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Shortcut Trick

Let the speed of Train A be S_A. Length L_A = S_A × 24.

Speed of Train B = 54 × (5/18) = 15 m/s.

When crossing Train B in opposite direction: (L_A + 150) = (S_A + 15) × 20.

Substitute L_A: 24S_A + 150 = 20S_A + 300 ⇒ 4S_A = 150.

S_A = 37.5 m/s = 37.5 × (18/5) = 135 km/h.

L_A = 37.5 × 24 = 900 m.

∴ The correct answer is 900 m and 135 km/h.

Alternate Method

Given:

Time to pass fixed pole = 24 sec

Time to cross Train B = 20 sec

Length of Train B = 150 m

Speed of Train B = 54 km/h

Formula Used:

Distance = Speed × Time

Relative Speed (Opposite) = S₁ + S₂

Calculations:

⇒ Let Speed of Train A = v m/s and Length of Train A = L m.

⇒ From first condition: L = v × 24 → (i)

⇒ Speed of Train B in m/s = 54 × (5/18) = 15 m/s.

⇒ From second condition (crossing in opposite direction):

⇒ Total Distance = L + 150

⇒ Relative Speed = v + 15

⇒ (L + 150) = (v + 15) × 20 → (ii)

⇒ Substituting (i) in (ii): 24v + 150 = 20v + 300

⇒ 4v = 150 ⇒ v = 37.5 m/s

⇒ Speed of Train A in km/h = 37.5 × (18/5) = 135 km/h.

⇒ Length of Train A = 37.5 × 24 = 900 m.

∴ The correct answer is 900 m and 135 km/h.

Additional Information

Relative Speed Concept

For objects moving in the same direction, Relative Speed = S₁ − S₂. For opposite directions, Relative Speed = S₁ + S₂.

Unit Conversion

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

Train Crossing a Pole vs Train

When a train crosses a pole/man, distance = length of train. When it crosses another train/platform, distance = sum of both lengths.

Shortcut Trick

In a trapezium (where AB || DC), the length of the segment EF joining the midpoints of the diagonals is half the difference of the parallel sides.

Length of EF = 1/2 × |AB − DC|

⇒ EF = 1/2 × |21 − 11|

⇒ EF = 1/2 × 10 = 5 cm

∴ The correct answer is 5 cm.

Alternate Method

Given: AB = 21 cm, DC = 11 cm (Parallel sides), E and F are midpoints of AC and BD.

Formula Used: Length of segment joining midpoints of diagonals = (1/2) × (Difference of parallel sides).

Calculations:

⇒ Parallel side 1 (AB) = 21 cm

⇒ Parallel side 2 (DC) = 11 cm

⇒ Difference of parallel sides = 21 − 11 = 10 cm

⇒ Length of segment EF = (1/2) × 10

⇒ EF = 5 cm

∴ The correct answer is 5 cm.

Additional Information

Median of a Trapezium

The segment joining the midpoints of the non-parallel sides is called the median, and its length is the average of the parallel sides: (AB + DC) ÷ 2.

Midpoint Segment Property

In any quadrilateral, the segment joining the midpoints of the diagonals lies on the median segment joining the midpoints of the non-parallel sides.

Isosceles Trapezium

In an isosceles trapezium, the non-parallel sides (AD and BC) are equal, and the diagonals (AC and BD) are also equal in length.

Shortcut Trick

Profit of 20% means Selling Price is 120% of Cost Price (CP).

Given 120% = ₹1,200 ⇒ 1% = ₹10.

For a Loss of 10%, the required Selling Price is (100 − 10)% = 90%.

Required Price = 90% × 10 = ₹900.

∴ The correct answer is ₹900.

Alternate Method

Given:

Initial Selling Price (SP₁) = ₹1,200

Initial Profit% = 20%

Required Loss% = 10%

Formula Used:

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

New Selling Price (SP₂) = CP × [(100 − Loss%) ÷ 100]

Calculations:

⇒ Cost Price (CP) = [1,200 ÷ (100 + 20)] × 100

⇒ CP = [1,200 ÷ 120] × 100

⇒ CP = 10 × 100 = ₹1,000

⇒ New Selling Price (SP₂) = 1,000 × [(100 − 10) ÷ 100]

⇒ SP₂ = 1,000 × (90 ÷ 100)

⇒ SP₂ = 1,000 × 0.9 = ₹900

∴ The correct answer is ₹900.

Additional Information

Profit and Loss Percentage

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

Marked Price and Discount

Marked Price (MP) is the label price. Discount is calculated on MP: Selling Price (SP) = MP − Discount.

Successive Profit/Loss

If an article is sold at a% profit and then b% profit, the Effective Profit % = [a + b + (a × b)/100]%.

Shortcut Trick

The filling rate of the second tap (19 hr) and the emptying rate of the third tap (19 hr) are identical.

When opened together, they neutralize each other's effect completely.

Net filling depends only on the first tap which fills the full cistern in 4 hours.

Time to fill one-fourth of the cistern = 4 hr × (1/4) = 1 hr.

∴ The correct answer is 1 hr.

Alternate Method

Given:

Time taken by Tap A (filling) = 4 hr

Time taken by Tap B (filling) = 19 hr

Time taken by Tap C (emptying) = 19 hr

Formula Used:

Efficiency = Total Work ÷ Time Taken

Time = Required Work ÷ Combined Efficiency

Calculations:

⇒ Total Efficiency of (A + B + C) = 19 + 4 − 4 = 19 units/hr

⇒ Required Work = 1/4 of total capacity = 76 ÷ 4 = 19 units

⇒ Time Taken = Required Work ÷ Total Efficiency

⇒ Time = 19 ÷ 19 = 1 hr

∴ The correct answer is 1 hr.

Additional Information

Inlet and Outlet Pipes

Inlet pipes fill the tank (positive efficiency), while outlet/leakage pipes empty the tank (negative efficiency).

Net Efficiency

If multiple pipes are open, the net efficiency is the algebraic sum of their individual efficiencies: E_net = E₁ + E₂ − E₃.

Fractional Work

The time taken to fill 1/n part of the tank is simply the time taken to fill the full tank divided by n.

Shortcut Trick

To find the fourth proportional (x) of three numbers a, b, and c, use the direct formula: x = (b × c) ÷ a.

Here, a = 37, b = 25, and c = 888.

x = (25 × 888) ÷ 37

x = 25 × 24 = 600.

∴ The correct answer is 600.

Alternate Method

Given: The numbers 37, 25, and 888 are the first three terms of a proportion, and x is the fourth proportional.

Formula Used: If four numbers a, b, c, d are in proportion (a : b :: c : d), then the product of extremes equals the product of means (a × d = b × c).

⇒ 37 ÷ 25 = 888 ÷ x

⇒ 37 × x = 25 × 888

⇒ x = (25 × 888) ÷ 37

⇒ x = 22200 ÷ 37

⇒ x = 600

∴ The correct answer is 600.

Additional Information

Fourth Proportional

For four non-zero quantities a, b, c, d, if a : b = c : d, then d is the fourth proportional to a, b, c.

Third Proportional

If a : b = b : c, then c is the third proportional to a and b, calculated as c = b² ÷ a.

Mean Proportional

The mean proportional (or geometric mean) between a and b is the value x such that a : x = x : b, which is x = √(a × b).

Shortcut Trick

Actual Increase = ₹220 − ₹200 = ₹20

Percentage Increase = (Increase ÷ Original Price) × 100

Percentage Increase = (20 ÷ 200) × 100 = 10%

∴ The correct answer is 10%.

Alternate Method

Given:

Initial Price of the book = ₹200

Final Price of the book = ₹220

Formula Used:

Percentage Increase = [(Final Value − Initial Value) ÷ Initial Value] × 100

Calculations:

⇒ Price Increase = ₹220 − ₹200 = ₹20

⇒ Percentage Increase = (20 ÷ 200) × 100

⇒ Percentage Increase = (1 ÷ 10) × 100

⇒ Percentage Increase = 10%

∴ The correct answer is 10%.

Additional Information

Percentage Change Formula

The formula for percentage change is [(New Value − Old Value) ÷ Old Value] × 100. If the result is positive, it is an increase; if negative, it is a decrease.

Fraction to Percentage

To convert any fraction into a percentage, multiply it by 100 and add the % symbol. For example, 1/5 × 100 = 20%.

Percentage Base Change

If a value increases from A to B, the base is A. If it decreases from B back to A, the base for calculation becomes B.

Shortcut Trick

Divisibility rule of 9 states that the sum of the digits must be divisible by 9.

Sum of digits of 5135n2 = 5 + 1 + 3 + 5 + n + 2 = 16 + n

The next higher multiple of 9 after 16 is 18.

⇒ 16 + n = 18 ⇒ n = 2

∴ The correct answer is 2.

Alternate Method

Given:

The number 5135n2 is completely divisible by 9.

Formula/Property Used:

A number is divisible by 9 if and only if the sum of its digits is a multiple of 9.

Calculation:

⇒ Sum of digits of the number = 5 + 1 + 3 + 5 + n + 2

⇒ Sum of digits = 16 + n

⇒ Since the number is divisible by 9, (16 + n) must be equal to 18, 27, 36, etc.

⇒ For the least value of n (where n is a single digit, 0 ≤ n ≤ 9):

⇒ 16 + n = 18

⇒ n = 18 − 16

⇒ n = 2

∴ The correct answer is 2.

Additional Information

Divisibility Rule of 3

A number is completely divisible by 3 if the sum of its digits is a multiple of 3.

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of digits at odd places and even places is either 0 or a multiple of 11.

Divisibility Rule of 8

A number is divisible by 8 if the number formed by the last three digits is divisible by 8.

Shortcut Trick

The given ratio of three numbers is 3 : 2 : 13.

Since the numbers in the ratio (3, 2, and 13) are all prime numbers, their LCM is simply the product of these numbers multiplied by their HCF.

HCF = LCM ÷ (Product of Ratio parts)

HCF = 2574 ÷ (3 × 2 × 13) = 2574 ÷ 78

HCF = 33

∴ The correct answer is 33.

Alternate Method

Given: Ratio of numbers = 3 : 2 : 13 | LCM = 2574

Formula Used: LCM of (ax, bx, cx) = x × LCM(a, b, c), where x is the HCF.

⇒ Let the three numbers be 3x, 2x, and 13x, where x is the HCF.

⇒ LCM(3x, 2x, 13x) = x × LCM(3, 2, 13)

⇒ Since 3, 2, and 13 are co-prime, their LCM = 3 × 2 × 13 = 78

⇒ Total LCM = 78x

⇒ 78x = 2574

⇒ x = 2574 ÷ 78

⇒ x = 33

∴ The correct answer is 33.

Additional Information

Relationship for Two Numbers

For any two numbers a and b, the product of the numbers is equal to the product of their HCF and LCM: a × b = HCF × LCM.

Co-prime Numbers

Numbers are called co-prime if their Highest Common Factor (HCF) is 1. The LCM of co-prime numbers is always their product.

HCF and LCM of Fractions

HCF = (HCF of Numerators) ÷ (LCM of Denominators) and LCM = (LCM of Numerators) ÷ (HCF of Denominators).

Given:

Equation: 3⁻²² + 3⁻¹⁹ + 3⁻¹⁷ + 3⁻¹⁶ = k × 3⁻²⁴

Formula used:

The exponent rule for division: a^m ÷ aⁿ = a^{m − n}

Calculations:

k × 3⁻²⁴ = 3⁻²² + 3⁻¹⁹ + 3⁻¹⁷ + 3⁻¹⁶

⇒ k = (3⁻²² + 3⁻¹⁹ + 3⁻¹⁷ + 3⁻¹⁶) ÷ 3⁻²⁴

⇒ k = 3^{(−22 − (−24))} + 3^{(−19 − (−24))} + 3^{(−17 − (−24))} + 3^{(−16 − (−24))}

⇒ k = 3² + 3⁵ + 3⁷ + 3⁸

⇒ k = 9 + 243 + 2187 + 6561

⇒ k = 9000

∴ The correct answer is option (4).

Shortcut Trick

Find two Factors of the product 144 that add up to the sum 26.

List factors: 18 × 8 = 144

Check sum: 18 + 8 = 26 (Matches given sum)

Required Positive Difference = 18 − 8 = 10

∴ The correct answer is 10.

Alternate Method

Given: Sum (a + b) = 26 and Product (ab) = 144

Formula Used: (a − b)² = (a + b)² − 4ab

⇒ (a − b)² = (26)² − 4 × 144

⇒ (a − b)² = 676 − 576

⇒ (a − b)² = 100

⇒ a − b = √100

⇒ a − b = 10

∴ The correct answer is 10.

Additional Information

Quadratic Equation Construction

If sum S and product P of two numbers are known, the numbers are roots of x² − Sx + P = 0.

Algebraic Identities

Useful relation: (a + b)² − (a − b)² = 4ab.

Finding Individual Numbers

Once sum (S) and difference (D) are found, the numbers are (S+D)÷2 and (S−D)÷2.