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Shortcut Trick

Convert all units to cm: 735 cm, 945 cm, and 1050 cm.

To find the greatest length, find the HCF using the difference method.

Difference 1 = 945 − 735 = 210; Difference 2 = 1050 − 945 = 105.

Since 105 divides 735 (105 × 7), 945 (105 × 9), and 1050 (105 × 10), the HCF is 105 cm.

105 cm = 100 cm + 5 cm = 1 m 5 cm.

∴ The correct answer is 1 m 5 cm.

Alternate Method

Given: Fabric lengths are 7 m 35 cm, 9 m 45 cm, and 10 m 50 cm.

Formula Used: The greatest length that can exactly measure several lengths is their Highest Common Factor (HCF).

⇒ Conversion to cm: (1 m = 100 cm)

⇒ L₁ = 7 × 100 + 35 = 735 cm

⇒ L₂ = 9 × 100 + 45 = 945 cm

⇒ L₃ = 10 × 100 + 50 = 1050 cm

⇒ Prime Factorization:

⇒ 735 = 3 × 5 × 7 × 7

⇒ 945 = 3 × 3 × 3 × 5 × 7

⇒ 1050 = 2 × 3 × 5 × 5 × 7

⇒ Common factors = 3, 5, 7

⇒ HCF = 3 × 5 × 7 = 105 cm

⇒ Converting back to meters: 105 cm = 100 cm + 5 cm = 1 m 5 cm

∴ The correct answer is 1 m 5 cm.

Additional Information

Highest Common Factor (HCF)

The HCF of two or more numbers is the largest number that divides each of them without leaving a remainder. In measurement problems, the greatest possible size is always the HCF.

Least Common Multiple (LCM)

The LCM is the smallest number that is a multiple of all the given numbers. It is used for finding simultaneous occurrences or the smallest common container.

Unit Conversion

Always ensure all measurements are in the same standard unit before performing calculations. 1 meter = 100 centimeters; 1 kilometer = 1000 meters.

Shortcut Trick

Assume the base average marks is 72.

Deviation for 3 additional subjects = 3 × (84 − 72) = +36.

Distribute this extra value across all 8 subjects = 36 ÷ 8 = 4.5.

New overall average = 72 + 4.5 = 76.5.

∴ The correct answer is 76.5.

Alternate Method

Given:

Number of subjects in first group (n₁) = 5

Average marks of first group (A₁) = 72

Number of additional subjects (n₂) = 3

Average marks of additional group (A₂) = 84

Formula Used:

Overall Average = (Total Marks in all subjects) ÷ (Total number of subjects)

Calculations:

⇒ Total marks in 5 subjects = 5 × 72 = 360

⇒ Total marks in 3 additional subjects = 3 × 84 = 252

⇒ Grand total marks = 360 + 252 = 612

⇒ Total number of subjects = 5 + 3 = 8

⇒ Overall Average = 612 ÷ 8

⇒ Overall Average = 76.5

Comparing this value with the given options, we find it matches Option (4).

∴ The correct answer is 76.5.

Additional Information

Weighted Average Formula

When two groups are combined, the new average is given by: Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Deviation Method

Overall Average = Assumed Average + (Total Deviation ÷ Total Count). This minimizes calculation steps for large datasets.

Concept of Net Change

In averages, the sum of deviations of all observations from their arithmetic mean is always zero.

Shortcut Trick

Direct identity: (Difference)² = (Sum)² − 4 × (Product)

Difference² = 67² − 4 × 1012

Difference² = 4489 − 4048 = 441

Difference = √441 = 21

∴ The correct answer is 21.

Alternate Method

Given:

Sum of two numbers (x + y) = 67

Product of two numbers (xy) = 1012

Formula Used:

The relationship between sum, product, and difference of two numbers is given by the algebraic identity: (x − y)² = (x + y)² − 4xy

Calculations:

⇒ Let the two positive numbers be x and y.

⇒ (x − y)² = (67)² − 4 × 1012

⇒ (x − y)² = 4489 − 4048

⇒ (x − y)² = 441

⇒ (x − y) = √441

⇒ x − y = 21

∴ The correct answer is 21.

Additional Information

Algebraic Identity for Squares

The square of a sum and square of a difference are related as: (a + b)² − (a − b)² = 4ab.

Quadratic Equation Roots

If the sum (S) and product (P) of two numbers are known, the numbers are the roots of the equation: t² − St + P = 0.

Difference of Squares

The product of the sum and the difference of two numbers equals the difference of their squares: (a + b)(a − b) = a² − b².

Shortcut Trick

The ratio of the first two numbers is derived from the fourth proportional: a : b = 20 : 36 = 5 : 9.

Sum of the ratio parts = 5 + 9 = 14.

Since the given sum is exactly 14, the actual numbers are 5 and 9.

The Geometric Mean = √(5 × 9) = √45 = 3√5.

∴ The correct answer is 3√5.

Alternate Method

Given:

The fourth proportional to a, b, and 20 is 36.

The sum of the first two numbers (a + b) is 14.

Formula Used:

If a, b, c, d are in proportion, then a/b = c/d.

Geometric Mean of two numbers a and b = √(a × b).

⇒ a/b = 20/36

⇒ a/b = 5/9

⇒ Let a = 5k and b = 9k

⇒ a + b = 14 ⇒ 5k + 9k = 14

⇒ 14k = 14 ⇒ k = 1

⇒ Therefore, a = 5 and b = 9

⇒ Geometric Mean = √(5 × 9) = √45

⇒ Geometric Mean = 3√5

∴ The correct answer is 3√5.

Additional Information

Fourth Proportional

In a proportion sequence a, b, c, d where a:b = c:d, the value d is the fourth proportional, calculated as d = (b × c) ÷ a.

Geometric Mean (GM)

For any two positive numbers x and y, the geometric mean is the square root of their product: GM = √(x × y).

Mean Proportional

The mean proportional between two numbers a and c is a number b such that a:b = b:c, which simplifies to b² = ac or b = √(ac).

Shortcut Trick

Identify prime numbers a such that a² is less than 20: only a = 2 and a = 3 are possible.

If a = 2, a² = 4. Possible primes b such that 4 + b is a prime < 20 are 3, 7, and 13.

If a = 3, a² = 9. The only prime b such that 9 + b is a prime < 20 is 2 (Sum = 11).

The set of all possible b values is {2, 3, 7, 13}.

Total count of b values = 4.

∴ The correct answer is 4.

Alternate Method

Given: a and b are prime numbers, and a² + b is a prime number < 20.

Step-by-step evaluation:

⇒ Prime numbers are {2, 3, 5, 7, 11, 13, 17, 19, ...}.

⇒ Since a² + b < 20, and b ≥ 2, we must have a² < 18. Thus a can only be 2 or 3.

⇒ For a = 2: 4 + b must be in {5, 7, 11, 13, 17, 19}.

⇒ 4 + b = 7 ⇒ b = 3; 4 + b = 11 ⇒ b = 7; 4 + b = 17 ⇒ b = 13.

⇒ For a = 3: 9 + b must be in {11, 13, 17, 19}.

⇒ 9 + b = 11 ⇒ b = 2; 9 + b = 13 ⇒ b = 4 (Not prime); 9 + b = 17 ⇒ b = 8 (Not prime); 9 + b = 19 ⇒ b = 10 (Not prime).

⇒ Distinct values of b = {2, 3, 7, 13}.

∴ The correct answer is 4.

Additional Information

Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The sequence begins 2, 3, 5, 7, 11, ...

Even Prime Property

The number 2 is the only even prime number. All other prime numbers are odd, which means their squares are also odd.

Parity in Sums

For the sum of two numbers to be an odd prime (like 7, 11, 17), one number must be even and the other must be odd.

Shortcut Trick

Total Work = LCM(10, 12, 15) = 60 units.

Efficiencies: A = 6, B = 5, C = 4.

Work done by A and B in 3 days = (6 + 5) × 3 = 33 units.

Remaining Work = 60 − 33 = 27 units.

Days required by B and C = 27 ÷ (5 + 4) = 3 days.

∴ The correct answer is 3 days.

Alternate Method

Given:

Time taken by A = 10 days

Time taken by B = 12 days

Time taken by C = 15 days

Calculations:

⇒ Work done by A and B together in 1 day = 6 + 5 = 11 units.

⇒ Work completed by A and B in 3 days = 11 × 3 = 33 units.

⇒ Remaining Work = Total Work − Completed Work = 60 − 33 = 27 units.

⇒ Combined Efficiency of B and C = 5 + 4 = 9 units/day.

⇒ Time required for B and C to complete the Remaining Work = 27 ÷ 9

⇒ Time = 3 days.

∴ The correct answer is 3 days.

Additional Information

Pipes and Cisterns

Similar to time and work, where filling pipes perform positive work and outlet/leakage pipes perform negative work.

Work and Wages

Wages are generally distributed in the ratio of Efficiency × Time (total work done by each individual).

Alternate Day Work

When people work on alternate days, work is calculated in cycles (e.g., 2-day or 3-day blocks) based on the rotation.

Man-Days Formula

The relationship between men, days, and hours is given by (M₁ × D₁ × H₁)/W₁ = (M₂ × D₂ × H₂)/W₂.

Given:

Expression = { (20/33) ÷ (20/7) } ÷ ( (3/5) × (10/9) + (2/11) ) + (7/4) ÷ (28/23) of (23/4)

Formula used:

The VBODMAS rule is applied to solve this expression, following the order: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Calculations:

First, solve the operations inside the brackets:

⇒ { (20/33) ÷ (20/7) } = (20/33) × (7/20) = 7/33

⇒ ( (3/5) × (10/9) + (2/11) ) = ( (1/1) × (2/3) + (2/11) ) = (2/3 + 2/11)

⇒ (22 + 6) / 33 = 28/33

Now, substitute these values back into the expression:

⇒ (7/33) ÷ (28/33) + (7/4) ÷ (28/23) of (23/4)

Next, evaluate the of operator:

⇒ (28/23) of (23/4) = (28/23) × (23/4) = 7

The expression now becomes:

⇒ (7/33) ÷ (28/33) + (7/4) ÷ 7

Perform the division operations from left to right:

⇒ (7/33 × 33/28) + (7/4 × 1/7)

⇒ 1/4 + 1/4

Finally, perform the addition:

⇒ 2/4 = 1/2

∴ The correct answer is option (3).

Shortcut Trick

Apply the divisibility rule of 11: The difference between the sum of digits at odd positions and the sum of digits at even positions must be 0 or a multiple of 11.

For the number 2145:

Sum of digits at odd places (1^st and 3^rd) = 2 + 4 = 6.

Sum of digits at even places (2^nd and 4^th) = 1 + 5 = 6.

Difference = 6 − 6 = 0.

∴ The correct answer is 2145.

Alternate Method

Given: Numbers to check are 3522, 4735, 2145, and 5281.

Method: Verify each number using the alternating sum property.

⇒ Sum of odd-placed digits (from left) = 2 + 4 = 6

⇒ Sum of even-placed digits (from left) = 1 + 5 = 6

⇒ Difference = |6 − 6| = 0

⇒ Since the result is 0, the number 2145 is divisible by 11.

Checking other options:

⇒ 3522: (3 + 2) − (5 + 2) = 5 − 7 = −2 (No)

⇒ 4735: (4 + 3) − (7 + 5) = 7 − 12 = −5 (No)

⇒ 5281: (5 + 8) − (2 + 1) = 13 − 3 = 10 (No)

∴ The correct answer is 2145.

Additional Information

Divisibility by 3 and 9

A number is divisible by 3 or 9 if the sum of its digits is divisible by 3 or 9 respectively.

Divisibility by 7

Subtract twice the last digit from the remaining number; if the result is divisible by 7, the original number is too.

Divisibility by 11 (General Case)

For any integer, the difference between the sum of alternating digits must be a multiple of 11 (e.g., ..., −22, −11, 0, 11, 22, ...).

Shortcut Trick

In such cases, the absolute increase in tax equals the absolute decrease in net income.

Tax Rate = [Reduction in Net Income / (Increase in Tax + Reduction in Net Income)] × 100

Rate = [2 / (18 + 2)] × 100

Rate = (2 / 20) × 100 = 10%

∴ The correct answer is 10%.

Alternate Method

Given:

Increase in Income Tax = 18%

Reduction in Net Income = 2%

Calculations:

⇒ Let Income Tax be T and Net Income be NI.

⇒ 18% of T = 2% of NI

⇒ 18/100 × T = 2/100 × NI

⇒ 18 × T = 2 × NI

⇒ T / NI = 2 / 18 = 1 / 9

⇒ Gross Income = Tax + Net Income = 1 + 9 = 10

⇒ Original Rate of Tax = (Income Tax / Gross Income) × 100

⇒ Rate = (1 / 10) × 100 = 10%

∴ The correct answer is 10%.

Additional Information

Net Income Formula

Net Income is the remaining income after deducting Income Tax from Gross Income. Formula: Net Income = Gross Income − Income Tax.

Tax and Net Income Relation

For a fixed Gross Income, any change in the amount of Income Tax results in an equal and opposite change in the amount of Net Income.

Tax Rate Calculation

The Tax Rate is always calculated as a percentage of the Gross Income. Formula: (Income Tax ÷ Gross Income) × 100.

Shortcut Trick

If an object travels different distances at the same uniform speed, the average speed for the entire journey is equal to that uniform speed itself.

Since the scooter travels both segments at 20 km/hr, the average speed is 20 km/hr.

∴ The correct answer is 20 km/hr.

Alternate Method

Given:

Distance 1 (d₁) = 40 km, Speed 1 (s₁) = 20 km/hr

Distance 2 (d₂) = 60 km, Speed 2 (s₂) = 20 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

Calculations:

⇒ Total Distance = 40 + 60 = 100 km

⇒ Time for first part (t₁) = 40 ÷ 20 = 2 hours

⇒ Time for second part (t₂) = 60 ÷ 20 = 3 hours

⇒ Total Time = 2 + 3 = 5 hours

⇒ Average Speed = 100 ÷ 5 = 20 km/hr

∴ The correct answer is 20 km/hr.

Additional Information

Average Speed for Equal Distances

If a journey is divided into two equal distances traveled at speeds x and y, the average speed = 2xy ÷ (x + y).

Relative Speed

When two objects move in the opposite direction, relative speed = s₁ + s₂. In the same direction, relative speed = |s₁ − s₂|.

Unit Conversion

To convert speed from km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.