Question

A boat goes 15 km upstream and 22 km downstream in 15 hours. In 14 hours, it can go 45 km upstream and 55 km downstream. What is the speed of the boat in still water?
(a). 3
(b). 7
(c). 6
(d). 8

Answer 1

Hint: Assume variables for the speed of the boat in still water and the speed of the water current. Now using the given information, form two equations in two variables and solve them to find the answer.

Step-by-step answer:
Let the speed of the boat in still water be x km/hr and the speed of the water current be y km/hr.
It is given that the boat goes 15 km upstream and 22 km downstream in 15 hours. Then, we have:
\[\dfrac{{15}}{{x - y}} + \dfrac{{22}}{{x + y}} = 5............(1)\]
It is also given that the boat goes 45 km upstream and 55 km downstream in 14 hours. Then, we have:
\[\dfrac{{45}}{{x - y}} + \dfrac{{55}}{{x + y}} = 14............(2)\]
Hence, we have two equations (1) and (2) in two variables. We try to solve them.
We divide equation (2) by 3 and subtract from equation (1) to get as follows:
\[\dfrac{{15}}{{x - y}} + \dfrac{{22}}{{x + y}} - \dfrac{1}{3}\left( {\dfrac{{45}}{{x - y}} + \dfrac{{55}}{{x + y}}} \right) = 5 - \dfrac{{14}}{3}\]
Simplifying, we have:
\[\dfrac{{15}}{{x - y}} + \dfrac{{22}}{{x + y}} - \dfrac{{15}}{{x - y}} - \dfrac{{55}}{{3(x + y)}} = \dfrac{{15 - 14}}{3}\]
\[\dfrac{{22}}{{x + y}} - \dfrac{{55}}{{3(x + y)}} = \dfrac{1}{3}\]
Multiplying the entire equation by (3) and cross multiplying, we have:
\[66 - 55 = x + y\]
Finding the value of x + y, we have:
\[x + y = 11.............(3)\]
Substituting equation (3) in equation (1), we have:
\[\dfrac{{15}}{{x - y}} + \dfrac{{22}}{{11}} = 5\]
Simplifying, we have:
\[\dfrac{{15}}{{x - y}} + 2 = 5\]
\[\dfrac{{15}}{{x - y}} = 5 - 2\]
\[\dfrac{{15}}{{x - y}} = 3\]
Finding the value of x – y, we have:
\[x - y = 5............(4)\]
Adding equations (3) and (4), we have:
\[x + y + x - y = 11 + 5\]
Canceling the y term, we have:
\[2x = 16\]
Solving for x, we have:
\[x = \dfrac{{16}}{2}\]
\[x = 8\]
Hence, the speed of the boat in still water is 8 km/hr.
Hence, the correct answer is option (d).

Note: You can also solve by finding the value of the speed of the water current and then substitute in the equation to find the value of the speed of the boat in still water. You can substitute both the values in the equations to verify your answers.