Question

A is faster than B. A and B each walk 24 km. The sum of their speeds is 7 kmph and the time taken by them is 14 hrs. Then the speed of A is equal to

Answer 1

Hint: Assume that the speed of A is x km/hr and the speed of B is y km/hr. The summation of their speeds is 7. So, \[\left( x+y \right)=7\] . Now, get the value of y in terms of x. It is given that both travel 24 km. Calculate the time taken by A and B to cover 24 km by using the formula, \[\text{Time=}\dfrac{\text{Distance}}{\text{Speed}}\] . It is given that the time taken by A and B to cover 24 km is 14 hrs. Now, make an equation using this information and in the equation put the value of y in terms of x. Solve it further and get the value of x. Consider only that value of x which satisfies the information that A is faster than B

Complete step by step answer:
First of all, let us assume that the speed of A is x km/hr and the speed of B is y km/hr.
The speed of A = x km/hr ……………………………………(1)
The speed of B = y km/hr ……………………………………(2)
It is given that A is faster than B and the sum of their speeds is 7 km/hr.
From equation (1) and equation (2), we have the speed of A and B. So, \[\left( x+y \right)=7\] .
 \[\Rightarrow \left( x+y \right)=7\]
\[\Rightarrow y=7-x\] ……………………………………(3)
It is also given that A and B travel 24 kilometers and time taken by them is 14 hrs.
The time taken by A and B = 14 hrs …………………………….(4)
The distance covered by A = 24 km ……………………………….(5)
The distance covered by B = 24 km ……………………………….(6)
We know the formula, \[\text{Time=}\dfrac{\text{Distance}}{\text{Speed}}\] …………………………..(7)
Using the formula, shown in equation (7), we can get the time taken by A to cover the distance 24 km.
From equation (1) and equation (5), we have the speed and the distance of A.
The time taken by A to cover 24 km = \[\dfrac{24}{x}\] ………………………………(8)
The time taken by B to cover 24 km = \[\dfrac{24}{y}\] ………………………………(9)
The time taken by A and B to cover 24 km = \[\dfrac{24}{x}+\dfrac{24}{y}\] ……………………………………..(10)
From equation (4), we also have the time taken by A and B to cover 24 km.
Now, on comparing equation (4) and equation (10), we get
\[\Rightarrow 14=\dfrac{24}{x}+\dfrac{24}{y}\]
 …………………………….(11)
Now, from equation (3) and equation (11), we get
\[\begin{align}
  & \Rightarrow 14=\dfrac{24}{x}+\dfrac{24}{7-x} \\
 & \Rightarrow 14=24\left( \dfrac{1}{x}+\dfrac{1}{7-x} \right) \\
 & \Rightarrow 7=12\left( \dfrac{7-x+x}{x\left( 7-x \right)} \right) \\
 & \Rightarrow 7=12\times \dfrac{7}{x\left( 7-x \right)} \\
 & \Rightarrow x\left( 7-x \right)=12 \\
 & \Rightarrow 7x-{{x}^{2}}=12 \\
 & \Rightarrow {{x}^{2}}-7x+12=0 \\
 & \Rightarrow {{x}^{2}}-3x-4x+12=0 \\
 & \Rightarrow x\left( x-3 \right)-4\left( x-3 \right)=0 \\
 & \Rightarrow \left( x-3 \right)\left( x-4 \right)=0 \\
\end{align}\]
So, \[x=3\] or \[x=4\] .
Taking \[x=3\] and putting the value of x in equation (4), we get
\[\Rightarrow y=7-3=4\]
Since the speed of A is greater than B so x must be greater than y.
But here, we got \[x=3\] and \[y=4\] . This is a contradiction. So, \[x=3\] is not possible.
Taking \[x=4\] and putting the value of x in equation (4), we get
\[\Rightarrow y=7-4=3\]
Since the speed of A is greater than B so x must be greater than y.
 Here, we got \[x=4\] and \[y=3\] , we can say that x is greater than y. So, \[x=4\] and \[y=3\] are the possible speeds.
Therefore, the speeds of A and B are 4 km/hr and 3 km/ hr respectively.
Hence, the speed of A is 4 km/hr.

Note: In this question, after getting \[x=3\] and \[y=4\] , one might take the speed of A and B as 3 km /hr and 4 km/hr and conclude it as an answer. This is wrong because it is given that A is faster than B. So, the speed of A must be greater than B.