Search Previous Year Questions

Shortcut Trick

A working for 16 days and B working for 44 days is equivalent to A and B working together for 16 days, and B working alone for 28 days (44 − 16).

Work done by A and B together in 16 days = 16 ÷ 30 = 8/15 of total work.

Remaining work = 1 − 8/15 = 7/15, which B completes alone in 28 days.

Total time taken by B to complete the entire task = 28 × (15 ÷ 7) = 60 days.

∴ The correct answer is 60 days.

Alternate Method

Given:

Time taken by A and B together = 30 days

A works for 16 days and B finishes the remaining work in 44 days.

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ Let Total Work = 30 units (LCM of 30).

⇒ Efficiency of (A + B) = 30 ÷ 30 = 1 unit/day.

⇒ Total work done = (A's work for 16 days) + (B's work for 44 days) = 30 units.

⇒ Grouping the work: (A + B) working for 16 days + B working alone for (44 − 16) days = 30 units.

⇒ 16 × (Efficiency of A + B) + 28 × (Efficiency of B) = 30

⇒ 16 × 1 + 28 × Efficiency of B = 30

⇒ 28 × Efficiency of B = 30 − 16

⇒ 28 × Efficiency of B = 14

⇒ Efficiency of B = 14 ÷ 28 = 0.5 units/day.

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

⇒ Time = 30 ÷ 0.5 = 60 days.

∴ The correct answer is 60 days.

Additional Information

Pipes & Cisterns

Follows the exact same principles as Time & Work. Inlet pipes have positive efficiency as they fill the tank, while outlet pipes have negative efficiency as they empty the tank.

Work & Wages

Wages are always distributed in direct proportion to the actual amount of work done by each person. If people work for the exact same number of days, wages can be shared directly in the ratio of their efficiencies.

Alternate Day Work

When individuals work on alternate days, calculate the total work done in one complete cycle (e.g., 2 days) and multiply it to get close to the total work, then find the remaining work and assign it to the next person in sequence.

Shortcut Trick

Let the initial mixture be 16x. The initial parts are Orange = 5x, Apple = 7x.

Removing 24 litres takes out (5 / 16) × 24 = 7.5L Orange and (7 / 16) × 24 = 10.5L Apple.

Resultant Apple − Resultant Orange = (7x − 10.5) − (5x − 7.5 + 10) = 14.

⇒ 2x − 13 = 14 ⇒ 2x = 27.

Initial quantity = 16x = 27 × 8 = 216.

∴ The correct answer is 216 litres.

Alternate Method

Given:

Ratio of Orange : Apple : Guava = 5 : 7 : 4

Quantity of mixture taken out = 24 litres

Quantity of added juices = 10 litres Orange, 6 litres Guava

Resultant Apple juice − Resultant Orange juice = 14 litres

Initial Mix 16x Litres O : A : G 5 : 7 : 4

− 24L

Remaining 16x − 24 Ratio intact 5 : 7 : 4

• 10L Org
• 6L Guv

Final Mix Apple − Org = 14 Litres

Calculations:

⇒ Total ratio parts initially = 5 + 7 + 4 = 16 parts

⇒ Let the initial quantity of mixture be 16x litres.

⇒ The initial quantities are Orange = 5x, Apple = 7x, and Guava = 4x.

⇒ Quantity of Orange juice removed = (5 / 16) × 24 = 7.5 litres

⇒ Quantity of Apple juice removed = (7 / 16) × 24 = 10.5 litres

After taking out 24 litres and adding the respective juices, the new volumes are:

⇒ Resultant Orange juice = 5x − 7.5 + 10 = 5x + 2.5

⇒ Resultant Apple juice = 7x − 10.5

According to the given condition, the difference is 14 litres:

⇒ (7x − 10.5) − (5x + 2.5) = 14

⇒ 7x − 10.5 − 5x − 2.5 = 14

⇒ 2x − 13 = 14

⇒ 2x = 27

⇒ x = 13.5

⇒ Initial quantity of mixture = 16x = 16 × 13.5 = 216 litres.

∴ The correct answer is 216 litres.

Additional Information

Constant Ratio Property

When any volume of a mixture is drawn out, the remaining mixture still contains the original ingredients in the exact same ratio. Only adding or removing an individual component selectively will alter this final ratio.

Replacement Method

When a certain volume of mixture is replaced completely by one of its pure ingredients, the remaining quantity of the pure liquid left can be found using the formula: Final = Initial × (1 − Removed/Total)ⁿ.

Rule of Alligation

Used to find the required ratio when two ingredients at different prices or concentrations are mixed. Ratio = (Dearer − Mean) : (Mean − Cheaper).

Shortcut Trick

For a constant distance, Speed is inversely proportional to Time.

Initial time = 1 hr 45 min = 105 min.

New time = 105 + 15 = 120 min.

Time ratio = T₁ : T₂ = 105 : 120 = 7 : 8

∴ Speed ratio = S₁ : S₂ = 8 : 7

Decrease in speed = (8 − 7) ÷ 8 = 1/8 = 0.125 (or 12.5%)

∴ The correct answer is 12.5% (0.125).

Alternate Method

Given:

⇒ Initial speed (S₁) = 60 km/h

⇒ Initial time (T₁) = 1 hour 45 minutes = 105 minutes = 1.75 hours

⇒ New time (T₂) = 105 minutes + 15 minutes = 120 minutes = 2 hours

Distance = 105 km (Constant) Case 1: S = 60 km/h, T = 105 min

Case 2: S = 52.5 km/h, T = 120 min

Speed Decrease = 7.5 km/h Percentage Decrease = 12.5% (0.125)

Calculations:

⇒ Distance covered = S₁ × T₁ = 60 × 1.75 = 105 km

⇒ New speed (S₂) = Distance ÷ T₂ = 105 ÷ 2 = 52.5 km/h

⇒ Decrease in speed = S₁ − S₂ = 60 − 52.5 = 7.5 km/h

⇒ Fraction decrease = Decrease ÷ Original Speed

⇒ Fraction decrease = 7.5 ÷ 60 = 1/8 = 0.125

⇒ Percentage decrease = 0.125 × 100% = 12.5%

∴ The correct answer is 12.5% (0.125).

Additional Information

Inverse Proportionality of Speed and Time

When distance is kept constant, speed is inversely proportional to the time taken (S ∝ 1/T). If the time increases by a ratio of a : b, the required speed decreases by the ratio of b : a.

Average Speed

Average Speed = Total Distance ÷ Total Time. It must be noted that it is strictly calculated using this formula and is not merely the simple arithmetic average of individual speeds unless the travel times are identical.

Percentage Change

Percentage Change = (New Value − Initial Value) ÷ (Initial Value) × 100. A negative end result simply indicates a percentage decrease.

Shortcut Trick

Speed ratio = 4 : 6 = 2 : 3, so Time ratio = 3 : 2

Difference in time parts = 3 − 2 = 1 part

1 part = 15 min (late) + 5 min (early) = 20 mins = 1/3 hours

Time taken at 4 km/h (3 parts) = 3 × 1/3 = 1 hour

Distance = Speed × Time = 4 km/h × 1 hour = 4 km

∴ The correct answer is 4 km.

Alternate Method

Given:

Speed 1 (S₁) = 4 km/h (reaches 15 minutes late)

Speed 2 (S₂) = 6 km/h (reaches 5 minutes early)

Home

School

Distance = D km

Speed = 4 km/h | Time = t + 15 min

Speed = 6 km/h | Time = t - 5 min

Formula Used:

Time = Distance ÷ Speed

Calculations:

⇒ Let the distance from home to school be D km.

⇒ Time taken to cover D km at 4 km/h = D / 4 hours

⇒ Time taken to cover D km at 6 km/h = D / 6 hours

⇒ Time difference = 15 minutes (late) − (−5 minutes early) = 20 minutes

⇒ Convert time difference into hours = 20 / 60 = 1 / 3 hours

⇒ D / 4 − D / 6 = 1 / 3

⇒ (3D − 2D) / 12 = 1 / 3

⇒ D / 12 = 1 / 3

⇒ D = 12 / 3

⇒ D = 4 km

∴ The correct answer is 4 km.

Additional Information

Direct Formula for Early/Late Problems

Distance = [ (S₁ × S₂) ÷ (S₁ − S₂) ] × Total Time Difference in hours.

Average Speed

When a certain distance is covered at speed x and the same distance is covered at speed y, Average Speed = (2xy) ÷ (x + y).

Relative Speed

If two bodies move in opposite directions at speeds S₁ and S₂, Relative Speed = S₁ + S₂. In the same direction, Relative Speed = |S₁ − S₂|.

Shortcut Trick

Circumference of the ground = Total Cost ÷ Rate per meter

Circumference = 3768 ÷ 120 = 31.4 m

Since Circumference = 2 × π × r

2 × 3.14 × r = 31.4 ⇒ 6.28 × r = 31.4 ⇒ r = 5 m

∴ The correct answer is 5 m.

Alternate Method

Given:

Total cost of fencing = ₹3,768

Rate of fencing = ₹120/m

Formula Used:

Total Cost = Circumference × Rate

Circumference of a circle = 2πr (where r is the radius)

O

r = ? Circumference = Total Cost ÷ Rate Perimeter = 31.4 m

Total Cost ₹3768

Rate ₹120/m

Calculation:

⇒ Circumference = Total Cost ÷ Rate

⇒ Circumference = 3768 ÷ 120 = 31.4 m

⇒ 2 × π × r = 31.4

Taking π ≈ 3.14:

⇒ 2 × 3.14 × r = 31.4

⇒ 6.28 × r = 31.4

⇒ r = 31.4 ÷ 6.28

⇒ r = 5 m

∴ The correct answer is 5 m.

Additional Information

Area of a Circle

The area enclosed by a circle is given by the formula πr². If a question asks for the cost of levelling or plowing a circular field, the area must be calculated instead of the perimeter.

Perimeter of a Semi-Circle

The perimeter of a semi-circle includes both the curved arc and the straight diameter. The direct formula is πr + 2r.

Difference Between Fencing and Paving

Fencing always relates to the outer boundary (circumference of 2πr), whereas paving, painting, or laying grass relates to the inner flat surface (area of πr²).

Shortcut Trick

The unused area when a circle is perfectly inscribed in a square is directly given by the formula: r²(4 − π)

Substitute the given radius r = 14 cm, and use a precise value for π ≈ 3.1416:

Unused Area = 14² × (4 − 3.1416)

Unused Area = 196 × 0.8584 ≈ 168.25 cm²

∴ The correct answer is 168.25 cm².

Alternate Method

Given:

Radius of the circular logo (r) = 14 cm

Formula Used:

Side of the circumscribing square (a) = 2 × r

Area of square = a²

Area of circle = πr²

Unused square space = Area of square − Area of circle

r = 14 cm

Side = 28 cm Unused space (shaded blue)

Calculations:

⇒ Side of the square (a) = 2 × 14 = 28 cm

⇒ Area of the square = 28 × 28 = 784 cm²

⇒ Area of the circular logo = 3.1416 × 14 × 14 ≈ 615.75 cm²

⇒ Unused square space = 784 − 615.75 = 168.25 cm²

∴ The correct answer is 168.25 cm².

Additional Information

Area of Circumscribed Square

When a circle of radius r perfectly fits (is inscribed) inside a square, the side of the square is exactly the diameter of the circle (2r). The area of the square evaluates to 4r².

Area of Inscribed Square

Conversely, if a square is inscribed inside a circle of radius r, its diagonal matches the circle's diameter (2r), and its area calculates to 2r².

Inscribed Area Ratio

The ratio of the area of an inscribed circle to the area of its circumscribing square is π : 4. This implies the circle covers approximately 78.5% of the total square space.

Shortcut Trick

The minute hand covers a complete circle (360°) in 60 minutes.

In 30 minutes, it covers exactly half of the circular path.

Distance covered in 30 minutes = Half of circumference = π × r

Distance = (22/7) × 14 = 44 cm

∴ The correct answer is 44 cm.

Alternate Method

r = 14 cm 30 min (180°) 12 6

Given:

Radius of the clock face (r) = 14 cm

Time duration = 30 minutes

Formula Used:

Arc length = (θ ÷ 360°) × 2πr

Angle covered by minute hand in 1 minute = 360° ÷ 60 = 6°

Calculations:

⇒ Angle covered in 30 minutes (θ) = 30 × 6° = 180°

⇒ Arc length = (180° ÷ 360°) × 2 × (22/7) × 14

⇒ Arc length = (1/2) × 2 × 22 × 2

⇒ Arc length = 1 × 44 = 44 cm

∴ The correct answer is 44 cm.

Additional Information

Area of a Sector

The region bounded by two radii and an arc is called a sector. Formula: Area = (θ ÷ 360°) × πr².

Speed of Clock Hands

The minute hand moves at a speed of 6° per minute, while the hour hand moves at 0.5° per minute.

Circumference of a Circle

The total boundary length of a circular path is its circumference. Formula: C = 2πr or C = πd.

Shortcut Trick

The given line is in the form y = mx + c, giving slope m = -3.

For perpendicular lines, the product of their slopes is always -1.

So, perpendicular slope = -1 ÷ (-3) = 1/3.

∴ The correct answer is 1/3.

Alternate Method

Given:

Equation of the line: y = -3x + 7

XYy = -3x + 7Slope (m1) = -3Perpendicular LineSlope (m2) = ?90°

Formula Used:

Slope-intercept form of a line: y = mx + c, where m is the slope.

Condition for two perpendicular lines: m₁ × m₂ = -1

Calculations:

⇒ The given line is y = -3x + 7

⇒ Comparing with y = mx + c, we get the slope of the line, m₁ = -3

⇒ Let the slope of the perpendicular line be m₂

⇒ According to the condition for perpendicular lines, m₁ × m₂ = -1

⇒ -3 × m₂ = -1

⇒ m₂ = 1/3

∴ The correct answer is 1/3.

Additional Information

Parallel Lines

Lines which are parallel to each other have equal slopes (m₁ = m₂).

General Equation of a Line

For a line given by Ax + By + C = 0, the slope is m = -A/B.

Angle Between Two Intersecting Lines

If m₁ and m₂ are the slopes, the acute angle θ between them is given by tan(θ) = |(m₁ − m₂) ÷ (1 + m₁ × m₂)|.

Shortcut Trick

If x − 1/x = a, then x² + 1/x² = a² + 2

If x² + 1/x² = b, then x⁴ + 1/x⁴ = b² − 2

Here, a = 3 ⇒ b = 3² + 2 = 9 + 2 = 11

Now, x⁴ + 1/x⁴ = 11² − 2 = 121 − 2 = 119

∴ The correct answer is 119.

Alternate Method

Given:

x − 1/x = 3

Formula Used:

(a − b)² = a² − 2ab + b²

(a + b)² = a² + 2ab + b²

Calculations:

Squaring both sides of the given equation:

⇒ (x − 1/x)² = 3²

⇒ x² − 2 × x × 1/x + 1/x² = 9

⇒ x² − 2 + 1/x² = 9

⇒ x² + 1/x² = 9 + 2 = 11

Squaring both sides again:

⇒ (x² + 1/x²)² = 11²

⇒ x⁴ + 2 × x² × 1/x² + 1/x⁴ = 121

⇒ x⁴ + 2 + 1/x⁴ = 121

⇒ x⁴ + 1/x⁴ = 121 − 2 = 119

∴ The correct answer is 119.

Additional Information

Square Identity 1

If x + 1/x = a, then x² + 1/x² = a² − 2

Cube Identity 1

If x + 1/x = a, then x³ + 1/x³ = a³ − 3a

Cube Identity 2

If x − 1/x = a, then x³ − 1/x³ = a³ + 3a

Power 5 Identity

x⁵ + 1/x⁵ = (x² + 1/x²) × (x³ + 1/x³) − (x + 1/x)

Shortcut Trick

Convert the percentage to a fraction to simplify the calculation directly.

2.5% = 2.5 ÷ 100 = 1 ÷ 40

Multiply the fraction by the given number:

Value = (1 ÷ 40) × 200 = 5

∴ The correct answer is 5.

Alternate Method

Given:

Base number = 200

Percentage = 2.5%

Formula Used:

Percentage Value = (Percentage ÷ 100) × Base Number

5 195

Total = 200 (100%)

2.5%

Calculations:

⇒ Value = (2.5 ÷ 100) × 200

⇒ Value = 2.5 × 2

⇒ Value = 5

∴ The correct answer is 5.

Additional Information

Percentage to Fraction Conversion

To convert any percentage to a fraction, divide it by 100 and simplify. Example: 2.5% = 2.5 ÷ 100 = 25 ÷ 1000 = 1 ÷ 40.

Successive Percentage

When a value changes by x% and then by y%, the net percentage change is given by the formula: Net Change = x + y + (x × y) ÷ 100.

Percentage Change

To find the percentage change between two values, use the formula: % Change = (New Value − Old Value) ÷ Old Value × 100.