A boat covers a distance of 18 km downstream and 14 km upstream in 9 hours and It covers a distance of 45 km downstream and 12 km upstream in 11 hours. What is the speed (in km/hr) of the current?a)11/2b)5c)7/2d)6Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Let's assume the speed of the boat in still water is 'b' km/hr and the speed of the current is 'c' km/hr.

1. Downstream Speed:
When the boat is moving downstream, its effective speed is increased by the speed of the current. Therefore, the speed of the boat downstream is (b + c) km/hr.

2. Upstream Speed:
When the boat is moving upstream, its effective speed is decreased by the speed of the current. Therefore, the speed of the boat upstream is (b - c) km/hr.

Now, let's solve the given equations:

Equation 1:
Time taken to cover 18 km downstream + Time taken to cover 14 km upstream = 9 hours
18/(b+c) + 14/(b-c) = 9

Equation 2:
Time taken to cover 45 km downstream + Time taken to cover 12 km upstream = 11 hours
45/(b+c) + 12/(b-c) = 11

To solve these equations, we can use the method of substitution. We can solve Equation 1 for (b + c) in terms of (b - c) and substitute it in Equation 2.

Solving Equation 1 for (b + c):
18/(b+c) + 14/(b-c) = 9
18(b - c) + 14(b + c) = 9(b + c)(b - c)
32b - 4c = 9(b^2 - c^2)
32b - 4c = 9b^2 - 9c^2

Simplifying further:
9c^2 - 32b + 4c = 0

Now, substitute (b + c) = (32/9) in Equation 2:
45/(b+c) + 12/(b-c) = 11
45(9/32) + 12/(32/9) = 11
405/32 + 108/32 = 11
(405 + 108)/32 = 11
513/32 = 11

Now, cross multiply:
513 = 11 * 32
513 = 352

Since the equation is not balanced, our assumption (b + c) = (32/9) is incorrect.

Let's try another assumption:
(b + c) = (32/9) * 2
(b + c) = 64/9

Now, substitute (b + c) = (64/9) in Equation 2:
45/(b+c) + 12/(b-c) = 11
45(9/64) + 12/(64/9) = 11
405/64 + 108/64 = 11
(405 + 108)/64 = 11
513/64 = 11

Cross multiply:
513 = 11 * 64
513 = 704

Again, the equation is not balanced.

Let's try (b + c) = (64/9) * 2:
(b + c) = 128/9

Now, substitute (b + c) = (128/9) in Equation 2:
45/(b+c) + 12/(b-c) = 11
45(9/128) + 12/(128/9) = 11
405

Answer

Given:

Condition 1:

A boat covers a distance in downstream = 18 km

A boat covers a distance in upstream = 14 km

Total time taken = 9 hour

Condition 2:

A boat covers a distance in downstream = 45 km

A boat covers a distance in upstream = 12 km

Total time taken = 11 hour

Concept used:

Time = Distance/Speed

If boat stream in downstream then speed of boat = speed of boat in still water + speed of current

If boat stream in upstream then speed of boat = speed of boat in still water - speed of current

Calculation:

Let speed of boat in still water = x km/hour

and speed of current = y km/hour

⇒ If boat stream in downstream then speed of boat = x + y

⇒ If boat stream in upstream then speed of boat = x - y

According to the question,

Equation (1) multiply with 5 and equation (2) multiply with 2

Subtracting equation (4) from equation (3)

⇒ x - y = 2 ----(5)

Putting the value of (x - y) in equation (1)

⇒ x + y = 9 ----(6)

now equation (5) - equation (6)

⇒ - 2y = - 7

⇒ y = 7/2 km/hour

Speed of the current = 7/2 km/hour.

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