Quantitative Aptitude PYQs

Shortcut Trick

Equate decimal places: Total numerator places = 13, Total denominator places = 12, resulting in a factor of 10⁻¹.

Identify powers: 6859 = 19³, 54872 = 38³ = (2 × 19)³, and 1156 = 34² = (2 × 17)².

Simplify the ratios: (19/38)³ × (34/17)² × (64000/1600) × 0.1 = (1/8) × 4 × 40 × 0.1 = 2.

Calculate the final root: √2 ≈ 1.414.

∴ The correct answer is 1.414.

Alternate Method

Given Expression: √( (0.6859 × 0.1156 × 0.64000) / (0.289 × 0.1600 × 0.54872) )

⇒ Count total decimal places in Numerator: 4 + 4 + 5 = 13.

⇒ Count total decimal places in Denominator: 3 + 4 + 5 = 12.

⇒ Remove decimals by adjusting powers of 10: √( (6859 × 1156 × 64000) / (289 × 1600 × 54872) × 10⁻¹ )

⇒ Simplify the large numbers using prime factorization: 6859 = 19³, 1156 = 34², 289 = 17², 54872 = 38³.

⇒ √( (19³ × 34² × 40) / (17² × 38³ × 10) )

⇒ √( (19³ × (2 × 17)² × 40) / (17² × (2 × 19)³ × 10) )

⇒ √( (19³ × 4 × 17² × 40) / (17² × 8 × 19³ × 10) )

⇒ Cancel common terms: √( (4 × 40) / (8 × 10) ) = √(160 / 80) = √2

⇒ Square root of 2 up to three decimal places = 1.414

∴ The correct answer is 1.414.

Additional Information

Cubes of Numbers

Memorizing cubes up to 20 is crucial for simplification: 19³ = 6859 and 38³ = 54872.

Square Root Constants

Standard square root values frequently used in exams: √2 ≈ 1.414 and √3 ≈ 1.732.

Power and Indices Rule

For any numbers a and b: (a × b)ⁿ = aⁿ × bⁿ. This helps in simplifying ratios like (34/17)² = 2² = 4.

Shortcut Trick

Relative Speed (opposite direction) = 55 + 65 = 120 km/h

Total distance to be covered = 160 + 140 = 300 m

Time = Total Distance ÷ Relative Speed

Time = 300 ÷ (120 × 5/18) = 300 ÷ (100/3) = 9 sec

∴ The correct answer is 9 sec.

Alternate Method

Given:

Length of Train 1 (L₁) = 160 m

Length of Train 2 (L₂) = 140 m

Speed of Train 1 (S₁) = 55 km/h

Speed of Train 2 (S₂) = 65 km/h

Formula Used:

Time taken to cross = (L₁ + L₂) ÷ (S₁ + S₂)

Calculations:

⇒ Total Distance = 160 + 140 = 300 m

⇒ Relative Speed = 55 + 65 = 120 km/h

⇒ Convert speed to m/s = 120 × (5/18) m/s

⇒ Relative Speed = 600/18 = 100/3 m/s

⇒ Time = 300 ÷ (100/3)

⇒ Time = (300 × 3) ÷ 100

⇒ Time = 900 ÷ 100 = 9 sec

∴ The correct answer is 9 sec.

Additional Information

Relative Speed in Opposite Directions

When two objects move towards each other or in opposite directions, their relative speed is the sum of their individual speeds: V_rel = S₁ + S₂.

Unit Conversion Factor

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

Crossing a Stationary Object

When a train crosses a pole or a standing man, the distance covered is equal to the length of the train itself.

Shortcut Trick

Given difference (x − y) = 20 and mean proportional √xy = 24.

Square of mean proportional: xy = 24² = 576.

Using the identity: (x + y)² = (x − y)² + 4xy

⇒ (x + y)² = 20² + 4 × 576 = 400 + 2304 = 2704.

⇒ Sum (x + y) = √2704 = 52.

∴ The correct answer is 52.

Alternate Method

Let the two positive numbers be 'a' and 'b' (where a > b).

Given:

Difference between numbers (a − b) = 20

Mean proportional between a and b = 24

Formula Used:

1. Mean Proportional = √(a × b)

2. (a + b)² = (a − b)² + 4ab

⇒ From mean proportional definition: √ab = 24

⇒ Squaring both sides: ab = 24 × 24 = 576

⇒ We know that (a + b)² = (a − b)² + 4ab

⇒ (a + b)² = (20)² + 4 × 576

⇒ (a + b)² = 400 + 2304

⇒ (a + b)² = 2704

⇒ a + b = √2704

⇒ a + b = 52

∴ The correct answer is 52.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is the square root of their product, expressed as x = √ab.

Third Proportional

If a, b, and c are in continued proportion (a:b = b:c), then c is called the third proportional, where c = b² ÷ a.

Algebraic Relationship

The relationship between the sum and difference of two numbers is given by the identity (a + b)² − (a − b)² = 4ab.

Shortcut Trick

Combined efficiency of P and Q = 1/16, and efficiency of P alone = 1/24.

Work done by P and Q together in 4 days = 4 ÷ 16 = 1/4 of total work.

Remaining work to be finished by P alone = 1 − 1/4 = 3/4.

More days required by P = (3/4) × 24 = 18 days.

∴ The correct answer is 18 days.

Alternate Method

Given: P and Q can complete the task in 16 days; P alone takes 24 days; they work together for 4 days.

Formula Used: Total Work = LCM of individual days; Efficiency = Total Work ÷ Time.

⇒ Total work (LCM of 16 and 24) = 48 units.

⇒ Efficiency of P + Q = 48 ÷ 16 = 3 units/day.

⇒ Efficiency of P alone = 48 ÷ 24 = 2 units/day.

⇒ Work completed by P and Q in 4 days = 3 units/day × 4 days = 12 units.

⇒ Remaining work = 48 − 12 = 36 units.

⇒ Additional days for P alone = 36 units ÷ 2 units/day = 18 days.

∴ The correct answer is 18 days.

Additional Information

Work and Efficiency Relationship

Efficiency is inversely proportional to the time taken (E ∝ 1/T). If a person is twice as efficient, they take half the time.

Combined Efficiency

The total efficiency of multiple workers is the sum of their individual efficiencies (E_Total = E₁ + E₂ + ...).

Pipes and Cisterns Analogy

The same LCM and efficiency method applies to tank-filling problems, where inlet pipes perform positive work and outlet pipes perform negative work.

Shortcut Trick

Net deviation from the overall Average Height (164 cm) must be zero.

Deviation for Class A = (160 − 164) × 24 = −96

Deviation for Class B = (168 − 164) × 20 = +80

Total deviation (A + B) = −16. To balance, Class C must provide a deviation of +16.

Deviation per student in Class C = 165 − 164 = +1.

Number of students in Class C = 16 ÷ 1 = 16.

∴ The correct answer is 16.

Alternate Method

Given:

Class A: n_A = 24, Avg_A = 160 cm

Class B: n_B = 20, Avg_B = 168 cm

Class C: n_C = x, Avg_C = 165 cm

Overall Average = 164 cm

Formula Used:

Overall Average = (n_A × Avg_A + n_B × Avg_B + n_C × Avg_C) ÷ (n_A + n_B + n_C)

Calculations:

⇒ 164 = (24 × 160 + 20 × 168 + x × 165) ÷ (24 + 20 + x)

⇒ 164 = (3840 + 3360 + 165x) ÷ (44 + x)

⇒ 164(44 + x) = 7200 + 165x

⇒ 7216 + 164x = 7200 + 165x

⇒ 7216 − 7200 = 165x − 164x

⇒ x = 16

∴ The correct answer is 16.

Additional Information

Weighted Average Formula

The combined average of multiple groups is calculated as A_w = (n₁a₁ + n₂a₂ + ... + n_ka_k) ÷ (n₁ + n₂ + ... + n_k).

Deviation Method

The sum of deviations of all observations from their arithmetic mean is always zero. This property helps in finding missing values or averages quickly.

Average Property

The combined average must always lie between the lowest and the highest individual group averages. In this case, 160 < 164 < 168.

Shortcut Trick

Total Volume = 30 × 20 × 8 = 4800 cm³

Remaining Volume = 90% of 4800 = 4320 cm³

Volume of 1 Cube = 4³ = 64 cm³

Number of Cubes = 4320 ÷ 64 = 67.5 → 67 complete cubes

∴ The correct answer is 67.

Alternate Method

Given:

Dimensions of the solid cuboid = 30 cm × 20 cm × 8 cm

Material lost during melting = 10%

Side of the identical cubes = 4 cm

Calculations:

⇒ Total volume of the cuboid = Length × Breadth × Height

⇒ Volume = 30 × 20 × 8 = 4800 cm³

⇒ Volume of material lost = 10% of 4800 = (10 ÷ 100) × 4800 = 480 cm³

⇒ Volume of remaining metal = 4800 − 480 = 4320 cm³

⇒ Volume of one small cube = (Side)³ = 4³ = 64 cm³

⇒ Number of cubes formed = Remaining Volume ÷ Volume of one cube

⇒ Number of cubes = 4320 ÷ 64

⇒ Number of cubes = 67.5

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

∴ The correct answer is 67.

Additional Information

Volume of Solid Shapes

For a cuboid, Volume = L × B × H. For a cube, Volume = a³. During melting and recasting, total volume remains constant unless a percentage loss is specified.

Material Loss Calculation

If x% of material is lost, the usable volume is (100 − x)% of the initial volume. Usable Volume = V_initial × (1 − x/100).

Complete Units Rule

When calculating the number of discrete items formed from a bulk material, always round down to the nearest whole number if the result is a fraction.

Shortcut Trick

Calculate total interest percentage: Interest = 2529.60 − 1860 = 669.60

Total Interest % = (669.60 / 1860) × 100 = 36%

Since the rate is 12% per year, Time (T) = Total % / Rate per annum

T = 36% / 12% = 3 years

∴ The correct answer is 3 years.

Alternate Method

Given:

Principal (P) = ₹1,860

Amount (A) = ₹2,529.60

Rate (R) = 12% p.a.

Formula Used: Simple Interest (SI) = (P × R × T) / 100

⇒ Simple Interest (SI) = Amount − Principal

⇒ SI = 2529.60 − 1860 = 669.60

⇒ 669.60 = (1860 × 12 × T) / 100

⇒ 669.60 = 18.6 × 12 × T

⇒ 669.60 = 223.2 × T

⇒ T = 669.60 / 223.2

⇒ T = 3 years

∴ The correct answer is 3 years.

Additional Information

Simple Interest Formula

The basic formula is SI = (P × R × T) / 100, where P is Principal, R is Rate, and T is Time.

Amount Calculation

Total Amount (A) = Principal (P) + Simple Interest (SI). It can also be expressed as A = P(1 + RT/100).

Finding Rate of Interest

If time and interest are known, the rate can be calculated as R = (SI × 100) / (P × T).

Shortcut Trick

Apply the Successive Percentage Change formula: Net % change = x + y + (xy/100).

Here, x = +25% (increase) and y = −20% (decrease).

Net change = 25 − 20 + [25 × (−20) / 100] = 5 − 5 = 0%.

Since the net change is 0%, the final salary remains the same as the initial salary.

∴ The correct answer is ₹58,000.

Alternate Method

Given:

Initial Salary = ₹58,000

First Change = 25% Increase

Second Change = 20% Decrease

Step 1: Calculate salary after 25% increase

⇒ Salary after increase = 58,000 + (25/100 × 58,000)

⇒ Salary after increase = 58,000 + 14,500 = ₹72,500

Step 2: Calculate salary after 20% decrease on the new amount

⇒ Final Salary = 72,500 − (20/100 × 72,500)

⇒ Final Salary = 72,500 − 14,500

⇒ Final Salary = ₹58,000

∴ The correct answer is ₹58,000.

Additional Information

Successive Percentage Formula

When a value is changed by x% and then by y%, the effective change is (x + y + xy/100)%. Use positive for increase and negative for decrease.

Multiplier Method

Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100). In this case: 58,000 × 1.25 × 0.80 = 58,000 × 1 = 58,000.

Base Change Property

An increase of 25% (1/4) followed by a decrease of 20% (1/5) results in no change because (5/4) × (4/5) = 1.

Shortcut Trick

The Total Surface Area (TSA) of a cube follows the formula 6 × a².

Since the side is 94 cm, the unit digit of 94² is 6 (4 × 4 = 16).

Multiplying by 6, the final unit digit must be 6 (6 × 6 = 36).

Estimating: 94² is slightly less than 100² (10,000), so TSA is slightly less than 60,000.

Calculation: 6 × 8836 = 53016.

∴ The correct answer is 53,016 cm².

Alternate Method

Given:

The side length (a) of the cube is 94 cm.

Formula Used:

Total Surface Area of a cube = 6 × (side)²

Calculations:

⇒ TSA = 6 × (94)²

⇒ TSA = 6 × (94 × 94)

⇒ 94² = (100 − 6)² = 10000 − 1200 + 36 = 8836

⇒ TSA = 6 × 8836

⇒ TSA = 53016 cm²

∴ The correct answer is 53,016 cm².

Additional Information

Volume of a Cube

The volume represents the total capacity of the cube and is calculated as a³.

Lateral Surface Area (LSA)

The LSA refers to the area of only the four side faces (excluding top and bottom), given by 4 × a².

Body Diagonal of a Cube

The length of the longest rod that can be placed inside a cube is a√3.

Face Diagonal of a Cube

The diagonal of any single square face of the cube is calculated as a√2.

Shortcut Trick

Total work = LCM(12, 30) = 60 units.

Efficiency of A = 5 units/day, B = 2 units/day.

Work done by both in 2 days = (5 + 2) × 2 = 14 units.

Remaining work = 60 − 14 = 46 units.

Time taken by B alone = 46 ÷ 2 = 23 days.

∴ The correct answer is 23 days.

Alternate Method

Given: A completes work in 12 days and B in 30 days.

⇒ Total work (LCM of 12 and 30) = 60 units.

⇒ Work completed by A and B together in 2 days = (5 + 2) × 2 = 14 units.

⇒ Remaining work to be done by B = 60 − 14 = 46 units.

⇒ Time taken by B to complete the remaining work = Remaining Work ÷ Efficiency of B

⇒ Time = 46 ÷ 2 = 23 days.

∴ The correct answer is 23 days.

Additional Information

Relationship between Efficiency and Time

Efficiency is inversely proportional to Time. If time ratio is x : y, the efficiency ratio is y : x.

Work & Wages

Wages are generally distributed in the ratio of the work done by each person, which is the product of Efficiency × Time.

Total Work Calculation

Total work can be assumed as any multiple of the given days, but LCM is preferred to keep efficiencies in whole numbers.

Shortcut Trick

Total difference in the daily wage bill = Excess + Shortfall = 4000 + 2000 = ₹6000.

Difference in daily pay per worker = 500 − 450 = ₹50.

Total temporary workers = Total Difference ÷ Difference per Worker = 6000 ÷ 50 = 120.

∴ The correct answer is 120.

Alternate Method

Given: Pay Rate₁ = ₹500, Pay Rate₂ = ₹450, Budget Excess = ₹4,000, Budget Shortfall = ₹2,000.

Formula Used: Total Wage Bill = Number of Workers × Daily Pay Rate

⇒ Let n be the number of workers and B be the company’s budget.

⇒ Case 1: 500 × n = B + 4000

⇒ Case 2: 450 × n = B − 2000

⇒ Subtracting Case 2 from Case 1: (500n − 450n) = (B + 4000) − (B − 2000)

⇒ 50n = 4000 + 2000

⇒ 50n = 6000

⇒ n = 6000 ÷ 50 = 120.

∴ The correct answer is 120.

Additional Information

Excess and Deficit Rule

When comparing two scenarios where one results in an excess (E) and the other in a shortfall or deficit (D), the total difference between the two conditions is calculated as (E + D).

Linear Equations in One Variable

These problems follow the standard form ax + b = c, where x represents the unknown quantity (like the number of workers) and can be solved by isolating the variable.

Rate-Quantity Relationship

The change in total value is equal to the quantity multiplied by the change in rate: ΔTotal = Quantity × ΔRate.

Given:

(32 ÷ 0.8 × x) − 72 = 720 ÷ (4 × 5 + 10)

Formula used:

Solve using the BODMAS rule (Bracket, Of, Division, Multiplication, Addition, Subtraction).

Calculation:

Let the missing value be x.

(32 ÷ 0.8 × x) − 72 = 720 ÷ (4 × 5 + 10)

⇒ (40 × x) − 72 = 720 ÷ (20 + 10)

⇒ 40 × x − 72 = 720 ÷ 30

⇒ 40 × x − 72 = 24

⇒ 40 × x = 24 + 72

⇒ 40 × x = 96

⇒ x = 96 ÷ 40

⇒ x = 2.4

∴ The correct answer is option (3).

Shortcut Trick

The Third Proportional of two numbers a and b is given by the formula b² ÷ a.

⇒ Third Proportional = (16 × 16) ÷ 8

⇒ 256 ÷ 8 = 32

∴ The correct answer is 32.

Alternate Method

Given:

First number (a) = 8

Second number (b) = 16

Formula Used:

If three numbers a, b, and c are in continued proportion, then:

a : b :: b : c

⇒ a ÷ b = b ÷ c

Calculations:

⇒ Let the third proportional be x.

⇒ 8 : 16 :: 16 : x

⇒ 8 ÷ 16 = 16 ÷ x

⇒ 1 ÷ 2 = 16 ÷ x

⇒ x = 16 × 2

⇒ x = 32

∴ The correct answer is 32.

Additional Information

Mean Proportional

The Mean Proportional between two numbers a and b is calculated as √(a × b).

Fourth Proportional

For three numbers a, b, and c, the fourth proportional (x) satisfies a : b = c : x, which means x = (b × c) ÷ a.

Continued Proportion

Three quantities are in continued proportion if the ratio of the first to the second is equal to the ratio of the second to the third.

Shortcut Trick

List integers that leave remainder 2 when divided by 5: 2, 7, 12, 17, 22, ...

List integers that leave remainder 3 when divided by 7: 3, 10, 17, 24, ...

Identify the first common number that appears in both sequences.

The smallest positive integer satisfying both conditions is 17.

∴ The correct answer is 17.

Alternate Method

Given:

Number (N) ÷ 5 → Remainder = 2

Number (N) ÷ 7 → Remainder = 3

Step-by-step Calculation:

⇒ Let N = 5k + 2 and N = 7m + 3

⇒ 5k + 2 = 7m + 3

⇒ 5k = 7m + 1

⇒ If m = 1, 5k = 8 (Not an integer)

⇒ If m = 2, 5k = 7(2) + 1 = 15

⇒ k = 15 ÷ 5 = 3

⇒ Smallest N = 5(3) + 2 = 17

∴ The correct answer is 17.

Additional Information

Chinese Remainder Theorem (CRT)

CRT provides a systematic way to solve systems of linear congruences with pairwise coprime moduli like 5 and 7.

General Form of Solution

The general solution for such problems is N = (LCM(d₁, d₂) × n) + x, where x is the smallest positive integer and n = 0, 1, 2...

Successive Division Method

For larger numbers, use N = d₁k + r₁ and substitute into the second congruence to solve for k iteratively.

Shortcut Trick

Ratio of the two complementary angles is 3 : 2 (since 2 × Angle₁ = 3 × Angle₂).

Total units = 3 + 2 = 5 units.

Given, total sum = 90º (complementary property).

5 units = 90º ⇒ 1 unit = 18º.

Larger angle = 3 units = 3 × 18º = 54º.

∴ The correct answer is 54º.

Alternate Method

Given:

Sum of two angles = 90º

Relationship: 2 × (One angle) = 3 × (Other angle)

Formula Used:

For Complementary Angles, θ₁ + θ₂ = 90º.

Calculations:

Let the two angles be x and y.

⇒ x + y = 90º    ---(i)

⇒ 2x = 3y    ---(ii)

⇒ x = 3y / 2

Substituting the value of x in equation (i):

⇒ (3y / 2) + y = 90º

⇒ (3y + 2y) / 2 = 90º

⇒ 5y / 2 = 90º

⇒ 5y = 180º

⇒ y = 36º

Now, finding the value of x:

⇒ x = 90º − 36º

⇒ x = 54º

Comparing 54º and 36º, the larger angle is 54º.

∴ The correct answer is 54º.

Additional Information

Complementary Angles

Two angles are called complementary if their sum is exactly 90º. Each angle is the complement of the other.

Supplementary Angles

Two angles are called supplementary if their sum is exactly 180º. They form a linear pair when placed adjacently on a straight line.

Adjacent Angles

Angles that have a common vertex and a common side but do not overlap are known as adjacent angles.

Shortcut Trick

Convert decimals into their simplest fractions: 5 = 5/1, 8.3 = 83/10, and 0.19 = 19/100.

Formula: LCM of fractions = LCM of Numerators ÷ HCF of Denominators.

LCM(5, 83, 19) = 5 × 83 × 19 = 7885 (since all are co-prime).

HCF(1, 10, 100) = 1.

Result = 7885 ÷ 1 = 7885.

∴ The correct answer is 7885.

Alternate Method

Given: Numbers are 5, 8.3, and 0.19.

Formula Used: To find the LCM of decimals, equalize the number of decimal places and find the LCM of the resulting integers.

⇒ Step 1: Make decimal places equal: 5.00, 8.30, 0.19.

⇒ Step 2: Treat them as integers: 500, 830, and 19.

⇒ Step 3: Find prime factors:

⇒ 500 = 2² × 5³

⇒ 830 = 2 × 5 × 83

⇒ 19 = 19 (Prime)

⇒ Step 4: LCM = Highest powers of all prime factors = 2² × 5³ × 83 × 19.

⇒ LCM = 4 × 125 × 83 × 19 = 500 × 1577 = 788500.

⇒ Step 5: Adjust decimal: 788500 ÷ 100 = 7885.

∴ The correct answer is 7885.

Additional Information

LCM of Fractions

The Least Common Multiple (LCM) of a set of fractions is given by: (LCM of Numerators) ÷ (HCF of Denominators).

HCF of Fractions

The Highest Common Factor (HCF) of a set of fractions is given by: (HCF of Numerators) ÷ (LCM of Denominators).

Co-prime Numbers

Numbers that have only 1 as a common factor are co-prime. The LCM of co-prime numbers is always equal to their product.

Shortcut Trick

When an object covers equal distances at different speeds, the Average Speed is the Harmonic Mean of the speeds.

Average Speed = 3 ÷ (1/v₁ + 1/v₂ + 1/v₃)

⇒ Average Speed = 3 ÷ (1/40 + 1/60 + 1/120)

⇒ Average Speed = 3 ÷ (6/120) = 3 × 20 = 60 km/hr.

∴ The correct answer is 60 km/hr.

Alternate Method

Given: Three segments of 40 km each. Speeds = 40 km/hr, 60 km/hr, and 120 km/hr.

Formula Used: Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Total Distance = 40 + 40 + 40 = 120 km

⇒ Time for 1^st segment (t₁) = 40 ÷ 40 = 1 hour

⇒ Time for 2^nd segment (t₂) = 40 ÷ 60 = 2/3 hour

⇒ Time for 3^rd segment (t₃) = 40 ÷ 120 = 1/3 hour

⇒ Total Time = 1 + 2/3 + 1/3 = 1 + 1 = 2 hours

⇒ Average Speed = 120 ÷ 2 = 60 km/hr

∴ The correct answer is 60 km/hr.

Additional Information

Two Equal Distance Segments

If a journey is divided into two equal halves with speeds x and y, the average speed is 2xy ÷ (x + y).

Unequal Distance Segments

For unequal distances d₁, d₂, ... with speeds v₁, v₂, ..., Average Speed = (d₁ + d₂ + ...) ÷ (d₁/v₁ + d₂/v₂ + ...).

Unit Conversion

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

Shortcut Trick

Use the Weighted Average method for calculation.

Ratio of goods quantity = 50% : (40% of 50%) : (Remaining 30%) = 50 : 20 : 30 = 5 : 2 : 3

Overall Profit x% = (5 × 36 + 2 × 14 + 3 × (−21)) ÷ (5 + 2 + 3)

x = (180 + 28 − 63) ÷ 10 = 145 ÷ 10 = 14.5%

∴ The correct answer is 14.5%.

Alternate Method

Given: 50% goods at 36% profit, 40% of remaining (50%) at 14% profit, and still remaining (30%) at 21% loss.

Formula Used: Overall Profit% = (Total Profit ÷ Total CP) × 100

Let the Total Cost Price (CP) of 100 units of goods be ₹100.

⇒ Case 1: CP = ₹50 | Profit = 36% of 50 = 50 × 0.36 = ₹18

⇒ Remaining goods = 100 − 50 = 50 units

⇒ Case 2: CP = 40% of 50 = 20 units | Profit = 14% of 20 = 20 × 0.14 = ₹2.8

⇒ Still Remaining goods = 50 − 20 = 30 units

⇒ Case 3: CP = 30 units | Loss = 21% of 30 = 30 × 0.21 = ₹6.3 (Loss is −6.3)

⇒ Total Profit = 18 + 2.8 − 6.3 = 14.5

⇒ x% = (14.5 ÷ 100) × 100 = 14.5%

∴ The correct answer is 14.5%.

Additional Information

Weighted Average Profit/Loss

Overall Profit % = (n₁p₁ + n₂p₂ + ... + n_kp_k) ÷ (n₁ + n₂ + ... + n_k), where n is quantity and p is profit percentage (negative for loss).

Successive Changes vs Parts

When goods are divided into parts, we use weighted averages. When one value changes multiple times, we use successive percentage formula: a + b + (ab/100).

Profit and Loss Basics

Profit % = (Profit ÷ CP) × 100 | SP = CP × (100 + P%) ÷ 100. Always calculate profit and loss on the Cost Price (CP) unless stated otherwise.

Shortcut Trick

The Total Sum is simply the product of the Average and the Total count of numbers.

Total Sum = Average × Number of items

Total Sum = 7 × 3 = 21

∴ The correct answer is 21.

Alternate Method

Given:

Average of three numbers = 7

Number of quantities (n) = 3

Formula Used:

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

Calculations:

⇒ 7 = (Sum of three numbers) ÷ 3

⇒ Sum of three numbers = 7 × 3

⇒ Sum of three numbers = 21

∴ The correct answer is 21.

Additional Information

Arithmetic Mean

The most common type of average, calculated by adding all numbers in a set and dividing by the count of numbers.

Weighted Average

Used when different values have different levels of importance or frequency: (w₁x₁ + w₂x₂) ÷ (w₁ + w₂).

Average of Consecutive Numbers

For a series in Arithmetic Progression (AP), the average is simply (First Term + Last Term) ÷ 2.

Shortcut Trick

Use the algebraic identity: (Difference)² = (Sum)² − 4 × (Product)

⇒ (Difference)² = 27² − 4 × 176

⇒ (Difference)² = 729 − 704 = 25

⇒ Difference = √25 = 5

∴ The correct answer is 5.

Alternate Method

Given: Sum of two numbers = 27, Product = 176.

Formula Used: Factorisation method to find the numbers x and y such that x + y = 27 and xy = 176.

⇒ Let the two positive numbers be x and y.

⇒ x + y = 27 and x × y = 176.

⇒ Factorizing 176 to find pairs whose sum is 27: 176 = 1 × 176, 2 × 88, 4 × 44, 8 × 22, 11 × 16.

⇒ Since 11 + 16 = 27, the numbers are 16 and 11.

⇒ Positive difference = 16 − 11 = 5.

∴ The correct answer is 5.

Additional Information

Algebraic Relationship

The core relationship between sum, product, and difference is given by (x + y)² − (x − y)² = 4xy.

Quadratic Equation Formulation

If sum (S) and product (P) of two numbers are known, the numbers are the roots of the quadratic equation t² − St + P = 0.

Nature of Roots

For the difference to be a rational number, the value (Sum² − 4 × Product) must be a perfect square.