒ Frequency of 26 = 2

⇒ Frequency of 28 = 2

⇒ Frequency of 23, 24, 25, 30, 31, 33 = 1 each

⇒ Maximum frequency = 3 (for the number 21)

∴ The correct answer is 21.

Additional Information

Measure of Central Tendency: Mode

The value which appears most often in a set of data. A data set can be unimodal (one mode), bimodal (two modes), or multimodal.

Empirical Relationship

For a moderately skewed distribution: Mode = 3 × Median − 2 × Mean.

Median

The middle value of a data set when it is arranged in ascending or descending order. If n is odd, Median = ((n+1)/2)^th term.

Shortcut Trick

Required percentage = (Marks of B / Marks of A) × 100

Required percentage = (50 / 80) × 100

Required percentage = 5 / 8 × 100 = 62.5%

∴ The correct answer is 62.5%.

Alternate Method

Given:

Marks obtained by B = 50

Marks obtained by A = 80

Formula Used:

Percentage = (Value / Base Value) × 100

Calculations:

⇒ Required Percentage = (Marks of B / Marks of A) × 100

⇒ Required Percentage = (50 / 80) × 100

⇒ Required Percentage = 5/8 × 100

⇒ Required Percentage = 0.625 × 100

⇒ Required Percentage = 62.5%

∴ The correct answer is 62.5%.

Additional Information

Percentage Change

Percentage change = (Difference / Original Value) × 100. Used to find how much a value increased or decreased.

Fraction to Percentage Conversion

Common fractions: 1/8 = 12.5%, 3/8 = 37.5%, 5/8 = 62.5%, and 7/8 = 87.5%.

Base Value Selection

In phrases like 'B is what % of A', the value following 'of' (which is A) always acts as the denominator.

Shortcut Trick

In right-angled ▵BDC, the angle at C is the complement of ∠DBC.

∠C = 90° − 80° = 10°

In ▵AEC, ∠AEB is the exterior angle at vertex E.

Using the Exterior Angle Theorem: ∠AEB = ∠CAE + ∠ACE

∠AEB = 37° + 10° = 47°

∴ The correct answer is 47°.

Alternate Method

Given:

In ▵ABC, BD ⊥ AC at D, ∠DBC = 80°, and ∠CAE = 37° where E is on BC.

Formula Used:

1. Sum of angles in a triangle = 180°
2. Exterior angle of a triangle = Sum of two opposite interior angles.

Calculations:

⇒ In ▵BDC, ∠BDC = 90° (since BD ⊥ AC).

⇒ ∠BCD + ∠DBC + ∠BDC = 180°

⇒ ∠BCD + 80° + 90° = 180°

⇒ ∠BCD = 180° − 170° = 10°

⇒ Now, consider ▵AEC. Point B lies on the extension of segment CE.

⇒ ∠AEB is the exterior angle of ▵AEC at vertex E.

⇒ ∠AEB = ∠CAE + ∠ACE

⇒ ∠AEB = 37° + 10°

⇒ ∠AEB = 47°

∴ The correct answer is 47°.

Additional Information

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

Angle Sum Property

The sum of the interior angles of any triangle is always 180°.

Linear Pair

Angles on a straight line add up to 180°. If E lies on BC, then ∠AEB + ∠AEC = 180°.

Shortcut Trick

Direct formula for Surface Area of a sphere = 4 × π × r²

Here, r = 10 cm.

Surface Area = 4 × π × (10)² = 4 × π × 100 = 400π

∴ The correct answer is 400π cm².

Alternate Method

Given:

The radius (r) of the sphere = 10 cm.

Formula Used:

Surface Area (A) = 4πr²

Calculations:

⇒ Surface Area = 4 × π × (10)²

⇒ Surface Area = 4 × π × (10 × 10)

⇒ Surface Area = 4 × π × 100

⇒ Surface Area = 400π cm²

∴ The correct answer is 400π cm².

Additional Information

Volume of a Sphere

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

Hemisphere Properties

For a hemisphere of radius r, the Curved Surface Area (CSA) is 2πr² and the Total Surface Area (TSA) is 3πr².

Ratio of Surface Area to Volume

The ratio of the surface area to the volume of a sphere is simplified as (4πr²) ÷ ((4/3)πr³) = 3/r.

Shortcut Trick

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

Here, let a = 5x and b = 3y.

⇒ Result = 4 × (5x) × (3y)

⇒ Result = 60xy

∴ The correct answer is 60xy.

Alternate Method

Given Expression: (5x + 3y)² − (5x − 3y)²

Formula Used: a² − b² = (a + b)(a − b)

Treating (5x + 3y) as 'a' and (5x − 3y) as 'b':

⇒ [(5x + 3y) + (5x − 3y)] × [(5x + 3y) − (5x − 3y)]

⇒ (5x + 5x + 3y − 3y) × (5x − 5x + 3y + 3y)

⇒ (10x) × (6y)

⇒ 60xy

∴ The correct answer is 60xy.

Additional Information

Sum of Squared Binomials

(a + b)² + (a − b)² = 2(a² + b²)

Square of a Binomial Expansion

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

Difference of Two Squares

a² − b² = (a − b)(a + b)

Shortcut Trick

Babita's first round votes = 40% of 12,000 = 4,800.

Growth in second round = 20%.

Second round votes = 4,800 × 1.2 = 5,760.

∴ The correct answer is 5,760.

Alternate Method

Given:

Total voters = 12,000

Babita's Round 1 percentage = 40%

Increase in Round 2 = 20% of Round 1 votes

Calculations:

⇒ Votes in Round 1 = (40 ÷ 100) × 12,000

⇒ Votes in Round 1 = 4,800

⇒ Increase in Round 2 = 20% of 4,800

⇒ Increase = (20 ÷ 100) × 4,800 = 960

⇒ Total votes in Round 2 = 4,800 + 960

⇒ Total votes in Round 2 = 5,760

∴ The correct answer is 5,760.

Additional Information

Percentage Increase Formula

New Value = Original Value × (1 + Rate/100). For 20% increase, Multiply by 1.2.

Successive Percentage Change

If a value changes by x% then y%, effective change = [x + y + (xy/100)]%.

Base Change Property

Always identify the correct base for percentage calculation; here the 20% gain is based on the 40% initial share, not the total population.

Shortcut Trick

To find the least number of beads such that no bead is left over, we calculate the LCM (Least Common Multiple) of the given numbers.

Given numbers: 24, 36, and 40.

LCM(24, 36, 40) = 360.

∴ The correct answer is 360 beads.

Alternate Method

Given: beads per necklace are 24, 36, or 40.

Formula Used: LCM by Prime Factorization Method.

⇒ Prime factors of 24 = 2 × 2 × 2 × 3 = 2³ × 3¹

⇒ Prime factors of 36 = 2 × 2 × 3 × 3 = 2² × 3²

⇒ Prime factors of 40 = 2 × 2 × 2 × 5 = 2³ × 5¹

⇒ LCM = Highest power of each prime factor = 2³ × 3² × 5¹

⇒ LCM = 8 × 9 × 5 = 360

∴ The correct answer is 360 beads.

Additional Information

Least Common Multiple (LCM)

The smallest positive integer that is divisible by each of the given numbers. Used for common timing or minimum quantity problems.

Highest Common Factor (HCF)

The largest positive integer that divides each of the numbers without a remainder. Used for maximum size or distribution problems.

Product Rule

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

Shortcut Trick

List the first 19 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67.

Check the unit digit of the sum: (2+3+5+7+1+3+7+9+3+9+1+7+1+3+7+3+9+1+7) → Sum of unit digits = 78.

The unit digit of the sum must be 8.

From the options, only 568 ends with 8.

∴ The correct answer is 568.

Alternate Method

Given: The first 19 prime numbers.

Formula Used: Sum of terms = S_n = a₁ + a₂ + ... + a_n

Calculation:

⇒ First 10 primes: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 129

⇒ Next 9 primes: 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 = 439

⇒ Total Sum = 129 + 439

⇒ Total Sum = 568

∴ The correct answer is 568.

Additional Information

Definition of Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Even Prime Number

2 is the only even prime number; all other prime numbers are odd.

Smallest Prime Numbers

The smallest prime number is 2, and the smallest odd prime number is 3. 1 is neither prime nor composite.

Shortcut Trick

Convert decimals: 5.12 = 512/100, 0.5488 = 5488/10000

Write in primes: 512 = 2⁹, 5488 = 2⁴×7³, 50625 = 3⁴×5⁴

Denominator: 100 × 10000 = 2⁶×5⁶

Combine powers ⇒ 2⁷ 3⁴ 5^-2 7³

∴ Answer = a + b + c + d = 12

Alternate Method

Given:

5.12 × 50625 × 0.5488 = 2^a 3^b 5^c 7^d

Formula used:

Convert decimals to fractions and use prime factorization with exponent rules

Calculation:

⇒ 5.12 = 512/100 = 2⁹ / (2²×5²) = 2⁷×5^-2

⇒ 50625 = 81 × 625 = 3⁴×5⁴

⇒ 0.5488 = 5488/10000 = (2⁴×7³) / (2⁴×5⁴) = 7³×5^-4

⇒ Multiply all: 2⁷ × 3⁴ × 5^-2 × 5⁴ × 7³ × 5^-4

⇒ = 2⁷ × 3⁴ × 5^-2+4-4 × 7³

⇒ = 2⁷ × 3⁴ × 5^-2 × 7³

⇒ So a = 7, b = 4, c = -2, d = 3

⇒ a + b + c + d = 7 + 4 − 2 + 3 = 12

∴ The correct answer is 12.

Additional Information

Decimal Conversion

Convert decimals into fractions to simplify multiplication and factorization.

Prime Factorization

Express numbers as products of primes like 2, 3, 5, 7 for exponent comparison.

Laws of Exponents

a^m × aⁿ = a^m+n, and a^-n = 1/aⁿ

Handling Negative Powers

Negative exponent indicates the factor is in denominator.

Shortcut Trick

Put θ = 60° to verify the expression quickly.

sec 60° = 2, cos 60° = 1/2, tan 60° = √3

Expression = (1 − 1/(1/2)) × (1 + 2)/(√3)²

⇒ (1 − 2) × (3 / 3) = −1 × 1 = −1

∴ The correct answer is -1.

Alternate Method

Given: (1 − 1/cosθ) × (1 + secθ) / tan²θ

Formula Used: secθ = 1/cosθ and tan²θ = sec²θ − 1

⇒ (1 − secθ) × (1 + secθ) / tan²θ

⇒ (1 − sec²θ) / tan²θ

⇒ Since sec²θ − tan²θ = 1, then 1 − sec²θ = −tan²θ

⇒ −tan²θ / tan²θ

⇒ −1

∴ The correct answer is -1.

Additional Information

Trigonometric Identities

Fundamental identity: sin²θ + cos²θ = 1. Also, 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.

Reciprocal Relations

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

Algebraic Identity Application

The difference of squares (a − b)(a + b) = a² − b² is frequently used to simplify trigonometric products.

Shortcut Trick

For four numbers a, b, c, d in proportion, the product of means equals the product of extremes: a × d = b × c.

Here, P × 11 = 22 × 8.

Divide by 11: P = 2 × 8 = 16.

∴ The correct answer is 16.

Alternate Method

Given: First proportional = P, Second = 22, Third = 8, Fourth = 11.

Formula Used: If a, b, c, d are in proportion, then a/b = c/d.

⇒ P / 22 = 8 / 11

⇒ P = (8 × 22) / 11

⇒ P = 8 × 2

⇒ P = 16

∴ The correct answer is 16.

Additional Information

Fourth Proportional

For three numbers a, b, and c, the fourth proportional d is calculated as d = (b × c) ÷ a.

Third Proportional

For two numbers a and b, the third proportional c is given by the relation a:b :: b:c, hence c = b² ÷ a.

Mean Proportional

The mean proportional between two numbers a and b is √(a × b).

Shortcut Trick

Radius increase = 37.5% = 3/8

Radius Ratio = 8 : 11 ⇒ Surface Area Ratio = 8² : 11² = 64 : 121

Increase in Area = 121 − 64 = 57 units

57 units = 35112 cm² ⇒ 1 unit = 35112 ÷ 57 = 616 cm²

Original Surface Area = 64 units × 616 = 39424 cm²

∴ The correct answer is 39,424 cm².

Alternate Method

Given: Increase in radius = 37.5%, Increase in surface area = 35,112 cm².

Formula Used: Surface Area of Sphere (A) = 4πr²

Calculations:

⇒ Let original radius (r) = 8x

⇒ New radius (R) = 8x + 37.5% of 8x = 8x + 3x = 11x

⇒ Original Area (A₁) = 4π(8x)² = 4π × 64x²

⇒ New Area (A₂) = 4π(11x)² = 4π × 121x²

⇒ Increase in Area = A₂ − A₁ = 4πx² × (121 − 64) = 4πx² × 57

⇒ 35112 = 57 × [4πx²]

⇒ 35112 = 57 × A₁ ÷ 64

⇒ A₁ = (35112 × 64) ÷ 57

⇒ A₁ = 616 × 64

⇒ A₁ = 39424 cm²

∴ The correct answer is 39,424 cm².

Additional Information

Volume of a Sphere

The amount of space inside a sphere is calculated as V = (4/3)πr³.

Successive Percentage for Area

If the radius increases by x%, the area increases by [2x + x²/100]%. Here, 2(37.5) + (37.5)²/100 = 75 + 14.0625 = 89.0625%.

Hemisphere Properties

Curved Surface Area (CSA) = 2πr² and Total Surface Area (TSA) = 3πr².

Shortcut Trick

Effective Cost Price (CP) = Purchase Price + Repair Costs.

CP = 850 + 200 = 1050.

Selling Price (SP) = 1200.

Profit % = (SP − CP) ÷ CP × 100.

Profit % = (150 ÷ 1050) × 100 = 1 ÷ 7 × 100 ≈ 14.29%.

∴ The correct answer is 14.29%.

Alternate Method

Given: Purchase Price = ₹850, Repair Cost = ₹200, SP = ₹1,200.

Formulas: Total CP = Purchase Price + Overhead Expenses | Profit = SP − CP | Profit% = (Profit ÷ CP) × 100.

⇒ Total Cost Price (CP) = 850 + 200 = ₹1,050.

⇒ Selling Price (SP) = ₹1,200.

⇒ Profit = SP − CP = 1200 − 1050 = ₹150.

⇒ Profit Percentage = (150 ÷ 1050) × 100.

⇒ Profit Percentage = (1 ÷ 7) × 100 = 14.2857...% ≈ 14.29%.

∴ The correct answer is 14.29%.

Additional Information

Overhead Expenses

Any extra money spent on repairs, transport, or labor after buying an article is always added to the purchase price to calculate the Effective Cost Price.

Percentage Conversion

The fraction 1/7 is approximately equal to 14.28%. In competitive exams, remembering these reciprocal values helps in faster calculation.

Shortcut Trick

The age difference between two people always remains constant throughout their lives.

Given: Difference after 24 years = 20 years → Present difference = 20 years.

Let Son's age = 1 unit, then Mother's age = 2 units.

Difference (2 − 1) = 1 unit → 1 unit = 20 years.

Present ages: Mother = 2 × 20 = 40 years, Son = 1 × 20 = 20 years.

∴ The correct answer is 40, 20 years.

Alternate Method

Given:

Present Age Ratio (Mother : Son) = 2 : 1

Age difference after 24 years = 20 years

⇒ Let the present age of the Son be x years.

⇒ Then, the present age of the Mother = 2x years.

⇒ After 24 years, Mother's age = 2x + 24 and Son's age = x + 24.

⇒ According to the question: (2x + 24) − (x + 24) = 20

⇒ 2x + 24 − x − 24 = 20

⇒ x = 20

⇒ Mother's present age = 2 × 20 = 40 years.

⇒ Son's present age = 20 years.

∴ The correct answer is 40, 20 years.

Additional Information

Constant Age Difference

The numerical difference between the ages of two persons never changes over time, regardless of how many years pass.

Ratio Balancing

If the age ratio changes over time, ensure the difference between ratio parts remains consistent with the actual age difference.