Quantitative Aptitude PYQs

Shortcut Trick

Net profit % = 80 − 25 − (80 × 25) ÷ 100 = 35%

Selling Price (SP) of 1 clock = 48000 ÷ 100 = Rs. 480

Cost Price (CP) × 1.35 = 480 ⇒ CP = 480 ÷ 1.35

CP = 3200 ÷ 9 ≈ Rs. 355.56

∴ The correct answer is Rs. 355.56.

Alternate Method

Given:

Total clocks = 100

Total amount paid by retailer = Rs. 48,000

Markup % = 80%

Discount % = 25%

Formula Used:

SP = CP × (1 + Markup%/100) × (1 − Discount%/100)

         CP = ?  SP = 480  MP  Markup: +80%  Discount: −25%  Net: +35%

Calculations:

⇒ SP of 1 clock = 48000 ÷ 100 = Rs. 480

⇒ 480 = CP × (1 + 80/100) × (1 − 25/100)

⇒ 480 = CP × 1.80 × 0.75

⇒ 480 = CP × 1.35

⇒ CP = 480 ÷ 1.35

⇒ CP = 48000 ÷ 135

⇒ CP = 3200 ÷ 9 ≈ 355.56

∴ The correct answer is Rs. 355.56.

Additional Information

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). Final Selling Price = MP × (1 − Discount%/100).

Effective Profit %

When a markup% and a discount% are applied, Effective profit % = Markup% − Discount% − (Markup% × Discount%) ÷ 100.

Successive Discount

When two discounts d₁% and d₂% are given successively, the equivalent net discount = d₁ + d₂ − (d₁ × d₂) ÷ 100.

Shortcut Trick

We are given tanA = 3/4.

In a right-angled triangle, tanA = Perpendicular ÷ Base.

This forms a standard Pythagorean triplet of (3, 4, 5).

Hence, Hypotenuse = √(3² + 4²) = 5.

Therefore, sinA = Perpendicular ÷ Hypotenuse = 3/5.

∴ The correct answer is 3/5.

Alternate Method

Given:

tanA = 3/4

Formula Used:

tanA = Perpendicular ÷ Base

sinA = Perpendicular ÷ Hypotenuse

Pythagoras Theorem: Hypotenuse² = Perpendicular² + Base²

A B C Base = 4x Perp = 3x Hyp = 5x

⇒ Let the Perpendicular (BC) = 3x and Base (AB) = 4x

⇒ In right triangle ABC, applying Pythagoras theorem:

⇒ AC² = AB² + BC²

⇒ AC² = (4x)² + (3x)²

⇒ AC² = 16x² + 9x² = 25x²

⇒ AC = √(25x²) = 5x

⇒ Now, sinA = Perpendicular ÷ Hypotenuse

⇒ sinA = BC ÷ AC = 3x ÷ 5x = 3/5

∴ The correct answer is 3/5.

Additional Information

Trigonometric Ratios

In a right-angled triangle, the primary trigonometric ratios are defined as: sin A = Perpendicular ÷ Hypotenuse, cos A = Base ÷ Hypotenuse, and tan A = Perpendicular ÷ Base.

Pythagorean Triplets

A set of three positive integers a, b, and c that satisfy the rule a² + b² = c². Common fundamental triplets include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Fundamental Trigonometric Identity

The most commonly used identity is sin²A + cos²A = 1, which acts as a direct link between the sine and cosine of the exact same angle.

Shortcut Trick

From the given trigonometric identity: 1 + cot²θ = cosec²θ

Rearranging this, we get: cosec²θ − cot²θ = 1

This is a universal identity valid for any angle θ.

Putting θ = 60°, the value directly becomes 1.

∴ The correct answer is 1.

Alternate Method

Given:

Equation: 1 + cot²θ = cosec²θ

Expression to find: cosec²60° − cot²60°

Formula Used:

Standard trigonometric values for 60°:

⇒ cosec 60° = 2/√3

⇒ cot 60° = 1/√3

Calculation:

⇒ cosec²60° − cot²60°

⇒ (2/√3)² − (1/√3)²

⇒ (4/3) − (1/3)

⇒ (4 − 1) / 3

⇒ 3 / 3

⇒ 1

∴ The correct answer is 1.

Additional Information

Fundamental Trigonometric Identities

There are three primary Pythagorean identities in trigonometry:

1. sin²θ + cos²θ = 1

2. sec²θ − tan²θ = 1

3. cosec²θ − cot²θ = 1

Standard Trigonometric Values for 60°

sin 60° = √3/2 | cos 60° = 1/2 | tan 60° = √3

cosec 60° = 2/√3 | sec 60° = 2 | cot 60° = 1/√3

Shortcut Trick

When the angle θ is given in radians, the formula for the area of a sector is: Area = 1/2 × r² × θ

Area = 1/2 × 10² × (3π ÷ 4)

Area = 50 × 0.75π = 37.5π cm²

∴ The correct answer is 37.5π cm².

Alternate Method

3π/4 rad 10 cm 10 cm O

Given:

Radius of the circle (r) = 10 cm

Central angle (θ) = 3π ÷ 4 radians

Formula Used:

Area of a sector = 1/2 × r² × θ (where θ is measured in radians)

Calculations:

⇒ Area = 1/2 × (10)² × (3π ÷ 4)

⇒ Area = 1/2 × 100 × (3π ÷ 4)

⇒ Area = 50 × (3π ÷ 4)

⇒ Area = 150π ÷ 4

⇒ Area = 37.5π cm²

∴ The correct answer is 37.5π cm².

Additional Information

Area of a Sector (Degree Measure)

If the central angle is given in degrees instead of radians, the formula changes to: Area = (θ ÷ 360) × πr².

Length of an Arc

The curved boundary length of the sector (Arc Length, L) is calculated using L = r × θ (when θ is in radians).

Perimeter of a Sector

The total boundary length of a sector can be found by adding the arc length and both radii: P = L + 2r.

Shortcut Trick

Using the complementary angle property, we know that cos(90° − θ) = sin θ.

Therefore, we are given that sin θ = x.

From the standard trigonometric identity, cos θ = √(1 − sin² θ).

Substituting the value of sin θ directly, we get cos θ = √(1 − x²).

∴ The correct answer is √(1 − x²).

Alternate Method

θ  90° − θ x 1 √(1 − x²) A B C

Given:

⇒ cos(90° − θ) = x

Formula Used:

⇒ cos(90° − θ) = sin θ

⇒ Base² + Perpendicular² = Hypotenuse²

Calculations:

⇒ Since cos(90° − θ) = sin θ, we have sin θ = x

⇒ sin θ = Perpendicular ÷ Hypotenuse = x ÷ 1

⇒ In right-angled ΔABC, let AB = x and AC = 1

⇒ BC² + AB² = AC²

⇒ BC² + x² = 1

⇒ BC = √(1 − x²)

⇒ Now, cos θ = Base ÷ Hypotenuse = BC ÷ AC

⇒ cos θ = √(1 − x²) ÷ 1 = √(1 − x²)

∴ The correct answer is √(1 − x²).

Additional Information

Complementary Angle Identities

The trigonometric ratios of complementary angles are related such that sin(90° − θ) = cos θ and cos(90° − θ) = sin θ.

Pythagorean Identities

The fundamental relation linking sine and cosine in any right-angled triangle is sin² θ + cos² θ = 1.

First Quadrant Rule

If θ is strictly an acute angle (0° < θ < 90°), it lies in the first quadrant where all trigonometric ratios are strictly positive, eliminating the need for negative square roots.

Shortcut Trick

Length of Transverse Common Tangent (TCT) = √(D² − (R + r)²)

⇒ TCT = √(10² − (6 + 2)²)

⇒ TCT = √(100 − 8²)

⇒ TCT = √(100 − 64) = √36

⇒ TCT = 6 cm

∴ The correct answer is 6 cm.

Alternate Method

Given:

Radius of the larger circle (R) = 6 cm

Radius of the smaller circle (r) = 2 cm

Distance between centers (D) = 10 cm

C₁ C₂ P Q 6 cm 2 cm 10 cm PQ = Transverse Common Tangent

Formula Used:

Length of a transverse common tangent (TCT) = √(D² − (R + r)²)

Calculation:

⇒ TCT = √(10² − (6 + 2)²)

⇒ TCT = √(100 − 8²)

⇒ TCT = √(100 − 64)

⇒ TCT = √36

⇒ TCT = 6 cm

∴ The correct answer is 6 cm.

Additional Information

Direct Common Tangent (DCT)

A common tangent that does not cross the line segment joining the centers of the two circles. Formula: DCT = √(D² − (R − r)²).

Number of Common Tangents Rule

When the distance between centers D > R + r, there are exactly 4 common tangents (2 direct and 2 transverse). If D = R + r, the circles touch externally and have exactly 3 common tangents.

Shortcut Trick

According to the circle theorem, angles subtended by the same arc or chord in the same segment of a circle are always equal.

Since points R and S lie on the same side of chord PQ, they fall within the same segment of the circle.

Therefore, the angle subtended at S is exactly equal to the angle subtended at R.

Angle at S = Angle at R = 50°

∴ The correct answer is 50°.

Alternate Method

P Q R S

50° 50°

Given:

Angle subtended by chord PQ at point R on the circumference = ∠PRQ = 50°

Points R and S are located on the circumference on the same side of chord PQ.

Concept Used:

Angles subtended by a chord at any point in the same segment of a circle are equal.

Calculation:

⇒ Both points R and S lie on the major arc formed by the chord PQ.

⇒ By the circle theorem, angles in the same segment are equal.

⇒ ∠PSQ = ∠PRQ

⇒ ∠PSQ = 50°

∴ The correct answer is 50°.

Additional Information

Angle at the Center Theorem

The angle subtended by an arc or chord at the center of the circle is always exactly double the angle subtended by it at any remaining point on the circumference.

Angles in a Semicircle

Any angle subtended by the diameter of a circle at any point on the circumference is always a right angle (90°).

Cyclic Quadrilateral

If all four vertices of a quadrilateral lie on the circumference of a circle, the sum of its opposite angles is always 180°.

Shortcut Trick

The perpendicular drawn from the center to a chord bisects it, forming a right-angled triangle.

Base of triangle = Half of the chord = 32 ÷ 2 = 16 cm.

Hypotenuse = Radius = 20 cm.

Distance from center = √(20² − 16²) = √(400 − 256) = √144 = 12 cm.

∴ The correct answer is 12 cm.

Alternate Method

Given:

Radius of the circle (r) = 20 cm

Length of the chord (AB) = 32 cm

Formula Used:

In a circle, the perpendicular distance from the center to a chord bisects the chord.

Pythagoras Theorem: (Hypotenuse)² = (Perpendicular)² + (Base)²

O    A B D 20 cm d = ? 16 cm 16 cm  Right-angled Triangle ODA

Calculations:

⇒ Let the perpendicular distance from the center O to the chord AB be OD = d.

⇒ The perpendicular bisects the chord, so AD = DB = 32 ÷ 2 = 16 cm.

⇒ In right-angled triangle ODA, (Radius OA)² = (Distance OD)² + (Base AD)²

⇒ (20)² = d² + (16)²

⇒ 400 = d² + 256

⇒ d² = 400 − 256

⇒ d² = 144

⇒ d = 12 cm

∴ The correct answer is 12 cm.

Additional Information

Perpendicular from Center to Chord

The straight line drawn from the center of a circle perpendicular to a chord bisects the chord. Conversely, a line drawn from the center to bisect a chord is perpendicular to the chord.

Equal Chords Property

Equal chords of a circle are always equidistant from the center. Conversely, chords that are equidistant from the center must be equal in length.

Pythagorean Triplets

Recognizing standard Pythagorean triplets like (3, 4, 5) or its common multiples such as (12, 16, 20) saves valuable calculation time in exams.

Shortcut Trick

The angle subtended by a chord at the center is always double the angle subtended by it at any point on the remaining part of the circle (major arc).

⇒ Angle at major arc = (Angle at center) ÷ 2

⇒ Angle = 60° ÷ 2 = 30°

∴ The correct answer is 30°.

Alternate Method

Given:

Angle subtended by chord AB at the center O, ∠AOB = 60°

Point C is any point on the major arc of the circle.

60°

30°

O

A

B

C

Formula Used:

According to the circle theorem, the angle subtended by an arc at the center is double the angle subtended by it at any remaining part of the circle.

⇒ ∠AOB = 2 × ∠ACB

Calculations:

⇒ 60° = 2 × ∠ACB

⇒ ∠ACB = 60° ÷ 2

⇒ ∠ACB = 30°

∴ The correct answer is 30°.

Additional Information

Angles in the Same Segment

Angles subtended by the same arc at any point on the remaining part of the circle are always equal.

Cyclic Quadrilateral

If a quadrilateral is inscribed in a circle, the sum of its opposite interior angles is always 180°.

Angle in a Semicircle

The angle subtended by the diameter at any point on the circumference of the circle is always a right angle (90°).

Shortcut Trick

The tangent to a circle, the radius, and the distance from the point to the center form a right-angled triangle.

The distance from the center forms the hypotenuse of this right-angled triangle.

Given the radius is 5 cm and the hypotenuse is 13 cm, we can use the common Pythagorean triplet (5, 12, 13).

Thus, the length of the tangent is directly 12 cm.

∴ The correct answer is 12 cm.

Alternate Method

     O P T 13 cm 5 cm Tangent PT = ?

Given:

Radius of the circle (OT) = 5 cm

Distance of point P from the center O (OP) = 13 cm

Formula Used:

Pythagoras Theorem: (Hypotenuse)² = (Perpendicular)² + (Base)²

Calculations:

⇒ Let PT be the tangent to the circle at point T.

⇒ A key property of circles is that the radius drawn to the point of tangency is perpendicular to the tangent line.

⇒ ∴ ∠OTP = 90°

⇒ In right-angled triangle OTP, applying Pythagoras theorem:

⇒ (OP)² = (OT)² + (PT)²

⇒ (13)² = (5)² + (PT)²

⇒ 169 = 25 + (PT)²

⇒ (PT)² = 169 − 25

⇒ (PT)² = 144

⇒ PT = √144 = 12 cm

∴ The correct answer is 12 cm.

Additional Information

Tangent to a Circle

A tangent is a straight line that touches the circle at exactly one distinct point. The radius intersecting this point of contact is always perpendicular to the tangent line.

Secant-Tangent Theorem

If a tangent segment and a secant segment are drawn to a circle from an exterior point, the square of the tangent length is equal to the product of the secant segment and its external portion: PT² = PA × PB.

Direct Common Tangent (DCT)

The length of a DCT between two non-intersecting circles with radii r₁ and r₂ and distance d between their centers is calculated as √[d² − (r₁ − r₂)²].

Transverse Common Tangent (TCT)

The length of a TCT between two circles with radii r₁ and r₂ and a central distance d is given by the formula √[d² − (r₁ + r₂)²].

Shortcut Trick

Evaluate square roots and solve directly:

⇒ ? ÷ 9 = 11 ÷ 11

⇒ ? ÷ 9 = 1

⇒ ? = 9

∴ The correct answer is 9.

Alternate Method

Given:

? ÷ √81 = √121 ÷ (10 + 1)

Formula Used:

Apply BODMAS and simplify standard perfect square roots.

√81 = 9

√121 = 11

Calculation:

Let the missing value (?) be x.

⇒ x ÷ √81 = √121 ÷ (10 + 1)

⇒ x ÷ 9 = 11 ÷ (11)

⇒ x ÷ 9 = 11 ÷ 11

⇒ x ÷ 9 = 1

⇒ x = 1 × 9

⇒ x = 9

∴ The correct answer is 9.

Additional Information

BODMAS Rule

Mathematical order of operations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

Perfect Squares

A number made by squaring a whole number. Knowing squares up to 30 helps in faster calculation (e.g., 9² = 81, 11² = 121).

Square Roots

The inverse operation of squaring. For example, √121 is 11 because 11 × 11 = 121.

Shortcut Trick

For a fixed principal and interest rate, Simple Interest (SI) is directly proportional to the time period.

⇒ SI ∝ Time

⇒ SI₁ : SI₂ = Time₁ : Time₂

⇒ SI₁ : SI₂ = 6 : 9 = 2 : 3

∴ The correct answer is 2:3.

Alternate Method

Given:

Time period 1 (T₁) = 6 years

Time period 2 (T₂) = 9 years

Principal (P) and Rate of Interest (R) are constant for both cases.

Formula Used:

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

6 Yrs

SI = 6PR ÷ 100 9 Yrs

SI = 9PR ÷ 100

Ratio → 6 : 9 → 2 : 3

Calculations:

⇒ SI for 6 years (SI₁) = (P × R × 6) ÷ 100

⇒ SI for 9 years (SI₂) = (P × R × 9) ÷ 100

⇒ Ratio = SI₁ : SI₂

⇒ Ratio = [(P × R × 6) ÷ 100] : [(P × R × 9) ÷ 100]

⇒ Cancelling common terms (P, R, and 100) from both sides:

⇒ Ratio = 6 : 9

⇒ Ratio = 2 : 3

∴ The correct answer is 2:3.

Additional Information

Proportionality in Simple Interest

When Principal and Time are constant, Simple Interest is directly proportional to the Rate of Interest (SI ∝ R). When Rate and Time are constant, SI is directly proportional to the Principal (SI ∝ P).

Total Amount Formula

Total Amount (A) = Principal (P) + Simple Interest (SI) = P × [1 + (R × T) ÷ 100].

SI vs CI Difference

The difference between Compound Interest and Simple Interest for exactly 2 years is calculated directly by the formula: D = P × (R ÷ 100)².

Shortcut Trick

Speed = Distance ÷ Time = 216 ÷ 3 = 72 km/h

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

Speed in m/s = 72 × (5/18) = 20 m/s

∴ The correct answer is 20 m/s.

Alternate Method

Train

Distance = 216 km Time = 3 hours Speed = ? m/s

Given:

⇒ Distance covered by the train = 216 km

⇒ Time taken = 3 hours

Formula Used:

⇒ Speed = Distance ÷ Time

⇒ Speed in m/s = Speed in km/h × (5/18)

Calculation:

⇒ Speed = 216 ÷ 3

⇒ Speed = 72 km/h

⇒ Convert km/h into m/s by multiplying with 5/18

⇒ Speed = 72 × (5/18) m/s

⇒ Speed = 4 × 5 = 20 m/s

∴ The correct answer is 20 m/s.

Additional Information

Unit Conversion

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

Average Speed

Average speed is defined as the ratio of total distance traveled to total time taken. Formula: Average Speed = Total Distance ÷ Total Time.

Relative Speed

If two bodies move in the same direction at speeds S₁ and S₂, their relative speed is |S₁ − S₂|. In opposite directions, it is S₁ + S₂.

Train Crossing an Object

When a train of length L crosses a stationary object of length x (like a platform or bridge), the total distance covered is L + x.

Shortcut Trick

For two objects positioned on opposite sides of a central tower of height h, the total distance between them is given by: D = h × (cot θ₁ + cot θ₂)

Here, the height of the lighthouse h = 100 m

The angles of elevation are θ₁ = 30° and θ₂ = 45°

⇒ Total distance = 100 × (cot 30° + cot 45°)

⇒ Total distance = 100 × (√3 + 1)

∴ The correct answer is 100(√3 + 1) m.

Alternate Method

30° 45° 100 m

x = 100√3 m y = 100 m

Ship A Ship B Lighthouse

Total Distance = 100(√3 + 1) m

Given:

Height of the lighthouse (h) = 100 m

Angle of elevation from Ship A = 30°

Angle of elevation from Ship B = 45°

Formula Used:

tan θ = Perpendicular ÷ Base

Calculations:

Let the lighthouse be represented by the central vertical line of height 100 m.

Let the distance of Ship A from the base of the lighthouse be x.

⇒ In the right-angled triangle on the left side:

⇒ tan 30° = 100 ÷ x

⇒ 1 ÷ √3 = 100 ÷ x

⇒ x = 100√3 m

Let the distance of Ship B from the base of the lighthouse be y.

⇒ In the right-angled triangle on the right side:

⇒ tan 45° = 100 ÷ y

⇒ 1 = 100 ÷ y

⇒ y = 100 m

Since the ships are exactly on opposite sides, the total distance between them is the sum of their individual distances from the lighthouse base.

⇒ Total Distance = x + y

⇒ Total Distance = 100√3 + 100

⇒ Total Distance = 100(√3 + 1) m

∴ The correct answer is 100(√3 + 1) m.

Additional Information

Same Side vs Opposite Sides Trick

For objects on opposite sides of a vertical tower, total distance = h × (cot θ₁ + cot θ₂). If both objects are on the same side, the distance between them is the difference: h × (cot θ₁ − cot θ₂).

Angles of Elevation and Depression

Angle of elevation is measured looking upwards from the horizontal plane, while depression is measured downwards. By the property of parallel lines, the angle of depression from the top of an object equals the angle of elevation from the ground.

Standard Trigonometric Ratios

tan 30° = 1 ÷ √3, tan 45° = 1, tan 60° = √3. Cotangent is the exact reciprocal of tangent: cot 30° = √3, cot 45° = 1, cot 60° = 1 ÷ √3.

Shortcut Trick

Area of the wall to be tiled = 2 m × 2.5 m = 5 m² = 50000 cm².

Area of 1 square tile = 25 cm × 25 cm = 625 cm².

Base number of tiles needed = 50000 ÷ 625 = 80 tiles.

Total tiles including 12% extra wastage = 80 × 1.12 = 89.6 tiles.

Since tiles can only be purchased in whole numbers, round up to 90.

∴ The correct answer is 90 tiles.

Alternate Method

Given:

Dimensions of the wall = 2 m (Length) and 2.5 m (Height)

Side length of one square tile = 25 cm

Extra allowance for cuts and wastage = 12%

(Note: The bathroom floor dimensions are distractor data, as only the wall is being tiled.)

WALL 2 m (200 cm) 2.5 m (250 cm)

Tiled with

TILE 25 cm 25 cm

Calculations:

⇒ First, convert wall dimensions from meters to centimeters (1 m = 100 cm).

⇒ Wall length = 200 cm and Wall height = 250 cm

⇒ Area of the wall = 200 × 250 = 50000 cm²

⇒ Area of 1 square tile = 25 × 25 = 625 cm²

⇒ Base number of tiles = Area of wall ÷ Area of 1 tile

⇒ Base number of tiles = 50000 ÷ 625 = 80 tiles

⇒ Additional tiles for wastage = 12% of 80 = 80 × (12 ÷ 100) = 9.6 tiles

⇒ Total tiles required = 80 + 9.6 = 89.6 tiles

⇒ Real-life items like whole tiles are rounded up to the nearest integer, which gives 90 tiles.

∴ The correct answer is 90 tiles.

Additional Information

Unit Conversion in Area

When working with mixed units, always convert the larger units to smaller units before calculating the area to avoid decimal placement errors. Remember that 1 m² = 10000 cm².

Wastage Allowance

In practical math problems, wastage or extra material required is always calculated on the base quantity. The final amount is mathematically formulated as: Base Quantity × (1 + Rate/100).

Handling Distractor Data

Examiners often include irrelevant information (such as the floor dimensions in this problem) to test reading comprehension and focus. Always isolate the specific subject undergoing the operation.

Shortcut Trick

When the investment time is constant, the Capital Ratio is exactly the same as the Profit Ratio.

Profit Ratio = 2 : 3

2 units of capital → Rs. 10,000

1 unit of capital → Rs. 5,000

3 units of capital → 3 × 5,000 = Rs. 15,000

∴ The correct answer is Rs. 15,000.

Alternate Method

Given:

Profit sharing ratio of the 1^st and 2^nd partner = 2 : 3

Capital of the 1^st partner = Rs. 10,000

Total profit = Rs. 15,000 (Redundant data, not needed for calculation)

Formula Used:

When the time period is the same for both partners: Ratio of Capitals = Ratio of Profits

Partnership Capital & Profit Ratio

Partner 1 Profit = 2 parts Capital = Rs. 10,000

:

Partner 2 Profit = 3 parts Capital = ?

Capital Ratio = Profit Ratio (Equal Time)

Calculations:

⇒ Let the capital invested by the second partner be x.

⇒ Capital Ratio = 10000 : x

⇒ Since Profit Ratio = 2 : 3

⇒ 10000 ÷ x = 2 ÷ 3

⇒ x = (10000 × 3) ÷ 2

⇒ x = 30000 ÷ 2

⇒ x = 15000

∴ The correct answer is Rs. 15,000.

Additional Information

Basic Partnership Principle

Profit is always shared in the ratio of the product of Capital and Time for each partner: P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Working vs Sleeping Partner

A working partner actively manages the business and usually receives a fixed salary or a specific percentage of the total profit before the remaining profit is distributed according to investments.

Identifying Redundant Data

Exam questions often include extra data (like the Total Profit of Rs. 15,000 here) to confuse candidates. Relying on direct ratios saves time and minimizes calculation errors.

Shortcut Trick

Interior angle + Exterior angle = 180°

Given Interior angle = k × Exterior angle

∴ (k + 1) × Exterior angle = 180° ⇒ Exterior angle = 180° ÷ (k + 1)

Number of sides (n) = 360° ÷ Exterior angle

n = 360° ÷ [180° ÷ (k + 1)] = 2k + 2

∴ The correct answer is 2k + 2.

Alternate Method

Interior = k × E

Exterior = E

Interior (I) + Exterior (E) = 180°

Given:

Interior Angle (I) = k × Exterior Angle (E)

Formula Used:

I + E = 180°

Number of sides (n) = 360° ÷ E

Calculation:

⇒ I + E = 180°

⇒ k × E + E = 180°

⇒ E × (k + 1) = 180°

⇒ E = 180° ÷ (k + 1)

⇒ n = 360° ÷ E

⇒ n = 360° ÷ [180° ÷ (k + 1)]

⇒ n = 360° × (k + 1) ÷ 180°

⇒ n = 2(k + 1)

⇒ n = 2k + 2

∴ The correct answer is 2k + 2.

Additional Information

Sum of Exterior Angles

The sum of all exterior angles of any convex polygon is always 360°, regardless of the number of sides.

Sum of Interior Angles

For a polygon of n sides, the sum of all interior angles is given by (n − 2) × 180°.

Each Interior Angle

For a regular polygon of n sides, each interior angle is [(n − 2) × 180°] ÷ n.

Number of Diagonals

The number of diagonals in an n-sided polygon is given by n(n − 3) ÷ 2.

Shortcut Trick

Let initial Income = 100.

Expense = 75% of 100 = 75, so Savings = 100 − 75 = 25.

New Income = 100 + 10% = 110.

New Expense = 75 × 1.20 = 90.

New Savings = 110 − 90 = 20.

% Change in Savings = [(20 − 25) ÷ 25] × 100 = −20%.

∴ The correct answer is 20% decrease.

Alternate Method

Given:

Initial expenditure = 75% of income

Increase in income = 10%

Increase in expenses = 20%

Formula Used:

Income = Expenditure + Savings

Percentage Change = [(New Value − Old Value) ÷ Old Value] × 100

Initial:

Income: 100

Expense: 75

Sav: 25

Final:

Income: 110 (+10%)

Expense: 90 (+20%)

Sav: 20

Savings reduced from 25 to 20 → 20% Decrease

Calculations:

⇒ Let the initial income be 100 units.

⇒ Initial expenditure = 75% of 100 = 75 units

⇒ Initial savings = Income − Expenditure = 100 − 75 = 25 units

⇒ New income = 100 + (10% of 100) = 110 units

⇒ New expenditure = 75 + (20% of 75) = 75 + 15 = 90 units

⇒ New savings = New income − New expenditure = 110 − 90 = 20 units

⇒ Change in savings = New savings − Initial savings = 20 − 25 = −5 units

⇒ Percentage change in savings = (−5 ÷ 25) × 100

⇒ Percentage change = −1 ÷ 5 × 100 = −20%

⇒ The negative sign indicates a decrease.

∴ The correct answer is 20% decrease.

Additional Information

Percentage Change

Percentage change is always calculated with respect to the initial base value. Formula: % Change = (Final Value − Initial Value) ÷ Initial Value × 100.

Successive Percentage Change

When a value is sequentially changed by a% and then b%, the net percentage change can be calculated directly using the formula: Net Change = a + b + (a × b) ÷ 100.

Income, Expense and Savings

The fundamental equation of personal finance is: Income = Expenditure + Savings. Any absolute change in total income must perfectly equal the sum of absolute changes in expenditure and savings.

Shortcut Trick

Side of the square = √225 = 15 cm.

When cut into two equal halves, the sides of each new rectangle are 15 cm and 7.5 cm.

Perimeter = 2 × (15 + 7.5) = 2 × 22.5 = 45 cm.

∴ The correct answer is 45 cm.

Alternate Method

Given:

Area of the square tile = 225 cm²

The square is cut into two equal rectangular tiles.

Formula Used:

Area of Square = (Side)²

Perimeter of Rectangle = 2 × (Length + Breadth)

15 cm 15 cm Area = 225

→

7.5 cm 15 cm

7.5 cm 15 cm

Perimeter = 2 × (15 + 7.5) = 45 cm

Calculations:

⇒ Side of the square tile = √225 = 15 cm

⇒ Since it is cut into two equal rectangular tiles, the cut will divide one side into two equal halves.

⇒ Length of the new rectangular tile (L) = 15 cm

⇒ Breadth of the new rectangular tile (B) = 15 ÷ 2 = 7.5 cm

⇒ Perimeter of each new rectangular tile = 2 × (15 + 7.5)

⇒ Perimeter = 2 × 22.5 = 45 cm

∴ The correct answer is 45 cm.

Additional Information

Area and Perimeter of a Square

Area = a² and Perimeter = 4a, where 'a' is the side of the square. A square is a regular quadrilateral with all equal sides and four right angles.

Area and Perimeter of a Rectangle

Area = L × B and Perimeter = 2 × (L + B), where L is length and B is breadth. Opposite sides of a rectangle are equal and parallel.

Diagonal of Square and Rectangle

Diagonal of a square = a√2. Diagonal of a rectangle = √(L² + B²).

Shortcut Trick

Value of 1 ratio part = 120 ÷ (2 + 3 + 5) = 120 ÷ 10 = 12 pencils.

Initial Red pencils = 2 × 12 = 24 and Initial Black pencils = 5 × 12 = 60.

After removal: New Red pencils = 24 − 10 = 14 and New Black pencils = 60 − 20 = 40.

New ratio (Red : Black) = 14 : 40 = 7 : 20.

∴ The correct answer is 7:20.

Alternate Method

Initial Total = 120 Pencils (Ratio 2 : 3 : 5)

Red: 24

Blue: 36

Black: 60

−10

−20

New Red: 14

New Blk: 40

New Ratio (Red : Black) = 14 : 40 = 7 : 20

Given:

Total number of pencils = 120

Initial ratio of Red, Blue, and Black pencils = 2 : 3 : 5

Red pencils removed = 10

Black pencils removed = 20

Calculations:

⇒ Sum of ratio parts = 2 + 3 + 5 = 10 parts

⇒ 10 parts = 120 pencils ⇒ 1 part = 12 pencils

⇒ Initial number of Red pencils = 2 × 12 = 24

⇒ Initial number of Black pencils = 5 × 12 = 60

⇒ New number of Red pencils = 24 − 10 = 14

⇒ New number of Black pencils = 60 − 20 = 40

⇒ New ratio of Red to Black pencils = 14 : 40

⇒ Dividing both sides by their highest common factor (2):

⇒ Simplified new ratio = 7 : 20

∴ The correct answer is 7:20.

Additional Information

Simplest Form of Ratio

A ratio a : b is considered to be in its simplest form when the Highest Common Factor (HCF) of a and b is exactly 1. It is simplified by dividing both parts by their HCF.

Properties of Proportions

When four quantities a, b, c, and d are in proportion (a:b = c:d), they satisfy the relation a × d = b × c. This states that the product of the extremes equals the product of the means.

Compound Ratio

When two or more ratios are given, their compound ratio is found by multiplying the first terms together and the second terms together. For ratios a : b and c : d, the compound ratio is ac : bd.