Quantitative Aptitude PYQs

Shortcut Trick

When persons work together for the exact same duration, the work done by them is directly proportional to their individual efficiency.

Let the efficiency of B = 100%, then the efficiency of A = 100% + 25% = 125%.

Ratio of work done by A to total work = Efficiency of A ÷ Total efficiency = 125 ÷ (125 + 100) = 125 ÷ 225 = 5 ÷ 9.

Percentage of work done by A = (5 ÷ 9) × 100 = 55.55...% ≈ 55.56%.

∴ The correct answer is 55.56%.

Alternate Method

Given:

A and B together can complete the task in 12 days.

A is 25% more efficient than B.

Formula Used:

Total Work = Efficiency × Time

Percentage of work = (Work done ÷ Total work) × 100

Calculations:

⇒ Let the efficiency of B be 4 units/day.

⇒ Efficiency of A = 4 + (25% of 4) = 5 units/day.

⇒ Combined efficiency of A and B = 5 + 4 = 9 units/day.

⇒ Total work = Combined efficiency × Time = 9 × 12 = 108 units.

⇒ Work done by A alone in 12 days = Efficiency of A × 12 = 5 × 12 = 60 units.

⇒ Percentage of work done by A = (Work done by A ÷ Total work) × 100.

⇒ Percentage of work done by A = (60 ÷ 108) × 100.

⇒ Percentage of work done by A = (5 ÷ 9) × 100 ≈ 55.56%.

∴ The correct answer is 55.56%.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to time. If person A is x times more efficient than B, the ratio of time taken by A to B is 1 : x.

Work and Wages

Wages are always distributed in proportion to the total work done by each individual. If all individuals work for the exact same duration, wages are strictly divided in the ratio of their efficiencies.

Percentage Change Formulation

If A is R% more than B, then the value of A can be directly calculated as A = B × (100 + R) ÷ 100.

Shortcut Trick

Total CP = Base Cost + Expenses = 50,000 + 5,000 + 3,000 = 58,000

SP = 63,000

Profit = SP − Total CP = 63,000 − 58,000 = 5,000

Profit % = (5,000 ÷ 58,000) × 100 = 500 ÷ 58 ≈ 8.62%

∴ The correct answer is 8.62% profit.

Alternate Method

Given:

Initial Cost Price (CP) = ₹50,000

Transportation Cost = ₹5,000

Storage Cost = ₹3,000

Selling Price (SP) = ₹63,000

Formula Used:

Total CP = Initial CP + Additional Expenses

Profit = SP − Total CP

Profit % = (Profit ÷ Total CP) × 100

Total CP = ₹58,000

SP = ₹63,000

Profit = ₹5,000 Profit % = (5,000 ÷ 58,000) × 100 ≈ 8.62%

Calculations:

⇒ Total CP = 50,000 + 5,000 + 3,000

⇒ Total CP = ₹58,000

⇒ Profit = 63,000 − 58,000

⇒ Profit = ₹5,000

⇒ Profit % = (5,000 ÷ 58,000) × 100

⇒ Profit % = (5 ÷ 58) × 100 = 500 ÷ 58

⇒ Profit % = 250 ÷ 29

⇒ Profit % ≈ 8.62%

∴ The correct answer is 8.62% profit.

Additional Information

Effective Cost Price

Any additional overheads like transportation, taxes, storage, or repair costs are always added to the base purchase price to calculate the true or effective Cost Price (CP).

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount. If SP is greater than CP, there is a profit. If SP is less than CP, there is a loss.

Successive Discounts

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

Shortcut Trick

Work done by Sujit and Rakesh in 1 day = 1/4 + 1/8 = 3/8

Work done in 2 days = 2 × (3/8) = 3/4

Work left = 1 − 3/4 = 1/4

Percentage of work left = 1/4 × 100 = 25%

∴ The correct answer is 25%.

Alternate Method

Given:

Time taken by Sujit = 4 days

Time taken by Rakesh = 8 days

They work together for 2 days.

Concept Used:

Total Work = LCM of individual days.

Efficiency = Total Work ÷ Number of Days

Calculations:

⇒ Combined efficiency of Sujit and Rakesh = 2 + 1 = 3 units/day

⇒ Work done by them in 2 days = 3 × 2 = 6 units

⇒ Remaining work = Total Work − Work done = 8 − 6 = 2 units

⇒ Percentage of work left = (Remaining work ÷ Total Work) × 100

⇒ Percentage of work left = (2 ÷ 8) × 100 = 25%

∴ The correct answer is 25%.

Additional Information

Total Work and Efficiency

Total work is always equal to the product of efficiency and the time taken. Formula: Work = Efficiency × Time.

Work and Wages

Wages are distributed in the ratio of the total work done by each individual. If individuals work for the same number of days, the wages are divided in the ratio of their efficiencies.

Pipes and Cisterns

The concept is identical to Time and Work. However, emptying pipes (leaks) have a negative efficiency, which subtracts from the overall filling rate.

Shortcut Trick

Let each equal distance be the LCM of speeds 30, 60, and 90, which is 180 km.

Time for each part: 180 ÷ 30 = 6 h, 180 ÷ 60 = 3 h, and 180 ÷ 90 = 2 h.

Total distance = 3 × 180 = 540 km; Total time = 6 + 3 + 2 = 11 h.

Average Speed = Total Distance ÷ Total Time = 540 ÷ 11 ≈ 49.09 km/h.

∴ The correct answer is 49.09 km/h.

Alternate Method

Given:

Speeds for three equal parts of the journey: 30 km/h, 60 km/h, and 90 km/h.

Formula Used:

Average Speed for 3 equal distances = 3xyz ÷ (xy + yz + zx), where x, y, and z are the individual speeds.

   Dist = d Dist = d Dist = d 30 km/h 60 km/h 90 km/h  Total Distance = 3d

Calculations:

⇒ Let x = 30 km/h, y = 60 km/h, and z = 90 km/h.

⇒ xy = 30 × 60 = 1800

⇒ yz = 60 × 90 = 5400

⇒ zx = 90 × 30 = 2700

⇒ xy + yz + zx = 1800 + 5400 + 2700 = 9900

⇒ 3xyz = 3 × 30 × 60 × 90 = 486000

⇒ Average Speed = 486000 ÷ 9900

⇒ Average Speed = 4860 ÷ 99 = 540 ÷ 11

⇒ Average Speed ≈ 49.09 km/h

∴ The correct answer is 49.09 km/h.

Additional Information

Average Speed for Two Equal Distances

If a person covers two equal halves of a journey at speeds x and y, the average speed over the entire journey is given directly by the formula 2xy ÷ (x + y).

General Average Speed Formula

When parts of the journey are unequal in distance, Average Speed must always be calculated as Total Distance ÷ Total Time taken.

Relative Speed

For two objects moving in the same direction, relative speed is the difference of their speeds (|x − y|). When moving towards each other in opposite directions, it is the sum of their speeds (x + y).

Shortcut Trick

The rectangular faces of a right triangular prism all share the exact same height.

Thus, their areas are strictly proportional to the side lengths of the triangular base.

The base sides are 3 cm, 4 cm, and the hypotenuse = √(3² + 4²) = 5 cm.

Area of largest face → corresponds to the largest side (5 cm).

Area of smallest face → corresponds to the smallest side (3 cm).

∴ The correct answer is 5:3.

Alternate Method

Given:

Base of the prism = Right-angled triangle with perpendicular sides 3 cm and 4 cm.

Height of the prism = 10 cm.

4 cm 3 cm 5 cm Base 30 cm² Small 40 cm² Medium 50 cm² Large 3 cm 4 cm 5 cm 10 cm

Formula Used:

Hypotenuse = √(Base² + Perpendicular²)

Area of rectangular face = Side of base × Height of prism

Calculations:

⇒ Hypotenuse of the base triangle = √(3² + 4²) = √(9 + 16) = √25 = 5 cm

⇒ The lengths of the 3 sides of the base are 3 cm, 4 cm, and 5 cm.

⇒ Area of the smallest rectangular face = Smallest side × Height = 3 × 10 = 30 cm²

⇒ Area of the largest rectangular face = Largest side × Height = 5 × 10 = 50 cm²

⇒ Required Ratio = Area of largest face : Area of smallest face

⇒ Required Ratio = 50 : 30 = 5 : 3

∴ The correct answer is 5:3.

Additional Information

Right Prism vs Oblique Prism

In a right prism, the lateral faces are always perpendicular to the bases, resulting in entirely rectangular lateral faces. In an oblique prism, the lateral edges are slanted and the lateral faces are parallelograms.

Total Surface Area of Prism

The total surface area (TSA) of any prism is given by: TSA = Lateral Surface Area + 2 × (Base Area). For a right prism, the Lateral Surface Area equals Perimeter of Base × Height.

Volume of a Prism

The volume of any given prism (both right and oblique) is calculated using the product of its base area and perpendicular height: Volume = Area of Base × Height.

Shortcut Trick

Initial juice = Total mixture − Soda = 80 − 30 = 50 L.

Since we are only adding soda, the quantity of juice remains constant at 50 L.

In the new mixture, the required ratio of Juice : Soda is 1 : 2.

If 1 unit of juice = 50 L, then 2 units of soda = 2 × 50 = 100 L.

Soda to be added = Final soda − Initial soda = 100 − 30 = 70 L.

∴ The correct answer is 70 L.

Alternate Method

Given:

Total initial mixture = 80 L

Initial amount of soda = 30 L

Initial amount of juice = 80 − 30 = 50 L

Target ratio of Juice to Soda = 1 : 2

Initial Mixture

Juice: 50 L

Soda: 30 L

• x L Soda

Final Ratio 1:2

Juice: 50 L

Soda: (30 + x) L

Result: x = 100 − 30 = 70 L

Calculations:

Let the amount of soda to be added be x L.

⇒ Quantity of juice in the new mixture = 50 L (remains unchanged)

⇒ Quantity of soda in the new mixture = (30 + x) L

⇒ New ratio = Juice ÷ Soda = 1 ÷ 2

⇒ 50 ÷ (30 + x) = 1 ÷ 2

⇒ Cross-multiplying, we get: 30 + x = 50 × 2

⇒ 30 + x = 100

⇒ x = 100 − 30

⇒ x = 70 L

∴ The correct answer is 70 L.

Additional Information

Constant Quantity Method

When only one component (e.g., soda) is added or removed from a mixture, the quantity of the other component (e.g., juice) remains constant. Always use the absolute amount of the unchanged component as a reference base to find the new required amounts.

Rule of Alligation

Alligation is a standard technique used to determine the exact ratio in which two ingredients at given prices or concentrations must be mixed to produce a mixture of a desired average price or concentration.

Ratio and Proportion Application

If two quantities A and B are in ratio a:b, and a certain amount x is added to B, the new ratio becomes a:(b+k). Equating the ratio fractions a ÷ (b+k) directly helps in determining the unknown addition without complex equations.

Shortcut Trick

sinA = Opposite ÷ Hypotenuse = 3 ÷ 5.

The sides of this right-angled triangle form a standard 3-4-5 Pythagorean triplet.

Adjacent side = √(5² − 3²) = √(25 − 9) = 4.

cosA = Adjacent ÷ Hypotenuse = 4/5.

∴ The correct answer is 4/5.

Alternate Method

Given:

sinA = 3/5

Angle A is an acute angle.

Formula Used:

sin²A + cos²A = 1

A Opposite = 3 Adjacent = 4 Hypotenuse = 5 3-4-5 Right Angled Triangle

Calculations:

⇒ cos²A = 1 − sin²A

⇒ cosA = √[1 − (3/5)²]

⇒ cosA = √(1 − 9/25)

⇒ cosA = √[(25 − 9) ÷ 25]

⇒ cosA = √(16/25)

⇒ cosA = 4/5

Since A is an acute angle, cosA must be positive.

∴ The correct answer is 4/5.

Additional Information

Trigonometric Ratios

sinθ = Perpendicular ÷ Hypotenuse, cosθ = Base ÷ Hypotenuse, and tanθ = Perpendicular ÷ Base.

Pythagorean Triplets

Common standard triplets like (3, 4, 5), (5, 12, 13), and (8, 15, 17) satisfy Pythagoras' Theorem a² + b² = c² and are highly useful for quick calculations.

ASTC Rule (Quadrant System)

In the 1^st quadrant (acute angles from 0° to 90°), all trigonometric ratios yield positive values.

Fundamental Identities

The three main identities are sin²θ + cos²θ = 1, sec²θ − tan²θ = 1, and cosec²θ − cot²θ = 1.

Shortcut Trick

The equation of a straight line in slope-intercept form is given by y = mx + c.

Given the slope m = 4, the equation becomes y = 4x + c.

Substitute the given point (3, -1) into the equation:

⇒ -1 = 4 × 3 + c

⇒ c = -1 − 12 = -13

∴ The correct answer is -13.

Alternate Method

Given:

Line passes through point (x₁, y₁) = (3, -1)

Slope of the line (m) = 4

Formula Used:

Point-slope form of linear equation: y − y₁ = m(x − x₁)

X Y O(0,0)

(3, -1)

(0, -13)

Δx = 3 Δy = 12

Calculations:

Substitute the given point and slope into the point-slope formula:

⇒ y − (-1) = 4 × (x − 3)

⇒ y + 1 = 4x − 12

⇒ y = 4x − 13

The y-intercept occurs at the point where the line crosses the Y-axis (i.e., x = 0):

⇒ y = 4 × 0 − 13

⇒ y = -13

∴ The correct answer is -13.

Additional Information

Slope-Intercept Form

The equation y = mx + c describes a line where 'm' is the slope and 'c' is the y-intercept.

Point-Slope Form

A line passing through (x₁, y₁) with slope m is given by y − y₁ = m(x − x₁).

x-intercept

The x-intercept is the point where the line crosses the X-axis. It is determined by substituting y = 0 into the linear equation.

Shortcut Trick

When two parallel lines are intersected by a transversal, any pair of corresponding angles are always equal.

Since one corresponding angle is given as 130°, the other corresponding angle must also be exactly equal to it.

Hence, the other angle measures 130°.

∴ The correct answer is 130°.

Alternate Method

Given:

⇒ Two parallel lines intersected by a transversal line.

⇒ The measure of one angle from a corresponding pair = 130°.

Concept Used:

⇒ Corresponding Angles Axiom: If a transversal intersects two parallel lines, then each pair of corresponding angles is equal in measure.

130° x

m n l

Line m || Line n, l is a transversal

Calculations:

⇒ Let the measure of the other corresponding angle be x.

⇒ According to the corresponding angles postulate:

⇒ x = Given angle

⇒ x = 130°

∴ The correct answer is 130°.

Additional Information

Alternate Interior Angles

When two parallel lines are cut by a transversal, the pairs of alternate interior angles are equal in measure.

Consecutive Interior Angles

The pairs of consecutive interior angles (angles on the same side of the transversal) are supplementary (their sum is 180°).

Vertically Opposite Angles

When two lines intersect, the angles that are opposite to each other at the intersection point are vertically opposite and always equal.

Shortcut Trick

In similar triangles, the corresponding angles are always equal.

Since ΔABC ~ ΔDEF, the order of the vertices directly indicates the corresponding angles.

⇒ ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F.

We are given that ∠B = 60°.

⇒ ∠E = 60°

∴ The correct answer is 60°.

Alternate Method

Given:

• ΔABC ~ ΔDEF

• ∠A = 50°

• ∠B = 60°

50°

60°

50°

E = ?

A B C

D E F

Property Used:

If two triangles are similar, their corresponding angles are equal and their corresponding sides are proportional in length.

Calculations:

⇒ ΔABC ~ ΔDEF indicates that the corresponding vertices match exactly as: A → D, B → E, and C → F.

⇒ By the property of similar triangles, ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F.

⇒ We are given that ∠B = 60°.

⇒ Therefore, substituting the value gives ∠E = 60°.

⇒ Additionally, ∠D = ∠A = 50°, and ∠F = ∠C = 180° − (50° + 60°) = 70°.

∴ The correct answer is 60°.

Additional Information

Properties of Similar Triangles

When two triangles are similar, their corresponding angles are strictly equal and the ratio of their corresponding sides is constant (AB/DE = BC/EF = AC/DF).

Angle Sum Property of a Triangle

The sum of all three interior angles of a triangle is always equal to 180°. (∠A + ∠B + ∠C = 180°)

Congruency vs Similarity

Congruent triangles have the exact same shape and size (ratio of sides = 1), whereas similar triangles possess the exact same shape but can proportionally differ in size.

Shortcut Trick

Given x = √5 + 2

Squaring both sides gives: x² = 5 + 4 + 4√5 = 9 + 4√5

Substitute x² directly into the expression: (9 + 4√5) − 4√5

The irrational terms cancel out perfectly, leaving 9.

∴ The correct answer is 9.

Alternate Method

Given:

x = √5 + 2

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ We need to find the value of x² − 4√5.

⇒ First, calculate x² by substituting x = √5 + 2.

⇒ x² = (√5 + 2)²

⇒ x² = (√5)² + (2)² + 2 × √5 × 2

⇒ x² = 5 + 4 + 4√5

⇒ x² = 9 + 4√5

⇒ Now, substitute the value of x² back into the original expression.

⇒ Value = (9 + 4√5) − 4√5

⇒ Value = 9 + (4√5 − 4√5)

⇒ Value = 9 + 0 = 9

∴ The correct answer is 9.

Additional Information

Algebraic Identities

• (a − b)² = a² + b² − 2ab

• a² − b² = (a − b)(a + b)

Conjugate Surds

For any irrational number of the form √a + b, its conjugate is √a − b. Their product is always a rational number: (√a + b)(√a − b) = a − b².

Simplification Rules

When solving algebraic expressions with surds, it is best to expand polynomial powers fully before adding or subtracting in order to effectively group and eliminate irrational terms.

Shortcut Trick

According to the geometric properties of a circle, equal chords are always equidistant from the center.

Since the lengths of both chords are equal (10 cm each), their perpendicular distances from the center will also be exactly equal.

The distance of the first chord from the center is given as 4 cm.

Therefore, the distance of the second chord from the center must also be 4 cm.

∴ The correct answer is 4 cm.

Alternate Method

O

4 cm M

d = ? N

A B C D

r r

AB = 10 cm, CD = 10 cm

Given:

⇒ Length of the first chord (AB) = 10 cm

⇒ Length of the second chord (CD) = 10 cm

⇒ Perpendicular distance of the first chord from the center (OM) = 4 cm

Formula Used:

⇒ Pythagoras Theorem: Hypotenuse² = Base² + Perpendicular²

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

Calculations:

⇒ Half of the first chord, AM = AB ÷ 2 = 10 ÷ 2 = 5 cm

⇒ In right-angled ΔOMA:

⇒ Radius (r)² = OM² + AM²

⇒ r² = 4² + 5²

⇒ r² = 16 + 25 = 41

⇒ Half of the second chord, CN = CD ÷ 2 = 10 ÷ 2 = 5 cm

⇒ Let the perpendicular distance of chord CD from the center be ON.

⇒ In right-angled ΔONC:

⇒ Radius (r)² = ON² + CN²

⇒ 41 = ON² + 5²

⇒ 41 = ON² + 25

⇒ ON² = 41 − 25 = 16

⇒ ON = √16 = 4 cm

∴ The correct answer is 4 cm.

Additional Information

Chords Equidistant from Center

If two chords are equidistant from the center of a circle, then their lengths are perfectly equal. The converse is also true.

Perpendicular Bisector Theorem

The perpendicular drawn from the center of a circle to any chord bisects the chord into two equal halves (i.e., AM = MB).

Longest Chord Property

The diameter is the longest chord of a circle, and its perpendicular distance from the center is exactly 0 cm.

Shortcut Trick

In a circle, the angle between two tangents drawn from an external point and the angle subtended by their chord of contact at the center are supplementary.

The sum of these two angles is exactly 180°.

∠APB + ∠AOB = 180°

∠APB = 180° − θ

∴ The correct answer is 180° − θ.

Alternate Method

θ

?

O

P

A

B

Given:

Angle subtended by the chord of contact AB at the center O = ∠AOB = θ

Method:

⇒ A radius is always perpendicular to the tangent at the point of contact.

⇒ ∠OAP = 90° (Since OA is a radius and PA is a tangent)

⇒ ∠OBP = 90° (Since OB is a radius and PB is a tangent)

⇒ In quadrilateral OAPB, the sum of all interior angles is 360°.

⇒ ∠OAP + ∠OBP + ∠AOB + ∠APB = 360°

⇒ 90° + 90° + θ + ∠APB = 360°

⇒ 180° + θ + ∠APB = 360°

⇒ ∠APB = 360° − 180° − θ

⇒ ∠APB = 180° − θ

∴ The correct answer is 180° − θ.

Additional Information

Cyclic Quadrilateral Property

Since the opposite angles ∠OAP and ∠OBP sum to 180°, the quadrilateral OAPB is cyclic. This means points O, A, P, and B lie on a single circle where the line segment OP acts as its diameter.

Lengths of Tangents

Tangents drawn from a single external point to a circle are always equal in length. Therefore, the tangent segment PA equals tangent segment PB.

Angle Subtended by Arc

The angle subtended by an arc at the center of the circle is exactly double the angle it subtends at any other point on the remaining part of the circle.

Shortcut Trick

From the algebraic identity: sin⁴ A + cos⁴ A = 1 − 2 × sin² A × cos² A

Given sin A = 5/13, which corresponds to the 5-12-13 Pythagorean triplet, so cos A = 12/13.

⇒ 1 − 2 × (5/13)² × (12/13)²

⇒ 1 − 2 × (25/169) × (144/169)

⇒ 1 − 7200 / 28561 = (28561 − 7200) / 28561 = 21361 / 28561

∴ The correct answer is 21361/28561.

Alternate Method

Given:

sin A = 5/13

A C B Base = 12 Perpendicular = 5 Hypotenuse = 13 5-12-13 Right Angled Triangle

⇒ In right-angled ΔABC, sin A = Perpendicular / Hypotenuse = 5 / 13

⇒ Perpendicular = 5, Hypotenuse = 13

⇒ Base = √(Hypotenuse² − Perpendicular²)

⇒ Base = √(13² − 5²) = √(169 − 25) = √144 = 12

⇒ cos A = Base / Hypotenuse = 12 / 13

⇒ Expression to calculate: sin⁴ A + cos⁴ A

⇒ (5/13)⁴ + (12/13)⁴

⇒ (625 / 28561) + (20736 / 28561)

⇒ (625 + 20736) / 28561

⇒ 21361 / 28561

∴ The correct answer is 21361/28561.

Additional Information

Trigonometric Identities

The foundational identity is sin² θ + cos² θ = 1. From this, we derive the useful powers sum rule: sin⁴ θ + cos⁴ θ = 1 − 2 sin² θ cos² θ.

Pythagorean Triplets

Sets of integers that perfectly satisfy Pythagoras' Theorem (a² + b² = c²). The most common triplets used in exams are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Basic Trigonometric Ratios

For any angle in a right-angled triangle, sin θ = Perpendicular / Hypotenuse, cos θ = Base / Hypotenuse, and tan θ = Perpendicular / Base.

Shortcut Trick

To add decimals quickly, align the decimal points and pad with zeros to equalize the decimal places.

Write the numbers as 3.450, 6.780, and 0.027.

Add the integer parts: 3 + 6 + 0 = 9.

Add the decimal parts: 450 + 780 + 27 = 1257 thousandths = 1.257.

Total sum = 9 + 1.257 = 10.257.

∴ The correct answer is 10.257.

Alternate Method

Given:

The expression: 3.45 + 6.78 + 0.027

Method Used:

Standard addition of decimals by aligning the decimal points.

Calculations:

⇒ First, equalize the decimal digits by adding trailing zeros.

⇒ 3.45 becomes 3.450

⇒ 6.78 becomes 6.780

⇒ 0.027 remains 0.027

⇒ Now, add them directly: 3.450 + 6.780 + 0.027

⇒ Sum = 10.257

∴ The correct answer is 10.257.

Additional Information

Decimal Addition & Subtraction

Always align the decimal points vertically. Add trailing zeros to ensure each number has the same amount of decimal places to prevent positional errors during addition.

BODMAS Rule

For combined mathematical operations, strictly follow the sequence of Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Fraction to Decimal Conversion

To convert fractions with denominators like 10, 100, or 1000 into decimals, shift the decimal point to the left according to the number of zeros present in the denominator.

Shortcut Trick

By memorizing base squares, we know 50² = 2500.

Checking perfect squares slightly below 50: 49² = 2401 and 48² = 2304.

The given number is 2378, which is strictly between 2304 and 2401.

The smallest number to be added to reach the next perfect square is 2401 − 2378.

Difference = 23.

∴ The correct answer is 23.

Alternate Method

Given:

The original number is 2378.

Concept Used:

To find the smallest number to be added to make a perfect square, find the closest perfect square that is strictly greater than the given number.

Calculations:

⇒ Let's estimate the square root of 2378.

⇒ We know that 40² = 1600 and 50² = 2500.

⇒ Since 2378 is much closer to 2500, we check the squares of numbers slightly below 50.

⇒ 49² = 49 × 49 = 2401

⇒ 48² = 48 × 48 = 2304

⇒ The number 2378 lies between 48² and 49² (2304 < 2378 < 2401).

⇒ To make it a perfect square, the given number must be increased to 2401.

⇒ Required number to be added = 2401 − 2378 = 23.

∴ The correct answer is 23.

Additional Information

Perfect Square Ending Digits

A perfect square never ends in 2, 3, 7, or 8. Any number ending in these digits cannot be a perfect square.

Digital Sum of Perfect Squares

The digital sum (sum of digits continuously reduced to a single digit) of a perfect square is always 1, 4, 7, or 9.

Smallest Number to Subtract

To find the smallest number to be subtracted to make a number a perfect square, find the square root using the long division method. The remainder obtained is the exact number to be subtracted.

Unit Digits of Squares

The unit digit of a perfect square depends entirely on the unit digit of its base number (e.g., if a number ends in 4 or 6, its square will always end in 6).

Shortcut Trick

SP of 1 paid cake = ₹75 × (100% − 20%) = ₹60.

Number of paid cakes out of the remaining 4500 = 4500 × (9 ÷ 10) = 4050 cakes.

Total Selling Price = 4050 × 60 = ₹243,000.

Overall Loss = Total CP − Total SP = ₹250,000 − ₹243,000 = ₹7000.

Loss % = (7000 ÷ 250000) × 100 = 2.8% loss.

∴ The correct answer is 2.8% loss.

Alternate Method

Given:

Total cakes produced = 5000

Total cost (CP) = ₹250,000

Cakes given for promotional event = 500

Market Price (MP) per cake = ₹75

Discount = 20%

Offer = Buy 9 get 1 free

Formula Used:

Selling Price = MP × (100 − Discount%) ÷ 100

Loss % = (Loss ÷ Total CP) × 100

Calculations:

Total Cakes = 5000 (Total CP = ₹250,000) Promo 500 Paid (Buy 9) 4050 cakes Free (Get 1) 450 Total SP = 4050 × ₹60 = ₹243,000 Loss = ₹7000 (2.8%)

⇒ Remaining cakes to be sold = 5000 − 500 = 4500

⇒ In the 'buy 9 get 1 free' offer, 10 cakes are distributed for the price of 9 cakes.

⇒ Number of sold (paid) cakes = 4500 × (9 ÷ 10) = 4050

⇒ Selling Price (SP) of 1 paid cake = ₹75 − (20% of 75) = 75 − 15 = ₹60

⇒ Total Selling Price = 4050 × 60 = ₹243,000

⇒ Overall Loss = Total CP − Total SP = 250,000 − 243,000 = ₹7000

⇒ Loss % = (7000 ÷ 250000) × 100 = 2.8%

∴ The correct answer is 2.8% loss.

Additional Information

Buy X Get Y Free Rule

The equivalent effective discount percentage in a "Buy X Get Y Free" scheme is calculated as: (Free Items ÷ Total Items) × 100.

Successive Discounts

When two successive discounts of a% and b% are offered, the net equivalent discount is evaluated using the formula: (a + b − ab ÷ 100)%.

Profit and Loss Calculation

Profit or loss percentage is always strictly calculated based on the Total Cost Price initially invested, unless the problem specifies an alternative base.

Shortcut Trick

The effective rate of Compound Interest for 2 years at 10% per annum can be found using successive percentage formula: A + B + (A × B) / 100.

Effective Rate = 10% + 10% + (10 × 10) / 100% = 21%.

Compound Interest = 21% of Principal

CI = (21 / 100) × 8000 = 1680.

∴ The correct answer is Rs. 1680.

Alternate Method

Given:

Principal (P) = Rs. 8000

Rate (R) = 10% per annum

Time (T) = 2 years

Formula Used:

Amount (A) = P × (1 + R / 100)^T

Compound Interest (CI) = A − P

Compound Interest Tree Method

P = 8000

Yr 1: 800 10%

Yr 2: 800 10%

• 80 10% of 800

Total CI = 800 + 800 + 80 = 1680

Calculations:

⇒ Amount = 8000 × (1 + 10 / 100)²

⇒ Amount = 8000 × (110 / 100)²

⇒ Amount = 8000 × (11 / 10) × (11 / 10)

⇒ Amount = 8000 × 121 / 100

⇒ Amount = 80 × 121

⇒ Amount = 9680

⇒ CI = 9680 − 8000 = 1680

∴ The correct answer is Rs. 1680.

Additional Information

Difference between CI and SI for 2 years

For a given principal P and rate R%, the exact difference between Compound Interest and Simple Interest for 2 years is calculated as: Difference = P(R / 100)².

Difference between CI and SI for 3 years

For a 3-year time period, the difference formula extends to: Difference = P(R / 100)² × (3 + R / 100).

Half-Yearly Compounding

When interest is compounded half-yearly, the interest rate is halved (R/2) and the time period in terms of cycles is doubled (2T). The formula becomes: Amount = P(1 + R / 200)^2T.

Shortcut Trick

If a sum becomes n₁ times in T₁ years and n₂ times in T₂ years at the same simple interest rate, we use the direct ratio:

(n₁ − 1) / T₁ = (n₂ − 1) / T₂

⇒ (4 − 1) / 8 = (12 − 1) / T₂

⇒ 3 / 8 = 11 / T₂

⇒ T₂ = 88 / 3 = 29 ¹/₃ years

∴ The correct answer is 29 ¹/₃ years.

Alternate Method

Given:

A sum becomes 4 times itself in 8 years at simple interest.

Formula Used:

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

Amount (A) = P + SI

 P 4P 12P 0 Years 8 Years ? Years            SI = 3P  SI = 11P

Calculations:

Let the principal sum be P and the rate of interest be R% per annum.

Case 1: The sum becomes 4 times in 8 years.

⇒ Amount (A₁) = 4P

⇒ Simple Interest (SI₁) = A₁ − P = 4P − P = 3P

⇒ 3P = (P × R × 8) / 100

⇒ R = 300 / 8 = 37.5%

Case 2: The sum becomes 12 times in T years.

⇒ Amount (A₂) = 12P

⇒ Simple Interest (SI₂) = A₂ − P = 12P − P = 11P

Using the derived rate of interest (R = 37.5%):

⇒ 11P = (P × 37.5 × T) / 100

⇒ T = 1100 / 37.5

⇒ T = 11000 / 375 = 440 / 15

⇒ T = 88 / 3 = 29 ¹/₃ years

∴ The correct answer is 29 ¹/₃ years.

Additional Information

Simple Interest (SI)

Calculated only on the initial principal sum. The interest amount remains exactly the same for every equal time period. SI = PRT/100.

Rule of Times

If a sum becomes 'x' times itself in 'T' years at Simple Interest, the Rate of Interest (R) can be found using: R = 100(x − 1) / T %.

Compound Interest (CI)

Unlike Simple Interest, Compound Interest is calculated on the initial principal as well as the accumulated interest of previous periods. Formula: A = P(1 + R/100)^T.

Shortcut Trick

An integer is a whole number with no fractional or decimal component.

By quickly evaluating the given options, we can see that √5 is an irrational number, while 3.2 and 3.4 are decimals.

The only expression that is a perfect square and evaluates to a whole number integer is √9, which equals exactly 3.

∴ The correct answer is Square root of 9.

Alternate Method

Given Options:

1. √5

2. √9

3. 3.2

4. 3.4

Evaluation:

⇒ √5 ≈ 2.236 (This is an irrational number, hence not an integer)

⇒ √9 = √(3 × 3) = 3 (This is a positive integer derived from the perfect square 9)

⇒ 3.2 = 32 ÷ 10 = 16/5 (This is a fraction/decimal, hence not an integer)

⇒ 3.4 = 34 ÷ 10 = 17/5 (This is a fraction/decimal, hence not an integer)

⇒ Out of all the given options, only √9 simplifies to a pure integer.

∴ The correct answer is Square root of 9.

Additional Information

Perfect Square

A perfect square is a number that can be expressed as the product of an integer with itself (e.g., 9 = 3²). The square root of a perfect square is always an integer.

Integers

The set of integers consists of all whole numbers, including positive numbers, negative numbers, and zero. They do not contain any decimal or fractional parts.

Irrational Numbers

These are real numbers that cannot be expressed as a simple fraction or ratio of two integers. Their decimal expansions are non-terminating and non-repeating (e.g., √5, π).