Quantitative Aptitude PYQs

Shortcut Trick

Substitute the coordinates (x, y) from each option directly into the equation y = 4x − 5.

For (1, −1): −1 = 4(1) − 5 ⇒ −1 = −1 (True)

For (0, −5): −5 = 4(0) − 5 ⇒ −5 = −5 (True)

For (2, 3): 3 = 4(2) − 5 ⇒ 3 = 8 − 5 ⇒ 3 = 3 (True)

Since all points satisfy the equation, the graph passes through all of them.

∴ The correct answer is All the above.

Alternate Method

Given:

The linear equation of the graph: y = 4x − 5

Concept Used:

A point (x, y) lies on a line if its coordinates completely satisfy the given linear equation when substituted into it.

Calculation:

(0, −5)

(1, −1)

(2, 3)

y = 4x − 5 X Y

⇒ Testing point (1, −1):

⇒ −1 = 4 × (1) − 5

⇒ −1 = −1 (Satisfies the equation)

⇒ Testing point (0, −5):

⇒ −5 = 4 × (0) − 5

⇒ −5 = −5 (Satisfies the equation)

⇒ Testing point (2, 3):

⇒ 3 = 4 × (2) − 5

⇒ 3 = 8 − 5 ⇒ 3 = 3 (Satisfies the equation)

⇒ As all the given points make the equation true, the graph of the line passes through all of them.

∴ The correct answer is All the above.

Additional Information

Slope-Intercept Form

The equation of a straight line is commonly expressed as y = mx + c, where m is the slope (steepness) of the line and c is the y-intercept (where the line intersects the Y-axis).

Finding Axis Intercepts

To find the x-intercept, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y. For the line y = 4x − 5, the y-intercept is −5 and the x-intercept is 5/4.

Collinear Points

If three or more points lie exactly on the same straight line, they are mathematically defined as collinear. This is confirmed if they satisfy the same single linear equation.

Shortcut Trick

Both sectors belong to the same circle, so their radii are equal (r = 9 cm).

The central angle of the first sector is 60°.

The central angle of the second sector is π/3 radians.

Convert π/3 radians to degrees: (π/3) × (180°/π) = 60°.

Since both the radius and the central angle are exactly the same, their areas must be identical.

∴ The correct answer is 1:1.

Alternate Method

Given:

Radius of both sectors (r) = 9 cm

Central angle of the first sector (θ₁) = 60°

Central angle of the second sector (θ₂) = π/3 radians

Formula Used:

Area of a sector (in degrees) = (θ ÷ 360°) × πr²

Area of a sector (in radians) = (1 ÷ 2) × r²θ

Sector 1

60° 9 cm

Sector 2

π/3 rad 9 cm

Area 1 : Area 2 = 1 : 1

Calculations:

Area of the first sector (A₁):

⇒ A₁ = (60° ÷ 360°) × π × (9)²

⇒ A₁ = (1 ÷ 6) × 81π = 13.5π cm²

Area of the second sector (A₂):

⇒ A₂ = (1 ÷ 2) × (9)² × (π ÷ 3)

⇒ A₂ = (1 ÷ 2) × 81 × (π ÷ 3) = 13.5π cm²

Ratio of the areas:

⇒ A₁ : A₂ = 13.5π : 13.5π

⇒ A₁ : A₂ = 1 : 1

∴ The correct answer is 1:1.

Additional Information

Area of a Sector (Degrees)

The area of a sector of a circle with central angle θ in degrees is calculated using the formula: Area = (θ ÷ 360°) × πr².

Area of a Sector (Radians)

When the central angle θ is given in radians, the formula simplifies to: Area = (1 ÷ 2) × r²θ.

Radian to Degree Conversion

To convert an angle from radians to degrees, multiply the value by (180° ÷ π). Therefore, π/3 radians is equal to 60°.

Shortcut Trick

By basic geometric definitions, the term parallelogram explicitly means a quadrilateral with two pairs of parallel opposite sides.

A direct geometric consequence of opposite sides being parallel is that they must also be equal in length.

Statements defining only one parallel pair describe a trapezium, while perpendicular diagonals describe a rhombus.

∴ The correct answer is Opposite sides are parallel and of equal length.

Alternate Method

Given:

Four statements describing the properties of quadrilaterals.

A B C D

Analysis of each option:

⇒ Statement 1: Only one pair of opposite sides is parallel. This defines a Trapezium.

⇒ Statement 2: All four interior angles are 90°. This defines a Rectangle or a Square.

⇒ Statement 3: Opposite sides are parallel and of equal length. This is the exact definition and core property of a Parallelogram.

⇒ Statement 4: The diagonals are perpendicular. This is a special property found in a Rhombus or a Square, but not in all standard parallelograms.

⇒ Therefore, the only universally true statement for any general parallelogram is that its opposite sides are parallel and equal.

∴ The correct answer is Opposite sides are parallel and of equal length.

Additional Information

Rhombus

A parallelogram in which all four sides are equal. The diagonals of a rhombus always bisect each other at exactly 90°.

Rectangle

A parallelogram where all four interior angles are exactly 90°. Its diagonals are equal in length and bisect each other.

Trapezium

A quadrilateral with strictly only one pair of parallel opposite sides. The parallel sides are called bases.

Square

A regular quadrilateral satisfying properties of both a rectangle and a rhombus. All sides are equal, all angles are 90°, and diagonals intersect at 90°.

Shortcut Trick

Scale factor of the similar triangles = 39 ÷ 13 = 3.

Area of 1^st triangle = 1/2 × 5 × √(13² − 5²) = 1/2 × 5 × 12 = 30 cm².

Ratio of areas = (Scale factor)² = 3² = 9.

Area of 2^nd triangle = 30 × 9 = 270 cm².

∴ The correct answer is 270 cm².

Alternate Method

5 cm 12 cm 13 cm Triangle 1

15 cm 36 cm 39 cm Triangle 2

Given:

Hypotenuse of first triangle = 13 cm

One leg of first triangle = 5 cm

Hypotenuse of second similar triangle = 39 cm

Formula Used:

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

Area of a Triangle = 1/2 × Base × Height

Calculations:

⇒ Let the unknown leg of the first triangle be x.

⇒ x² + 5² = 13²

⇒ x² + 25 = 169

⇒ x² = 144 ⇒ x = 12 cm

⇒ The second triangle is similar to the first triangle.

⇒ Ratio of their corresponding sides = 39 ÷ 13 = 3

⇒ Therefore, the legs of the second triangle will also be 3 times the legs of the first triangle.

⇒ Base of the second triangle = 5 × 3 = 15 cm

⇒ Height of the second triangle = 12 × 3 = 36 cm

⇒ Area of second triangle = 1/2 × 15 × 36

⇒ Area of second triangle = 15 × 18 = 270 cm²

∴ The correct answer is 270 cm².

Additional Information

Properties of Similar Triangles

If two triangles are similar, the ratio of their corresponding sides, medians, altitudes, and perimeters is exactly equal to the scale factor (k).

Area Theorem for Similar Triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides: Area₁ ÷ Area₂ = (Side₁ ÷ Side₂)².

Pythagorean Triples

Memorizing common Pythagorean triples helps in fast calculations. Some common primitive triples are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Shortcut Trick

Identify the equal corresponding parts between ΔABD and ΔCDB directly from the question parameters.

We are given Side AB = Side CD and ∠ABD = ∠CDB.

The third equal part is the common diagonal, so Side BD = Side DB.

Since the given equal angle is included strictly between the two equal sides, we apply the Side-Angle-Side rule.

Therefore, the triangles are congruent by the SAS criterion.

∴ The correct answer is Yes, by SAS.

Alternate Method

θ θ

A B C D

Given:

⇒ Quadrilateral ABCD with diagonal BD.

⇒ AB = CD

⇒ ∠ABD = ∠CDB

Congruence Check:

⇒ Consider ΔABD and ΔCDB:

⇒ AB = CD (Given Side)

⇒ ∠ABD = ∠CDB (Given Included Angle)

⇒ BD = DB (Common Side)

⇒ Here, two sides and the included angle of ΔABD are strictly equal to the corresponding two sides and included angle of ΔCDB.

⇒ By Side-Angle-Side (SAS) congruence criterion, the two triangles are congruent (ΔABD ≅ ΔCDB).

∴ The correct answer is Yes, by SAS.

Additional Information

SAS (Side-Angle-Side) Congruence

If two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle, the triangles are exactly congruent.

SSS (Side-Side-Side) Congruence

If all three sides of one triangle are identical in length to the corresponding three sides of another triangle, then the two triangles are congruent.

ASA (Angle-Side-Angle) Congruence

If two angles and the included side of one triangle are equal to the corresponding two angles and the included side of another triangle, the triangles are congruent.

RHS (Right Angle-Hypotenuse-Side)

If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and corresponding side of another right-angled triangle, they are congruent.

Shortcut Trick

Apply the Digital Sum (DS) method to solve instantly.

DS of 27³ = 0; DS of 10³ = 1; DS of 29³ = 8; DS of 246 = 3.

DS of expression = 0 + 1 − 8 + 3 = −4. (Add 9 for positive DS) ⇒ 5.

Checking options, DS of −3460 = −(13) → −4 ⇒ 5, matching perfectly.

∴ The correct answer is -3460.

Alternate Method

Given:

27³ + 10³ − 29³ + 246

Calculation:

⇒ 27³ = 27 × 27 × 27 = 19683

⇒ 10³ = 10 × 10 × 10 = 1000

⇒ 29³ = 29 × 29 × 29 = 24389

Substituting the values into the given expression:

⇒ 19683 + 1000 − 24389 + 246

⇒ 20683 − 24389 + 246

⇒ 20929 − 24389

⇒ −3460

∴ The correct answer is -3460.

Additional Information

Digital Sum Method

The digital sum of a number is the single-digit value obtained by repeatedly adding its digits. It is highly useful in checking calculations because the digital sum of an expression equals the digital sum of its final answer. If a digital sum is negative, add 9 to it.

Unit Digit Concept

The unit digit of an expression can be determined by operating only on the unit digits of the numbers. For example, the unit digit of 27³ is the same as the unit digit of 7³, which is 3.

BODMAS Rule

Simplification should always follow the BODMAS order: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

Shortcut Trick

For a right-angled triangle, if sin x = 3/5, the sides form a standard Pythagorean triplet (3, 4, 5).

Thus, Perpendicular = 3, Hypotenuse = 5, and Base = 4.

We know tan x = Perpendicular ÷ Base = 3/4.

Expression = (1 + 3/4) ÷ (1 − 3/4) = (7/4) ÷ (1/4) = 7.

∴ The correct answer is 7.

Alternate Method

Given:

sin x = 3/5

x ∈ (0, π/2), which means x lies in the first quadrant, so all trigonometric ratios are positive.

Formula Used:

sin x = Perpendicular / Hypotenuse

tan x = Perpendicular / Base

Hypotenuse² = Perpendicular² + Base² (Pythagoras Theorem)

x

C B A

Base (B) = 4 Perpendicular (P) = 3 Hypotenuse (H) = 5

Calculations:

⇒ Let Perpendicular (P) = 3 units and Hypotenuse (H) = 5 units.

⇒ Base (B) = √(H² − P²)

⇒ B = √(5² − 3²) = √(25 − 9) = √16 = 4 units.

⇒ tan x = P / B = 3 / 4

Now, substitute the value of tan x into the given expression:

⇒ (1 + tan x) ÷ (1 − tan x)

⇒ (1 + 3/4) ÷ (1 − 3/4)

⇒ [(4 + 3) / 4] ÷ [(4 − 3) / 4]

⇒ (7 / 4) × (4 / 1)

⇒ 7

∴ The correct answer is 7.

Additional Information

Trigonometric Quadrant Rule (ASTC)

In the 1^st quadrant (0 to π/2), All ratios are positive. In the 2^nd quadrant, Sine & Cosec are positive. In the 3^rd, Tangent & Cot are positive. In the 4^th, Cosine & Sec are positive.

Pythagorean Triplets

Sets of integers that satisfy a² + b² = c². Standard base triplets include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Multiples of these base numbers are also proper triplets.

Basic Trigonometric Ratios

sin θ = Perpendicular / Hypotenuse | cos θ = Base / Hypotenuse | tan θ = Perpendicular / Base.

Shortcut Trick

The radius, tangent, and the line joining the external point to the center form a right-angled triangle.

Using the Pythagoras theorem: Distance = √(Radius² + Tangent²)

⇒ Distance = √(6² + 10²)

⇒ Distance = √(36 + 100) = √136

⇒ Distance ≈ 11.66 cm

The closest approximate value among the options is 12 cm.

∴ The correct answer is 12 cm.

Alternate Method

Given:

Length of the tangent (PT) = 10 cm

Radius of the circle (OT) = 6 cm

Formula Used:

According to the tangent-radius theorem, the tangent at any point of a circle is perpendicular to the radius through the point of contact.

In right-angled ΔOTP (where ∠OTP = 90°):

OP² = OT² + PT² (Pythagoras Theorem)

O P T

6 cm 10 cm d = ?

Calculations:

⇒ OP² = 6² + 10²

⇒ OP² = 36 + 100

⇒ OP² = 136

⇒ OP = √136

⇒ OP ≈ 11.66 cm

Approximating to the nearest whole number, the distance from the external point to the center is 12 cm.

∴ The correct answer is 12 cm.

Additional Information

Properties of Tangent

A tangent to a circle is a line that touches the circle at exactly one point, and it is always perpendicular to the radius at that specific point of tangency (∠OTP = 90°).

Number of Tangents

From any external point outside a circle, exactly two tangents can be drawn. The lengths of these two tangents to the points of tangency are always strictly equal.

Pythagorean Theorem Check

While 6, 8, and 10 form a famous Pythagorean triplet (where 10 is the hypotenuse), in this case, the tangent (10) and radius (6) are the perpendicular legs, making the hypotenuse √136 instead of an integer.

Shortcut Trick

For a right-angled triangle, the inradius can be directly calculated using the lengths of the Perpendicular (P), Base (B), and Hypotenuse (H).

Inradius (r) = (P + B − H) ÷ 2

Hypotenuse = √(5² + 12²) = 13

r = (5 + 12 − 13) ÷ 2 = 4 ÷ 2 = 2

∴ The correct answer is 2 units.

Alternate Method

r r Base = 12 Height = 5 Hypotenuse = 13

Given:

Legs of the right triangle are a = 5 and b = 12.

Formula Used:

Hypotenuse (c) = √(a² + b²)

Area of triangle (A) = 1/2 × Base × Height

Semi-perimeter (s) = (a + b + c) ÷ 2

Inradius (r) = Area ÷ s

Calculations:

⇒ c = √(5² + 12²)

⇒ c = √(25 + 144) = √169 = 13

⇒ Area = 1/2 × 5 × 12 = 30

⇒ s = (5 + 12 + 13) ÷ 2 = 30 ÷ 2 = 15

⇒ Inradius (r) = 30 ÷ 15 = 2

∴ The correct answer is 2 units.

Additional Information

Circumradius of a Right Triangle

The circumcenter of a right-angled triangle lies exactly at the midpoint of its hypotenuse. Therefore, the Circumradius (R) is always exactly half of the hypotenuse: R = Hypotenuse ÷ 2.

Inradius of any General Triangle

For any triangle, regardless of its angles, the radius of the inscribed circle (inradius) is given by the universal formula r = Area ÷ s, where s is the semi-perimeter.

Pythagorean Triplets

A set of three positive integers a, b, and c that satisfy a² + b² = c². Common triplets frequently tested in exams include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Shortcut Trick

If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Here, a = 0.5, b = 0.1, and c = −0.6, which gives a + b + c = 0.5 + 0.1 − 0.6 = 0.

The numerator becomes 3 × 0.5 × 0.1 × (−0.6).

Expression = [3 × 0.5 × 0.1 × (−0.6)] ÷ (3 × 0.5 × 0.1 × 0.6).

All values cancel out leaving exactly −1.

∴ The correct answer is −1.

Alternate Method

Given:

(0.5³ + 0.1³ − 0.6³) ÷ (3 × 0.5 × 0.1 × 0.6)

Formula Used:

If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Calculations:

⇒ Let a = 0.5, b = 0.1, and c = −0.6

⇒ Sum of terms: a + b + c = 0.5 + 0.1 + (−0.6) = 0.6 − 0.6 = 0

⇒ Since a + b + c = 0, the numerator a³ + b³ + c³ directly equals 3abc.

⇒ Numerator = 3 × 0.5 × 0.1 × (−0.6)

⇒ Substitute this evaluated numerator back into the full expression.

⇒ Expression = [3 × 0.5 × 0.1 × (−0.6)] ÷ (3 × 0.5 × 0.1 × 0.6)

⇒ Cancel out the common terms (3 × 0.5 × 0.1) from both the numerator and the denominator.

⇒ Expression = (−0.6) ÷ 0.6

⇒ Expression = −1

∴ The correct answer is −1.

Additional Information

Standard Identity for Cubes

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

Sum of Two Cubes

a³ + b³ = (a + b)(a² − ab + b²)

Difference of Two Cubes

a³ − b³ = (a − b)(a² + ab + b²)

Shortcut Trick

For any two variables X and Y, if X + Y = A and X − Y = B, then X = (A + B) ÷ 2 and Y = (A − B) ÷ 2.

Here, treating X = √a and Y = √b:

⇒ √a = (5 + 1) ÷ 2 = 3 ⇒ a = 3² = 9

⇒ √b = (5 − 1) ÷ 2 = 2 ⇒ b = 2² = 4

∴ The correct answer is a = 9, b = 4.

Alternate Method

Given:

√a + √b = 5 .....(i)

√a − √b = 1 .....(ii)

Calculation:

Adding equation (i) and equation (ii), we get:

⇒ (√a + √b) + (√a − √b) = 5 + 1

⇒ 2√a = 6

⇒ √a = 3

Squaring both sides to find the value of a:

⇒ a = 3² = 9

Substituting the value of √a back into equation (i):

⇒ 3 + √b = 5

⇒ √b = 5 − 3

⇒ √b = 2

Squaring both sides to find the value of b:

⇒ b = 2² = 4

∴ The correct answer is a = 9, b = 4.

Additional Information

Properties of Surds

If √x = y, squaring both sides removes the square root to give x = y². Surds can be treated like standard algebraic variables when adding or subtracting like terms.

Difference of Squares

The identity a − b = (√a + √b)(√a − √b) links linear factors to square root terms. For this question, multiplying the equations gives a − b = 5 × 1 = 5, which perfectly matches 9 − 4 = 5.

Linear Equations

When provided with the sum and difference of the same two unknowns, adding the equations eliminates the second variable while subtracting them eliminates the first variable.

Shortcut Trick

A perfect square number can never end in the digits 2, 3, 7, or 8.

Looking at the given options, 100588, 100688, and 100788 all end in the digit 8.

Thus, they are instantly eliminated as they cannot be perfect squares.

Only 100489 ends in a valid perfect square unit digit (9).

∴ The correct answer is 100489.

Alternate Method

Given:

The smallest 6-digit number is 100000.

Method Used:

Find the approximate square root of the smallest 6-digit number, and square the next higher whole number to get the least perfect square.

Calculation:

⇒ Let us estimate the square root of 100000.

⇒ We know that 300² = 90000

⇒ Let us check 316: 316² = 99856 (This is a 5-digit number)

⇒ Since 99856 is less than 100000, 316 is the largest integer whose square is a 5-digit number.

⇒ To find the smallest 6-digit perfect square, we must square the next integer, which is 317.

⇒ Least 6-digit perfect square = 317²

⇒ 317² = (300 + 17)²

⇒ 317² = 300² + 2 × 300 × 17 + 17²

⇒ 317² = 90000 + 10200 + 289

⇒ 317² = 100489

∴ The correct answer is 100489.

Additional Information

Unit Digit Rule for Perfect Squares

Perfect squares can only end in the digits 0, 1, 4, 5, 6, or 9. Any number ending with 2, 3, 7, or 8 is never a perfect square.

Digital Root of Perfect Squares

The sum of the digits (reduced to a single digit) of a perfect square must always be 1, 4, 7, or 9. This serves as a quick check to eliminate non-squares.

Last Two Digits Check

The last two digits of any perfect square will always be the same as the last two digits of the square of some integer from 1 to 24 (e.g., 17² = 289, so 89 is a valid ending).

Shortcut Trick

Convert percentages into standard fractions directly: 12.5% = 1/8 and 50% = 1/2.

According to the given condition: (1/8) × p = (1/2) × q

Multiply both sides by 8: p = 4 × q

Rearranging the variables gives the ratio: p / q = 4 / 1

∴ The correct answer is 4 : 1.

Alternate Method

Given:

12.5% of p = 50% of q

Formula Used:

x% of A = (x ÷ 100) × A

Visualizing 12.5% of p = 50% of q p 12.5% Total p (100% = 8 × 12.5%) q 50% Total q (2 × 50%) = Ratio p : q = 320 : 80 = 4 : 1

Calculations:

⇒ (12.5 ÷ 100) × p = (50 ÷ 100) × q

⇒ 12.5 × p = 50 × q

⇒ p ÷ q = 50 ÷ 12.5

⇒ p ÷ q = 500 ÷ 125

⇒ p ÷ q = 4 ÷ 1

⇒ p : q = 4 : 1

∴ The correct answer is 4 : 1.

Additional Information

Percentage Definition

A percentage signifies a value out of 100. The basic equation to find x% of a number y is written mathematically as (x ÷ 100) × y.

Percentage to Fraction Shortcuts

Remembering standard fractions saves calculation time in exams: 12.5% = 1/8, 25% = 1/4, 37.5% = 3/8, 50% = 1/2, 62.5% = 5/8, and 87.5% = 7/8.

Properties of Proportions

If two quantities satisfy the equality equation a × x = b × y, the numerical ratio between the variables x and y directly equals b : a.

Shortcut Trick

Mean Cost Price (CP) = (Price₁ × Ratio₁ + Price₂ × Ratio₂) ÷ Total Ratio

⇒ Mean CP = (40 × 2 + 60 × 3) ÷ (2 + 3) = (80 + 180) ÷ 5 = ₹52/kg

Given Selling Price (SP) = ₹57.20/kg

⇒ Profit = SP − CP = 57.20 − 52.00 = ₹5.20/kg

⇒ Profit % = (Profit ÷ CP) × 100 = (5.20 ÷ 52) × 100 = 10% Gain

∴ The correct answer is 10% Gain.

Alternate Method

Given:

Cost Price (CP) of 1^st variety of rice = ₹40/kg

Cost Price (CP) of 2^nd variety of rice = ₹60/kg

Ratio of mixture = 2 : 3

Selling Price (SP) of the mixed variety = ₹57.20/kg

₹40/kg ₹60/kg

Mean CP ₹52/kg

2 Parts 3 Parts

SP = ₹57.20 → 10% Profit

Formula Used:

Mean Cost Price = (Quantity₁ × Price₁ + Quantity₂ × Price₂) ÷ (Quantity₁ + Quantity₂)

Profit Percentage = (Profit ÷ Cost Price) × 100

Calculations:

⇒ Let the exact quantity of the first variety be 2 kg and the second variety be 3 kg.

⇒ Total Cost Price of the mixture = (2 × 40) + (3 × 60)

⇒ Total Cost Price = 80 + 180 = ₹260

⇒ Total Quantity = 2 + 3 = 5 kg

⇒ Cost Price (CP) per kg of the mixed variety = 260 ÷ 5 = ₹52

⇒ The Selling Price (SP) of the mixture is ₹57.20 per kg.

⇒ Since SP > CP, the grocer makes a gain.

⇒ Gain per kg = SP − CP = 57.20 − 52.00 = ₹5.20

⇒ Gain Percent = (Gain ÷ CP) × 100

⇒ Gain Percent = (5.20 ÷ 52) × 100 = 0.10 × 100 = 10%

∴ The correct answer is 10% Gain.

Additional Information

Alligation Method

It is a rule that establishes the relationship between the quantities and prices of different ingredients. Ratio = (Dearer Price − Mean Price) : (Mean Price − Cheaper Price).

Weighted Average

When multiple elements are mixed in differing proportions, the net average value per unit is found using the weighted average formula: Total Value ÷ Total Units.

Profit and Loss Concept

Gain or Loss is strictly calculated over the overall average Cost Price (CP). If SP > CP, there is a profit, and Profit % = [(SP − CP) ÷ CP] × 100.

Shortcut Trick

Using the successive discount multiplying factor method:

Final SP = MP × (100 − D₁)% × (100 − D₂)%

Final SP = 600 × (90 ÷ 100) × (80 ÷ 100)

Final SP = 600 × 0.9 × 0.8 = 432

∴ The correct answer is ₹432.

Alternate Method

Given:

Original Marked Price (MP) = ₹600

First successive discount = 10%

Second successive discount = 20%

Formula Used:

Selling Price (SP) = MP − Discount Amount

Discount = Rate% of Base Price

₹600 Marked Price

−10%

₹540 After 1^st Disc.

−20%

₹432 Final SP

Step-by-Step Discount Reduction

Calculations:

⇒ First discount amount = 10% of 600 = (10 ÷ 100) × 600 = ₹60

⇒ Price after first discount = 600 − 60 = ₹540

⇒ Second discount amount = 20% of 540 = (20 ÷ 100) × 540 = ₹108

⇒ Final Selling Price = 540 − 108 = ₹432

∴ The correct answer is ₹432.

Additional Information

Equivalent Successive Discount

When two successive discounts of a% and b% are given, the single equivalent discount is calculated as: D = a + b − (a × b) ÷ 100.

Marked Price vs Selling Price

Discounts are always evaluated on the Marked Price (MP). Hence, the base value changes after every applied discount, making Selling Price (SP) = MP − Discount.

Fraction Method for Discounts

A 10% discount implies the new price is 90% (or 9/10) of the original, while a 20% discount implies the new price is 80% (or 4/5) of the base value. Multiplying these fractions provides direct and accurate results.

Shortcut Trick

MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%)

120 ÷ 100 = (100 + 8) ÷ (100 − D)

6 ÷ 5 = 108 ÷ (100 − D)

100 − D = (108 × 5) ÷ 6 = 90

D = 10

∴ The correct answer is 10%.

Alternate Method

Given:

Markup Percentage = 20%

Profit Percentage = 8%

Formula Used:

MP = CP × (1 + Markup% ÷ 100)

SP = CP × (1 + Profit% ÷ 100)

Discount % = (MP − SP) ÷ MP × 100

CP 100

SP 108

MP 120

Markup = +20%

Profit = +8%

Discount = ?

Calculation:

⇒ Let the Cost Price (CP) of the commodity be 100 units.

⇒ The Marked Price (MP) is set at a 20% premium over the Cost Price.

⇒ Marked Price (MP) = 100 × (1 + 20 ÷ 100) = 120 units

⇒ The transaction ultimately yields an 8% profit on the Cost Price.

⇒ Selling Price (SP) = 100 × (1 + 8 ÷ 100) = 108 units

⇒ Discount given = Marked Price − Selling Price

⇒ Discount = 120 − 108 = 12 units

⇒ Discount % = (Discount ÷ MP) × 100

⇒ Discount % = (12 ÷ 120) × 100

⇒ Discount % = 10%

∴ The correct answer is 10%.

Additional Information

Direct Relation Between CP and MP

The direct ratio between Cost Price and Marked Price is always defined as: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%). This allows for rapid calculation skipping the SP entirely.

Successive Discounts

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

Dishonest Dealer Concept

When a dealer pretends to sell goods at CP but uses a manipulated weight, the actual profit percentage is calculated as: [(True Weight − False Weight) ÷ False Weight] × 100.

Shortcut Trick

We can solve this quickly using the value-putting method by finding the integer factors of 6.

The possible factors of 6 are (1, 6) and (2, 3).

Let us test a = 2 and b = 3 ⇒ a³ + b³ = 2³ + 3³ = 8 + 27 = 35.

This matches the given condition perfectly.

Therefore, a + b = 2 + 3 = 5.

∴ The correct answer is 5.

Alternate Method

Given:

a³ + b³ = 35

ab = 6

Formula Used:

(a + b)³ = a³ + b³ + 3ab(a + b)

Calculations:

Let the value of (a + b) be x.

⇒ x³ = 35 + 3 × 6 × x

⇒ x³ = 35 + 18x

⇒ x³ − 18x − 35 = 0

By hit and trial method, if we substitute x = 5:

⇒ 5³ − 18(5) − 35

⇒ 125 − 90 − 35 = 0

Since it perfectly satisfies the equation, we get a + b = 5.

∴ The correct answer is 5.

Additional Information

Algebraic Identity for Cubes

(a − b)³ = a³ − b³ − 3ab(a − b)

Sum of Cubes

a³ + b³ = (a + b)(a² − ab + b²)

Difference of Cubes

a³ − b³ = (a − b)(a² + ab + b²)

Shortcut Trick

Total Simple Interest (SI) generated in 4 years = 2520 − 2000 = ₹520.

Interest per year = 520 ÷ 4 = ₹130.

Rate % = (Interest per year ÷ Principal) × 100

Rate % = (130 ÷ 2000) × 100 = 6.5%.

∴ The correct answer is 6.5%.

Alternate Method

Given:

Principal (P) = ₹2,000

Amount (A) = ₹2,520

Time (T) = 4 years

Formula Used:

Simple Interest (SI) = Amount − Principal

Rate (R) = (SI × 100) ÷ (P × T)

Principal ₹2,000 Amount ₹2,520

Time = 4 Years

• SI = ₹520

Calculation:

⇒ SI = 2520 − 2000 = ₹520

⇒ R = (520 × 100) ÷ (2000 × 4)

⇒ R = 52000 ÷ 8000

⇒ R = 52 ÷ 8

⇒ R = 6.5%

∴ The correct answer is 6.5%.

Additional Information

Simple Interest (SI) vs Compound Interest (CI)

SI is calculated only on the principal amount for the entire period, while CI is calculated on the principal and also on the accumulated interest of previous periods.

Principal & Amount

Principal is the initial sum of money borrowed or invested. Amount is the total money paid back after the given time, which includes Principal and Interest (A = P + SI).

Finding Time (T)

If Principal (P), Rate (R), and Simple Interest (SI) are given, the time period can be calculated directly using the formula: T = (SI × 100) ÷ (P × R).

Shortcut Trick

Simple Interest for 1 year from the 1^st scheme = 6% of ₹8,000 = ₹480

Simple Interest for 1 year from the 2^nd scheme = 8% of ₹5,000 = ₹400

Total Simple Interest generated in 1 year = 480 + 400 = ₹880

Total time duration = Total Interest Received ÷ 1 Year Interest = 1760 ÷ 880 = 2 years

∴ The correct answer is 2 years.

Alternate Method

Given:

Principal of 1^st investment (P₁) = ₹8,000

Rate of 1^st investment (R₁) = 6%

Principal of 2^nd investment (P₂) = ₹5,000

Rate of 2^nd investment (R₂) = 8%

Total Simple Interest (SI) = ₹1,760

Formula Used:

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

Scheme 1 ₹8,000 @ 6% for T years

Scheme 2 ₹5,000 @ 8% for T years

Total SI = ₹1,760

Calculations:

Let the duration of the investments be T years.

⇒ Total SI = SI₁ + SI₂

⇒ 1760 = [(8000 × 6 × T) ÷ 100] + [(5000 × 8 × T) ÷ 100]

⇒ 1760 = 480T + 400T

⇒ 1760 = 880T

⇒ T = 1760 ÷ 880

⇒ T = 2 years

∴ The correct answer is 2 years.

Additional Information

Compound Interest (CI) Formula

When interest is compounded annually, the total amount is A = P × (1 + R ÷ 100)^T and Compound Interest = A − P.

Simple Interest vs Compound Interest

For the first year (or the first compounding period), Simple Interest and Compound Interest are exactly the same when calculated on the same principal at the same rate of interest.

Effective Total Simple Interest

If multiple principals P₁, P₂... are invested at rates R₁, R₂... for the same time duration T, the Total SI = (P₁R₁ + P₂R₂ + ...) × T ÷ 100.

Shortcut Trick

In any 30°-60°-90° right-angled triangle, the side opposite the 30° angle is exactly half of the hypotenuse.

⇒ Ramp Length = 2 × Height

⇒ Ramp Length = 2 × 1.5

⇒ Ramp Length = 3 m

∴ The correct answer is 3 m.

Alternate Method

A B C 1.5 m L (Ramp Length) 30°

Given:

Height of the platform (Opposite side) = 1.5 m

Angle of elevation of the ramp from the ground (θ) = 30°

Formula Used:

sin(θ) = Opposite ÷ Hypotenuse

Calculations:

Let the length of the ramp be L.

⇒ sin(30°) = 1.5 ÷ L

⇒ 1 ÷ 2 = 1.5 ÷ L

⇒ L = 1.5 × 2

⇒ L = 3 m

∴ The correct answer is 3 m.

Additional Information

Trigonometric Ratios

For a right-angled triangle, sin(θ) = Opposite ÷ Hypotenuse, cos(θ) = Adjacent ÷ Hypotenuse, and tan(θ) = Opposite ÷ Adjacent.

Angle of Elevation vs Depression

The angle of elevation is measured upwards from the horizontal line of sight, whereas the angle of depression is measured downwards. In parallel configurations, they are mathematically equal alternate interior angles.

30°-60°-90° Triangle Theorem

The sides of a 30°-60°-90° right triangle are always in the precise ratio 1 : √3 : 2, making geometrical height and distance calculations much faster without full trigonometric tables.