Quantitative Aptitude PYQs

Shortcut Trick

Identify the given corresponding parts between △PQR and △STU directly:

1. PQ = ST (Side)
2. ∠QPR = ∠TSU (Included Angle)
3. PR = SU (Side)

Since the equal angle is formed strictly between the two corresponding equal sides, it satisfies the Side-Angle-Side property.

∴ The correct answer is SAS.

Alternate Method

P Q R

S T U

S S A

S S A

△PQR △STU

Given:

⇒ In △PQR and △STU,

⇒ PQ = ST (Given Side)

⇒ ∠QPR = ∠TSU (Given Angle)

⇒ PR = SU (Given Side)

Concept Used:

⇒ Two triangles are defined as congruent if two sides and the included angle of one triangle are respectively equal to the corresponding two sides and the included angle of the other.

⇒ In △PQR, ∠QPR is directly between sides PQ and PR. Similarly, in △STU, ∠TSU is directly between sides ST and SU.

⇒ Thus, the Side-Angle-Side criteria is perfectly satisfied.

⇒ By SAS Congruence, △PQR ≅ △STU.

∴ The correct answer is SAS.

Additional Information

SSS (Side-Side-Side) Congruence

If three sides of one triangle are respectively equal to the corresponding three sides of another triangle, they are congruent.

ASA (Angle-Side-Angle) Congruence

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

AAS (Angle-Angle-Side) Congruence

If two angles and a non-included side of one triangle are respectively equal to the corresponding angles and non-included side of another triangle, they are congruent.

RHS (Right angle-Hypotenuse-Side) Congruence

If in two right-angled triangles, the hypotenuse and one standard side are respectively equal, the triangles are congruent.

Shortcut Trick

Express the terms inside the square root as perfect squares by finding factors of 6 that add up to 5 (which are 3 and 2).

5 + 2√6 = (√3 + √2)² and 5 − 2√6 = (√3 − √2)²

Numerator = √(√3 + √2)² + √(√3 − √2)² = (√3 + √2) + (√3 − √2) = 2√3

Given expression = 2√3 ÷ √2 = √2 × √3 = √6

∴ The correct answer is √6.

Alternate Method

Given:

(√(5 + 2√6) + √(5 − 2√6)) ÷ √2

Formula Used:

(a + b)² = a² + b² + 2ab

(a − b)(a + b) = a² − b²

Calculation:

⇒ Let the numerator be N = √(5 + 2√6) + √(5 − 2√6)

⇒ Squaring both sides to simplify the surds:

⇒ N² = (√(5 + 2√6) + √(5 − 2√6))²

⇒ N² = (5 + 2√6) + (5 − 2√6) + 2√((5 + 2√6)(5 − 2√6))

⇒ N² = 10 + 2√(5² − (2√6)²)

⇒ N² = 10 + 2√(25 − 24)

⇒ N² = 10 + 2√1

⇒ N² = 12

⇒ N = √12 = 2√3

⇒ Now, substitute N back into the original expression:

⇒ Expression = N ÷ √2

⇒ Expression = 2√3 ÷ √2

⇒ Expression = √2 × √3 = √6

∴ The correct answer is √6.

Additional Information

Perfect Squares of Surds

Any expression of the form x ± 2√y can often be written as (√a ± √b)². To do this, find two numbers a and b such that their sum is x (a + b = x) and their product is y (a × b = y).

Conjugate Surds

Expressions like √a + √b and √a − √b are called conjugate pairs. Multiplying them together always yields a rational number using the difference of squares identity: (√a + √b)(√a − √b) = a − b.

Simplification Rules

Always resolve square roots containing inner surds either by converting them into perfect squares or by assigning a variable and squaring the entire equation to eliminate the outermost root.

Shortcut Trick

The angle subtended by a chord at the center of a circle is exactly double the angle subtended by it at any point on the remaining part of the circle.

Required Angle = Angle at center ÷ 2

Required Angle = 70° ÷ 2 = 35°

∴ The correct answer is 35°.

Alternate Method

Given:

The angle subtended by the chord at the center = 70°

Formula Used:

Angle at the circumference = Angle at the center ÷ 2

70°

35°

O

A

B

C

Angle subtended by chord AB

Calculations:

⇒ Let the chord be AB and the center of the circle be O.

⇒ The angle subtended by chord AB at the center is ∠AOB = 70°.

⇒ Let C be any point on the major arc (remaining part) of the circle.

⇒ According to the central angle theorem, ∠ACB = ∠AOB ÷ 2

⇒ ∠ACB = 70° ÷ 2

⇒ ∠ACB = 35°

∴ The correct answer is 35°.

Additional Information

Angle in a Semicircle

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

Angles in the Same Segment

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

Cyclic Quadrilateral

A quadrilateral inscribed in a circle is called a cyclic quadrilateral. The sum of its opposite interior angles is always 180°.

Alternate Segment Theorem

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

Shortcut Trick

For any regular polygon, the angle subtended by one side at the center is given directly by dividing the full circular angle by the number of sides:

⇒ Central Angle = 360° ÷ n

Here, a regular pentagon has n = 5 sides.

⇒ Central Angle = 360° ÷ 5 = 72°

∴ The correct answer is 72°.

Alternate Method

Given:

A circle is inscribed in a regular pentagon.

Number of sides in a pentagon (n) = 5

Formula Used:

Angle subtended by one side at the center = Total angle around the center ÷ Number of sides

72°

O A B C D E

Calculations:

⇒ In a regular pentagon, all 5 sides are equal in length and therefore subtend equal angles at the geometric center.

⇒ The center of the inscribed circle perfectly coincides with the center of the regular pentagon.

⇒ The total angle entirely around the center point O is 360°.

⇒ Angle subtended by one side (like CD) = 360° ÷ 5

⇒ Angle = 72°

∴ The correct answer is 72°.

Additional Information

Exterior Angle of a Regular Polygon

The exterior angle of any regular polygon is calculated as 360° ÷ n. Notice that the angle subtended by a side at the center is exactly equal to its exterior angle.

Interior Angle of a Regular Polygon

The interior angle is derived using the formula: (n − 2) × 180° ÷ n. For a regular pentagon, the interior angle is 108°.

Incenter and Circumcenter Identity

For any regular polygon, the center of the inscribed circle (incircle) and the circumscribed circle (circumcircle) coincide at the exact same geometric point.

Shortcut Trick

Given, sin x = Perpendicular / Hypotenuse = 1 / √10

Base = √(H² − P²) = √(10 − 1) = √9 = 3

We know, sec x = Hypotenuse / Base

⇒ sec x = √10 / 3

∴ The correct answer is √10/3.

Alternate Method

Given:

sin x = 1 / √10

Formula Used:

sin x = Perpendicular (P) / Hypotenuse (H)

sec x = Hypotenuse (H) / Base (B)

H² = P² + B²

x Base (B) = 3 Perpendicular (P) = 1 Hypotenuse (H) = √10

Calculations:

⇒ By comparing, Perpendicular (P) = 1 and Hypotenuse (H) = √10

⇒ According to Pythagoras theorem: (√10)² = 1² + B²

⇒ 10 = 1 + B²

⇒ B² = 9

⇒ Base (B) = 3

Now, substitute the values into the formula for sec x:

⇒ sec x = H / B

⇒ sec x = √10 / 3

∴ The correct answer is √10/3.

Additional Information

Trigonometric Ratios

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

Pythagorean Identities

The primary Pythagorean identity is sin² θ + cos² θ = 1. Other forms include 1 + tan² θ = sec² θ and 1 + cot² θ = cosec² θ.

Reciprocal Identities

The reciprocal of cos θ is sec θ (sec θ = 1 / cos θ). Similarly, cosec θ = 1 / sin θ and cot θ = 1 / tan θ.

Shortcut Trick

Target return on selling 120 items is 140% of Total CP (to get a 40% profit).

Relative return per single item = 140% ÷ 120.

Actual return on selling 135 items = (140% ÷ 120) × 135 = 140% × 1.125 = 157.5% of Total CP.

Overall Profit % = 157.5% − 100% = 57.5%.

∴ The correct answer is 57.5%.

Alternate Method

Given:

Total portable speakers ordered = 150

Cost price per speaker = ₹400

Target profit = 40% on total investment by selling 120 speakers

Actual speakers sold = 135

Formula Used:

Profit % = [(Total SP − Total CP) ÷ Total CP] × 100

Total CP: ₹60,000 Actual SP: ₹94,500 Break-even Profit: ₹34,500

Calculations:

⇒ Total Cost Price (CP) = 150 × 400 = ₹60,000

⇒ Target Total Selling Price (SP for 120 speakers) = Total CP + (40% of Total CP)

⇒ Target Total SP = 60,000 + 24,000 = ₹84,000

⇒ Selling Price per speaker = 84,000 ÷ 120 = ₹700

⇒ Actual Total Selling Price (for 135 speakers) = 135 × 700 = ₹94,500

⇒ Actual Profit = Actual Total SP − Total CP

⇒ Actual Profit = 94,500 − 60,000 = ₹34,500

⇒ Actual Profit % = (34,500 ÷ 60,000) × 100

⇒ Actual Profit % = 57.5%

∴ The correct answer is 57.5%.

Additional Information

Profit & Loss Base Rule

Profit and Loss are always calculated on the Cost Price (CP) unless explicitly specified otherwise. Formula: Profit % = (Net Profit ÷ CP) × 100.

Successive Discounts

If two successive discounts of a% and b% are offered on an item, the single equivalent discount is given by (a + b − a×b÷100)%.

Dishonest Dealer Concept

When a dealer professes to sell goods at Cost Price but uses false weights, the profit is driven entirely by the missing weight. Profit % = (Error ÷ True Value Used) × 100.

Shortcut Trick

The set of real numbers represents all the numbers that can be plotted on a continuous number line.

Every point on this number line corresponds to exactly one number, which is either a rational number or an irrational number.

Therefore, the real number system is formed by combining both rational and irrational numbers.

∴ The correct answer is Both Rational and Irrational Numbers.

Alternate Method

Given:

The set of real numbers and four classification options.

Concept Used:

⇒ Real numbers include any value that represents a distance along a line, meaning they cover the entire number spectrum used in standard calculations.

Detailed Classification:

⇒ Real Numbers = Rational Numbers + Irrational Numbers.

⇒ Rational numbers are those that can be written as a fraction p/q (where q is not 0). This includes natural numbers, whole numbers, and integers.

⇒ Irrational numbers are numbers that cannot be expressed as a simple fraction, meaning their decimal values are non-terminating and non-repeating (like √2 or π).

⇒ Since no real number can fall outside of these two distinct categories, the complete set of real numbers comprises both rational and irrational numbers.

∴ The correct answer is Both Rational and Irrational Numbers.

Additional Information

Rational Numbers

Numbers that can be expressed in the form p/q, where p and q are integers and q is not equal to 0. Their decimal expansions are either terminating or repeating.

Irrational Numbers

Numbers that cannot be written as a simple fraction. Their decimal expansions are strictly non-terminating and non-repeating (e.g., √3, √5, π).

Integers and Whole Numbers

Integers include all whole numbers and their negative counterparts (..., −2, −1, 0, 1, 2, ...). Whole numbers start from 0 (0, 1, 2, 3, ...). Both groups are subsets of rational numbers.

Complex Numbers

Numbers expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit (√−1). The set of real numbers is a subset of complex numbers where b = 0.

Shortcut Trick

Speed = Distance ÷ Time

Speed in m/s = 100 ÷ 50 = 2 m/s

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

Speed in km/h = 2 × (18/5) = 7.2 km/h

∴ The correct answer is 7.2 km/h.

Alternate Method

Given:

Distance crossed (D) = 100 m

Time taken (T) = 50 s

Formula Used:

Speed (S) = Distance (D) ÷ Time (T)

Conversion: 1 m/s = 18/5 km/h

      Distance = 100 m Time = 50 s  Start  End  Speed = ?

Calculation:

⇒ First, calculate the speed in m/s.

⇒ S = 100 ÷ 50 = 2 m/s

⇒ Now, convert the speed to km/h.

⇒ S = 2 × (18 ÷ 5) km/h

⇒ S = 36 ÷ 5 = 7.2 km/h

∴ The correct answer is 7.2 km/h.

Additional Information

Unit Conversions

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

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

Average Speed

Average speed over a journey is given by Total Distance Traveled ÷ Total Time Taken. It is not always the average of the different speeds.

Relative Speed

If two objects are moving in opposite directions, their relative speed is the sum of their individual speeds (u + v). If they are moving in the same direction, it is the difference between their speeds (u − v).

Shortcut Trick

Volume of cube = a³ = 1728 ⇒ Side of cube (a) = 12 cm.

For an inscribed sphere, the diameter is equal to the side of the cube ⇒ Radius (r) = 12 ÷ 2 = 6 cm.

Total Surface Area of sphere = 4πr² = 4 × π × 6² = 144π cm².

∴ The correct answer is 144π cm².

Alternate Method

    r = 6 cm a = 12 cm Cross-section: Sphere in a Cube

Given:

⇒ Volume of the cube = 1728 cm³

Formula Used:

⇒ Volume of a cube = a³ (where a is the side length)

⇒ Total Surface Area of a sphere = 4πr² (where r is the radius)

Calculation:

⇒ Let the side length of the cube be a.

⇒ a³ = 1728

⇒ a = 12 cm

⇒ When a sphere is inscribed inside a cube, its diameter is strictly equal to the side length of the cube.

⇒ Diameter of the sphere (d) = 12 cm

⇒ Radius of the sphere (r) = d ÷ 2

⇒ r = 12 ÷ 2 = 6 cm

⇒ Total Surface Area of the sphere = 4 × π × (6)²

⇒ Total Surface Area = 4 × π × 36

⇒ Total Surface Area = 144π cm²

∴ The correct answer is 144π cm².

Additional Information

Cube Formulas

Volume = a³ | Total Surface Area = 6a² | Lateral Surface Area = 4a² | Diagonal = a√3

Sphere Formulas

Volume = (4 ÷ 3)πr³ | Total Surface Area = 4πr²

Largest Sphere Inscribed in a Cube

The diameter of the inscribed sphere touches all six faces and is equal to the side of the cube: 2r = a ⇒ r = a ÷ 2.

Largest Cube Inscribed in a Sphere

The body diagonal of the inscribed cube is equal to the diameter of the sphere: a√3 = 2r.

Shortcut Trick

Inner area of the open box = (L × W) + 2H(L + W)

Area = (2.5 × 1.8) + 2 × 1.2(2.5 + 1.8) = 4.5 + 10.32 = 14.82 m²

Total fabric area needed with 15% extra = 14.82 × 1.15 = 17.043 m²

Area of 1 square fabric piece = 0.5 m × 0.5 m = 0.25 m²

Number of pieces required = 17.043 ÷ 0.25 = 68.172

∴ The correct answer is 69 pieces.

Alternate Method

        L = 2.5 m W = 1.8 m H = 1.2 m   50 × 50 cm Fabric

Given:

Inner Length of the box (L) = 2.5 m

Inner Width of the box (W) = 1.8 m

Inner Height of the box (H) = 1.2 m

Side of one square fabric piece (a) = 50 cm = 0.5 m

Extra fabric allowance for cuts and overlaps = 15%

Formula Used:

Total inner surface area = Area of the floor + Area of the 4 side walls

Total Area = (L × W) + 2 × (L × H + W × H)

Calculations:

⇒ Total Area = (2.5 × 1.8) + 2 × (2.5 × 1.2 + 1.8 × 1.2)

⇒ Total Area = 4.5 + 2 × (3.0 + 2.16)

⇒ Total Area = 4.5 + 2 × (5.16)

⇒ Total Area = 4.5 + 10.32 = 14.82 m²

⇒ Adding 15% extra material: 14.82 × (1 + 15/100) = 14.82 × 1.15

⇒ Total fabric area needed = 17.043 m²

⇒ Area of a single fabric piece = 0.5 m × 0.5 m = 0.25 m²

⇒ Number of pieces required = Total fabric area ÷ Area of 1 piece

⇒ Number of pieces = 17.043 ÷ 0.25 = 68.172

⇒ To fully cover the inside, we must purchase the closest integer number of complete pieces, which is 69.

∴ The correct answer is 69 pieces.

Additional Information

Total Surface Area (TSA) of a Cuboid

For a closed cuboid box, TSA = 2(lb + bh + hl). It represents the total geometric surface encompassing all 6 faces.

Lateral Surface Area (LSA) of a Cuboid

LSA = 2h(l + b). It represents the total area of only the four surrounding side walls, ignoring the top and bottom sections.

Area and Unit Conversion

Always standardize your length units before applying formulas. 1 m² = 10,000 cm². If solving purely in cm, 14.82 m² = 148,200 cm².

Shortcut Trick

Working with values in lakhs to simplify the calculation:

Profit Ratio of M : N = (2 × 4 + 3 × 8) : (3 × 4 + 2.5 × 8)

⇒ M : N = (8 + 24) : (12 + 20) = 32 : 32 = 1 : 1

N's share = (1 ÷ 2) × 1,80,000 = Rs. 90,000

∴ The correct answer is Rs. 90,000.

Alternate Method

Given:

M's initial investment = Rs. 2,00,000

N's initial investment = Rs. 3,00,000

Total profit at the end of the year = Rs. 1,80,000

Formula Used:

Profit Ratio = (Capital of A × Time period of A) : (Capital of B × Time period of B)

M's Investment Timeline: 2L × 4 months 3L × 8 months N's Investment Timeline: 3L × 4 months 2.5L × 8 months Effective Ratio = 32L : 32L = 1 : 1

Calculations:

⇒ Effective capital of M for 1 year = (2,00,000 × 4) + (2,00,000 + 1,00,000) × 8

⇒ Effective capital of M = 8,00,000 + (3,00,000 × 8) = 8,00,000 + 24,00,000 = 32,00,000

⇒ Effective capital of N for 1 year = (3,00,000 × 4) + (3,00,000 − 50,000) × 8

⇒ Effective capital of N = 12,00,000 + (2,50,000 × 8) = 12,00,000 + 20,00,000 = 32,00,000

⇒ Ratio of profit sharing (M : N) = 32,00,000 : 32,00,000 = 1 : 1

⇒ Total ratio parts = 1 + 1 = 2

⇒ N's share of profit = (1 ÷ 2) × 1,80,000

⇒ N's share = Rs. 90,000

∴ The correct answer is Rs. 90,000.

Additional Information

Partnership Profit Principle

When investments are made for different time periods, the profit is distributed in the ratio of the product of the invested capital and the duration. Ratio = (C₁ × T₁) : (C₂ × T₂).

Active vs Sleeping Partner

An active partner usually receives a fixed percentage of total profit or a monthly salary for managing the business, while the remaining profit is distributed among all partners based strictly on their capital-time investment ratios.

Compound Partnership

A partnership where partners invest varying amounts over varying periods. The equivalent capital for one unit of time (e.g., 1 month) must be calculated first to determine the accurate profit ratio.

Shortcut Trick

Area of original square = 10² = 100

Area of removed square = 4² = 16

Remaining Area = 100 − 16 = 84

Ratio = 84 ÷ 100 = 21:25

∴ The correct answer is 21:25.

Alternate Method

Given:

Side of the original square = 10 cm

Side of the removed square = 4 cm

  10 cm    10 cm    4 cm    4 cm Remaining 84 cm² Cut (16 cm²)

Formula Used:

Area of a square = Side × Side

Remaining Area = Total Original Area − Removed Area

Calculations:

⇒ Area of the original square = 10 × 10 = 100 cm²

⇒ Area of the removed square = 4 × 4 = 16 cm²

⇒ Remaining Area = 100 − 16 = 84 cm²

⇒ Ratio of remaining area to original area = 84 / 100

⇒ Simplifying the ratio by dividing by 4 = 21 / 25

∴ The correct answer is 21:25.

Additional Information

Area and Perimeter of a Square

The area of a square is equal to the square of its side (Area = a²). The perimeter is equal to four times the side length (Perimeter = 4a).

Diagonal of a Square

The length of the diagonal of a square with side 'a' can be determined using the Pythagoras theorem, giving the result: Diagonal = a√2.

Area Subtraction Principle

When a smaller geometric figure is cut and removed from a larger one, the resulting remaining area is strictly the difference between the area of the original figure and the area of the cut figure.

Shortcut Trick

Ratio of pond area to park area = (π × r² / 2) / (L × B)

Required % = [π × 20² / (2 × 60 × 40)] × 100

⇒ [400π / 4800] × 100 = (25π / 3)%

⇒ (25 × 22) / (3 × 7) = 550 / 21 ≈ 26.19%

∴ The correct answer is 26.19%.

Alternate Method

Given:

Length of the rectangular park (L) = 60 m

Breadth of the rectangular park (B) = 40 m

Diameter of the semicircular pond = 40 m ⇒ Radius (r) = 20 m

Formula Used:

Area of rectangle = Length × Breadth

Area of semicircle = πr² / 2

Diameter = 40 m Length = 60 m Breadth = 40 m Semicircular Pond  Park Area = 2400 m²

Calculations:

⇒ Area of the park = 60 × 40 = 2400 m²

⇒ Area of the semicircular pond = (22/7) × (20 × 20) / 2

⇒ Area of pond = (22/7) × 200 = 4400/7 m²

⇒ Required Percentage = (Area of pond / Area of park) × 100

⇒ Required Percentage = [(4400/7) / 2400] × 100

⇒ Required Percentage = (4400 / 16800) × 100

⇒ Required Percentage = 4400 / 168 = 550 / 21 ≈ 26.19%

∴ The correct answer is 26.19%.

Additional Information

Area of Rectangle

The area of a rectangle is the product of its length and breadth. Formula: A = L × B.

Area of Circle & Semicircle

The area of a full circle is πr². A semicircle is exactly half of a circle, so its area is πr² / 2.

Percentage Covered Area

To find what percent of a total area is occupied by an inner region, divide the inner area by the total area and multiply by 100.

Shortcut Trick

Side of regular hexagon (a) = Perimeter ÷ 6 = 72 ÷ 6 = 12 cm.

Area of regular hexagon = 3√3 ÷ 2 × a²

⇒ Area = 1.5 × 1.732 × (12)²

⇒ Area = 2.598 × 144 = 374.112 ≈ 374.12 cm²

∴ The correct answer is 374.12 cm².

Alternate Method

Given:

Perimeter of regular hexagon = 72 cm

Formula Used:

A regular hexagon consists of 6 equilateral triangles.

Area of regular hexagon = 6 × (√3 ÷ 4) × a² = (3√3 ÷ 2) × a²

Perimeter = 6 × a

 a = 12 cm a = 12 cm Perimeter = 72 cm  Center

Calculations:

⇒ 6 × a = 72

⇒ a = 72 ÷ 6 = 12 cm

⇒ Area = (3√3 ÷ 2) × (12)²

⇒ Area = (3√3 ÷ 2) × 144

⇒ Area = 3√3 × 72 = 216√3

⇒ Value of √3 ≈ 1.732

⇒ Area = 216 × 1.732

⇒ Area ≈ 374.112 cm²

∴ The correct answer is 374.12 cm².

Additional Information

Inradius of a Regular Hexagon

The radius of the inscribed circle that touches all sides is r = (√3 ÷ 2) × a.

Circumradius of a Regular Hexagon

The radius of the circumscribed circle that passes through all vertices is exactly equal to its side length, R = a.

Interior Angle

Each interior angle of a regular hexagon measures 120°, and the sum of all interior angles is 720°.

Shortcut Trick

Total monthly income is always taken as 100%.

Combined percentage for household and savings = 60% + 15% = 75%

Remaining income percentage = Total Income − Combined percentage

Remaining income percentage = 100% − 75% = 25%

∴ The correct answer is 25%.

Alternate Method

Given:

Percentage of income spent on household expenses = 60%

Percentage of income spent on savings = 15%

Formula Used:

Remaining Value = Total Value − Sum of Parts

Household (60%) 15% Savings Other (?) Remaining Total Monthly Income = 100%

Calculation:

⇒ Let the total monthly income of the woman be 100 units.

⇒ Expenditure on household items = 60% of 100 = 60 units

⇒ Amount allocated for savings = 15% of 100 = 15 units

⇒ Total units accounted for = 60 + 15 = 75 units

⇒ Remaining units spent on other items = 100 − 75 = 25 units

⇒ Percentage of income spent on other items = (25 ÷ 100) × 100% = 25%

∴ The correct answer is 25%.

Additional Information

Total Percentage Base Concept

When a complete amount is divided into various categories, the total sum of the fractions or percentages always equals 1 or 100% respectively. All divisions are calculated based on the total initial amount.

Successive vs Direct Percentages

If expenses are directly given as a percentage of the total income, they are simply added together. However, if an expense is a percentage of the remaining income, use the successive formula: Final Amount = Initial × (1 − R₁/100) × (1 − R₂/100).

Shortcut Trick

Let B's initial salary = 100, then A's initial salary = 140.

B's new salary after a 20% increase = 120.

A's new salary is 25% more than B's new salary = 120 × 1.25 = 150.

Increase in A's salary = 150 − 140 = 10.

Percentage increase (x) = (10 ÷ 140) × 100 = 50 ÷ 7 ≈ 7.14%.

∴ The correct answer is 7.14%.

Alternate Method

B = 100 +20% B' = 120 A = 140 + x% = ? A' = 150 +40% +25%

Given:

A's salary is 40% more than B's.

B's salary increases by 20%.

A's new salary is 25% more than B's new salary.

Calculations:

⇒ Let B's initial salary = 100

⇒ A's initial salary = 100 + (40% of 100) = 140

⇒ B's new salary = 100 + (20% of 100) = 120

⇒ A's new salary = 120 + (25% of 120) = 120 + 30 = 150

⇒ Increase in A's salary = 150 − 140 = 10

⇒ Percentage increase in A's salary (x) = (Increase ÷ Initial Salary) × 100

⇒ x = (10 ÷ 140) × 100

⇒ x = 100 ÷ 14 = 50 ÷ 7

⇒ x ≈ 7.14%

∴ The correct answer is 7.14%.

Additional Information

Percentage Increase / Decrease

Percentage Change = (Difference ÷ Initial Base Value) × 100. Always ensure to use the original value as the denominator unless otherwise specified.

Successive Percentage Change

If a value is changed successively by a% and b%, the net effective percentage change can be calculated using the formula: (a + b + ab ÷ 100)%.

Comparison Percentage

If Quantity A is x% more than Quantity B, then Quantity B is strictly [x ÷ (100 + x)] × 100% less than Quantity A.

Shortcut Trick

According to the properties of averages, if a constant value is added to every quantity in a dataset, the average of the dataset increases by the exact same constant.

New Average = Old Average + Added Value

New Average = 45 + 5

New Average = 50

∴ The correct answer is 50.

Alternate Method

Given:

Average of a certain number of quantities = 45

Value added to each quantity = 5

Formula Used:

Average = Sum of all quantities ÷ Number of quantities

45 50 Old Average = 45 New Average = 50 + 5 to every quantity

Calculations:

⇒ Let there be 'n' quantities and let them be denoted as x₁, x₂, ..., x_n.

⇒ (x₁ + x₂ + ... + x_n) ÷ n = 45

⇒ Sum of all old quantities = 45n

⇒ When 5 is added to each quantity, the new quantities become (x₁ + 5), (x₂ + 5), ..., (x_n + 5).

⇒ New sum = (x₁ + 5) + (x₂ + 5) + ... + (x_n + 5)

⇒ New sum = (x₁ + x₂ + ... + x_n) + (5 × n)

⇒ New sum = 45n + 5n = 50n

⇒ New Average = New sum ÷ n

⇒ New Average = 50n ÷ n = 50

∴ The correct answer is 50.

Additional Information

Properties of Averages (Addition & Subtraction)

If a constant 'k' is added to or subtracted from each observation in a dataset, the new average will also respectively increase or decrease by 'k'. (New Average = Old Average + or − k)

Properties of Averages (Multiplication & Division)

If each observation in a dataset is multiplied or divided by a non-zero constant 'k', the new average will also be multiplied or divided by 'k' respectively. (New Average = Old Average × or ÷ k)

Combined Average

When combining two groups where group 1 has n₁ items with an average of A₁, and group 2 has n₂ items with an average of A₂, the combined average is calculated as (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Shortcut Trick

Total Volume of the prism = Base Area × Height = 45 × 14 = 630 cm³

Since 10% is hollowed out, the remaining solid fraction is (100% − 10%) = 90%.

Solid Volume = 90% of 630 = 0.9 × 630 = 567 cm³

∴ The correct answer is 567 cm³.

Alternate Method

Base Area = 45 cm² Height = 14 cm Solid Part = 90% Volume

Given:

Base area of the triangular prism = 45 cm²

Height of the prism = 14 cm

Percentage of prism hollowed out = 10%

Formula Used:

Volume of a prism = Base Area × Height

Calculations:

⇒ Total volume of the prism = 45 × 14

⇒ Total volume = 630 cm³

⇒ Since 10% of the prism is hollowed out, the solid part forms (100% − 10%) = 90% of the total volume.

⇒ Volume of the solid part = 90% of 630

⇒ Volume of the solid part = (90 ÷ 100) × 630

⇒ Volume of the solid part = 0.9 × 630 = 567 cm³

∴ The correct answer is 567 cm³.

Additional Information

Volume of a Prism

The volume of any prism is the product of its base area and height: Volume = Base Area × Height.

Surface Area of a Prism

The total surface area of a regular prism is calculated as: Total Surface Area = Lateral Surface Area + (2 × Base Area).

Volume of a Pyramid

A pyramid having the exact same base and height as a prism will always have one-third of the prism's volume: Volume = (1 ÷ 3) × Base Area × Height.

Percentage Decrease

When a quantity decreases or is reduced by x%, the remaining quantity is (100 − x)% of the original value.

Shortcut Trick

Using the Net Deviation Method:

The given average bill per person is ₹210.

4 friends paid ₹200 each, creating a deficit of ₹10 per person (−10 × 4 = −₹40).

To balance this deficit and maintain the average, the 5^th friend must pay the average plus this deficit amount.

x = 210 + 40 = 250

∴ The correct answer is ₹250.

Alternate Method

Given:

Total number of friends = 5

Amount paid by 4 friends = ₹200 each

Average bill per person = ₹210

Formula Used:

Average = Sum of all values ÷ Total number of values

Sum of all values = Average × Total number of values

Total Bill = 5 × 210 = ₹1050 4 Friends = ₹800 5^th Friend = x x = 1050 − 800 = ₹250

Calculations:

⇒ Total bill amount = 210 × 5 = ₹1050

⇒ Total amount paid by 4 friends = 200 × 4 = ₹800

⇒ Amount paid by the 5^th friend (x) = Total bill − Total amount paid by 4 friends

⇒ x = 1050 − 800

⇒ x = 250

∴ The correct answer is ₹250.

Additional Information

Net Deviation Method

The sum of the deviations of all observations from their true average is always zero. This property helps find missing individual values rapidly.

Weighted Average

If two distinct groups have quantities n₁ and n₂ with their respective averages A₁ and A₂, the combined average is calculated as (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Constant Change Property

If every term in a series is increased, decreased, multiplied, or divided by a constant value 'k', the new average undergoes the exact same mathematical operation by 'k'.

Shortcut Trick

Using the direct ratio relation between Cost Price (CP) and Marked Price (MP):

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

CP : MP = (100 − 40) : (100 + 20) = 60 : 120 = 1 : 2

Given CP (1 unit) = Rs. 720

Marked Price (2 units) = 720 × 2 = Rs. 1440

∴ The correct answer is Rs. 1,440.

Alternate Method

Given:

Cost Price (CP) = Rs. 720

Discount (D%) = 40%

Profit (P%) = 20%

Formulas Used:

Selling Price (SP) = CP + Profit

Selling Price (SP) = MP − Discount

CP = 720 SP = 864 MP = 1440 +20% Profit -40% Discount CP / MP = (100 - D%) / (100 + P%)

Calculations:

⇒ Profit = 20% of CP = (20 / 100) × 720

⇒ Profit = Rs. 144

⇒ Selling Price (SP) = CP + Profit = 720 + 144

⇒ SP = Rs. 864

⇒ We know that SP = MP × (100 − D%) / 100

⇒ 864 = MP × (100 − 40) / 100

⇒ 864 = MP × (60 / 100)

⇒ 864 = MP × 0.6

⇒ MP = 864 / 0.6

⇒ MP = 1440

∴ The correct answer is Rs. 1,440.

Additional Information

Mark-up Percentage

It is the percentage by which the Marked Price (MP) is raised above the Cost Price (CP). Formula: Mark-up % = [(MP − CP) / CP] × 100.

Successive Discounts

When multiple discounts x% and y% are given consecutively, the effective single discount is calculated as: [x + y − (x × y) / 100]%.

Dishonest Dealer Concept

A dealer claims to sell at CP but uses false weights. The profit percentage in such cases is based on the actual goods given: Profit % = (True Weight − False Weight) / False Weight × 100.