Quantitative Aptitude PYQs

Shortcut Trick

Effective Selling Price percentage = 100% − 20% = 80%

Selling Price = 80% of 1000

Selling Price = 0.8 × 1000 = 800

∴ The correct answer is ₹800.

Alternate Method

Given:

Marked Price (MP) = ₹1000

Discount = 20%

Formula Used:

Selling Price (SP) = Marked Price × [1 − (Discount % ÷ 100)]

Calculations:

⇒ Discount Amount = 20% of 1000

⇒ Discount Amount = (20 ÷ 100) × 1000 = ₹200

⇒ Selling Price = Marked Price − Discount Amount

⇒ Selling Price = 1000 − 200 = ₹800

∴ The correct answer is ₹800.

Additional Information

Marked Price (MP)

The price printed on the label or tag of an item is known as the Marked Price or List Price.

Discount Percentage

Discount is always calculated on the Marked Price. Formula: Discount % = (Discount ÷ MP) × 100.

Successive Discounts

If two discounts d₁% and d₂% are given, the single equivalent discount is [d₁ + d₂ − (d₁ × d₂ ÷ 100)]%.

Shortcut Trick

The Area of a circle can be directly calculated from the diameter (d) using the formula: Area = (π ÷ 4) × d²

Given diameter = 30 m

Area = (3.14 ÷ 4) × 30² = 0.785 × 900

Area = 706.5 m²

∴ The correct answer is 706.5 m².

Alternate Method

Given:

Diameter (d) of the circular garden = 30 m

Formula Used:

Radius (r) = Diameter ÷ 2

Area of a circle = π × r² (Taking π ≈ 3.14)

Calculations:

⇒ Radius (r) = 30 ÷ 2 = 15 m

⇒ Area = 3.14 × (15)²

⇒ Area = 3.14 × 225

⇒ Area = 706.5 m²

∴ The correct answer is 706.5 m².

Additional Information

Circumference of a Circle

The distance around the circle, calculated as C = 2 × π × r or C = π × d.

Area of a Semi-circle

Half the area of a full circle, given by Area = (π × r²) ÷ 2.

Relation between Area and Circumference

Area can also be expressed in terms of circumference (C) as Area = C² ÷ (4 × π).

Shortcut Trick

Total area of circle = πr² = (22/7) × 14 × 14 = 616 cm².

Ratio of Sector Area to Total Area = 154 ÷ 616 = 1/4.

Since the total angle of a circle is 360°, the Sector Angle = (1/4) × 360° = 90°.

∴ The correct answer is 90°.

Alternate Method

Given:

Area of the sector = 154 cm²

Radius (r) = 14 cm

Value of π = 22/7

Formula Used:

Area of a sector = (θ ÷ 360) × πr²

Calculations:

⇒ 154 = (θ ÷ 360) × (22 ÷ 7) × 14 × 14

⇒ 154 = (θ ÷ 360) × 22 × 2 × 14

⇒ 154 = (θ ÷ 360) × 616

⇒ θ ÷ 360 = 154 ÷ 616

⇒ θ ÷ 360 = 1/4

⇒ θ = 360 ÷ 4

⇒ θ = 90°

∴ The correct answer is 90°.

Additional Information

Length of Arc

The length of the arc (l) of a sector is given by: l = (θ ÷ 360) × 2πr.

Perimeter of a Sector

The total perimeter of a sector includes the arc length and two radii: Perimeter = l + 2r.

Relationship between Area and Arc Length

Area of a sector can also be calculated as: Area = (1 ÷ 2) × l × r.

Area of Circle

The total area of a circle is πr², which corresponds to a central angle of 360°.

Shortcut Trick

Since the speed is constant, distance is directly proportional to time.

Ratio of Time = 2 hr : 5 hr = 2 : 5

Therefore, ratio of Distance = 2 : 5

2 units → 120 km

1 unit → 60 km

5 units → 5 × 60 = 300 km

∴ The correct answer is 300 km.

Alternate Method

Given:

Distance₁ = 120 km

Time₁ = 2 hr

Time₂ = 5 hr

Formula used:

Speed = Distance ÷ Time

Distance = Speed × Time

Calculations:

⇒ Speed of the train = 120 ÷ 2

⇒ Speed = 60 km/hr

⇒ Distance traveled in 5 hr = Speed × Time₂

⇒ Distance = 60 × 5

⇒ Distance = 300 km

∴ The correct answer is 300 km.

Additional Information

Basic Speed Formula

Speed is defined as the rate of change of distance with time: Speed = Distance / Time.

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.

Direct Proportionality

When speed is constant, distance (D) is directly proportional to time (T), expressed as D₁/T₁ = D₂/T₂.

Shortcut Trick

Speed in m/s = Distance (Length) ÷ Time

Speed = 200 ÷ 20 = 10 m/s

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

Speed = 10 × (18/5) = 2 × 18 = 36 km/h

∴ The correct answer is 36 km/h.

Alternate Method

Given:

Length of the train (Distance) = 200 m

Time taken to pass the pole = 20 sec

Formula Used:

Speed = Distance ÷ Time

Speed (km/h) = Speed (m/s) × (18/5)

Calculations:

⇒ Speed in m/s = 200 ÷ 20

⇒ Speed = 10 m/s

⇒ To convert into km/h: Speed = 10 × (18/5)

⇒ Speed = 2 × 18

⇒ Speed = 36 km/h

∴ The correct answer is 36 km/h.

Additional Information

Unit Conversion Shortcut

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

Passing a Point Object

When a train passes a pole, a tree, or a stationary person, the distance covered is equal to the length of the train itself.

Passing a Stationary Object with Length

When a train passes a bridge, platform, or another stationary train, the total distance covered = Length of train + Length of the object.

Shortcut Trick

Expenditure percentage = (Value ÷ Total) × 100

Ratio of Spent to Salary = 200 : 1000 = 1 : 5

Percentage = (1 ÷ 5) × 100 = 20%

∴ The correct answer is 20%.

Alternate Method

Given:

Total Salary of the man = ₹1000

Amount Spent (Expenditure) = ₹200

Formula Used:

Percentage = (Part Value ÷ Whole Value) × 100

Calculations:

⇒ Percentage of Salary spent = (₹200 ÷ ₹1000) × 100

⇒ Percentage = (200 ÷ 1000) × 100

⇒ Percentage = (2 ÷ 10) × 100

⇒ Percentage = 0.2 × 100

⇒ Percentage = 20%

∴ The correct answer is 20%.

Additional Information

Income, Expenditure, and Savings Relationship

Total Income = Expenditure + Savings. In this case, Savings = ₹1000 − ₹200 = ₹800 (which is 80% of the salary).

Percentage to Fraction Conversion

To convert a percentage to a fraction, divide by 100. Example: 20% = 20 ÷ 100 = 1/5.

Percentage Change Formula

Percentage Increase/Decrease = [(New Value − Original Value) ÷ Original Value] × 100.

Shortcut Trick

When a value is increased and then decreased by the same percentage x%, the net change is always a decrease of (x² / 100)%.

Here, x = 20%.

Net percentage change = (20 × 20 / 100)% = 400 / 100 = 4% decrease.

∴ The correct answer is 4% decrease.

Alternate Method

Given: Initial increase = 20%, subsequent decrease = 20%.

Formula: Net Change % = a + b + (a × b / 100), where a and b are percentage changes.

⇒ Let the initial number be 100.

⇒ After 20% increase: 100 + (20/100) × 100 = 120.

⇒ After 20% decrease: 120 − (20/100) × 120 = 120 − 24 = 96.

⇒ Net change = 100 − 96 = 4 units decrease.

⇒ Percentage net change = (4 / 100) × 100 = 4%.

∴ The correct answer is 4% decrease.

Additional Information

Successive Percentage Change Formula

The net effect of two successive changes a% and b% is calculated as [a + b + (ab / 100)]%. Use positive for increase and negative for decrease.

Base Value Property

When a value is increased and decreased by the same percentage, the net result is always a decrease because the second percentage is calculated on a larger base value.

Fraction Method

20% = 1/5. Increase (5 → 6), Decrease (5 → 4). Combined ratio: (5 × 5) → (6 × 4) = 25 → 24. Loss = 1/25 = 4%.

Shortcut Trick
Loss of 20% implies the Selling Price (SP) is 80% of the Cost Price (CP).
80% of CP = ₹720
1% of CP = 720 ÷ 80 = 9
100% (Total CP) = 9 × 100 = ₹900
∴ The correct answer is ₹900.

Alternate Method
Given:
Selling Price (SP) = ₹720
Loss Percentage = 20%
Formula Used:
Cost Price (CP) = [SP × 100] ÷ [100 − Loss%]

⇒ CP = [720 × 100] ÷ [100 − 20]
⇒ CP = [720 × 100] ÷ 80
⇒ CP = 9 × 100
⇒ CP = 900
The calculated Cost Price of the article is ₹900. This matches the value provided in Option (2).
∴ The correct answer is ₹900.

Additional Information
Profit and Loss Basic Formulas
Profit = SP − CP (when SP > CP) and Loss = CP − SP (when CP > SP).
Selling Price with Profit
SP = CP × (100 + Profit%) ÷ 100. This formula is used to find the selling price when a gain is made.
Percentage Change
Profit/Loss Percentage is always calculated on the Cost Price (CP) unless stated otherwise: (Profit or Loss ÷ CP) × 100.

Shortcut Trick

Difference in Amount for 2 years (4 − 2) = 900 − 750 = ₹150.

Since Simple Interest (SI) remains constant every year, SI for the first 2 years is also ₹150.

Initial Principal = Amount after 2 years − SI for 2 years = 750 − 150 = ₹600.

∴ The correct answer is ₹600.

Alternate Method

Given:

Amount after 2 years (A₂) = ₹750

Amount after 4 years (A₄) = ₹900

Formula Used:

Simple Interest (SI) = Amount − Principal

SI for 'n' years = (Amount after T+n years) − (Amount after T years)

Calculations:

⇒ SI for 2 years = A₄ − A₂

⇒ SI for 2 years = 900 − 750 = ₹150

⇒ Principal (P) = Amount after 2 years − SI for 2 years

⇒ P = 750 − 150

⇒ P = ₹600

∴ The correct answer is ₹600.

Additional Information

Simple Interest Property

Simple Interest is calculated only on the Principal amount and remains constant for every equal time interval if the rate is unchanged.

Formula for Rate of Interest

Rate (R) = (SI × 100) ÷ (P × T). In this case, R = (150 × 100) ÷ (600 × 2) = 12.5% per annum.

Relationship between Amount and Principal

The Amount (A) in simple interest is a linear function of time (T): A = P + (P × R × T) ÷ 100.

Shortcut Trick

Convert Speed to m/s: 54 × (5/18) = 15 m/s.

Total Distance (Train + Platform) = Speed × Time = 15 × 20 = 300 m.

Train Length = Total Distance − Platform Length = 300 − 200 = 100 m.

∴ The correct answer is 100 m.

Alternate Method

Given:

Speed of train = 54 km/h

Length of Platform = 200 m

Time taken = 20 sec

Formula Used:

Distance = Speed × Time

When a Train crosses a Platform, the total distance covered is (Length of train + Length of platform).

Calculations:

⇒ Speed in m/s = 54 × (5/18) m/s

⇒ Speed = 3 × 5 = 15 m/s

⇒ Total Distance = Speed × Time

⇒ (Train Length + Platform Length) = 15 × 20

⇒ Train Length + 200 = 300

⇒ Train Length = 300 − 200

⇒ Train Length = 100 m

∴ The correct answer is 100 m.

Additional Information

Unit Conversion

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

Crossing a Point Object

When a Train crosses a stationary point object (like a pole or a person), the distance covered is equal to the Length of the Train.

Relative Speed

When two objects move in the Same Direction, Relative Speed = S₁ − S₂. In the Opposite Direction, Relative Speed = S₁ + S₂.

Shortcut Trick

Downstream speed (D) = 24 ÷ 3 = 8 km/h

Upstream speed (U) = 24 ÷ 4 = 6 km/h

Boat speed in still water = (D + U) ÷ 2 = (8 + 6) ÷ 2 = 7 km/h

Stream speed = (D − U) ÷ 2 = (8 − 6) ÷ 2 = 1 km/h

∴ The correct answer is 7 km/h, 1 km/h.

Alternate Method

Given:

Distance (d) = 24 km

Time taken downstream (t₁) = 3 hours

Time taken upstream (t₂) = 4 hours

Formula Used:

Speed = Distance ÷ Time

Speed of boat (u) = (Downstream Speed + Upstream Speed) ÷ 2

Speed of stream (v) = (Downstream Speed − Upstream Speed) ÷ 2

Calculations:

⇒ Downstream Speed (D) = 24 ÷ 3 = 8 km/h

⇒ Upstream Speed (U) = 24 ÷ 4 = 6 km/h

⇒ Speed of boat (u) = (8 + 6) ÷ 2

⇒ u = 14 ÷ 2 = 7 km/h

⇒ Speed of stream (v) = (8 − 6) ÷ 2

⇒ v = 2 ÷ 2 = 1 km/h

∴ The correct answer is 7 km/h, 1 km/h.

Additional Information

Downstream Motion

When the boat moves in the direction of the stream, the effective speed is the sum of boat speed and stream speed (u + v).

Upstream Motion

When the boat moves against the direction of the stream, the effective speed is the difference between boat speed and stream speed (u − v).

Still Water Speed Calculation

The speed of the boat in still water is the arithmetic mean of the downstream and upstream speeds: u = 1/2 × (D + U).

Stream Speed Calculation

The speed of the water current is half the difference between downstream and upstream speeds: v = 1/2 × (D − U).

Shortcut Trick

The given runs 34, 36, 38, 40, 42 are in an Arithmetic Progression (common difference = 2).

For an odd number of terms in an AP, the average is always the middle term.

The middle term of the series (34, 36, 38, 40, 42) is 38.

∴ The correct answer is 38 runs.

Alternate Method

Given:

Runs in 5 innings: 34, 36, 38, 40, 42

Total number of innings = 5

Formula Used:

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

Calculations:

⇒ Sum of runs = 34 + 36 + 38 + 40 + 42

⇒ Sum of runs = 190

⇒ Average = 190 ÷ 5

⇒ Average = 38

∴ The correct answer is 38 runs.

Additional Information

Average of Arithmetic Progression (AP)

For any series in AP, the average can also be calculated as (First Term + Last Term) ÷ 2. In this case, (34 + 42) ÷ 2 = 76 ÷ 2 = 38.

Sum of Observations

Total Sum = Average × Number of items. This formula is vital for finding missing values in a data set when the average is known.

Consecutive Numbers Average

The average of n consecutive even or odd numbers is simply the middle value if n is odd, or the average of the two middle values if n is even.

Shortcut Trick

Mean proportional a = √(18 × 8) = 12

Third proportional b = (16 × 16) ÷ 12 = 64/3

Required Ratio = 3(12) : 2(64/3) = 36 : 128/3

Multiply by 3 → 108 : 128 = 27 : 32

∴ The correct answer is 27:32.

Alternate Method

Given:

Mean proportional a of 18 and 8.

Third proportional b of 12 and 16.

Formula Used:

For two numbers x and y:

1. Mean Proportional = √(x × y)
2. Third Proportional = y² ÷ x

Calculations:

⇒ a = √(18 × 8)

⇒ a = √144 = 12

⇒ b = (16 × 16) ÷ 12

⇒ b = 256 ÷ 12 = 64/3

⇒ Value of 3a = 3 × 12 = 36

⇒ Value of 2b = 2 × (64/3) = 128/3

⇒ Required Ratio = 3a : 2b

⇒ Ratio = 36 : (128/3)

⇒ Ratio = (36 × 3) : 128

⇒ Ratio = 108 : 128

⇒ Ratio = 27 : 32

∴ The correct answer is 27:32.

Additional Information

Mean Proportional

The mean proportional between two numbers x and y is the square root of their product, expressed as b = √(xy).

Third Proportional

If x, y, and z are in continued proportion (x:y :: y:z), then z is the third proportional to x and y, calculated as z = y²/x.

Fourth Proportional

If x:y :: z:w, then w is the fourth proportional to x, y, and z, given by w = (y × z)/x.

Compound Ratio

The compound ratio of (p:q) and (r:s) is (p × r) : (q × s).

Shortcut Trick

Volume of Pyramid = (1/3) × 10² × 24 = 800 cm³

Volume of Sphere = (4/3) × π × r³ = 800

⇒ π × r³ = 600

⇒ r = ³√(600/π) cm

∴ The correct answer is ³√(600/π) cm.

Alternate Method

Given:

Side of square base (a) = 10 cm

Height of pyramid (h) = 24 cm

Formula Used:

Volume of Regular Pyramid = (1/3) × Area of Base × Height

Volume of Sphere = (4/3) × π × r³

Calculations:

⇒ Area of square base = 10 × 10 = 100 cm²

⇒ Volume of Pyramid = (1/3) × 100 × 24

⇒ Volume of Pyramid = 100 × 8 = 800 cm³

⇒ According to the question, Volume of Pyramid = Volume of Sphere

⇒ 800 = (4/3) × π × r³

⇒ r³ = (800 × 3) / (4 × π)

⇒ r³ = 2400 / (4 × π)

⇒ r³ = 600 / π

⇒ r = ³√(600 / π) cm

∴ The correct answer is ³√(600/π) cm.

Additional Information

Volume of Regular Pyramid

The volume is always (1/3) × Base Area × Perpendicular Height, regardless of the base shape.

Sphere Properties

Volume = (4/3)πr³ and Total Surface Area = 4πr².

Slant Height of Pyramid

For a square pyramid, the slant height (l) is given by l = √(h² + (a/2)²), where h is height and a is base side.

Shortcut Trick

Convert time to years: 18 months = 1.5 years.

Total interest percentage = Rate × Time = 10% × 1.5 = 15%.

Total Amount = 115% of Principal = 1.15 × 25,000.

Amount = ₹28,750.

∴ The correct answer is ₹28,750.

Alternate Method

Given:

Principal (P) = ₹25,000

Rate of Interest (R) = 10% per annum

Time (T) = 18 months

Formula Used:

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

Total Amount (A) = Principal + Simple Interest

⇒ Time in years (T) = 18 ÷ 12 = 1.5 years

⇒ SI = (25000 × 10 × 1.5) ÷ 100

⇒ SI = 250 × 15

⇒ SI = ₹3,750

⇒ Total Amount = 25000 + 3750

⇒ Total Amount = ₹28,750

∴ The correct answer is ₹28,750.

Additional Information

Simple Interest vs Compound Interest

In Simple Interest, the interest is calculated only on the initial principal. In Compound Interest, interest is calculated on the principal plus the accumulated interest of previous periods.

Time Conversion Rule

Always convert time into years before applying the standard formula (SI = PRT/100). If time is in days, divide by 365; if in months, divide by 12.

Effective Rate of Interest

The effective interest rate for the entire duration in Simple Interest is simply the product of Rate (R) and Time (T).