Dear Reader,
Below are three problems where you have to calculate distances based on data in questions.

Question 1

Government of Uttar Pradesh was contemplating making changes in the water transport system of the State. The Ganga river has shores on its two sides known as Point One and Point Two. Boat A proceeded from Point One to Point Two and Boat B started from Point Two to Point One.(simultaneously) The two boats travel at their specific speeds. Both the boats meet at Spot I - 600 m from Point One. They continued the journey and in their return journey (after touching the opposite shore) they meet again at Spot II-400 m from Point Two. Find the distance between Point One and Point Two.

a) 1400 m b) 1200 m c) 1600 m d) none of these

Answer : a) 1400 m

Solution :

Please refer to the diagram below.

Let the distance between Point One and Point Two be D.
Let the speeds of the boats be V1 and V2 respectively.
I meet :
During I meet, Boat A travels 600m from Point I while Boat B travels (D - 600)m from Point II.
Therefore, Time taken for Boat A to travel 600m = Time taken for Boat B to travel (D - 600)
Or 600 / V1 = (D - 600) / V2
Or 600 / (D - 600) = V1 / V2 ...(1)

II meet :
After crossing spot I, both the boats proceed in their respective directions, reach shores and return back to cross each other at Spot II which is 400m from Point II.
From Spot I to Spot II, Boat A would had travelled a distance of (D - 600) + 400 m
From Spot I to Spot II, Boat B would had travelled a distance of 600 + (D - 400) m
Time taken by Boat A to travel from Spot I to Spot II will be the same as that of Boat B from Spot I to Spot II

Therefore, V1 / (D - 600) + 400 = V2 / 600 + (D - 400)
Or (D - 600) + 400 / 600 + (D - 400) = V1/V2 ...(2)
From equations I and II, we get,
600 / (D - 600) = (D - 600) + 400 / 600 + (D - 400)
600 / (D - 600) = D - 200 / D + 200
By Cross- Multiplying,
600 (D + 200) = (D - 200) (D - 600)
600D + 120000 = D² - 200D - 600D + 120000
D² - 1400D = 0
By Factorizing,
D(D - 1400) = 0
D = 1400

Question 2

Godavari river is one of the important rivers helping irrigation in Andhra Pradesh. The Godavari river has shores on its two sides known as Point Number One and Point Number Two. Boat A started from Point One to Point Two and Boat B started from Point Two to Point One.(simultaneously) The two boats travel at their specific speeds. Both the boats meet 300 m from Point One.(Spot A) They continued the journey and in their return journey (after touching the opposite shore) they meet again at Point Two. Find the distance between Point One and Point Two.

a) 1400 m b)900 m c)800 m d) none of these

Answer : b) 900 m

Solution :

Let the distance between point 1 and point 2 be D.
Let Spot A represent the point of first meet and Spot B represent the point of second meet.

Proceeding similar to the solution for Question I we will reach at the equation.
300 / (D - 300) = (D - 300) / 300 + D
(Pls refer to first question's solution to know how we arrived at the above equation)

By Cross- Multiplying,
300D + 90000 = D² - 600D + 90000
D² - 900D = 0
By Factorizing,
D(D - 900) = 0
D = 900

Question 3

Kaveri river has shores on its two sides known as P and Q. Boat A started from P to Point Q and Boat B started from Q to P.(simultaneously) The two boats travel at their specific speeds. Both the boats meet 1000 m from P--Point One . They continued the journey and in their return journey (after touching the opposite shore) they meet again at Point Two which is situate 600 m from Q. Find the distance between P and Q.

a) 1800 m b)1900 m c) 1300 m d) 2400

Answer : d) 2400

Solution :

Let the distance between the shores be D.

Proceeding similar to the solution for Question I we will reach at the equation.
1000 / (D - 1000) = (D - 1000) + 600 / 1000 + (D - 600) (Pls refer to first question's solution to know how we arrived at the above equation)

Simplifying, we get : 1000 / (D - 1000) = D - 400 / D + 400
By Cross- Multiplying,
1000 (D + 400) = (D - 400) (D - 1000)
1000D + 400000 = D² - 400D - 1000D + 400000
D² - 2400D = 0
By Factorizing,
D(D - 2400) = 0
D = 2400