Quantitative Aptitude PYQs

Shortcut Trick

Since the cylinder and hemisphere have equal volumes on the same base radius, their height ratio h : r = 2 : 3.

Total height in terms of ratio units = 2 + 3 = 5 units.

Given total combined height = 30 cm.

Radius of hemisphere (3 units) = (30 ÷ 5) × 3 = 18 cm.

∴ The correct answer is 18 cm.

Alternate Method

Given:

Total combined height = h + r = 30 cm

Volume of the Cylinder = Volume of the Hemisphere

Formula Used:

Volume of Cylinder = πr²h

Volume of Hemisphere = (2/3)πr³

30 cm

h

r

r

Cylinder and Hemisphere

Calculations:

⇒ πr²h = (2 ÷ 3) × πr³

⇒ h = (2 ÷ 3)r

⇒ Substituting the value of h in the combined height equation:

⇒ (2 ÷ 3)r + r = 30

⇒ (5 ÷ 3)r = 30

⇒ r = 30 × 3 ÷ 5

⇒ r = 18 cm

∴ The correct answer is 18 cm.

Additional Information

Volume and Surface Area of a Cylinder

The volume is given by V = πr²h. The total surface area is 2πr(r + h), where r is the base radius and h is the height.

Volume and Surface Area of a Hemisphere

The volume is given by V = (2/3)πr³. The curved surface area is 2πr², and the total surface area is 3πr².

Ratio-Unitary Method in Geometry

When two 3D shapes have equal volumes and identical base radii, equating their volume formulas directly provides the ratio of their heights, significantly speeding up complex fractional calculations.

Shortcut Trick

The volume of any pyramid is directly one-third of the product of its base area and height.

Volume = (1 ÷ 3) × Base Area × Height

Volume = (1 ÷ 3) × 60 × 9 = 180 cm³

∴ The correct answer is 180 cm³.

Alternate Method

Given:

Base Area of the pyramid = 60 cm²

Height of the pyramid = 9 cm

Formula Used:

Volume of a Pyramid = (1 ÷ 3) × Base Area × Height

h = 9 cm Base Area = 60 sq. cm

Calculations:

⇒ Volume = (1 ÷ 3) × 60 × 9

⇒ Volume = 20 × 9

⇒ Volume = 180 cm³

∴ The correct answer is 180 cm³.

Additional Information

Prism vs Pyramid Volume

The volume of a prism is directly calculated as Base Area × Height, whereas a pyramid's volume is exactly one-third of the prism's volume having the same base and height.

Total Surface Area of a Pyramid

Total Surface Area = Lateral Surface Area + Base Area. For a regular pyramid, Lateral Surface Area = (1 ÷ 2) × Base Perimeter × Slant Height.

Cone as a Pyramid

A cone is a specific type of pyramid with a circular base. Its volume formula is exactly (1 ÷ 3) × πr²h, which follows the same foundational logic.

Shortcut Trick

For an equilateral triangle circumscribing a circle of radius r, the remaining area can be directly calculated as:

⇒ Remaining Area = r² × (3√3 − π)

⇒ Remaining Area = 7² × (3 × 1.732 − 3.142)

⇒ Remaining Area ≈ 49 × (5.196 − 3.142) = 49 × 2.054

⇒ Remaining Area ≈ 100.64 cm²

∴ The correct answer is 100.6 cm².

Alternate Method

Given:

Radius of the inscribed circular disc (r) = 7 cm

Formula Used:

Inradius of equilateral triangle = a ÷ 2√3 (where a is the side length)

Area of equilateral triangle = (√3 ÷ 4) × a²

Area of a circle = π × r²

r = 7 cm

a = 14√3 cm Remaining Area = Area of Triangle − Area of Circle

Calculations:

⇒ a ÷ 2√3 = 7

⇒ a = 14√3 cm

⇒ Area of the equilateral triangle = (√3 ÷ 4) × (14√3)²

⇒ Area of the equilateral triangle = (√3 ÷ 4) × 196 × 3 = 147√3 cm²

⇒ Taking √3 ≈ 1.732

⇒ Area of the triangle ≈ 147 × 1.732 = 254.604 cm²

⇒ Area of the circle = (22 ÷ 7) × 7² = 154 cm²

⇒ Remaining Area = Area of the triangle − Area of the circle

⇒ Remaining Area = 254.604 − 154 = 100.604 cm²

∴ The correct answer is 100.6 cm².

Additional Information

Circumcircle of an Equilateral Triangle

The circumradius (R) of an equilateral triangle is exactly double its inradius. Formula: R = a ÷ √3 = 2r.

Area Ratio of Incircle and Circumcircle

For any equilateral triangle, the ratio of the area of the incircle to the area of the circumcircle is always 1:4.

Height of an Equilateral Triangle

The height or median (h) divides the triangle symmetrically. Formula: h = (√3 ÷ 2) × a.

Shortcut Trick

According to the geometric properties of a triangle, the centroid divides the triangle into three smaller triangles of equal area.

These three triangles are ΔGAB, ΔGBC, and ΔGCA.

Since their areas are equal, the area of ΔGAB is exactly one-third of the total area of ΔABC.

Ratio = Area(ΔGAB) : Area(ΔABC) = 1 : 3.

∴ The correct answer is 1:3.

Alternate Method

A B C D E F G Area(ΔGAB) = 1/3 × Area(ΔABC)

Given:

⇒ AD, BE, and CF are the medians of ΔABC.

⇒ G is the centroid where all three medians intersect.

Property Used:

⇒ The medians of a triangle divide it into six smaller triangles of equal area.

⇒ Area(ΔAFG) = Area(ΔBFG) = Area(ΔBDG) = Area(ΔCDG) = Area(ΔCEG) = Area(ΔAEG)

Calculation:

⇒ Let the area of one small triangle, Area(ΔAFG) = 1 unit

⇒ Total Area(ΔABC) = 6 units

⇒ Area(ΔGAB) = Area(ΔAFG) + Area(ΔBFG)

⇒ Area(ΔGAB) = 1 + 1 = 2 units

⇒ Ratio = Area(ΔGAB) ÷ Area(ΔABC)

⇒ Ratio = 2 ÷ 6

⇒ Ratio = 1 ÷ 3

∴ The correct answer is 1:3.

Additional Information

Centroid Division Property

The centroid divides each median of the triangle in the ratio 2:1, with the longer segment always originating from the vertex (e.g., AG:GD = 2:1).

Apollonius Theorem

It establishes a relationship between the lengths of the sides of a triangle and its median. For median AD on side BC: AB² + AC² = 2 × (AD² + BD²).

Medians and Area Rule

A new triangle formed by taking the lengths of the three medians of an original triangle as its sides will have an area exactly equal to 3/4 of the area of the original triangle.

Shortcut Trick

Area of △ADE : Area of trapezoid DECB = 4 : 21

Total Area of △ABC = Area(△ADE) + Area(DECB) = 4 + 21 = 25 parts

Ratio of corresponding sides AD : AB = √(Area of △ADE) : √(Area of △ABC) = √4 : √25 = 2 : 5

Since point D is on AB, DB = AB − AD = 5 − 2 = 3 parts

∴ The correct answer is 2:3.

Alternate Method

A B C D E Area = 4 units Area = 21 units

Given:

DE is parallel to BC (DE || BC).

Area of △ADE : Area of trapezoid DECB = 4 : 21.

Formula Used:

If two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Area(△ADE) ÷ Area(△ABC) = (AD ÷ AB)²

Calculations:

⇒ Area of △ABC = Area of △ADE + Area of trapezoid DECB

⇒ Area of △ABC = 4 + 21 = 25 units

⇒ In △ADE and △ABC, DE || BC

⇒ ∠ADE = ∠ABC (Corresponding angles)

⇒ ∠AED = ∠ACB (Corresponding angles)

⇒ ∴ △ADE ~ △ABC (by AA similarity criterion)

⇒ Area(△ADE) ÷ Area(△ABC) = (AD ÷ AB)²

⇒ 4 ÷ 25 = (AD ÷ AB)²

⇒ AD ÷ AB = √(4 ÷ 25) = 2 ÷ 5

⇒ Let AD = 2x and AB = 5x

⇒ Since point D lies on AB, AB = AD + DB

⇒ DB = AB − AD

⇒ DB = 5x − 2x = 3x

⇒ Ratio of AD to DB = 2x : 3x = 2 : 3

∴ The correct answer is 2:3.

Additional Information

Area of Similar Triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides, altitudes, medians, or angle bisectors.

Basic Proportionality Theorem (Thales Theorem)

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. ⇒ AD ÷ DB = AE ÷ EC.

AA Similarity Criterion

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are structurally similar.

Shortcut Trick

Perimeter of larger triangle = 3 + 4 + 5 = 12.

Area of larger triangle = 1/2 × 3 × 4 = 6.

For similar triangles, the ratio of their areas equals the square of the ratio of their perimeters.

Area of smaller triangle = 6 × (6 ÷ 12)²

⇒ Area = 6 × (1/2)² = 6 ÷ 4 = 1.5.

∴ The correct answer is 1.5 sq. units.

Alternate Method

Given:

⇒ Sides of the larger right-angled triangle = 3, 4, and 5

⇒ Perimeter of the inscribed smaller similar triangle = 6

Formula Used:

⇒ Area of a right triangle = 1/2 × Base × Perpendicular Height

⇒ For two similar triangles: Area₁ / Area₂ = (Perimeter₁ / Perimeter₂)²

3 4 5

2 1.5 2.5

Calculations:

⇒ The perimeter of the larger triangle = 3 + 4 + 5 = 12

⇒ Area of the larger triangle = 1/2 × 3 × 4 = 6 sq. units

⇒ The ratio of their perimeters (Smaller : Larger) = 6 / 12 = 1/2

⇒ According to the similarity theorem, the ratio of their areas is the square of the scale factor.

⇒ Area Ratio = (1 / 2)² = 1 / 4

⇒ Let the area of the smaller triangle be A.

⇒ A / 6 = 1 / 4

⇒ A = 6 × (1 / 4) = 1.5 sq. units

∴ The correct answer is 1.5 sq. units.

Additional Information

Similarity Area Theorem

For any two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides, altitudes, or perimeters: Area₁ / Area₂ = (P₁ / P₂)².

Medial Triangle Property

The triangle formed by connecting the midpoints of the sides of a given triangle is called the medial triangle. It perfectly represents a similar inscribed triangle whose area is exactly 1/4^th of the original area, and perimeter is half of the original.

Right Angled Triangle Basics

The area of a right-angled triangle is simply 1/2 × Base × Perpendicular Height. The three sides strictly follow the Pythagorean theorem relation: a² + b² = c².

Shortcut Trick For Direct Common Tangent (DCT) and distance between centers (d): d2 = DCT2 + (R − r)2 ⇒ d = √(152 + 82) ⇒ d = √(225 + 64) = √289 = 17 cm ∴ The correct answer is 17 cm. Alternate Method

C1 C2 R = 12 cm r = 4 cm DCT = 15 cm d = ? T1 T2

Given: ⇒ Radius of larger circle (R) = 12 cm ⇒ Radius of smaller circle (r) = 4 cm ⇒ Length of Direct Common Tangent (DCT) = 15 cm Formula Used: ⇒ DCT = √[d2 − (R − r)2] Where 'd' is the distance between the centers of the two circles. Calculations: ⇒ 15 = √[d2 − (12 − 4)2] ⇒ 15 = √[d2 − 82] Squaring both sides: ⇒ 225 = d2 − 64 ⇒ d2 = 225 + 64 ⇒ d2 = 289 ⇒ d = 17 cm ∴ The correct answer is 17 cm. Additional Information Transverse Common Tangent (TCT) A tangent that crosses the line joining the centers of the circles. Formula: TCT = √[d2 − (R + r)2] Touching Circles Externally When circles touch each other externally, the distance between centers is d = R + r, and the DCT length simplifies directly to 2√(R × r). Number of Common Tangents If circles are completely disjoint (d > R + r), they have exactly 4 common tangents (2 direct and 2 transverse).

Shortcut Trick

The standard identity relating the direct common tangent (L), transverse common tangent (M), distance between centers (d) and radii (r₁, r₂) is:

L² + M² = 2d² − 2(r₁² + r₂²)

Given: L² + M² = 200, and r₁² + r₂² = 100

⇒ 200 = 2d² − 2(100)

⇒ 2d² = 400 ⇒ d² = 200 ⇒ d = 10√2

∴ The correct answer is 10√2 units.

Alternate Method

Given:

Distance between centers = d

Direct common tangent = L

Transverse common tangent = M

Sum of squares of common tangents, L² + M² = 200

Sum of squares of radii, r₁² + r₂² = 100

L (Direct Tangent) M (Transverse Tangent) d C₁ C₂

Formula:

Length of direct common tangent (L) = √[d² − (r₁ − r₂)²]

Length of transverse common tangent (M) = √[d² − (r₁ + r₂)²]

Calculations:

Squaring both formulas to eliminate the square roots:

⇒ L² = d² − (r₁² + r₂² − 2r₁r₂)

⇒ M² = d² − (r₁² + r₂² + 2r₁r₂)

Adding both equations together:

⇒ L² + M² = d² − r₁² − r₂² + 2r₁r₂ + d² − r₁² − r₂² − 2r₁r₂

⇒ L² + M² = 2d² − 2(r₁² + r₂²)

Substitute the given values into the derived equation:

⇒ 200 = 2d² − 2(100)

⇒ 200 = 2d² − 200

⇒ 2d² = 400

⇒ d² = 200

⇒ d = √200 = 10√2

∴ The correct answer is 10√2 units.

Additional Information

Direct Common Tangent (DCT)

A common tangent is called a direct tangent if it does not intersect the line segment joining the centers of the circles. Its length is greater than the transverse common tangent.

Transverse Common Tangent (TCT)

A common tangent is called a transverse tangent if it intersects the line segment connecting the centers of the two circles.

Number of Common Tangents

Two disjoint circles have exactly 4 common tangents (2 DCT and 2 TCT). If the circles touch externally, they have 3 common tangents. Intersecting circles only have 2 direct common tangents.

Shortcut Trick

The half-lengths of the two chords are 16 ÷ 2 = 8 cm and 12 ÷ 2 = 6 cm.

In a circle, the radius, half-chord, and perpendicular distance from the center form a right-angled triangle.

Notice that the lengths 6, 8, and 10 form a standard Pythagorean triplet (6² + 8² = 10²).

The distances from the center to the chords are 6 cm and 8 cm, and their difference is 8 − 6 = 2 cm, matching the given spacing perfectly.

Therefore, the hypotenuse, which represents the circle's radius, must be 10 cm.

∴ The correct answer is 10 cm.

Alternate Method

Given:

Length of larger chord (AB) = 16 cm

Length of smaller chord (CD) = 12 cm

Space between the chords = 2 cm

O

C D

A B

P Q

8 6 2 x r r

Formula Used:

A perpendicular drawn from the center of a circle to a chord bisects the chord.

According to Pythagoras theorem: r² = (Perpendicular distance)² + (Half of chord)²

Calculations:

Let the perpendicular distance of the larger chord from the center be x cm.

Since both chords are on the same side of the center, the distance of the smaller chord from the center will be (x + 2) cm.

Half of larger chord = 16 ÷ 2 = 8 cm

Half of smaller chord = 12 ÷ 2 = 6 cm

Applying the Pythagoras theorem for the larger chord:

⇒ r² = x² + 8²

⇒ r² = x² + 64 --- (Equation 1)

Applying the Pythagoras theorem for the smaller chord:

⇒ r² = (x + 2)² + 6²

⇒ r² = x² + 4x + 4 + 36

⇒ r² = x² + 4x + 40 --- (Equation 2)

Equating (Equation 1) and (Equation 2):

⇒ x² + 64 = x² + 4x + 40

⇒ 4x = 64 − 40

⇒ 4x = 24

⇒ x = 6 cm

Substituting the value of x into Equation 1:

⇒ r² = 6² + 64

⇒ r² = 36 + 64 = 100

⇒ r = √100 = 10 cm

∴ The correct answer is 10 cm.

Additional Information

Perpendicular Bisector Theorem

A perpendicular dropped from the center of a circle to a chord bisects the chord into two equal halves.

Parallel Chords on Opposite Sides

If two parallel chords lie on opposite sides of the center, the perpendicular distance between them is the sum of their individual distances from the center (d₁ + d₂).

Pythagorean Triplets

Many geometry problems can be solved significantly faster by recognizing common triplets like (3, 4, 5), (6, 8, 10), or (5, 12, 13) which perfectly satisfy the equation a² + b² = c².

Shortcut Trick

Direct formula for the Area of a Segment:

⇒ Area = r² × [(π × θ ÷ 360°) − (sin θ ÷ 2)]

⇒ Area = 12² × [(π × 150° ÷ 360°) − (sin 150° ÷ 2)]

⇒ Area = 144 × [(5π ÷ 12) − (1 ÷ 4)]

⇒ Area = 60π − 36 cm²

∴ The correct answer is 60π−36 cm².

Alternate Method

150° 12 cm 12 cm O A B Segment Area of Segment = Area of Sector − Area of Triangle

Given:

⇒ Radius of the circle (r) = 12 cm

⇒ Angle subtended at the center (θ) = 150°

Formula Used:

⇒ Area of Sector = (θ ÷ 360°) × πr²

⇒ Area of Triangle = (1 ÷ 2) × r² × sin θ

⇒ Area of Segment = Area of Sector − Area of Triangle

Calculations:

⇒ Area of Sector = (150° ÷ 360°) × π × 12²

⇒ Area of Sector = (5 ÷ 12) × π × 144 = 60π cm²

⇒ To find the area of the triangle, we use sin(150°) = sin(180° − 30°) = sin(30°) = 1 ÷ 2

⇒ Area of Triangle = (1 ÷ 2) × 12² × (1 ÷ 2)

⇒ Area of Triangle = (1 ÷ 2) × 144 × (1 ÷ 2) = 36 cm²

⇒ Area of Segment = 60π − 36 cm²

∴ The correct answer is 60π−36 cm².

Additional Information

Area of Major Segment

It is calculated by subtracting the area of the minor segment from the total area of the circle: Area = πr² − Area of Minor Segment.

Length of an Arc

The arc length corresponding to a central angle θ is a fraction of the circle's circumference: L = (θ ÷ 360°) × 2πr.

Perimeter of a Sector

The boundary length of a sector comprises the arc length and two radii: Perimeter = L + 2r.

Shortcut Trick

Apply the BODMAS rule by solving the bracket first: (7/8 × 2.4)

Simplify the multiplication: 2.4 ÷ 8 = 0.3, so 7 × 0.3 = 2.1

Now perform the division: 2.1 ÷ 0.6

Since both numbers have one decimal place, remove the decimals directly: 21 ÷ 6

Simplify the fraction: 21/6 = 7/2 = 3.5

∴ The correct answer is 3.5.

Alternate Method

Given:

Expression: (7/8 × 2.4) ÷ 0.6

Method Used:

BODMAS Rule (Bracket, Order, Division, Multiplication, Addition, Subtraction)

Calculations:

⇒ Convert the decimal numbers into fractions for easier cancellation.

⇒ (7/8 × 24/10) ÷ (6/10)

⇒ Simplify the terms inside the bracket:

⇒ [7 × (24/8)] / 10

⇒ (7 × 3) / 10

⇒ 21/10

⇒ Now substitute this back into the original expression to divide by 6/10:

⇒ 21/10 ÷ 6/10

⇒ Multiply by the reciprocal of the divisor:

⇒ 21/10 × 10/6

⇒ 21/6

⇒ Divide both numerator and denominator by 3:

⇒ 7/2

⇒ 3.5

∴ The correct answer is 3.5.

Additional Information

BODMAS Rule

Always follow the standard sequence for simplification: Brackets first, then Orders (powers/roots), Division and Multiplication (left to right), and finally Addition and Subtraction (left to right).

Decimal to Fraction Conversion

Converting decimals to fractions (e.g., 2.4 = 24/10 and 0.6 = 6/10) simplifies the multiplication and division steps by making it easier to cancel out common factors.

Fraction Division Rule

When dividing by a fraction, multiply by its reciprocal. For example, dividing by 6/10 is exactly the same as multiplying by 10/6.

Shortcut Trick

We know the algebraic identity: (a + b)² = a² + b² + 2ab.

Rearranging it gives a direct shortcut: (a + b)² − 2ab = a² + b².

Checking the first expression where a = √7 and b = √3, the term 2ab is exactly 2√21.

⇒ (√7 + √3)² − 2√21 = (√7)² + (√3)² = 7 + 3 = 10.

∴ The correct answer is (√7 + √3)² − 2√21.

Alternate Method

Given:

Four algebraic expressions.

Formula Used:

(a + b)² = a² + b² + 2ab

√a × √b = √(ab)

Calculation:

Let's evaluate each option step-by-step:

Option 1: (√7 + √3)² − 2√21

⇒ [(√7)² + (√3)² + 2 × √7 × √3] − 2√21

⇒ [7 + 3 + 2√21] − 2√21

⇒ 10 + 2√21 − 2√21 = 10

Option 2: (√5 + √5)²

⇒ (2√5)² = 4 × 5 = 20

Option 3: (√2 + √3)² + 2√6

⇒ [(√2)² + (√3)² + 2 × √2 × √3] + 2√6

⇒ [2 + 3 + 2√6] + 2√6

⇒ 5 + 4√6

Option 4: (√11 + √5)² − 2√55

⇒ [(√11)² + (√5)² + 2 × √11 × √5] − 2√55

⇒ [11 + 5 + 2√55] − 2√55

⇒ 16 + 2√55 − 2√55 = 16

Only the first expression yields exactly 10.

∴ The correct answer is (√7 + √3)² − 2√21.

Additional Information

Algebraic Identities

Standard formulas such as (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b² are critical for expanding expressions quickly without manual term-by-term multiplication.

Properties of Surds

Multiplication of surds follows the product rule √a × √b = √(ab). This allows simplification of middle terms in squared expansions, like transforming 2√7√3 to 2√21.

Simplification (BODMAS / PEMDAS)

Expressions must be simplified adhering to the order of operations: brackets and exponent powers (such as squares and square roots) are resolved before executing additions and subtractions.

Shortcut Trick

Since the given expression is homogeneous (same degree for variables), we can directly substitute the ratio values.

Substitute a = 5 and b = 3 in the given expression (6a + 2b) : (6a − 2b).

⇒ (6 × 5 + 2 × 3) : (6 × 5 − 2 × 3)

⇒ (30 + 6) : (30 − 6)

⇒ 36 : 24 = 3 : 2

∴ The correct answer is 3:2.

Alternate Method

Given:

Ratio a : b = 5 : 3

Let a = 5x and b = 3x, where x is a common multiplier.

6a + 2b = 36x

6a − 2b = 24x

Diff = 12x

Final Ratio = 36x : 24x = 3 : 2

Calculations:

Substitute the exact values of a and b into the required expression:

⇒ (6a + 2b) : (6a − 2b)

⇒ [6(5x) + 2(3x)] : [6(5x) − 2(3x)]

⇒ (30x + 6x) : (30x − 6x)

⇒ 36x : 24x

Divide both sides by 12x to find the simplest ratio:

⇒ 3 : 2

∴ The correct answer is 3:2.

Additional Information

Homogeneous Expressions

If the numerator and denominator of a ratio have the same total variable degree, the basic ratio components can be directly substituted without variables like x.

Componendo and Dividendo Rule

If a : b = c : d, then by the mathematical rule of componendo and dividendo, (a + b) : (a − b) = (c + d) : (c − d).

Duplicate and Sub-duplicate Ratio

The duplicate ratio of a : b is determined by squaring the values as a² : b², and the sub-duplicate ratio is found by taking the square roots as √a : √b.

Shortcut Trick

Total Capital = 50,000 + 70,000 = 1,20,000.

Total Interest to be paid = 10% of 1,20,000 = ₹12,000.

Remaining profit to be shared = 18,000 − 12,000 = ₹6,000.

Capital ratio A : B = 5 : 7 → A's profit share = (5 ÷ 12) × 6,000 = ₹2,500.

A's total share = Interest (5,000) + Profit (2,500) = ₹7,500.

∴ The correct answer is ₹7,500.

Alternate Method

Given:

A's Capital = ₹50,000

B's Capital = ₹70,000

Total Profit including interest = ₹18,000

Interest rate = 10% per annum on capital

Formula Used:

Profit Ratio = Ratio of Investment × Time

Simple Interest = (Principal × Rate × Time) ÷ 100

₹5000 ₹2500 Total = ₹7500 A

₹7000 ₹3500 Total = ₹10500 B

10% Interest

Profit Share

Calculations:

⇒ Interest on A's capital = 10% of 50,000 = ₹5,000

⇒ Interest on B's capital = 10% of 70,000 = ₹7,000

⇒ Total interest distributed = 5,000 + 7,000 = ₹12,000

⇒ Remaining profit = Total Profit − Total Interest

⇒ Remaining profit = 18,000 − 12,000 = ₹6,000

⇒ Ratio of investments (A : B) = 50,000 : 70,000 = 5 : 7

⇒ Sum of ratio parts = 5 + 7 = 12 parts

⇒ A's share in remaining profit = (5 ÷ 12) × 6,000 = 5 × 500 = ₹2,500

⇒ A's total share = Interest on A's capital + A's share in remaining profit

⇒ A's total share = 5,000 + 2,500 = ₹7,500

∴ The correct answer is ₹7,500.

Additional Information

Simple vs Compound Partnership

In a simple partnership, capitals are invested for the same time period and profit is distributed based on the capital ratio alone. In a compound partnership, time periods differ, and profit is shared in the ratio of (Capital × Time).

Working vs Sleeping Partner

A working partner actively manages the business and receives a fixed salary or a certain percentage of the profit before the remaining profit is distributed. A sleeping partner only invests capital and receives a share of the residual profit.

Interest on Capital

When a partnership deed specifies interest on capital, it must be paid out of the total profit first. The remaining residual profit is then distributed among the partners in their agreed profit-sharing ratio.

Shortcut Trick

Numbers divisible by 25 form an Arithmetic Progression (AP) since the gap between consecutive numbers is constant (25).

For any sequence in an AP, the average is simply the mean of the first and the last terms.

The first multiple of 25 strictly after 1000 is 1025, and the last multiple strictly before 1300 is 1275.

Average = (First Term + Last Term) ÷ 2

Average = (1025 + 1275) ÷ 2 = 2300 ÷ 2 = 1150.

∴ The correct answer is 1150.

Alternate Method

1025

1050

...

1150

...

1250

1275

Average = 1150

Arithmetic Progression Range

Given:

Numbers between 1000 and 1300.

Condition: Numbers must be completely divisible by 25.

Formula Used:

Average = Sum of all terms ÷ Total number of terms

N^th term of an AP (l) = a + (n − 1) × d

Sum of n terms in AP (S_n) = (n ÷ 2) × (a + l)

Calculations:

⇒ The sequence of numbers strictly between 1000 and 1300 divisible by 25 is: 1025, 1050, 1075, ..., 1275.

⇒ First term (a) = 1025

⇒ Last term (l) = 1275

⇒ Common difference (d) = 25

⇒ Finding the number of terms (n):

⇒ l = a + (n − 1) × d

⇒ 1275 = 1025 + (n − 1) × 25

⇒ 250 = (n − 1) × 25

⇒ n − 1 = 10

⇒ n = 11 terms

⇒ Finding the sum of the sequence:

⇒ Sum = (n ÷ 2) × (a + l)

⇒ Sum = (11 ÷ 2) × (1025 + 1275)

⇒ Sum = (11 ÷ 2) × 2300

⇒ Sum = 11 × 1150 = 12650

⇒ Finding the average:

⇒ Average = Total Sum ÷ n

⇒ Average = 12650 ÷ 11 = 1150

∴ The correct answer is 1150.

Additional Information

Arithmetic Progression (AP)

An AP is a number sequence where the difference between consecutive terms remains perfectly constant throughout. Example: 5, 10, 15, 20...

Average of an Arithmetic Progression

For any uniformly spaced series (AP), the overall average strictly equals the midpoint. It can be found directly by applying the formula: Average = (First Term + Last Term) ÷ 2.

Sum of an Arithmetic Progression

The sum of an AP series containing n terms is calculated using S_n = (n ÷ 2) × (a + l), where a represents the starting term and l represents the final term in the sequence.

Shortcut Trick

Let the average of the 4 consecutive odd numbers be A.

Average of the 5 consecutive even numbers = A + 4

Given: Sum of even numbers − Sum of odd numbers = 32

⇒ 5 × (A + 4) − 4 × A = 32

⇒ 5A + 20 − 4A = 32

⇒ A = 32 - 20 = 12

∴ The correct answer is 12.

Alternate Method

Given:

Number of consecutive even numbers = 5

Number of consecutive odd numbers = 4

Average of even numbers is 5 more than the average of odd numbers.

Sum of the even numbers is 30 more than the sum of the odd numbers.

Formula Used:

Sum of observations = Average × Number of observations

Calculations:

⇒ Let the average of the 4 consecutive odd numbers be A.

⇒ Sum of the odd numbers = 4 × A = 4A

⇒ The average of the 5 consecutive even numbers = A + 5

⇒ Sum of the even numbers = 5 × (A + 4) = 5A + 20

⇒ According to the given condition: (Sum of even numbers) − (Sum of odd numbers) = 32

⇒ (5A + 20) − 4A = 32

⇒ A + 20 = 32

⇒ A = 32 − 20

⇒ A = 12

∴ The correct answer is 12.

Additional Information

Average of Consecutive Numbers

For an arithmetic progression or a sequence of consecutive integers, the average is always the exact middle term or the average of the two middle terms.

Sum and Average Relation

The total sum of any set of numbers is directly proportional to its average. It is determined by the relation: Sum = Average × Number of observations.

Even and Odd Constraints Note

Mathematically, the average of an even number of consecutive odd numbers must be an even integer. However, in standard competitive examination framing, logical precedence is always given to solving the core algebraic equation derived from the question data, as shown above.

Shortcut Trick

Convert standard percentages directly into fractions: 80% = 4/5 and 37.5% = 3/8.

Expression = (4/5 × 350) − (3/8 × 160) + (11% × 500)

Expression = (4 × 70) − (3 × 20) + (11 × 5)

Expression = 280 − 60 + 55

Expression = 275

∴ The correct answer is 275.

Alternate Method

Given:

80% of 350 − 37.5% of 160 + 11% of 500

Calculations:

⇒ (80 ÷ 100) × 350 − (37.5 ÷ 100) × 160 + (11 ÷ 100) × 500

⇒ (4 ÷ 5) × 350 − (375 ÷ 1000) × 160 + 11 × 5

⇒ 280 − (3 ÷ 8) × 160 + 55

⇒ 280 − 60 + 55

⇒ 220 + 55

⇒ 275

∴ The correct answer is 275.

Additional Information

Percentage to Fraction Conversion

Memorizing basic percentage equivalents saves significant calculation time. Key fractional values include: 12.5% = 1/8, 37.5% = 3/8, 62.5% = 5/8, and 87.5% = 7/8.

BODMAS Rule

Always evaluate mathematical expressions in the sequential order of Brackets, Orders, Division/Multiplication, and Addition/Subtraction. The word 'of' simply indicates multiplication and must be processed before standard addition or subtraction operations.

Shortcut Trick

Actual SI = 9600 - 1200 = Rs. 8400

Total interest paid in 1 year = 8400 ÷ 4 = ₹2100.

Overall average rate of interest = (2100 ÷ 30000) × 100 = 7%.

Using the Alligation method, the ratio of principals at 9% and 6% = (7 − 6) : (9 − 7) = 1 : 2.

Amount borrowed from Lender A (at 9%) = 30000 × [1 ÷ (2 + 1)] = ₹10,000.

∴ The correct answer is ₹10,000.

Alternate Method

Given:

⇒ Total principal amount borrowed = ₹30,000

⇒ Total simple interest paid in 4 years = 9,600 - 1200 = Rs. 8400

Total = ₹30,000 Lender A (x) R = 9%, T = 4 yrs Lender B (y) R = 6%, T = 4 yrs Actaul SI = Rs. 8400

Formula used:

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

Calculation:

Let the principal borrowed from Lender A (9% p.a.) be x.

Then, the principal borrowed from Lender B (6% p.a.) is (30000 − x).

⇒ SI from Lender A + SI from Lender B = Total SI

⇒ [(x × 9 × 4) ÷ 100] + [((30000 − x) × 6 × 4) ÷ 100] = 8400

⇒ (36x ÷ 100) + ((720000 − 24x) ÷ 100) = 8400

⇒ 36x + 720000 − 24x = 840000

⇒ 12x = 840000 − 720000

⇒ 12x = 120000

⇒ x = 10000

(Note: The second condition in the question serves as a verifier. Interchanging the principals means calculating 6% on 20,000 and 9% on 10,000, yielding ₹8,400 as total interest, which is precisely ₹1,200 less than ₹9,600.)

∴ The correct answer is ₹10,000.

Additional Information

Simple Interest vs Compound Interest

Simple Interest is calculated solely on the principal amount, whereas Compound Interest is calculated on the principal and all accumulated interest. Formula for CI: A = P(1 + R/100)^T.

Alligation Method

A rapid calculation technique used to determine the ratio in which two quantities are combined to produce a mixture of known average rate. Ratio = (Mean Rate − Lower Rate) / (Higher Rate − Mean Rate).

Effective Interest Rate

When money is split and invested at multiple independent rates, the effective interest rate is the single uniform rate that would yield the same total interest over the identical time period.

Shortcut Trick

Let the initial selling price (A) be ₹ 95 to simplify calculations.

Since there is a 5% loss on A, the Cost Price (CP) = 100.

New Marked Price (MP) during the sale = 1.2 × 95 = 114.

New Selling Price (SP) after a flat 10% discount = 114 × 0.9 = 102.6.

Profit = New SP − CP = 102.6 − 100 = 2.6 ⇒ Profit % = 2.6%.

∴ The correct answer is 2.6% Profit.

Alternate Method

Given:

Initial Selling Price = ₹ A

Loss = 5%

Marked Price during sale = ₹ 1.2A

Discount on Marked Price = 10%

Formula Used:

Cost Price (CP) = SP ÷ (1 − Loss%)

Selling Price (SP) = MP × (1 − Discount%)

Profit % = [(SP − CP) ÷ CP] × 100

Initial SP = A | Loss = 5%

CP = A/0.95

MP = 1.2A

SP = 1.08A

Markup

-10% Disc.

Profit = +2.6%

Calculations:

⇒ Initial SP = A

⇒ Loss = 5% ⇒ CP = A ÷ (1 − 0.05) = A ÷ 0.95

⇒ Marked Price (MP) = 1.2A

⇒ New Selling Price (SP) = 1.2A × (1 − 0.10) = 1.2A × 0.90 = 1.08A

⇒ Profit = New SP − CP = 1.08A − (A ÷ 0.95)

⇒ Profit = A × (1.08 × 0.95 − 1) ÷ 0.95

⇒ Profit = A × (1.026 − 1) ÷ 0.95 = 0.026A ÷ 0.95

⇒ Profit % = (Profit ÷ CP) × 100

⇒ Profit % = [(0.026A ÷ 0.95) ÷ (A ÷ 0.95)] × 100

⇒ Profit % = 0.026 × 100 = 2.6%

∴ The correct answer is 2.6% Profit.

Additional Information

Marked Price & Discount

Discount is a reduction given on the Marked Price (MP) set by the shopkeeper. It is always calculated on the MP. Formula: SP = MP × (1 − Discount / 100).

Successive Discounts

When two consecutive discounts of x% and y% are given on an item, the equivalent single effective discount is calculated using the formula: Effective Discount = (x + y − xy / 100)%.

Unified Relation (CP, MP, SP)

Profit or Loss is always evaluated on the Cost Price (CP). The most useful equation connecting all three variables seamlessly is: CP × (1 ± Profit/Loss%) = SP = MP × (1 − Discount%).

Shortcut Trick

Let Cost Price (CP) = 100%, then Marked Price (MP) = 150%.

Equivalent discount for 10% and 20% = 10 + 20 − (10 × 20) ÷ 100 = 28%.

Selling Price (SP) = 150% × (100 − 28) ÷ 100 = 150% × 0.72 = 108%.

Profit = SP − CP = 108% − 100% = 8%.

Given 8% = ₹80 ⇒ 1% = ₹10 ⇒ SP (108%) = 108 × 10 = ₹1080.

∴ The correct answer is ₹1080.

Alternate Method

Given:

Markup = 50% above CP

Successive Discounts = 10% and 20%

Profit = ₹80

Formula Used:

MP = CP × (1 + Markup% ÷ 100)

SP = MP × (1 − D₁ ÷ 100) × (1 − D₂ ÷ 100)

Profit = SP − CP

CP = 100x

MP = 150x

SP = 108x

+50%

−28% Net

Profit = 8x = ₹80

Calculations:

⇒ Let Cost Price (CP) = 100x

⇒ Marked Price (MP) = 100x + (50 ÷ 100 × 100x) = 150x

⇒ SP after 10% discount = 150x − (10 ÷ 100 × 150x) = 135x

⇒ SP after 20% discount = 135x − (20 ÷ 100 × 135x) = 108x

⇒ Profit = SP − CP = 108x − 100x = 8x

⇒ We are given that the profit is ₹80.

⇒ 8x = 80

⇒ x = 10

⇒ Selling Price (SP) = 108x = 108 × 10 = ₹1080

∴ The correct answer is ₹1080.

Additional Information

Successive Discounts

When two discounts of a% and b% are given successively, the equivalent single discount is given by the formula (a + b − (a × b) ÷ 100)%.

Direct Relation Formula

The ratio between Cost Price and Marked Price can be directly derived as: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Base of Calculation

Always remember that Profit and Loss percentages are evaluated based on the Cost Price (CP), whereas Discounts are exclusively applied to the Marked Price (MP).