Quantitative Aptitude PYQs

Shortcut Trick

Net filling efficiency = 1/8 − 1/40 = (5 − 1)/40 = 4/40 = 1/10 per minute.

Time to fill full tank = 10 minutes.

Time to fill one-half of the tank = 10 ÷ 2 = 5 minutes.

∴ The correct answer is 5 minutes.

Alternate Method

Given: Filling pipe time = 8 min, Emptying pipe time = 40 min.

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

⇒ Combined efficiency = 5 − 1 = 4 units/min

⇒ One-half tank capacity = 40 ÷ 2 = 20 units

⇒ Time required = 20 ÷ 4 = 5 minutes

∴ The correct answer is 5 minutes.

Additional Information

Net Work Done

When one pipe fills (rate x) and another empties (rate y), the net rate is (1/x − 1/y) per unit time.

Pipes and Cisterns Logic

Inlet pipes have positive efficiency, while outlet (leakage) pipes have negative efficiency.

Partial Filling

If only a fraction k of the tank needs to be filled, the time taken is k × (Total Capacity ÷ Net Efficiency).

Shortcut Trick

Let the Base be b. Then the Altitude (h) = (2/3)b.

Area = 1/2 × Base × Altitude ⇒ 84 = 1/2 × b × (2/3)b.

84 = 1/3 × b² ⇒ b² = 252.

b = √252 = √(36 × 7) = 6√7 cm.

∴ The correct answer is 6√7 cm.

Alternate Method

Given:

⇒ Area of the triangle = 84 cm²

⇒ Altitude (h) = 2/3 of Base (b)

Formula Used:

⇒ Area of a triangle = 1/2 × Base × Altitude

Calculations:

⇒ Area = 1/2 × b × h

⇒ 84 = 1/2 × b × (2/3)b

⇒ 84 = (1/3) × b²

⇒ b² = 84 × 3

⇒ b² = 252

⇒ b = √252

⇒ b = √(36 × 7)

⇒ b = 6√7 cm

∴ The correct answer is 6√7 cm.

Additional Information

Heron's Formula

For a triangle with sides a, b, and c, Area = √[s(s−a)(s−b)(s−c)], where s (semi-perimeter) = (a + b + c) / 2.

Equilateral Triangle Area

The area of an equilateral triangle with side 'a' is given by (√3 / 4) × a².

Right-Angled Triangle

Area = 1/2 × Product of the two sides containing the right angle.

Shortcut Trick

Mean proportional of X and 125 is 25.

By property: X × 125 = 25²

⇒ X × 125 = 625

⇒ X = 625 ÷ 125 = 5

∴ The correct answer is 5.

Alternate Method

Given: First term = X, Third term = 125, Mean Proportional = 25.

Formula Used: If b is the mean proportional between a and c, then b = √(a × c) or b² = a × c.

Calculations:

⇒ 25 = √(X × 125)

Squaring both sides:

⇒ 25² = X × 125

⇒ 625 = X × 125

⇒ X = 625 ÷ 125

⇒ X = 5

∴ The correct answer is 5.

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 :: b:c, then c is called the third proportional to a and b, calculated as c = b² ÷ a.

Fourth Proportional

In the proportion a:b :: c:d, d is the fourth proportional, where d = (b × c) ÷ a.

Shortcut Trick

Assume a base mean of 50 and calculate the deviations: −21, +40, 0, −24, −14, +27, +36, −17, −9.

Net deviation ∑d = (−21 − 24 − 14 − 17 − 9) + (40 + 27 + 36) = −85 + 103 = +18.

Correction in mean = Net deviation ÷ Count = 18 ÷ 9 = +2.

Actual Mean = Base + Correction = 50 + 2 = 52.

∴ The correct answer is 52.

Alternate Method

Given: Observations = 29, 90, 50, 26, 36, 77, 86, 33, 41

Formula Used: Arithmetic Mean = Sum of all observations ÷ Total number of observations

⇒ Total number of observations (n) = 9

⇒ Sum = 29 + 90 + 50 + 26 + 36 + 77 + 86 + 33 + 41

⇒ Sum = 468

⇒ Mean = 468 ÷ 9

⇒ Mean = 52

∴ The correct answer is 52.

Additional Information

Properties of Arithmetic Mean

The sum of the deviations of all observations from their arithmetic mean is always zero (∑(x_i − x̄) = 0).

Combined Mean Formula

If two groups have sizes n₁, n₂ and means x̄₁, x̄₂, their combined mean is (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂).

Effect of Operations

If every observation is increased, decreased, multiplied, or divided by a constant 'k', the mean also changes by the same constant.

Shortcut Trick

Sum of ages after 6 years = 42 + (2 × 6) = 54 years.

Ratio of A : B after 6 years = 2 : 1.

Total units (2 + 1) = 3 units → 54 years.

1 unit (B's age after 6 years) = 54 ÷ 3 = 18 years.

B's present age = 18 − 6 = 12 years.

∴ The correct answer is 12 years.

Alternate Method

Given: Sum of A and B = 42 years. After 6 years, A = 2 × B.

Formula Used: Linear equation in one variable.

Let B's present age be x years.

⇒ A's present age = 42 − x

⇒ After 6 years, A's age = (42 − x) + 6 = 48 − x

⇒ After 6 years, B's age = x + 6

⇒ According to question: 48 − x = 2 × (x + 6)

⇒ 48 − x = 2x + 12

⇒ 3x = 48 − 12

⇒ 3x = 36

⇒ x = 12

∴ The correct answer is 12 years.

Additional Information

Age Ratio Property

When adding years to a sum of ages, multiply the number of years by the number of people: Total Sum_future = Sum_present + (n × years).

Constant Difference

The difference between the ages of two people always remains constant, regardless of the number of years passed.

Shortcut Trick

Walking time = (Distance ÷ Speed) × 60 = (6 ÷ 12) × 60 = 30 minutes.

Rest stops = Distance − 1 = 6 − 1 = 5 stops (No rest after the last km).

Rest time = 5 × 7 = 35 minutes.

Total time = 30 + 35 = 65 minutes.

∴ The correct answer is 65 minutes.

Alternate Method

Given: Speed = 12 km/hr, Distance = 6 km, Rest duration = 7 mins after every 1 km.

Formula Used: Time = Distance ÷ Speed; Total Time = Walking Time + Resting Time.

⇒ Walking time = (6 ÷ 12) hours

⇒ Walking time in minutes = 0.5 × 60 = 30 minutes

⇒ Number of rest intervals = (6 ÷ 1) − 1 = 5

⇒ Total rest duration = 5 × 7 = 35 minutes

⇒ Total time taken = 30 + 35 = 65 minutes

∴ The correct answer is 65 minutes.

Additional Information

Rest Stops Calculation

In problems involving stops, if a person stops every n km, the number of stops is (Total Distance ÷ n), but if the division is exact, subtract 1 because the last stop is at the destination.

Average Speed with Stoppages

Average speed including stoppages = Total Distance ÷ (Walking Time + Stopping Time). Stoppage time per hour = (Fast Speed − Slow Speed) ÷ Fast Speed.

Shortcut Trick

For any point P inside a rectangle ABCD, the British Flag Theorem states:

PA² + PC² = PB² + PD²

⇒ 100² + 85² = 76² + PD²

⇒ 10000 + 7225 = 5776 + PD²

⇒ 17225 − 5776 = PD² ⇒ PD² = 11449

⇒ PD = √11449 = 107 cm

∴ The correct answer is 107 cm.

Alternate Method

Given:

PA = 100 cm, PB = 76 cm, PC = 85 cm

Formula Used:

In a rectangle ABCD, if P is any point in its plane, then PA² + PC² = PB² + PD².

A
B
C
D
P
100 cm
76 cm

Shortcut Trick

The total number always represents 100%.

Since adding 20% to 84 gives the full 100%, the value 84 must represent 80% of the number.

We need to find 75% of the same number.

Required Value = (84 ÷ 80) × 75 = 1.05 × 75 = 78.75

∴ The correct answer is 78.75.

Alternate Method

Given:

20% of a number + 84 = The same number

Formula Used:

Total Number = 100%

20% Added Value = 84 (80% part) Total Number (100%) = 105

Calculations:

⇒ Let the number be x.

⇒ 20% of x + 84 = x

⇒ 84 = x − 20% of x

⇒ 84 = 100% of x − 20% of x

⇒ 84 = 80% of x

⇒ x = (84 × 100) ÷ 80

⇒ x = 8400 ÷ 80 = 105

⇒ Now, we need to find 75% of the same number (x).

⇒ 75% of 105 = (75 ÷ 100) × 105

⇒ 75% of 105 = 0.75 × 105 = 78.75

∴ The correct answer is 78.75.

Additional Information

Percentage Base Rule

In any percentage calculation, the original or total quantity is always considered as the 100% base value.

Percentage to Fraction Conversion

To simplify percentage calculations, convert them into fractions by dividing the percentage value by 100 (e.g., 75% = 75 ÷ 100 = 3/4).

Percentage Change

Percentage change is calculated using the formula: ((Final Value − Initial Value) ÷ Initial Value) × 100.

Shortcut Trick
Increase in average = (New score − Old average) ÷ Total innings
Increase = (231 − 97) ÷ 7
Increase = 134 ÷ 7 ≈ 19.14
Rounding to the nearest whole number, the increase is 19.
∴ The correct answer is 19 runs.
Alternate Method
Given:
Average for 6 innings = 97
Runs in 7^th inning = 231

Formula Used:
Average = Total runs ÷ Number of innings
Calculations:
⇒ Total runs in 6 innings = 97 × 6 = 582
⇒ Total runs in 7 innings = 582 + 231 = 813
⇒ New average = 813 ÷ 7 ≈ 116.14
⇒ Increase in average = 116.14 − 97 = 19.14
⇒ Rounded to nearest whole number = 19
∴ The correct answer is 19 runs.
Additional Information
Basic Average Formula
The average (Arithmetic Mean) is calculated by dividing the sum of all observations by the total number of observations.
Net Change in Average
When a new value is added, the new average increases if the new value > old average, and decreases if new value < old average.
Deviation Method
Change in average = Total deviation ÷ New total count. This avoids large multiplications during exams.

Shortcut Trick

To find the Square of 2.99, use the Algebraic Identity (a − b)² = a² − 2ab + b².

Let a = 3 and b = 0.01.

(3 − 0.01)² = 3² − (2 × 3 × 0.01) + (0.01)²

= 9 − 0.06 + 0.0001 = 8.9401.

Comparing with the given choices, this matches Option 4.

∴ The correct answer is 8.9401.

Alternate Method

Given: Find the value of (2.99)².

Formula Used: Direct multiplication of decimal numbers.

⇒ (2.99)² = 2.99 × 2.99

⇒ 2.99 = 299/100

⇒ (299/100) × (299/100)

⇒ (299 × 299) / (100 × 100)

⇒ 89401 / 10000

⇒ 8.9401

This result is represented by Option 4.

∴ The correct answer is 8.9401.

Additional Information

Square of a Number (a + b)²

Another useful identity is (a + b)² = a² + 2ab + b², which helps in calculating squares of numbers slightly above a base.

Decimal Multiplication Rule

When multiplying two decimals, the number of decimal places in the product is the sum of the decimal places in the numbers being multiplied.

Difference of Squares

The identity a² − b² = (a − b)(a + b) is helpful for mental math and fast simplification of products.

Shortcut Trick

Expand brackets mentally by distributing the coefficients: a³ + 2a², −3a³ + 9a, and −5a² − 25a.

Combine cubic terms: (1 − 3)a³ = −2a³.

Combine square terms: (2 − 5)a² = −3a².

Combine linear terms: (9 − 25)a = −16a.

∴ The correct answer is −2a³ − 3a² − 16a.

Alternate Method

Given Expression: a²(a + 2) − 3a(a² − 3) − 5a(a + 5)

Formula Used: Distributive Property of multiplication over addition/subtraction: x(y + z) = xy + xz.

Step-by-step Calculation:

⇒ a²(a) + a²(2) − [3a(a²) − 3a(3)] − [5a(a) + 5a(5)]

⇒ (a³ + 2a²) − (3a³ − 9a) − (5a² + 25a)

⇒ a³ + 2a² − 3a³ + 9a − 5a² − 25a

⇒ (a³ − 3a³) + (2a² − 5a²) + (9a − 25a)

⇒ −2a³ − 3a² − 16a

∴ The correct answer is −2a³ − 3a² − 16a.

Additional Information

Like Terms

Terms having the same variable raised to the same power are called like terms.

Only like terms can be added or subtracted.

BODMAS in Algebra

Always simplify the brackets first using the distributive law before performing addition or subtraction across terms.

Exponent Rules

When multiplying terms with the same base, add their exponents: a^m × aⁿ = a^m+n.

Shortcut Trick

Difference in annual Interest Rate = 7% − 3.5% = 3.5%

Total Interest % difference for 4 years = 3.5% × 4 = 14%

Required Difference = 14% of ₹4,60,000

⇒ 0.14 × 4,60,000 = ₹64,400

∴ The correct answer is ₹64,400.

Alternate Method

Given: Principal (P) = ₹4,60,000, Time (T) = 4 years, Rates = 3.5% and 7%

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

⇒ SI paid to Bank A = (4,60,000 × 3.5 × 4) / 100

⇒ SI_A = 4,600 × 14 = ₹64,400

⇒ SI paid to Bank B = (4,60,000 × 7 × 4) / 100

⇒ SI_B = 4,600 × 28 = ₹1,28,800

⇒ Positive Difference = SI_B − SI_A = 1,28,800 − 64,400

⇒ Difference = ₹64,400

∴ The correct answer is ₹64,400.

Additional Information

Simple Interest Difference Rule

When Principal and Time are constant, the difference in SI is given by: ΔSI = P × (R₂ − R₁) × T / 100.

Total Amount Formula

The total amount (A) to be repaid is the sum of the Principal and the Simple Interest: A = P + SI.

Effect of Rate on Interest

Simple Interest is directly proportional to the Rate. Doubling the rate while keeping P and T constant will double the interest earned.

Shortcut Trick

For equal distances, use the Harmonic Mean formula for average speed.

Average Speed = (2 × x × y) ÷ (x + y)

Average Speed = (2 × 33 × 57) ÷ (33 + 57) = 3762 ÷ 90

Average Speed = 41.8 km/hr

∴ The correct answer is 41.8 km/hr.

Alternate Method

Given:

Speed during onward journey (x) = 33 km/hr

Speed during return journey (y) = 57 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Let the distance between home and mall be D km.

⇒ Total Distance = D + D = 2D

⇒ Time 1 = D ÷ 33

⇒ Time 2 = D ÷ 57

⇒ Average Speed = 2D ÷ (D/33 + D/57)

⇒ Average Speed = 2 ÷ ((57 + 33) / (33 × 57))

⇒ Average Speed = (2 × 33 × 57) ÷ 90

⇒ Average Speed = 3762 ÷ 90 = 41.8 km/hr

∴ The correct answer is 41.8 km/hr.

Additional Information

Average Speed (Different Distances)

When distances are not equal, use Total Distance (d₁ + d₂ + ...) ÷ Total Time (t₁ + t₂ + ...).

Relative Speed

Same direction: S_relative = S₁ − S₂. Opposite direction: S_relative = S₁ + S₂.

Unit Conversion

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

Shortcut Trick

In a 30°-60°-90° triangle, the side opposite to 60° is √3 times the side opposite to 30°.

Given: Side opposite to 60° (Height) = 279 m.

Distance (Side opposite to 30°) = 279 ÷ √3 = (93 × 3) ÷ √3 = 93√3 m.

∴ The correct answer is 93√3 m.

Alternate Method

Given:

Height of lighthouse (h) = 279 m

Angle of depression = 60° ⇒ Angle of elevation from ship = 60°

Formula Used:

tan(θ) = Perpendicular ÷ Base

Calculations:

⇒ tan(60°) = Height ÷ Distance

⇒ √3 = 279 ÷ Distance

⇒ Distance = 279 ÷ √3

⇒ Distance = (93 × 3) ÷ √3

⇒ Distance = 93√3 m

∴ The correct answer is 93√3 m.

Additional Information

Angle of Depression and Elevation

The angle of depression from an observer to an object is numerically equal to the angle of elevation from the object to the observer due to alternate interior angles.

Standard Trigonometric Ratios

For height and distance problems: tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3.

30-60-90 Triangle Ratio

The ratio of sides opposite to 30°, 60°, and 90° is always 1 : √3 : 2 respectively.

Shortcut Trick

Base Income = 100, Savings = 17, Expenditure = 83

Increase in Income = 38% × 100 = 38

Increase in Expenditure = 20% × 83 = 16.6

Increase in Savings (Units) = 38 − 16.6 = 21.4

Actual Increase = (61,500 ÷ 100) × 21.4 = 615 × 21.4 = ₹13,161

∴ The correct answer is increase by ₹13,161.

Alternate Method

Given:

Initial Income = ₹61,500

Initial Savings = 17% of 61,500

Initial Expenditure = 83% of 61,500

⇒ Initial Savings = 61,500 × (17 ÷ 100) = ₹10,455

⇒ Initial Expenditure = 61,500 − 10,455 = ₹51,045

⇒ New Income = 61,500 × 1.38 = ₹84,870

⇒ New Expenditure = 51,045 × 1.20 = ₹61,254

⇒ New Savings = 84,870 − 61,254 = ₹23,616

⇒ Change in Savings = 23,616 − 10,455 = ₹13,161

∴ The correct answer is increase by ₹13,161.

Additional Information

Income-Expenditure Relation

The basic formula is Income = Expenditure + Savings. Any change in two components directly affects the third.

Percentage Change

To find the new value after an x% increase, multiply the original value by (1 + x/100).

Alligation in Income-Exp

Percentage change in Income is the weighted average of percentage changes in Expenditure and Savings based on their initial ratio.

Shortcut Trick

Profit per magazine = Selling Price − Cost Price = 23 − 20 = 3.

Profit percentage = (Profit ÷ Cost Price) × 100.

Profit % = (3 ÷ 20) × 100 = 3 × 5 = 15%.

∴ The correct answer is 15%.

Alternate Method

Given:

Cost Price (CP) of magazine = ₹20

Selling Price (SP) of magazine = ₹23

Formula: Profit% = [(SP − CP) ÷ CP] × 100

⇒ Profit = 23 − 20 = 3

⇒ Profit % = (3 ÷ 20) × 100

⇒ Profit % = 0.15 × 100 = 15%

∴ The correct answer is 15%.

Additional Information

Profit and Loss Calculation

Profit occurs when Selling Price (SP) is greater than Cost Price (CP). Profit = SP − CP.

Percentage Basis

Profit or loss percentage is always calculated on the Cost Price (CP) unless stated otherwise in the problem.

Shortcut Trick

Assume the number is the LCM of denominators (4 and 5) = 20 units.

Three-fourth of 20 = 15 units.

Two-fifth of 20 = 8 units.

Difference (15 − 8) = 7 units → 63.

1 unit = 9 ∴ 20 units = 20 × 9 = 180.

∴ The correct answer is 180.

Alternate Method

Given: Difference between 3/4 and 2/5 of a number is 63.

Formula: Let the number be 'x'.

⇒ (3/4)x − (2/5)x = 63

⇒ [(15 − 8) / 20]x = 63

⇒ (7 / 20)x = 63

⇒ x = (63 × 20) / 7

⇒ x = 9 × 20

⇒ x = 180

∴ The correct answer is 180.

Additional Information

Fraction to Percentage

To solve quickly, convert fractions to percentages: 3/4 = 75% and 2/5 = 40%. The difference 35% of x = 63.

LCM Method for Fractions

Always assume the total quantity as the LCM of denominators to avoid decimal or fractional calculations during steps.

Algebraic Translation

The word 'exceeds' translates to a subtraction equation (Larger Value − Smaller Value = Difference).

Shortcut Trick

Shift vertex B(3, −2) to the origin (0, 0) by subtracting (3, −2) from all vertices.

New coordinates: A'(1, −4), B'(0, 0), and C'(2, 4).

Area of triangle = 1/2 |x₁y₂ − x₂y₁|

Area = 1/2 |(1 × 4) − (2 × −4)| = 1/2 |4 + 8| = 6 sq. units.

∴ The correct answer is 6 sq. units.

Alternate Method

Given: Vertices of the triangle are A(4, −6), B(3, −2), and C(5, 2).

Formula Used: Area = 1/2 |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

⇒ Area = 1/2 |4(−2 − 2) + 3(2 − (−6)) + 5(−6 − (−2))|

⇒ Area = 1/2 |4(−4) + 3(2 + 6) + 5(−6 + 2)|

⇒ Area = 1/2 |−16 + 3(8) + 5(−4)|

⇒ Area = 1/2 |−16 + 24 − 20|

⇒ Area = 1/2 |−12|

⇒ Area = 6 sq. units.

∴ The correct answer is 6 sq. units.

Additional Information

Collinearity of Points

Three points are collinear if the area of the triangle formed by them is zero.

Centroid of a Triangle

The centroid G(x, y) of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is calculated as ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).

Distance Formula

The distance between two points (x₁, y₁) and (x₂, y₂) is given by √((x₂ − x₁)² + (y₂ − y₁)²).

Shortcut Trick

Factor out perfect squares to express all terms as multiples of √3.

√75 = 5√3, √108 = 6√3, √6075 = 45√3

Expression = (5√3 − 6√3 + 45√3)²

Sum inside bracket = 44√3

Square the sum = (44√3)² = 1936 × 3 = 5808

∴ The correct answer is 5808.

Alternate Method

Given:

(√75 − √108 + √6075)²

Calculations:

⇒ Simplifying each surd independently:

⇒ √75 = √(25 × 3) = 5√3

⇒ √108 = √(36 × 3) = 6√3

⇒ √6075 = √(2025 × 3) = 45√3

⇒ Substituting the simplified surds back into the expression:

⇒ (5√3 − 6√3 + 45√3)²

⇒ Grouping the terms together:

⇒ [(5 − 6 + 45)√3]²

⇒ [44√3]²

⇒ Squaring both parts inside the bracket:

⇒ 44² × (√3)²

⇒ 1936 × 3 = 5808

∴ The correct answer is 5808.

Additional Information

Properties of Surds

The multiplication property √(a × b) = √a × √b helps extract perfect squares to simplify complex roots easily.

Perfect Squares

Memorising squares of numbers up to 50 is extremely useful. For example, knowing 45² = 2025 instantly identifies √6075 as 45√3.

BODMAS / PEMDAS Rule

Always simplify the operations inside the Brackets first before applying the exponent (power of 2) to the overall expression.

Shortcut Trick

Net percentage change formula: Net % = a + b + (a × b) ÷ 100

Here, markup a = +29% and discount b = −7%.

Net Gain% = 29 − 7 + (29 × (−7)) ÷ 100

Net Gain% = 22 − 203 ÷ 100 = 22 − 2.03

Net Gain% = 19.97%.

∴ The correct answer is 19.97% gain.

Alternate Method

Given: Markup = 29% above CP, Discount = 7% on MP.
Formulas: MP = CP × (100 + Markup%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100 | Profit% = (SP − CP) ÷ CP × 100.

⇒ Let Cost Price (CP) = 100 units.

⇒ Marked Price (MP) = 100 + 29% of 100 = 129 units.

⇒ Selling Price (SP) = 129 − 7% of 129 = 129 − (7 × 1.29) = 129 − 9.03 = 119.97 units.

⇒ Gain = SP − CP = 119.97 − 100 = 19.97 units.

⇒ Gain % = (19.97 ÷ 100) × 100 = 19.97%.

∴ The correct answer is 19.97% gain.

Additional Information

Relationship between CP, MP, and Discount

The ratio of Marked Price to Cost Price can be expressed as: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Successive Percentage Change

When a value is increased by x% and then decreased by y%, the net effect is x − y − (xy ÷ 100)%. If the result is positive, it is a gain; if negative, it is a loss.