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The logic followed to solve this problem is based on linear equations and basic arithmetic.

1. Formulating the Equation:

Let the original number be x.

According to the question, one-third of the number increased by 55 results in 77.

The equation is: (x / 3) + 55 = 77

2. Solving for the Original Number (x):

First, subtract 55 from both sides of the equation to isolate the term with x:

x / 3 = 77 − 55

x / 3 = 22

Now, multiply both sides by 3 to find the value of x:

x = 22 × 3

x = 66

So, the original number is 66.

 

Key Points

3. Calculating the Sum of the Digits:

The digits of the original number 66 are 6 and 6.

Sum of the digits = 6 + 6 = 12

 

Therefore, the sum of the digits of the original number is 12.

Shortcut Trick

Total Distance = Train Length + Bridge Length = 240 + 420 = 660 m

Speed in m/s = Total Distance ÷ Time = 660 ÷ 22 = 30 m/s

Convert to km/h: 30 × 18/5 = 6 × 18 = 108 km/h

∴ The correct answer is 108 km/h.

Alternate Method

Given:

Length of the Train (L₁) = 240 m

Length of the Bridge (L₂) = 420 m

Time taken (t) = 22 sec

Formula Used:

Speed = Total Distance ÷ Time

Total Distance = (Length of Train + Length of Bridge)

Calculations:

⇒ Total Distance = 240 + 420 = 660 m

⇒ Speed = 660 ÷ 22

⇒ Speed = 30 m/s

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

⇒ Speed = 30 × (18 ÷ 5)

⇒ Speed = 6 × 18 = 108 km/h

∴ The correct answer is 108 km/h.

Additional Information

Unit Conversion Rules

1 km/h = 5/18 m/s | 1 m/s = 18/5 km/h. Multiply by these factors to switch between units quickly.

Train Crossing a Static Object

When a train crosses a platform, bridge, or another stationary train, the distance is the sum of both lengths. When it crosses a pole or man, distance is just the train length.

Relative Speed

If two objects move in the opposite direction, relative speed = S₁ + S₂. If they move in the same direction, relative speed = |S₁ − S₂|.

Shortcut Trick

Total numbers in the range [11, 19] = 9.

Total possible pairs = ⁹C₂ = (9 × 8) / 2 = 36.

Non-coprime pairs from even numbers {12, 14, 16, 18} = ⁴C₂ = 6.

Non-coprime pairs from multiples of 3 {12, 15, 18} = (12, 15) and (15, 18) = 2 unique pairs.

Total non-coprime pairs = 6 + 2 = 8.

∴ The correct answer is 28 pairs.

Alternate Method

Given: The range of numbers is from 11 to 19 (both inclusive).

Formula: Two numbers are coprime if their Highest Common Factor (HCF) or Greatest Common Divisor (GCD) is 1.

⇒ Total numbers (n) = 9

⇒ Total possible pairs = n(n − 1) / 2 = (9 × 8) / 2 = 36

⇒ Identifying non-coprime pairs (GCD > 1):

⇒ Pairs with factor 2: (12, 14), (12, 16), (12, 18), (14, 16), (14, 18), (16, 18) = 6 pairs

⇒ Pairs with factor 3: (12, 15), (15, 18) = 2 pairs

⇒ Total non-coprime pairs = 6 + 2 = 8

⇒ Number of coprime pairs = Total pairs − Non-coprime pairs

⇒ Coprime pairs = 36 − 8 = 28

∴ The correct answer is 28 pairs.

Additional Information

Coprime Numbers

Two numbers are coprime if their only common factor is 1. This means HCF(a, b) = 1.

Consecutive Integers

Any two consecutive integers are always coprime to each other (e.g., 14 and 15).

Prime Numbers and Coprimality

A prime number is coprime to every other number that is not a multiple of itself.

Twin Primes

Prime numbers that have a difference of 2 are called twin primes (e.g., 11 and 13, 17 and 19).

Shortcut Trick

Base Average = 20

Deviation of 11 = 11 − 20 = −9

Deviation of 21 = 21 − 20 = +1

Net deviation for first two numbers = −9 + 1 = −8

To maintain the average, 3^rd number deviation must be +8 ⇒ 20 + 8 = 28

∴ The correct answer is 28.

Alternate Method

Given:

Average of three numbers = 20

First number = 11

Second number = 21

Formula Used:

Sum of numbers = Average × Total count of numbers

Calculations:

⇒ Sum of three numbers = 20 × 3

⇒ Sum of three numbers = 60

⇒ Sum of the two given numbers = 11 + 21

⇒ Sum of two numbers = 32

⇒ Third number = (Sum of three numbers) − (Sum of two numbers)

⇒ Third number = 60 − 32

⇒ Third number = 28

∴ The correct answer is 28.

Additional Information

Arithmetic Mean Formula

The average or mean is calculated as the sum of all observations divided by the total number of observations: A = (Σx_i) ÷ n.

Concept of Net Deviation

The sum of deviations of all observations from their arithmetic mean is always zero. This is a powerful property used to find missing values in a dataset.

Weighted Average

When two groups with averages A₁ and A₂ and sizes n₁ and n₂ are combined, the combined average is (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Shortcut Trick

Interest for 12 months = 8%, so Interest for 6 months = 4%.

Total repayment Amount = 100% (Principal) + 4% (Interest) = 104%.

Total amount = 1.04 × 14,750 = 15,340.

∴ The correct answer is ₹ 15340.

Alternate Method

Given: Principal (P) = ₹14,750, Rate (R) = 8% p.a., Time (T) = 6 months.

Formula Used: SI = (P × R × T) / 100; Amount (A) = P + SI

⇒ Time (T) in years = 6 / 12 = 0.5 years

⇒ Simple Interest (SI) = (14750 × 8 × 0.5) / 100

⇒ SI = (14750 × 4) / 100 = 147.5 × 4 = 590

⇒ Total Amount (A) = Principal + Simple Interest

⇒ Total Amount = 14750 + 590 = 15340

∴ The correct answer is ₹ 15340.

Additional Information

Simple Interest (SI)

It is calculated only on the principal amount of a loan or deposit. SI = (P × R × T) / 100.

Time Conversion Rule

While calculating interest, the time (T) must always be in years. Months are divided by 12, and days are usually divided by 365.

Amount in SI

The total repayable amount is the sum of the original principal and the interest earned: A = P[1 + (RT/100)].

Shortcut Trick

The equation (sin²θ − 3sinθ + 2) / cos²θ = 1 simplifies to (2 − sinθ) / (1 + sinθ) = 1.

⇒ 2 − sinθ = 1 + sinθ ⇒ 2sinθ = 1 ⇒ sinθ = 1/2.

For sinθ = 1/2 in the range 0° < θ < 90°, we get θ = 30°.

Substitute θ = 30° into the required expression: cot(1.5 × 30°) + cos(3 × 30°) = cot 45° + cos 90°.

⇒ 1 + 0 = 1.

∴ The correct answer is 1.

Alternate Method

Given: (sin²θ − 3sinθ + 2) / cos²θ = 1 where 0° < θ < 90°.

Formula Used: cos²θ = 1 − sin²θ | Quadratic Factorization

⇒ sin²θ − 3sinθ + 2 = cos²θ

⇒ sin²θ − 3sinθ + 2 = 1 − sin²θ

⇒ 2sin²θ − 3sinθ + 1 = 0

⇒ 2sin²θ − 2sinθ − sinθ + 1 = 0

⇒ 2sinθ(sinθ − 1) − 1(sinθ − 1) = 0

⇒ (2sinθ − 1)(sinθ − 1) = 0

⇒ sinθ = 1/2 or sinθ = 1

⇒ Since 0° < θ < 90°, θ = 30°

⇒ Value = cot(3/2 × 30°) + cos(3 × 30°)

⇒ Value = cot 45° + cos 90° = 1 + 0 = 1

∴ The correct answer is 1.

Additional Information

Trigonometric Quadrant Rule

In the first quadrant (0° to 90°), all trigonometric ratios are positive.

Standard Value of cot 45°

The value of cot 45° is always 1, as it is the reciprocal of tan 45°.

Complementary Angle Identities

sin(90° − θ) = cosθ and tan(90° − θ) = cotθ.

Shortcut Trick

Total daily work of (A + B + C) = 1/6 + 1/42 = 8/42 = 4/21.

Given A = B + C, so total daily work = A + (B + C) = 2A.

2A = 4/21 ⇒ A efficiency = 2/21.

B efficiency = (A + B) − A = 1/6 − 2/21 = 3/42 = 1/14.

∴ The correct answer is 14 days.

Alternate Method

Given:

Time taken by (A + B) = 6 days

Time taken by C = 42 days

Efficiency condition: Efficiency of A = Efficiency of (B + C)

Calculations:

⇒ Total Efficiency of (A + B + C) = 7 + 1 = 8 units/day

⇒ Since Efficiency A = B + C, then A + (B + C) = 8 becomes 2 × A = 8

⇒ Efficiency of A = 8 ÷ 2 = 4 units/day

⇒ Efficiency of B = Efficiency of (A + B) − Efficiency of A

⇒ Efficiency of B = 7 − 4 = 3 units/day

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

⇒ Time taken by B = 42 ÷ 3 = 14 days

∴ The correct answer is 14 days.

Additional Information

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken. If a person is twice as efficient as another, they will take half the time to complete the same work.

Combined Work Formula

If X takes 'a' days and Y takes 'b' days, they together take (a × b) ÷ (a + b) days.

Work and Wages

Wages are distributed in the ratio of work done or the ratio of efficiencies when all persons work for the same duration.

Shortcut Trick

The relationship a : b = c : x represents a proportion where x is the fourth proportional.

Logic: Product of Extremes (a × x) = Product of Means (b × c).

⇒ 12 × x = 18 × 27

⇒ x = (18 × 27) ÷ 12 = 486 ÷ 12 = 40.5

∴ The correct answer is 40.5.

Alternate Method

Given:

Values: a = 12, b = 18, c = 27

Formula Used:

For a proportion a : b = c : d, the ratio of the first two terms equals the ratio of the last two terms: a/b = c/d

Calculations:

⇒ a/b = c/x

⇒ 12/18 = 27/x

⇒ 2/3 = 27/x (Simplifying 12/18 by dividing by 6)

⇒ 2 × x = 3 × 27

⇒ 2x = 81

⇒ x = 81 ÷ 2

⇒ x = 40.5

∴ The correct answer is 40.5.

Additional Information

Fourth Proportional

When four quantities a, b, c, d are in proportion (a:b = c:d), d is the fourth proportional given by d = (b × c) ÷ a.

Third Proportional

In a continued proportion a:b = b:c, c is called the third proportional and is calculated as c = b² ÷ a.

Mean Proportional

The mean proportional between two numbers a and b is √(a × b), such that a:x = x:b.

Compounded Ratio

The compounded ratio of two ratios a:b and c:d is defined as (a × c) : (b × d).

Shortcut Trick

Total Area (m²) = 2,500 × (1/2 × 0.50 × 0.35) = 2,500 × 0.0875 = 218.75 m².

Total Cost = 218.75 × ₹5 = ₹1093.75.

Courtyard Area = 218.75 + 6.25 = 225 m².

Side of Square = √225 = 15 m.

∴ The correct answer is ₹1093.75, 15 m.

Alternate Method

Given:

Number of slabs = 2,500

Diagonals of rhombus (d₁, d₂) = 50 cm, 35 cm

Polishing rate = ₹5 per m²

Additional area = 6.25 m²

Calculations:

⇒ Area of 1 rhombus slab = 1/2 × d₁ × d₂

⇒ Area of 1 slab = 1/2 × 50 × 35 = 875 cm²

⇒ Total area of 2,500 slabs = 2,500 × 875 = 2,187,500 cm²

⇒ Total area in m² = 2,187,500 ÷ 10,000 = 218.75 m²

⇒ Total polishing cost = 218.75 × 5 = ₹1093.75

⇒ Area of square courtyard = Total area + 6.25

⇒ Area of square = 218.75 + 6.25 = 225 m²

⇒ Side of square = √Area = √225 = 15 m

∴ The correct answer is ₹1093.75, 15 m.

Additional Information

Area of Rhombus

The area of a rhombus can also be calculated as Base × Height if the side and altitude are given.

Unit Conversion

1 m² = 10,000 cm². Always convert units to a common system (SI) before calculating costs.

Properties of Square

Area of a square = side². Therefore, the side length is the square root of the total area.

Shortcut Trick

If John is twice as old as 10 years ago, then the 10-year gap represents half his current age.

⇒ John's present age = 10 × 2 = 20 years.

Sum of ages 5 years ago was 25, so the present sum increases by 5 for each person (5 × 2 = 10).

⇒ Present sum = 25 + 10 = 35 years.

⇒ Jane's age = 35 − 20 = 15 years.

Difference = 20 − 15 = 5 years.

∴ The correct answer is 5 years.

Alternate Method

Given: Sum of John and Jane 5 years ago = 25 years. John's current age = 2 × (Age 10 years ago).

Formula Used: Present Age = Past Age + Time Interval; Difference = Age₁ − Age₂.

⇒ Let John's present age be x.

⇒ According to the condition: x = 2(x − 10)

⇒ x = 2x − 20 ⇒ x = 20 years.

⇒ Let Jane's present age be y.

⇒ Sum 5 years ago: (x − 5) + (y − 5) = 25

⇒ (20 − 5) + (y − 5) = 25

⇒ 15 + y − 5 = 25 ⇒ y + 10 = 25

⇒ y = 15 years.

⇒ Difference between present ages = 20 − 15 = 5 years.

∴ The correct answer is 5 years.

Additional Information

Constant Age Difference

The difference between the ages of two individuals remains the same throughout their lives, regardless of how many years pass.

Sum of Ages in the Future/Past

If the sum of ages of 'n' people is S now, then after 't' years, the sum will be S + (n × t).

Age Ratios

When ages are given in ratios, always use a constant multiplier 'k' (e.g., 3k and 4k) to represent the actual ages before applying time shifts.