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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).