Quantitative Aptitude PYQs

Shortcut Trick

Total Selling Price = 1800 + 1700 + 1600 = Rs. 5100

Total Cost Price = Rs. 4500

Profit = Total Selling Price − Total Cost Price = 5100 − 4500 = Rs. 600

∴ The correct answer is Rs. 600 profit.

Alternate Method

Given:

Total Cost Price (CP) of 3 watches = Rs. 4500

Selling Prices (SP) of watches = Rs. 1800, Rs. 1700, and Rs. 1600

Formula Used:

Total SP = SP₁ + SP₂ + SP₃

Profit = Total SP − Total CP (if SP > CP)

Profit Calculation Total CP = Rs. 4500 SP1: 1800 SP2: 1700 SP3: 1600 Total SP = Rs. 5100 Profit Rs. 600

Calculations:

⇒ Total SP = 1800 + 1700 + 1600 = Rs. 5100

⇒ Since Total SP > Total CP (5100 > 4500), an overall profit is earned.

⇒ Overall Profit = Total SP − Total CP

⇒ Profit = 5100 − 4500

⇒ Profit = Rs. 600

∴ The correct answer is Rs. 600 profit.

Additional Information

Profit & Loss Basics

A profit is made when the Selling Price (SP) is greater than the Cost Price (CP). Profit = SP − CP. Conversely, a loss occurs when CP > SP.

Profit Percentage

Profit percentage is always calculated on the base of Cost Price unless stated otherwise. Profit % = (Profit ÷ CP) × 100.

Marked Price & Discount

Marked Price (MP) is the list price labeled on the article. Any discount is calculated directly on the MP. Selling Price = MP − Discount.

Shortcut Trick

Using the MDH formula: (M₁ × D₁) ÷ W₁ = (M₂ × D₂) ÷ W₂

⇒ (35 × 8) ÷ 1 = (16 × D₂) ÷ 4

⇒ 280 = 4 × D₂

⇒ D₂ = 280 ÷ 4 = 70

∴ The correct answer is 70 Days.

Alternate Method

Given:

Initial workers (M₁) = 35 men, Time taken (D₁) = 8 days

New workers (M₂) = 16 men, New work (W₂) = 4 × Original work

⇒ Total Original Work = 35 × 8 = 280 units

⇒ New Total Work required = 4 × 280 = 1120 units

⇒ Combined efficiency of 16 men = 16 units per day

⇒ Time required by 16 men = New Total Work ÷ Efficiency

⇒ Time required = 1120 ÷ 16 = 70 Days

∴ The correct answer is 70 Days.

Additional Information

MDH Formula

(M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂, where M is Men, D is Days, H is Hours, and W is the Work done.

Work Efficiency Rule

Efficiency is inversely proportional to time. A person taking less time has a higher efficiency: E₁ × T₁ = E₂ × T₂.

Group Work Concept

If multiple people work together uniformly, their combined efficiency is simply the sum of their individual efficiencies: E_Total = E₁ + E₂ + ... + E_n.

Shortcut Trick

If A and B together can do a work in 'x' days and A alone can do it in 'y' days, then B alone takes (x × y) ÷ (y − x) days.

Time taken by Dhiraj = (12 × 20) ÷ (20 − 12)

⇒ 240 ÷ 8 = 30 days

∴ The correct answer is 30 days.

Alternate Method

Given:

Time taken by Dhaval and Dhiraj together = 12 days

Time taken by Dhaval alone = 20 days

Concept Used:

Total Work = LCM of given days

Efficiency = Total Work ÷ Time taken

Calculations:

⇒ Combined efficiency of Dhaval and Dhiraj = 5 units/day

⇒ Efficiency of Dhaval alone = 3 units/day

⇒ Efficiency of Dhiraj alone = 5 − 3 = 2 units/day

⇒ Time taken by Dhiraj alone = Total Work ÷ Efficiency of Dhiraj

⇒ Time taken by Dhiraj alone = 60 ÷ 2 = 30 days

∴ The correct answer is 30 days.

Additional Information

Pipes & Cisterns

Similar to time and work, filling a tank is considered positive work and emptying is considered negative work. Net efficiency = (Inlet Efficiency) − (Outlet Efficiency).

Work & Wages

Wages are distributed in the direct ratio of the amount of work done by each person. If they work for the same number of days, wages are distributed in the ratio of their efficiencies.

Alternate Day Work

When persons work on alternate days, find the total work done in a complete cycle (e.g., 2 days for 2 persons). Multiply this cycle value to get close to the total work, then calculate for the remaining days separately.

Shortcut Trick

Supervisor's Salary = Old Average + (New Total Persons × Increase in Average)

⇒ Total persons = 40 + 1 = 41

⇒ Increase in average = 24500 − 24000 = Rs. 500

⇒ Supervisor's Salary = 24000 + (41 × 500) = Rs. 44,500

∴ The correct answer is Rs. 44,500.

Alternate Method

Given:

Average salary of 40 workers = Rs. 24,000

New average after adding the supervisor = Rs. 24,500

40 Workers Avg: Rs. 24,000 41 Persons Avg: Rs. 24,500 +Rs. 500 Avg Supervisor = 24000 + (41 × 500) = Rs. 44,500

Formula Used:

Total Value = Average × Number of Entities

Calculations:

⇒ Total salary of 40 workers = 40 × 24000 = Rs. 9,60,000

⇒ Total number of people including supervisor = 40 + 1 = 41

⇒ Total salary of 41 people = 41 × 24500 = Rs. 10,04,500

⇒ Supervisor's salary = Total salary of 41 people − Total salary of 40 workers

⇒ Supervisor's salary = 1004500 − 960000 = 44500

∴ The correct answer is Rs. 44,500.

Additional Information

1. Replacement Method in Averages

When a new person replaces an old person: New Person's Value = Old Person's Value ± (Total Entities × Change in Average).

2. Combined (Weighted) Average

If group A has n₁ items with average A₁, and group B has n₂ items with average A₂, then Combined Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

3. Shift in Origin and Scale

If every observation is increased, decreased, multiplied, or divided by a constant value 'k', the new average also changes by applying the exact same arithmetic operation with 'k' on the old average.

Shortcut Trick

When two right prisms have the same height, the ratio of their volumes is strictly equal to the ratio of their base areas.

Ratio = Area of regular hexagonal base : Area of square base

Ratio = [6 × (√3 / 4) × a²] : a²

Since the side length a = 6 cm is identical for both, a² cancels out completely.

Ratio = (3√3 / 2) : 1 = 3√3 : 2

∴ The correct answer is 3√3 : 2.

Alternate Method

Given:

Side of the regular hexagonal base (a) = 6 cm

Side of the square base (b) = 6 cm

Height of both right prisms (h) = 10 cm

Formula Used:

Volume of a prism = Area of base × height

Area of a regular hexagon = (3√3 / 2) × (side)²

Area of a square = (side)²

Hexagonal Base a = 6 cm Square Base b = 6 cm Ratio of Volumes = Ratio of Base Areas

Calculation:

⇒ Volume of the hexagonal prism (V₁) = [(3√3 / 2) × 6²] × 10

⇒ V₁ = (3√3 / 2) × 36 × 10

⇒ V₁ = 54√3 × 10 = 540√3 cm³

⇒ Volume of the square prism (V₂) = [6²] × 10

⇒ V₂ = 36 × 10 = 360 cm³

⇒ Ratio of their volumes = V₁ : V₂

⇒ Ratio = 540√3 : 360

⇒ Dividing both sides by 180, we get:

⇒ Ratio = 3√3 : 2

∴ The correct answer is 3√3 : 2.

Additional Information

Volume of a Prism

The volume of any right prism is calculated by multiplying its base area by its height. Formula: V = Base Area × height.

Total Surface Area of a Prism

The total surface area encompasses all the faces of the 3D shape. Formula: TSA = Lateral Surface Area + 2 × Base Area, where Lateral Surface Area = Base Perimeter × height.

Area of a Regular Hexagon

A regular hexagon behaves exactly like 6 adjacent equilateral triangles joined at the center. Area = 6 × (√3 / 4) × side² = (3√3 / 2) × side².

Shortcut Trick

Initial Ratio of Kerosene to Diesel = 4 : 5

Final Ratio of Kerosene to Diesel = 4 : 6

Since no kerosene is added, its ratio parts remain the same (4 parts).

Increase in Diesel parts = 6 − 5 = 1 part.

Total initial parts = 9 parts → 45 L ⇒ 1 part = 5 L.

∴ The correct answer is 5 L.

Alternate Method

  Initial Mixture (45 L)   K: 4 parts D: 5 parts  + Diesel Final Mixture   K: 4 parts D: 6 parts

Given:

Initial mixture volume = 45 L

Initial Kerosene to Diesel ratio = 4 : 5

Final Kerosene to Diesel ratio = 4 : 6

Calculations:

⇒ Quantity of Kerosene = 45 × (4 / 9) = 20 L

⇒ Quantity of Diesel = 45 × (5 / 9) = 25 L

Let the amount of diesel added be x L.

⇒ New ratio = 20 / (25 + x) = 4 / 6

⇒ 20 × 6 = 4 × (25 + x)

⇒ 120 = 100 + 4x

⇒ 4x = 20

⇒ x = 5 L

∴ The correct answer is 5 L.

Additional Information

Constant Component Strategy

When one component is added or removed, the quantity of the other component remains unchanged. Equating its ratio parts directly yields the change in the total mixture.

Rule of Alligation

Used to find the ratio in which two ingredients are combined: Ratio = (Quantity of Cheaper) / (Quantity of Dearer) = (Price of Dearer − Mean Price) / (Mean Price − Price of Cheaper).

Replacement in Mixture

When a part of a mixture is replaced by one of its pure components, the ratio of ingredients in the remaining mixture initially remains the same as the original mixture.

Shortcut Trick

We need to find the value of sinA × secA.

We know that secA = 1 / cosA, so sinA × secA = sinA / cosA = tanA.

Given sinA = 5/13, we have Perpendicular = 5 and Hypotenuse = 13. By triplet (5, 12, 13), Base = 12.

Since angle A lies in the 2^nd quadrant, the tangent function is negative.

Therefore, tanA = −(Perpendicular / Base) = −5/12.

∴ The correct answer is −5/12.

Alternate Method

Given:

sinA = 5/13 and angle A lies in the 2^nd quadrant.

X X' Y Y' A 13 5 -12 2^nd Quadrant Origin (0,0)

Formula Used:

sin²A + cos²A = 1

secA = 1 / cosA

Calculations:

⇒ (5/13)² + cos²A = 1

⇒ 25/169 + cos²A = 1

⇒ cos²A = 1 − 25/169

⇒ cos²A = 144/169

⇒ cosA = ±12/13

Since angle A lies in the 2^nd quadrant, the cosine function must be negative.

⇒ cosA = −12/13

⇒ secA = 1 / (−12/13) = −13/12

Now, substitute the values to find sinA × secA:

⇒ sinA × secA = (5/13) × (−13/12)

⇒ sinA × secA = −5/12

∴ The correct answer is −5/12.

Additional Information

Trigonometric Signs (ASTC Rule)

1^st Quadrant: All positive | 2^nd Quadrant: Sin & Cosec positive | 3^rd Quadrant: Tan & Cot positive | 4^th Quadrant: Cos & Sec positive.

Pythagorean Triplets

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

Reciprocal Trigonometric Identities

Formulas linking basic and reciprocal functions: secA = 1 / cosA, cosecA = 1 / sinA, and cotA = 1 / tanA.

Shortcut Trick

Horizontal difference (dx) = 4 − 1 = 3

Vertical difference (dy) = 6 − 2 = 4

The differences form the legs of a right-angled triangle. Using the standard Pythagorean triplet (3, 4, 5), the hypotenuse (distance) is directly 5.

∴ The correct answer is 5 units.

Alternate Method

Given:

The coordinates of the two points are A(1, 2) and B(4, 6).

Formula Used:

Distance d = √[(x₂ − x₁)² + (y₂ − y₁)²]

A (1, 2) B (4, 6) C (4, 2)

dx = 3 dy = 4 d = ?

Calculations:

⇒ Let (x₁, y₁) = (1, 2) and (x₂, y₂) = (4, 6)

⇒ d = √[(4 − 1)² + (6 − 2)²]

⇒ d = √[3² + 4²]

⇒ d = √[9 + 16]

⇒ d = √25

⇒ d = 5

∴ The correct answer is 5 units.

Additional Information

Midpoint Formula

The coordinates of the exact midpoint of a segment joining (x₁, y₁) and (x₂, y₂) is given by M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2).

Section Formula

The coordinates of a point dividing the line segment internally in the ratio m:n are given as ((mx₂ + nx₁) ÷ (m + n), (my₂ + ny₁) ÷ (m + n)).

Distance from Origin

The distance of any point (x, y) directly from the origin (0, 0) simplifies to d = √(x² + y²).

Shortcut Trick

By the Exterior Angle Theorem, ∠A + ∠B = Exterior ∠C.

Given the ratio ∠A:∠B = 2:5, their sum represents 7 parts.

7 parts = 105° ⇒ 1 part = 15°.

Therefore, ∠A = 30° and ∠B = 75°.

Interior ∠C = 180° − 105° = 75°.

The largest interior angle out of 30°, 75°, and 75° is 75°.

∴ The correct answer is 75°.

Alternate Method

Given:

In triangle ABC, side BC is extended to D.

Exterior angle at C = 105°

Ratio of interior angles A and B = 2 : 5

Formula Used:

Exterior angle of a triangle = Sum of opposite interior angles

Sum of interior angles on a straight line (Linear Pair) = 180°

        A B C D  2x 5x 105°

Calculations:

⇒ Let the interior angles at A and B be 2x and 5x respectively.

⇒ According to the exterior angle theorem, ∠A + ∠B = Exterior ∠C

⇒ 2x + 5x = 105°

⇒ 7x = 105°

⇒ x = 105° ÷ 7 = 15°

⇒ Interior ∠A = 2 × 15° = 30°

⇒ Interior ∠B = 5 × 15° = 75°

⇒ Now, the interior angle at C can be found using the linear pair property on line BD.

⇒ Interior ∠C = 180° − Exterior ∠C

⇒ Interior ∠C = 180° − 105° = 75°

⇒ The interior angles of the triangle are 30°, 75°, and 75°.

⇒ The largest interior angle of the triangle is 75°.

∴ The correct answer is 75°.

Additional Information

Exterior Angle Theorem

The exterior angle of a triangle is equal to the sum of the two opposite interior angles. This property is highly useful in solving angle-chasing geometry problems without needing to find the third interior angle first.

Linear Pair of Angles

When a straight line is extended, the interior and exterior angles at a vertex form a linear pair, meaning their sum is always exactly 180°.

Isosceles Triangle Property

When two angles of a triangle are equal (like 75° and 75° in this case), the sides opposite to those angles are also equal. Such a triangle is classified as an isosceles triangle.

Shortcut Trick

When two triangles are congruent, their corresponding angles and sides are perfectly equal.

The order of the vertices in the congruence statement directly indicates the corresponding parts.

For △ABC ≅ △DEF, the first letter A corresponds to the first letter D.

∴ ∠A = ∠D

Since ∠A = 50°, we directly get ∠D = 50° without any complex calculations.

∴ The correct answer is 50°.

Alternate Method

Given:

⇒ △ABC ≅ △DEF

⇒ ∠A = 50°

A B C 50° D E F ? ≅

Concept Used:

⇒ Corresponding Parts of Congruent Triangles are Congruent (CPCTC).

⇒ The position of the letters in the congruence notation maps out the equal corresponding parts.

Calculations:

⇒ From the given statement △ABC ≅ △DEF, we can establish the exact mapping between the vertices:

⇒ Vertex A ↔ Vertex D

⇒ Vertex B ↔ Vertex E

⇒ Vertex C ↔ Vertex F

⇒ This mapping means the corresponding angles are equal: ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F.

⇒ We are given that ∠A = 50°.

⇒ Substituting this value into our correspondence relation, we get:

⇒ ∠D = 50°

∴ The correct answer is 50°.

Additional Information

Congruence Criteria for Triangles

Two triangles are congruent if they satisfy any of the five major criteria: SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), or RHS (Right angle-Hypotenuse-Side).

CPCTC Rule

CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent". It simply implies that once two triangles are proven to be congruent, all their remaining corresponding sides and angles must be perfectly equal.

Difference Between Congruence and Similarity

Congruent triangles have exactly the same shape and exactly the same size. Similar triangles have the same shape but can have proportionally different sizes. All congruent triangles are similar, but similar triangles are not always congruent.

Shortcut Trick

Given, x + 1/x = 5

We know the identity, x − 1/x = √((x + 1/x)² − 4) = √(5² − 4) = √21

Divide the numerator and denominator of the given expression 2x / (x² − 1) by x to get 2 / (x − 1/x)

Substitute the value to get 2 / √21 = 2√21 / 21

∴ The correct answer is 2√21 / 21.

Alternate Method

Given:

x + 1/x = 5

Formula Used:

(a − b)² = (a + b)² − 4ab

Calculation:

⇒ (x − 1/x)² = (x + 1/x)² − 4 × x × 1/x

⇒ (x − 1/x)² = 5² − 4

⇒ (x − 1/x)² = 25 − 4

⇒ (x − 1/x)² = 21

⇒ x − 1/x = √21

We need to find the value of 2x / (x² − 1)

Dividing the numerator and the denominator by x:

⇒ (2x / x) / ((x² − 1) / x)

⇒ 2 / (x − 1/x)

Substituting the value of (x − 1/x):

⇒ 2 / √21

Multiplying the numerator and denominator by √21 to rationalize:

⇒ (2 × √21) / (√21 × √21)

⇒ 2√21 / 21

∴ The correct answer is 2√21 / 21.

Additional Information

Higher Power Algebraic Identities

If x + 1/x = k, then x² + 1/x² = k² − 2 and x³ + 1/x³ = k³ − 3k.

Algebraic Difference Formula

If x + 1/x = k, then x − 1/x = √(k² − 4). Conversely, if x − 1/x = k, then x + 1/x = √(k² + 4).

Rationalization of Surds

When a fraction has an irrational root in its denominator, it is simplified using rationalization. This is done by multiplying both the numerator and the denominator by the appropriate surd to produce a rational denominator.

Shortcut Trick

The perpendicular drawn from the center to a chord bisects it.

Half of the chord = 24 ÷ 2 = 12 cm.

Using Pythagoras theorem: Radius = √(distance² + half-chord²)

Radius = √(7² + 12²) = √(49 + 144) = √193

√193 is approximately 13.89 cm, which rounds to 14 cm.

∴ The correct answer is 14 cm.

Alternate Method

Given:

Length of the chord (AB) = 24 cm

Perpendicular distance from the center (OC) = 7 cm

Formula Used:

Radius² = (Perpendicular distance)² + (Half of the chord)²

  O C A B 7 cm 12 cm r

Calculations:

⇒ The perpendicular from the center of a circle to a chord bisects it.

⇒ Therefore, AC = CB = 24 ÷ 2 = 12 cm.

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

⇒ OA² = OC² + AC²

⇒ Radius² = 7² + 12²

⇒ Radius² = 49 + 144

⇒ Radius² = 193

⇒ Radius = √193

⇒ Radius ≈ 13.89 cm

⇒ Approximating to the closest whole number, we get 14 cm.

∴ The correct answer is 14 cm.

Additional Information

Perpendicular to a Chord Theorem

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

Equal Chords Property

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

Longest Chord (Diameter)

The diameter is the longest chord of a circle. Because it passes directly through the center, its perpendicular distance from the center is 0.

Shortcut Trick

Angle subtended by a chord at the centre is always double the angle subtended at the major arc.

⇒ Central angle = 2 × 30° = 60°.

When a chord subtends 60° at the centre, the triangle formed by the chord and the two radii is an equilateral triangle.

⇒ Radius of the circle = Length of the chord = 10 cm.

∴ The correct answer is 10 cm.

Alternate Method

Given:

Length of chord AB = 10 cm

Angle subtended at the major arc by chord AB (∠ACB) = 30°

Formula Used:

Angle subtended by an arc at the centre = 2 × Angle subtended at the remaining part of the circle.

Sum of angles in a triangle = 180°.

       O C A B 10 cm 30° 60° r r

Calculations:

⇒ Let the centre of the circle be O and the radius be r.

⇒ Angle at the centre ∠AOB = 2 × ∠ACB

⇒ ∠AOB = 2 × 30° = 60°

⇒ In ΔAOB, OA = OB = r (Radii of the same circle)

⇒ Since OA = OB, the angles opposite to equal sides must be equal: ∠OAB = ∠OBA

⇒ In ΔAOB: ∠AOB + ∠OAB + ∠OBA = 180°

⇒ 60° + 2 × ∠OAB = 180°

⇒ 2 × ∠OAB = 120°

⇒ ∠OAB = 60°

⇒ Since all three interior angles are 60°, ΔAOB is an equilateral triangle.

⇒ Therefore, all sides of the triangle are equal: OA = OB = AB

⇒ Radius (r) = AB = 10 cm

∴ The correct answer is 10 cm.

Additional Information

Angle at Centre Theorem

The angle subtended by an arc or chord at the centre of a circle is strictly double the angle subtended by it at any point on the remaining part of the circle.

Equilateral Triangle Property

If a central angle formed by a chord is 60°, the isosceles triangle bounded by the two radii and the chord inherently becomes equilateral. Hence, the length of the chord is perfectly equal to the radius.

Angles in Same Segment

Angles subtended by the same arc at any two distinct points on the major (or minor) segment of a circle are identical in magnitude.

Shortcut Trick

Use the fundamental trigonometric identity: sin²A + cos²A = 1.

Rearranging the formula gives: cos A = √(1 − sin²A).

Substitute sin A = m / n ⇒ cos A = √(1 − (m / n)²) = √((n² − m²) / n²).

Since A ∈ (0, π/2), cos A is positive, yielding √(n² − m²) / n.

∴ The correct answer is √(n² − m²) / n.

Alternate Method

Given:

sin A = m / n

Angle A is in the first quadrant, A ∈ (0, π/2)

Formula Used:

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

sin A = Perpendicular / Hypotenuse

cos A = Base / Hypotenuse

   A  P Q R  √(n² − m²) m n

Calculations:

⇒ Comparing sin A = m / n with Perpendicular / Hypotenuse, we get Perpendicular = m and Hypotenuse = n.

⇒ By Pythagoras theorem: n² = Base² + m²

⇒ Base² = n² − m²

⇒ Base = √(n² − m²)

⇒ We know that cos A = Base / Hypotenuse.

⇒ Substituting the corresponding values, we get cos A = √(n² − m²) / n.

∴ The correct answer is √(n² − m²) / n.

Additional Information

Trigonometric Identities

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

Signs of Trigonometric Functions

In the first quadrant (from 0 to π/2 radians), all six primary trigonometric functions always yield positive values.

Basic Ratios

For any given right-angled triangle: tan θ = Perpendicular / Base, sec θ = Hypotenuse / Base, and cosec θ = Hypotenuse / Perpendicular.

Shortcut Trick

Convert decimals directly to improper fractions for a quick calculation.

Earnings = 52.50 = 105/2

Time = 3.5 = 7/2

Hourly Wage = (105/2) ÷ (7/2) = 105 ÷ 7 = 15

Expressed as a fraction, it is 15/1.

∴ The correct answer is 15/1.

Alternate Method

Given:

Total earnings = 52.50

Total time worked = 3.5 hours

Formula Used:

Hourly Wage = Total Earnings ÷ Total Time

Calculations:

⇒ Hourly Wage = 52.50 ÷ 3.5

⇒ Multiply the numerator and the denominator by 10 to remove decimals.

⇒ Hourly Wage = 525 ÷ 35

⇒ Divide both numbers by 5:

⇒ Hourly Wage = 105 ÷ 7

⇒ Divide both numbers by 7:

⇒ Hourly Wage = 15

⇒ Since the options are in a fraction form, we write 15 with a denominator of 1.

⇒ Hourly Wage = 15/1

∴ The correct answer is 15/1.

Additional Information

Unitary Method

It is a mathematical technique for solving a problem by finding the value of a single unit, and then calculating the required value by multiplying the single unit value.

Converting Decimals to Fractions

Count the decimal places, write the digits without the decimal point as the numerator, and place 1 followed by that many zeros as the denominator, then simplify the fraction.

Simplification Rules (BODMAS)

Always follow the Brackets, Order, Division, Multiplication, Addition, and Subtraction sequence for solving mixed arithmetic expressions accurately.

Shortcut Trick

Expand the left side mentally using the formula (a − b)² = a² + b² − 2ab.

(√5 − 1)² = 5 + 1 − 2√5 = 6 − 2√5

Comparing 6 − 2√5 with ? − 2√5 directly gives ? = 6.

∴ The correct answer is 6.

Alternate Method

Given:

(√5 − 1)² = ? − 2√5

Formula Used:

(a − b)² = a² − 2ab + b²

Calculations:

⇒ (√5)² − 2 × √5 × 1 + 1² = ? − 2√5

⇒ 5 − 2√5 + 1 = ? − 2√5

⇒ 6 − 2√5 = ? − 2√5

⇒ ? = 6

∴ The correct answer is 6.

Additional Information

Algebraic Identity (a + b)²

The square of a sum is given by (a + b)² = a² + 2ab + b².

Difference of Squares

The product of the sum and difference of two terms is given by (a + b)(a − b) = a² − b².

Properties of Surds

When a square root is squared, it results in the original number: (√a)² = a. Multiplication of surds follows √a × √b = √(ab).

Shortcut Trick

The effective compound interest rate for 3 years at 10% p.a. is directly known to be 33.1%.

CI = 33.1% of Principal

CI = (33.1 / 100) × 25000

CI = 33.1 × 250 = 8275

∴ The correct answer is ₹8,275.

Alternate Method

Given:

Principal (P) = ₹25,000

Rate (R) = 10% p.a.

Time (T) = 3 years

Formula Used:

Amount (A) = P(1 + R/100)^T

Compound Interest (CI) = A − P

  Compound Interest Progression  25,000 Year 0  27,500 Year 1  30,250 Year 2  33,275 Year 3  + ₹2,500 (10%)  + ₹2,750 (10%)  + ₹3,025 (10%) Total Compound Interest = 33275 − 25000 = ₹8,275

Calculations:

⇒ Amount (A) = 25000 × (1 + 10/100)³

⇒ A = 25000 × (11/10)³

⇒ A = 25000 × (1331/1000)

⇒ A = 25 × 1331 = 33275

⇒ CI = 33275 − 25000 = 8275

∴ The correct answer is ₹8,275.

Additional Information

CI vs SI Difference

For 2 years: Difference = P(R/100)²

For 3 years: Difference = P(R/100)²(3 + R/100)

Half-Yearly Compounding

If interest is compounded half-yearly, the Rate becomes R/2 and the Time period becomes 2T.

Rule of 72

A quick rule of thumb: the time taken to double an investment at compound interest is approximately 72 / R years.

Shortcut Trick

Effective interest rate for case 1 = 3 × 6% = 18%

Effective interest rate for case 2 = 5 × 8% = 40%

Given 18% of Principal = ₹360

⇒ 1% of Principal = 360 ÷ 18 = ₹20

⇒ 40% of Principal = 20 × 40 = ₹800

∴ The correct answer is ₹800.

Alternate Method

Given:

Simple Interest (SI₁) = ₹360

Time (T₁) = 3 years, Rate (R₁) = 6% per annum

Time (T₂) = 5 years, Rate (R₂) = 8% per annum

Formula Used:

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

Case 1 3 years @ 6%

SI = ₹360 18% of Principal

Case 2 5 years @ 8%

SI = ? 40% of Principal

Calculations:

From Case 1, we can find the Principal (P):

⇒ 360 = (P × 6 × 3) ÷ 100

⇒ 360 = (18 × P) ÷ 100

⇒ P = (360 × 100) ÷ 18

⇒ P = ₹2000

Now, we will calculate Simple Interest for Case 2 using the same Principal:

⇒ SI₂ = (2000 × 8 × 5) ÷ 100

⇒ SI₂ = (2000 × 40) ÷ 100

⇒ SI₂ = 80000 ÷ 100 = ₹800

∴ The correct answer is ₹800.

Additional Information

Direct Ratio Method

Since the principal is constant, Simple Interest is directly proportional to the product of Rate and Time. SI₁ ÷ SI₂ = (R₁ × T₁) ÷ (R₂ × T₂).

Amount in Simple Interest

Amount = Principal + Simple Interest. The formula becomes Amount = P(1 + RT ÷ 100).

Compound vs Simple Interest

For the first year, SI and CI are equal if calculated annually. From the second year onwards, CI will always be greater than SI for a given rate and principal.

Shortcut Trick

Any decimal number that terminates (has an end) is always a rational number.

0.875 terminates after three decimal places, meaning it can be written exactly as 875 ÷ 1000.

Since it can be expressed in p/q form where both are integers, it is directly classified as a rational number.

∴ The correct answer is Rational Number.

Alternate Method

Given:

The decimal number 0.875.

Method Used:

Let us verify the definitions of the options provided:

⇒ Whole Number: Non-negative basic counting numbers (0, 1, 2, 3, ...).

⇒ Integer: Whole numbers and their negative counterparts (..., −2, −1, 0, 1, 2, ...).

⇒ Irrational Number: A number that cannot be written as a simple fraction (non-terminating and non-repeating).

⇒ Rational Number: A number that can be written in the form p/q, where p and q are integers and q ≠ 0.

Calculations:

⇒ Clearly, 0.875 is neither a whole number nor an integer because it has a fractional part.

⇒ Convert 0.875 to a fraction.

⇒ 0.875 = 875 / 1000

⇒ Simplify the fraction by dividing both numerator and denominator by their highest common factor, 125.

⇒ 0.875 = 7 / 8

⇒ Here, p = 7 and q = 8. Both are integers and the denominator q is not equal to zero.

⇒ Therefore, it perfectly fits the definition of a rational number.

∴ The correct answer is Rational Number.

Additional Information

Real Numbers

Real numbers consist of both rational and irrational numbers. Every valid point on the standard number line represents a real number.

Repeating Decimals

Decimals that are non-terminating but repeating (like 0.333...) can also be expressed as fractions (1/3). They are strictly classified as Rational Numbers.

Irrational Numbers Example

Famous irrational numbers include π (≈ 3.14159...) and √2 (≈ 1.414...). Their decimal expansions never terminate and never repeat their patterns.

Shortcut Trick

Speed = Distance ÷ Time

Speed = 300 ÷ 25 = 12 m/s

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

Speed in km/h = 12 × (18 ÷ 5) = 216 ÷ 5 = 43.2

∴ The correct answer is 43.2 km/hr.

Alternate Method

Given:

Distance of the race (d) = 300 m

Time taken (t) = 25 seconds

Start End Distance = 300 m Time = 25 s Speed = ? km/h

Formula Used:

Speed = Distance ÷ Time

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

Calculations:

⇒ Speed of the athlete = 300 ÷ 25

⇒ Speed = 12 m/s

⇒ To convert the speed into km/h, multiply by 18 ÷ 5

⇒ Speed = 12 × (18 ÷ 5) km/h

⇒ Speed = 216 ÷ 5 km/h

⇒ Speed = 43.2 km/h

∴ The correct answer is 43.2 km/hr.

Additional Information

Unit Conversions for Speed

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

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

Average Speed

Average Speed = Total Distance Covered ÷ Total Time Taken.

Relative Speed

If two bodies move in the same direction, Relative Speed = S₁ − S₂.

If two bodies move in opposite directions, Relative Speed = S₁ + S₂.