Given:

The time taken by a boat to cover a certain distance upstream is equal to 4/7 of time taken by it cover three times the same distance downstream.

The speed of the stream is 7.5 km/h

Concept used:

Downstream Speed  = The speed of the boat in still water + The speed of the stream

Upstream Speed  = The speed of the boat in still water - The speed of the stream

Formula used:

Time = Distance/Speed

Calculation:

Let the distance and speed of the boat in still water be D km and x km/hr respectively.

Downstream speed =  (x + 7.5) km/hr

Upstream speed = (x - 7.5) km/hr

According to the question

D/(x - 7.5) = 3D/ (x + 7.5) × 4/7

⇒ 1/(x - 7.5) = 12/(7x + 52.5)

⇒ 7x + 52.5 = 12x - 90

⇒ 142.5 = 5x

⇒ x = 28.5

Downstream speed = (28.5 + 7.5) = 36 km/hr

Upstream speed = (28.5 - 7.5) = 21 km/hr

Now, the time taken by the boat to go 54 km downstream &  42 km upstream.

⇒  54/36 + 42/21

⇒ 1.5 + 2 = 3.5 hours

∴ The boat will take 3.5 hours to go 54 km downstream and 42 km upstream.

Given:

The perpendicular drawn from point A (4, 1) divides the join of B (2, -1) and C (6, 5).

Concept used:

Straight Line equation using two points:

[Where (x₁,y₁) and (x₂,y₂) are given points]

If two straight lines are perpendicular, their slopes are opposite reciprocals of one another, or the product of their slopes is (−1).

General equation of a straight line: ( y - y1) = m(x - x1) (where M is the slope)

Calculation:

According to the concept,

Equation of BC:

⇒ 3x - 2y = 8      ....(1)

The slope of equation (1) = 3/2

The slope of the straight line that is perpendicular to equation (1)

⇒ (-1) ÷ (3/2) = -2/3

Now using the slope line formula:

( y - y1) = m(x - x1)

Substitute the value of x1 = 4 and y1 = 1

⇒ y - 1 = (-2/3)(x - 4)

⇒ 2x + 3y = 11

∴ The Correct answer is 2x + 3y = 11.

Given:

Selling price = Rs.1,190

Discount = 15%

The formula used

Selling Price = Marked Price - Discount

calculation

The marked price is Rs. x

Discount = 15% of x

Selling price = x - 15% of x

1190 = x - (15x/100)

⇒1190 = (100x - 15x)/100

⇒1190 = 85x/100

⇒x = (1190 × 100)/85

⇒x = 1400

New Discount = 12.5%

New selling price = 1400 - 12.5% ​​of 1400

⇒1400 - 175

⇒1225

∴ The item was sold for Rs.1225.

Given:

Monthly income = Rs 12,000

Monthly Expenditure = Rs 10,000

Increase in income after a year = 12%

Increase in expenditure after a year = 5%

Concept Used:

New Amount = Original Amount + Percentage of Original Amount

Savings = Income – Expenditure

Calculation:

Savings before increase = 12000 – 10000 = Rs 2,000

Increase in income after a year = 12000 + 12% of 12000

Hence, the percentage increase in savings is 47%.

Given:

Pipes A and B can fill a tank in 10 hours and x hours respectively.

Pipe C alone can empty the full tank in 15 hours. When all three pipes are opened simultaneously for 8 hours, 60% of the tank can be filled. 

Concept Used:

Work done = Time × Efficiency

Calculation:

Let the LCM of (10, x, and 15) = 150x

So, the total work is 150x.

Efficiency of A = 150x/10 = 15x

Efficiency of B = 150x/x = 150

Efficiency of C = 150x/15 = - 10x (-ve sign shows that the pipe C is emptying the tank)

Total work done by them in 1 hour when opened simultaneously = (15x) + (150) + (-10x) = 5x + 150

When opened for 8 hours they fill 60% of the tank,

⇒ (5x + 150) × 8 = 60% of 150x

⇒ 40x + 1200 = 90x

⇒ 50x = 1200

⇒ x = 24

Total work = 150 × 24 = 3600

Efficiency of B and C together = (150 - 10x) units

⇒ {150 - 10(24)} units

⇒ (150 - 240) units

⇒ - 90 units (-ve sign shows that the tank will be emptied)

Time taken by B and C together to empty two-fifths of the tank = {3600 × (2/5)}/90

⇒ 1440/90 = 16 hours

Hence, the correct answer is 16 hours.

Given:

Total interest earned = Rs.9280.

Initial ratio = 5:4.

Simple Interest Rate = 10% for 2 years.

Compound Interest (Part 1) = 20% for 2 years.

Simple Interest (Part 2) = 30% for 3 years.

Formula:

Simple Interest = P × R × T / 100.

Compound Interest = P(1 + R/100)^T - P.

Where P = Principal, R = Rate, T = Time.

Calculation:

⇒ Total sum received after 2 years at 10% simple interest

⇒ K + (K × 10 × 2 / 100) = K(1 + 0.20) = 1.2K

⇒ Divided amounts ratio (5:4), Total parts = 9.

⇒ First part = (5/9) × 1.2K, Second part = (4/9) × 1.2K.

⇒ Compound Interest for first part for 2 years at 20% = (5/9) × 1.2K [(1 + 0.20)² - 1].

⇒ Simple Interest for second part for 3 years at 30% = (4/9) × 1.2K × 0.30 × 3.

⇒ Total interest = (5/9) × 1.2K × 0.44 + (4/9) × 1.2K × 0.90.

⇒ 9280 = 0.44 × (5/9) × 1.2K + 0.90 × (4/9) × 1.2K.

⇒ 9280 = 0.2933K + 0.4800K.

⇒ 9280 = 0.7733K.

⇒ K = 9280 / 0.7733.

⇒ K = Rs. 12,000.

Hence, the value of K is Rs. 12,000.

Shortcut Trick

Principal = Rs. K,  Time = 2 years, Interest = 10%

Simple Interest = PRT/100 = 20K/100 = 1K/5

Amount = SI + Principal

= 1K/5 + K = 6K/5

He divide this amount in the ratio 5 : 4.

According to question,

5/9 × 6K/5 × [(1 + 20/100)2 - 1] + 4/9 × 6K/5 × 30/100 × 3 = 9280

After solving this we get,

K = 12000

Given:

Red balls = 5

Blue balls = m

Concept used:

Probability = Number of favorable outcomes / Number of possible outcomes

Calculation:

Probability of getting red ball = 5/(5 + m)

Probability of getting blue ball = m/(5 + m)

According to the question,

⇒ m = 20

∴ The number of blue balls are 20.

Given:

The distance between two pillars of length 16 m and 9 m is x meters.

Two angles of elevation of their respective top from the bottom of the other are complementary to each other.

Concept used:

If two angles of elevation are complimentary to each other then H = √ab

Where a and b are the Length of the pillars.

Calculation:

AB and CD are two pillars of Length 16 m and 9m.

Let the angle of elevation at B and D be θ and (90 - θ)

Distance between two pillars BD be x metres

IN Δ ABD

Tanθ = AB/BD = 16/x       - - - -(i)

In Δ BDC

Tan(90 - θ) = CD/BD = 9/x

Cotθ = 9/x                       - - - - (ii)

Multiply equation i and ii

⇒ Tanθ × Cotθ = (16/x) (9/x)

⇒ 144/x² = 1

⇒ x² = 144

⇒ x = 12 m

Alternate Method

If two angles of elevation are complimentary to each other then x = √ab

Where a and b are the Length of the pillars.

x = √144

x = 12 m

Given:

The average age of Anil, Sheela and Rohit at the time of marriage of the Rohit = 46 years.

Average age of all 5 member after marriage of Rohit and Nisha  = 37.8 years.

Concept used:

Average = Sum of observation / Number of observation

Sum of observation = Average x Number of observation

Calculation:

Total age of Anil, Sheela and Rohit = 46 x 3 = 138 years

After 5 year of marriage,

Total age of 5 member(Anil + Sheela + Rohit + Nisha + Child)  = 37.8 x 5 = 189 year  

Total age of (Anil, Sheela and Rohit) = 138 + 5 + 5 + 5  = 153 year 

And child was born after 2 year of marriage,

Hence, age of child after 5 year of marriage = 3 year

Now, After 5 years,

Total age of (Anil, Sheela and Rohit + Nisha + Child) = 189 year

⇒ 153 + Nisha + 3 = 189

⇒ Age of Nisha = 33 year

Hence Age of Nisha at the time of marriage = 33 - 5 = 28 year.

∴ Age of Nisha at the time of marriage is 28 year.