Quantitative Aptitude PYQs

Shortcut Trick

Mean Cost Price (CP) = SP × [100 ÷ (100 + Profit%)] = 84 × (100 ÷ 120) = ₹70/kg.

By applying the rule of alligation on the Cost Prices:

Ratio = (CP of Type 2 − Mean CP) : (Mean CP − CP of Type 1)

Ratio = (90 − 70) : (70 − 60)

Ratio = 20 : 10 = 2 : 1

∴ The correct answer is 2:1.

Alternate Method

Given:

Cost price of Type 1 pulse = ₹60/kg

Cost price of Type 2 pulse = ₹90/kg

Selling price of the mixture = ₹84/kg

Profit on selling = 20%

Formula Used:

Cost Price (CP) = Selling Price (SP) × [100 / (100 + Profit%)]

Alligation Ratio = (Dearer CP − Mean CP) : (Mean CP − Cheaper CP)

Calculations:

⇒ Mean CP of the mixture = 84 × [100 / (100 + 20)]

⇒ Mean CP = 84 × (100 / 120) = ₹70/kg

Type 1 Type 2 ₹60 ₹90 Mean CP ₹70 90 − 70 70 − 60 20 10 Ratio = 20 : 10 = 2 : 1

⇒ Using the alligation rule:

⇒ Ratio of Type 1 to Type 2 = (90 − 70) : (70 − 60)

⇒ Ratio = 20 : 10 = 2 : 1

⇒ Alternate algebraic check: Let x kg of Type 1 and y kg of Type 2 be mixed.

⇒ 60x + 90y = 70(x + y)

⇒ 90y − 70y = 70x − 60x

⇒ 20y = 10x ⇒ x : y = 2 : 1

∴ The correct answer is 2:1.

Additional Information

Alligation Rule

It is a mathematical rule to determine the ratio in which two ingredients at given prices must be mixed to produce a mixture of a desired average price. Ratio = (Dearer Price − Mean Price) : (Mean Price − Cheaper Price).

Cost Price vs Selling Price

The rule of alligation must always be applied on equivalent values. If applying on prices, always use the Cost Price (CP) for all components including the mean value, never directly on Selling Price if a profit/loss percentage is involved.

Replacement Method

If from a vessel containing 'a' units of pure liquid, 'b' units are drawn off and replaced by water 'n' times, the remaining pure liquid is calculated as: Remaining = a × (1 − b/a)ⁿ.

Shortcut Trick

Efficiency ratio of A : B = 3 : 1, so Time ratio A : B = 1 : 3.

Difference in time ratio = 3 − 1 = 2 units.

2 units = 60 days ⇒ 1 unit = 30 days (A's time) and 3 units = 90 days (B's time).

Time taken together = (30 × 90) ÷ (30 + 90) = 2700 ÷ 120 = 22.5 days.

∴ The correct answer is 22.5 days.

Alternate Method

Given:

A is 3 times as efficient as B.

A completes the job in 60 days less than B.

Formula Used:

Total Work = Efficiency × Time

Calculation:

⇒ Let time taken by A be x days and by B be 3x days.

⇒ Difference in time = 3x − x = 2x = 60 days

⇒ x = 30 days (Time taken by A)

⇒ 3x = 90 days (Time taken by B)

⇒ Total efficiency of A and B = 3 + 1 = 4 units/day

⇒ Time taken by A and B together = Total Work ÷ Total Efficiency

⇒ Time taken = 90 ÷ 4 = 22.5 days

∴ The correct answer is 22.5 days.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to time. If efficiency ratio of A and B is x:y, then the time taken ratio will be y:x.

Work and 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.

Pipes & Cisterns

The concept of pipes filling or emptying a tank is identical to Time and Work. An inlet pipe has positive efficiency while an outlet pipe has negative efficiency.

Shortcut Trick

Distance covered = 80 × (4/5) = 64 km.

Time = 4 hrs 48 mins = 4 + (48 ÷ 60) = 24/5 hours.

Speed = Distance ÷ Time = 64 × (5/24) = 40/3 = 13.33 km/h.

∴ The correct answer is 13.33 km/h.

Alternate Method

Given:

Total Distance = 80 km

Time taken = 4 hours 48 minutes

Fraction of distance completed = 4/5

Formula Used:

Speed = Distance ÷ Time

Start Current End

64 km in 24/5 hrs

Total Distance = 80 km

Speed = ?

Calculations:

⇒ Distance covered = 80 × (4/5) = 64 km

⇒ Time taken in hours = 4 + (48 ÷ 60)

⇒ Time taken = 4 + 4/5 = 24/5 hours

⇒ Speed = 64 ÷ (24/5)

⇒ Speed = 64 × (5/24)

⇒ Speed = 40/3 = 13.33 km/h

∴ The correct answer is 13.33 km/h.

Additional Information

Unit Conversion

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 = Total Distance ÷ Total Time. If equal distances are covered at speeds x and y, Average Speed = 2xy ÷ (x + y).

Relative Speed

If two objects move in opposite directions, Relative Speed = S₁ + S₂. If moving in the same direction, Relative Speed = |S₁ − S₂|.

Shortcut Trick

From the given condition, O = 1.8M and O = 2.5N.

Investment ratio M : N : O = (1 ÷ 1.8) : (1 ÷ 2.5) : 1 = 25 : 18 : 45.

Let the investments be 25 units, 18 units, and 45 units.

Total Simple Interest = (25 × 10%) + (18 × 12%) + (45 × 15%) = 2.5 + 2.16 + 6.75 = 11.41 units.

Since 11.41 units = ₹2282, 1 unit = 2282 ÷ 11.41 = ₹200.

Amount in Scheme N = 18 units = 18 × 200 = ₹3600.

∴ The correct answer is ₹3600.

Alternate Method

Scheme M25x10%I = 2.5xScheme N18x12%I = 2.16xScheme O45x15%I = 6.75x++Total Interest = 11.41x

Given:

Rate of interest for Scheme M (R_M) = 10% p.a.

Rate of interest for Scheme N (R_N) = 12% p.a.

Rate of interest for Scheme O (R_O) = 15% p.a.

Total Interest accrued in 1 year = ₹2282

Formula Used:

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

Calculations:

Let the amount invested in Scheme O be O.

⇒ O = 180% of M = 1.8 M   ⇒   M = O ÷ 1.8 = 5O / 9

⇒ O = 250% of N = 2.5 N   ⇒   N = O ÷ 2.5 = 2O / 5

Ratio of investments M : N : O = 5/9 : 2/5 : 1

Multiplying by 45 (LCM of 9 and 5) to remove fractions:

⇒ M : N : O = 25 : 18 : 45

Let the investments be 25x, 18x, and 45x respectively.

Calculating Simple Interest for 1 year for each scheme:

⇒ I_M = (25x × 10 × 1) ÷ 100 = 2.5x

⇒ I_N = (18x × 12 × 1) ÷ 100 = 2.16x

⇒ I_O = (45x × 15 × 1) ÷ 100 = 6.75x

Total interest from all three schemes:

⇒ Total Interest = I_M + I_N + I_O

⇒ 2.5x + 2.16x + 6.75x = 2282

⇒ 11.41x = 2282

⇒ x = 2282 ÷ 11.41 = 200

Amount invested in Scheme N:

⇒ N = 18x = 18 × 200 = ₹3600

∴ The correct answer is ₹3600.

Additional Information

Simple Interest (SI)

Interest calculated uniformly only on the initial principal amount throughout the investment period. The formula is I = (P × R × T) ÷ 100, where P is principal, R is rate, and T is time.

Fraction and Ratio Simplification

If a relation is given as A = x% of B, convert it to A/B = x/100 to find the direct ratio. For multiple components, equate all parts using the LCM of their denominators to create a flat ratio without fractions.

Total Interest across Multiple Schemes

When a capital sum is divided into parts P₁, P₂, and P₃ invested at different rates, the total simple interest accrued is just the additive sum of individual interests: I = I₁ + I₂ + I₃.

Shortcut Trick When the angle of elevation changes from β to α over a distance d, the height H is found using: d = H × (cot β − cot α). Here d = 10 m, α = 60° and β = 30°: 10 = H × (cot 30° − cot 60°) 10 = H × (√3 − 1/√3) ⇒ 10 = H × (2/√3) H = 5√3 m. ∴ The correct answer is 5√3 m. Alternate Method Given: Distance between the two points, PQ = 10 m. Angle of elevation from point P, ∠APB = 60°. Angle of elevation from point Q, ∠AQB = 30°.

30°

60°

A

B

P

Q H

10 m

x

Formula Used: tan(θ) = Perpendicular ÷ Base Calculations: Let the height of the pole AB be H and the distance PB be x. In right-angled triangle APB: ⇒ tan(60°) = AB ÷ PB ⇒ √3 = H ÷ x ⇒ x = H ÷ √3     ... (Equation 1) In right-angled triangle AQB: ⇒ tan(30°) = AB ÷ QB ⇒ 1 ÷ √3 = H ÷ (PQ + PB) ⇒ 1 ÷ √3 = H ÷ (10 + x) ⇒ 10 + x = H√3 Substitute the value of x from Equation 1: ⇒ 10 + (H ÷ √3) = H√3 ⇒ 10 = H√3 − (H ÷ √3) ⇒ 10 = (3H − H) ÷ √3 ⇒ 10 = 2H ÷ √3 ⇒ 2H = 10√3 ⇒ H = 5√3 m ∴ The correct answer is 5√3 m. Additional Information Distance Formula in Heights & Distances When moving a distance d directly towards or away from a vertically standing object of height H, causing the angle of elevation to change between α and β (where α > β), the relation is given by d = H × (cot β − cot α). Angle of Elevation vs Angle of Depression The angle of elevation is the angle formed above the horizontal line of sight when looking up. The angle of depression is the angle formed below the horizontal line when looking down. Geometrically, they are equal for two observers looking at each other due to alternate interior angles. Key Trigonometric Values Most height and distance questions are solved using three standard tangent values: tan(30°) = 1/√3, tan(45°) = 1, and tan(60°) = √3.

Shortcut Trick

The radius (r) of a sphere inscribed in a right circular cone of base radius R, height H, and slant height L is given directly by the formula:

⇒ r = (R × H) ÷ (R + L)

⇒ L = √(R² + H²) = √(7² + 24²) = √625 = 25 cm

⇒ r = (7 × 24) ÷ (7 + 25) = 168 ÷ 32

⇒ r = 5.25 cm

∴ The correct answer is 5.25 cm.

Alternate Method

V A B O C P

r r R = 7 H = 24 L = 25

Given:

Radius of the cone (R) = 7 cm

Height of the cone (H) = 24 cm

Let the radius of the inscribed sphere be r.

Formula Used:

Slant height of cone (L) = √(R² + H²)

Using similar triangles ΔVPC ≈ ΔVOB:

⇒ CP ÷ VC = OB ÷ VB

Calculation:

⇒ L = √(7² + 24²) = √(49 + 576) = 25 cm

In ΔVOB, OB = 7 cm, VO = 24 cm, and VB = 25 cm.

In ΔVPC, CP = r and VC = (24 − r).

Equating the ratios from the similar triangles:

⇒ r ÷ (24 − r) = 7 ÷ 25

⇒ 25 × r = 7 × (24 − r)

⇒ 25r = 168 − 7r

⇒ 32r = 168

⇒ r = 168 ÷ 32 = 5.25 cm

∴ The correct answer is 5.25 cm.

Additional Information

Volume & Surface Area of Cone

Volume = ⅓ × π × R² × H | Curved Surface Area (CSA) = π × R × L.

Incircle of a Triangle

The radius of a circle inscribed inside a triangle (inradius) is calculated using the formula: r = Area ÷ Semi-perimeter.

Similar Triangles Property

When two triangles are similar, the ratio of their corresponding sides is always equal. This property frequently solves inscription problems in geometry.

Shortcut Trick

A regular octagon has 8 equal sides.

Side length = Total Perimeter ÷ Number of sides

Side length = 80 ÷ 8 = 10 cm

∴ The correct answer is 10 cm.

Alternate Method

Side = a Regular Octagon Perimeter = 8a

Given:

Perimeter of the regular octagon = 80 cm

Formula Used:

Perimeter of a regular polygon = Number of sides × Length of one side

For a regular octagon, Perimeter = 8 × a (where a is the side length)

Calculations:

⇒ 8 × a = 80

⇒ a = 80 ÷ 8

⇒ a = 10 cm

∴ The correct answer is 10 cm.

Additional Information

Sum of Interior Angles

The sum of interior angles of an n-sided polygon is given by (n − 2) × 180°. For an octagon, it is (8 − 2) × 180° = 1080°.

Interior and Exterior Angle

Each interior angle of a regular octagon is 1080° ÷ 8 = 135°. Each exterior angle is 360° ÷ 8 = 45°.

Number of Diagonals

The total number of diagonals in an n-sided polygon is n(n − 3) ÷ 2. For an octagon, it is 8(8 − 3) ÷ 2 = 20 diagonals.

Area of Regular Octagon

The area of a regular octagon with side a is 2a²(1 + √2).

Shortcut Trick

Total area of the wheel = Area of 1 sector × Number of sectors

⇒ Total area = 45 × 8 = 360 cm²

Area of a circle = πr² = 360

⇒ r² = 360 × (7 ÷ 22) = 114.54

⇒ r = √114.54 ≈ 10.7 cm

∴ The correct answer is 10.7 cm.

Alternate Method

45 cm² r Total Area = 8 × 45 = 360 cm²

Given:

Number of equal sectors in the wheel = 8

Area of one sector = 45 cm²

Formula Used:

Total Area of a circle = πr²

Total Area = Area of one sector × Total number of sectors

Calculations:

⇒ Total Area of the wheel = 45 × 8 = 360 cm²

⇒ πr² = 360

⇒ (22 ÷ 7) × r² = 360

⇒ r² = (360 × 7) ÷ 22

⇒ r² = 2520 ÷ 22 = 114.54

⇒ r = √114.54

⇒ r ≈ 10.7 cm

∴ The correct answer is 10.7 cm.

Additional Information

Area of a Sector

The area of a sector of a circle with radius r and central angle θ (in degrees) is given by Area = (θ ÷ 360°) × πr².

Length of an Arc

The length of an arc that subtends an angle θ at the center is given by L = (θ ÷ 360°) × 2πr.

Central Angle of Equal Sectors

When a circle is divided into 'n' equal sectors, the central angle of each sector formed is θ = 360° ÷ n.

Shortcut Trick

The Lateral Surface Area (LSA) of any regular prism is directly proportional to its base perimeter and height.

Base Perimeter = LSA ÷ Height

Base Perimeter = 240 ÷ 8 = 30 cm

∴ The correct answer is 30 cm.

Alternate Method

Given:

Lateral Surface Area (LSA) of the prism = 240 cm²

Height of the prism (h) = 8 cm

Height = 8 cm LSA = 240 cm² Base Perimeter = ?

Formula Used:

Lateral Surface Area of a Prism = Perimeter of the base × Height

Calculations:

⇒ LSA = Base Perimeter × Height

⇒ 240 = Base Perimeter × 8

⇒ Base Perimeter = 240 ÷ 8

⇒ Base Perimeter = 30 cm

∴ The correct answer is 30 cm.

Additional Information

Total Surface Area of a Prism

TSA = Lateral Surface Area + 2 × (Area of Base)

Volume of a Prism

Volume = Area of Base × Height

Prism vs Pyramid

A prism is a 3D shape with two identical and parallel bases connected by rectangular faces. A pyramid has only one base and its lateral faces are triangles meeting at a single top vertex.

Shortcut Trick

Since both expressions are equal to y, equate them directly:

2x + 1 = -x + 4

3x = 3 ⇒ x = 1

Substitute x = 1 back into the second equation:

y = -1 + 4 = 3

∴ The correct answer is (1, 3).

Alternate Method

Given:

Equation 1: y = 2x + 1

Equation 2: y = -x + 4

X Y O

y = 2x + 1

y = -x + 4

(1, 3)

1 3

Calculations:

To find the exact point of intersection visually represented above, we solve the system algebraically using the substitution method.

Substitute the value of y from Equation 2 into Equation 1:

⇒ -x + 4 = 2x + 1

Add x to both sides to group the variables:

⇒ 4 = 3x + 1

Subtract 1 from both sides:

⇒ 3 = 3x

⇒ x = 1

Now, substitute x = 1 into Equation 1 to find the corresponding y-coordinate:

⇒ y = 2(1) + 1

⇒ y = 2 + 1

⇒ y = 3

The system of equations shares a common solution at the coordinate pair (1, 3).

∴ The correct answer is (1, 3).

Additional Information

System of Linear Equations

A system of equations consists of multiple equations evaluated simultaneously. For a two-variable system, the solution represents the exact point where their lines cross on a coordinate plane.

Methods of Solving

Substitution: Isolate one variable and substitute its expression into the second equation.

Elimination: Add or subtract the equations to permanently cancel out one variable.

Types of Solutions

Unique Solution: The lines have different slopes and intersect exactly once (m₁ ≠ m₂).

No Solution: The lines are perfectly parallel and never intersect (m₁ = m₂, c₁ ≠ c₂).

Infinite Solutions: The lines overlap each other completely (m₁ = m₂, c₁ = c₂).

Shortcut Trick

Convert each surd into its simplest form by extracting perfect square factors.

√98 = 7√2 | √32 = 4√2 | √50 = 5√2

Expression = 7√2 − 4√2 + 5√2 = 8√2

∴ The correct answer is 8√2.

Alternate Method

Given:

The expression to simplify is √98 − √32 + √50

Method Used:

Prime factorisation method to simplify the square roots.

Calculations:

⇒ √98 = √(2 × 49) = √(2 × 7 × 7) = 7√2

⇒ √32 = √(2 × 16) = √(2 × 4 × 4) = 4√2

⇒ √50 = √(2 × 25) = √(2 × 5 × 5) = 5√2

Substituting these simplified values into the original expression:

⇒ 7√2 − 4√2 + 5√2

⇒ (7 − 4 + 5)√2

⇒ (3 + 5)√2

⇒ 8√2

∴ The correct answer is 8√2.

Additional Information

Properties of Surds

For any positive numbers a and b, √(a × b) = √a × √b. This property allows us to separate perfect squares from non-perfect squares under the root.

Addition and Subtraction of Surds

Only like surds (surds having the exact same number under the root) can be added or subtracted directly. For example, x√a + y√a = (x + y)√a.

BODMAS Rule

Always follow the order of operations. Roots (Orders) are simplified first, followed by addition and subtraction from left to right as they have the same priority level.

Shortcut Trick When two circles touch each other internally, they share exactly one point of contact. A common tangent is a straight line that touches both circles without crossing inside them. Since one circle is inside the other, any line touching the smaller circle elsewhere will cut through the larger circle. Thus, a tangent can only be drawn exactly at their single shared point of contact. ∴ The correct answer is 1. Alternate Method

C₁ C₂ P Common Tangent

Given: Two circles are touching each other internally. Concept Used: ⇒ If two circles touch internally, the distance between their centers is equal to the difference of their radii: C1C2 = R − r. ⇒ Because the smaller circle lies completely inside the larger circle, no transverse common tangents can be drawn across them. ⇒ The circles share only one common point, which is their point of contact (let's call it P). ⇒ A common tangent must pass through point P and be perpendicular to the straight line joining the centers C1 and C2. ⇒ Because there is only one point of contact, exactly one common tangent exists. ∴ The correct answer is 1. Additional Information Circles completely non-intersecting and outside each other Distance between centers > R + r. There are exactly 4 common tangents (2 direct and 2 transverse). Circles touching externally Distance between centers = R + r. There are exactly 3 common tangents (2 direct and 1 transverse at the point of contact). Circles intersecting at two points R − r < Distance between centers < R + r. There are exactly 2 common tangents (both are direct). One circle completely inside another Distance between centers < R − r. There are 0 common tangents.

Shortcut Trick

The perpendicular from the center bisects the chord, forming a right-angled triangle with the radius.

Half of the chord length = √(Radius² − Distance²)

Half chord = √(26² − 10²) = √(676 − 100) = √576 = 24 cm.

Total length of chord AB = 2 × 24 = 48 cm.

∴ The correct answer is 48 cm.

Alternate Method

Given:

Radius of the circle (OA) = 26 cm

Distance of the chord from the center (OM) = 10 cm

Formula Used:

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

By Pythagoras theorem: Hypotenuse² = Base² + Perpendicular²

O

A B

M 10 cm

26 cm

Right Angled Triangle OMA

Calculations:

⇒ In right-angled triangle OMA,

⇒ OA² = OM² + AM²

⇒ 26² = 10² + AM²

⇒ 676 = 100 + AM²

⇒ AM² = 576

⇒ AM = √576 = 24 cm

⇒ Since OM bisects the chord AB, Length of AB = 2 × AM

⇒ AB = 2 × 24 = 48 cm

∴ The correct answer is 48 cm.

Additional Information

Perpendicular from Center to Chord

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

Equal Chords Property

Chords that are equal in length are always equidistant from the center of the circle, which is why chord CD is identically 10 cm from the center.

Pythagorean Triplets

The values (10, 24, 26) form a standard Pythagorean triplet. It is easily derived by multiplying the base triplet (5, 12, 13) by 2.

Shortcut Trick

Area of a segment = r² × [(πθ ÷ 360°) − (sinθ ÷ 2)]

⇒ Area = 6² × [(π × 60° ÷ 360°) − (sin60° ÷ 2)]

⇒ Area = 36 × [(π ÷ 6) − (√3 ÷ 4)]

⇒ Area = (6π − 9√3) sq. cm

∴ The correct answer is (6π−9√3) sq. cm.

Alternate Method

 60° O A B 6 cm 6 cm Segment

Given:

Radius of the circle (r) = 6 cm

Central angle of the sector (θ) = 60°

Formula Used:

Area of sector = (θ ÷ 360°) × πr²

Area of equilateral triangle = (√3 ÷ 4) × side²

Calculations:

⇒ Area of sector OAB = (60° ÷ 360°) × π × 6²

⇒ Area of sector OAB = (1 ÷ 6) × 36π = 6π sq. cm

⇒ Since two sides (OA and OB) are equal radii and the included angle is 60°, the remaining angles are also 60°. Thus, ▵OAB is an equilateral triangle with side 6 cm.

⇒ Area of triangle OAB = (√3 ÷ 4) × 6²

⇒ Area of triangle OAB = (√3 ÷ 4) × 36 = 9√3 sq. cm

⇒ Area of corresponding segment = Area of sector − Area of triangle

⇒ Area of corresponding segment = (6π − 9√3) sq. cm

∴ The correct answer is (6π−9√3) sq. cm.

Additional Information

Area of a Sector

The area of a sector of a circle with radius r and central angle θ is directly given by the formula A = (θ ÷ 360°) × πr².

Length of an Arc

The length of an arc corresponding to a central angle θ can be calculated using the formula L = (θ ÷ 360°) × 2πr.

Area of Triangle in a Sector

The area of the triangle formed by two respective radii and their corresponding chord is obtained generally by A = (1 ÷ 2) × r² × sinθ.

Shortcut Trick

To convert a pure recurring decimal to a fraction, write the repeating digits over the same number of 9s.

Here, the repeating sequence is 12, which has two digits.

Fraction = 12 / 99

Simplifying by dividing both numerator and denominator by 3 gives 4 / 33.

∴ The correct answer is 4/33.

Alternate Method

Given:

Decimal value = 0.121212...

Method Used:

Let the repeating decimal be x. Multiply it by 10ⁿ (where n is the number of repeating digits) and subtract the original x to solve for the fraction.

Calculation:

⇒ Let x = 0.121212... — (Equation 1)

⇒ Multiply by 100 since two digits (12) are repeating.

⇒ 100x = 12.121212... — (Equation 2)

⇒ Subtracting Equation 1 from Equation 2:

⇒ 100x − x = 12.121212... − 0.121212...

⇒ 99x = 12

⇒ x = 12 / 99

⇒ Dividing numerator and denominator by 3:

⇒ x = 4 / 33

∴ The correct answer is 4/33.

Additional Information

Pure Recurring Decimal

A decimal in which all digits after the decimal point repeat continuously (e.g., 0.333...). It is converted by placing the repeating part over an equal number of 9s.

Mixed Recurring Decimal

A decimal where some digits after the decimal point do not repeat, followed by a repeating sequence (e.g., 0.12333...).

Converting Mixed Recurring Decimal

Fraction = (Total number formed after decimal − Non-repeating part) ÷ (Number of 9s equal to repeating digits followed by 0s equal to non-repeating digits).

Calculation:

(3.2 - 1$\frac35$) + (2$\frac14$ ÷ 0.5)

⇒ (3.2 - $8\over5$) + ($9\over4$ ÷ 0.5)

⇒ (3.2 - 1.6) + (2.25 ÷ 0.5)

⇒ 1.6 + 4.5

⇒ 6.1

Hence the correct answer is option (2).

Shortcut Trick

Marketing department's share = ₹36,000 × [5 ÷ (4 + 5 + 3)] = ₹36,000 × (5/12) = ₹15,000

Remaining amount after deductions = ₹15,000 − ₹3,000 − ₹1,500 = ₹10,500

Ratio of top performers = 5 : 4, so the smaller share gets 4 parts out of 9.

Smaller share = ₹10,500 × (4/9) = ₹4,666.67

∴ The correct answer is ₹4,666.67.

Alternate Method

Total: 36k 5/12 Mktg: 15k - 4.5k Rem: 10.5k 5 parts 4 parts Smaller Share = 4/9 × 10500 = 4666.67

Given:

Total bonus pool = ₹36,000

Bonus ratio for Sales, Marketing, and Operations = 4 : 5 : 3

Marketing deductions = ₹3,000 (team building) and ₹1,500 (intern)

Top performers' allocation ratio = 5 : 4

Calculations:

⇒ Total initial ratio parts = 4 + 5 + 3 = 12 parts

⇒ Marketing department's share = (5 ÷ 12) × 36,000 = ₹15,000

⇒ Total fixed deductions = 3,000 + 1,500 = ₹4,500

⇒ Remaining amount for top performers = 15,000 − 4,500 = ₹10,500

⇒ The remaining amount is split into 5 + 4 = 9 parts.

⇒ The smaller share corresponds to the 4 parts ratio.

⇒ Value of the smaller share = (4 ÷ 9) × 10,500

⇒ Value = 42,000 ÷ 9 = ₹4,666.66...

∴ The correct answer is ₹4,666.67.

Additional Information

Ratio Distribution Rule

When a quantity Q is divided in the ratio a : b : c, the share of the first part is calculated directly as [a ÷ (a+b+c)] × Q.

Sequential Allocations

When solving hierarchical distribution problems, calculate the pool for each subsequent stage by subtracting any fixed amounts before applying the new ratio constraints.

Proportions in Partnerships

If profits, revenues, or bonuses are distributed among groups, the respective shares behave similarly to partnership accounts, where total allocation strictly bounds internal distributions.

Shortcut Trick

For continued fractions, always calculate from the bottommost nested denominator upwards.

Bottommost expression = 2 + 1/2 = 5/2

Middle expression = 2 + 1 / (5/2) = 2 + 2/5 = 12/5

Final top expression = 1 + 1 / (12/5) = 1 + 5/12 = 17/12

∴ The correct answer is 17/12.

Alternate Method

Given:

x = 1 + 1 / (2 + 1 / (2 + 1/2))

Calculations:

⇒ First, we simplify the innermost fraction at the bottom: 2 + 1/2

⇒ 2 + 1/2 = (4 + 1) / 2 = 5/2

⇒ Substitute this back into the expression:

⇒ x = 1 + 1 / (2 + 1 / (5/2))

⇒ Taking the reciprocal: 1 / (5/2) = 2/5

⇒ x = 1 + 1 / (2 + 2/5)

⇒ Now, simplify the new bottom fraction: 2 + 2/5

⇒ 2 + 2/5 = (10 + 2) / 5 = 12/5

⇒ Substitute this back into the expression:

⇒ x = 1 + 1 / (12/5)

⇒ Taking the reciprocal: 1 / (12/5) = 5/12

⇒ x = 1 + 5/12

⇒ x = (12 + 5) / 12 = 17/12

∴ The correct answer is 17/12.

Additional Information

Continued Fractions

A continued fraction is evaluated by working from the bottommost nested fraction upwards to the main fraction.

Reciprocal Rule

The reciprocal of a fraction a/b is b/a. This property is used repeatedly when resolving 1 ÷ (a/b).

BODMAS Rule

For nested expressions, calculations must always resolve the deepest levels first according to the standard order of operations.

Shortcut Trick

Profit ratio is directly proportional to Equivalent Capital (Capital × Time).

Kapil's equivalent capital = (10 × 6) + (6 × 6) = 60 + 36 = 96 units

Manish's equivalent capital = 8 × 9 = 72 units

Profit Ratio = 96 : 72 = 4 : 3

∴ The correct answer is 4:3.

Alternate Method

Given:

Kapil's initial investment = ₹1,00,000

Manish's investment after 3 months = ₹80,000

Kapil withdrew ₹40,000 after 3 more months (i.e., at the end of 6 months).

Formula Used:

Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂)

₹1,00,000 × 6m

₹60,000 × 6m

Kapil

₹80,000 × 9m

Manish

0m 3m 6m 12m

Profit Ratio = (100k × 6 + 60k × 6) : (80k × 9) = 960 : 720 = 4:3

Calculations:

⇒ Kapil invested ₹1,00,000 for the first 6 months.

⇒ After 6 months, his capital becomes = ₹1,00,000 − ₹40,000 = ₹60,000.

⇒ Kapil invested ₹60,000 for the remaining 6 months of the year.

⇒ Kapil's equivalent capital = (1,00,000 × 6) + (60,000 × 6)

⇒ Kapil's equivalent capital = 6,00,000 + 3,60,000 = 9,60,000

⇒ Manish joined after 3 months, so his money was invested for (12 − 3) = 9 months.

⇒ Manish's equivalent capital = 80,000 × 9 = 7,20,000

⇒ Profit Ratio = Kapil's equivalent capital : Manish's equivalent capital

⇒ Profit Ratio = 9,60,000 : 7,20,000

⇒ Profit Ratio = 96 : 72 = 4 : 3

∴ The correct answer is 4:3.

Additional Information

Compound Partnership

When partners invest for different time periods or change their capital mid-way, profit is distributed based on the product of Capital and Time (Equivalent Capital).

Simple Partnership

When all partners invest their capital for the exact same time period, the profit is divided purely in the ratio of their invested capitals.

Working vs Sleeping Partner

A working partner actively manages the business and usually receives a fixed salary or commission from the profit before the remaining profit is divided among all partners based on investment.

Shortcut Trick

Profit ratio of partners = Ratio of their (Capital × Time).

A's Profit : B's Profit = (60000 × 10) : (90000 × 8) = 600000 : 720000 = 5 : 6

Total profit ratio parts = 5 + 6 = 11 units, which corresponds to the total profit of ₹40,000.

A's share = (5 ÷ 11) × 40000 = 200000 ÷ 11 ≈ ₹18,181.82

∴ The correct answer is ₹18,181.82.

Alternate Method

Given:

A's investment = ₹60,000 | Time period = 10 months

B's investment = ₹90,000 | Time period = 8 months

Total Profit = ₹40,000

Formula Used:

Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂)

Partner A₹60,000× 10 months= 600,000Partner B₹90,000× 8 months= 720,000Profit Ratio → 600k : 720k = 5 : 6Total Profit = ₹40,000

Calculations:

⇒ Profit Ratio of A to B = (60000 × 10) : (90000 × 8)

⇒ Profit Ratio = 600000 : 720000

⇒ Simplifying the ratio by dividing by 120000, we get 5 : 6

⇒ Total parts in the profit ratio = 5 + 6 = 11

⇒ A's Share = (A's ratio ÷ Total ratio) × Total Profit

⇒ A's Share = (5 ÷ 11) × 40000

⇒ A's Share = 200000 ÷ 11 ≈ 18181.8181...

∴ The correct answer is ₹18,181.82.

Additional Information

Compound Partnership

When partners invest different amounts for different time periods, the profit ratio is calculated as the product of their investment and time: (Capital₁ × Time₁) : (Capital₂ × Time₂).

Simple Partnership

If all partners invest their capital for the exact same period of time, their profit is shared precisely in the ratio of their initial investments.

Active vs Sleeping Partner

An active partner manages the business and usually receives a fixed salary or a certain percentage of the total profit before the remaining profit is distributed based on the normal capital ratio. A sleeping partner only invests the capital.

Finding the Time Ratio

If the profit ratio and the investment capital ratio are both known, the time ratio can be directly found using: Time Ratio = Profit Ratio ÷ Capital Ratio.