1) The distance between two stations is 300 km. Two motorcyclists start simultaneously from these stations and move towards each other. The speed of one of them is 7 km/h more than that of the other. If the distance between them after 2 hours of their start is 34 km. find the speed of each motorcyclist.
2) The difference between the ages of two cousins is 10 years. 15 years ago, if the elder one was twice as old as the younger one, find their present ages.
Answers & Comments
Answer:
[tex]\qquad \:\boxed{\begin{aligned}& \:\sf \:Speed\:of\:first\:motorcyclist=63 \: km \: per \: hr \: \\ \\& \:\sf \: Speed\:of\:first\:motorcyclist=70 \: km \: per \: hr \end{aligned}} \qquad \: \\ \\ [/tex]
[tex]\qquad \:\boxed{\begin{aligned}& \:\sf \: Present\:age\:of\:younger\:cousin=25 \: years \: \\ \\& \:\sf \: Present\:age\:of\:elder\:cousin=35 \: years\end{aligned}} \qquad \: \\ \\ [/tex]
Step-by-step explanation:
[tex]\large\underline{\sf{Solution-1}}[/tex]
Given that, the distance between two stations is 300 km. Two motorcyclists start simultaneously from these stations and move towards each other. The speed of one of them is 7 km/h more than that of the other
[tex]\sf \: Let\:assume \: speed\:of\:first\:motorcyclist = x \: km \: per \: hr \\ \\ [/tex]
So,
[tex]\sf \: Speed\:of\:second\:motorcyclist = (x + 7)\: km \: per \: hr \\ \\ [/tex]
Now,
Distance covered by first motorcyclist in 2 hours at the speed of x km per hour = 2 × x = 2x km
Also,
Distance covered by second motorcyclist in 2 hours at the speed of (x + 7) km per hour = 2 × (x + 7) = 2x + 14 km
Now according to statement, it is given that If the distance between them after 2 hours of their start is 34 km.
[tex]\sf \: 2x + 2x + 14 + 34 = 300 \\ \\ [/tex]
[tex]\sf \: 4x + 48 = 300 \\ \\ [/tex]
[tex]\sf \: 4x = 300 - 48\\ \\ [/tex]
[tex]\sf \: 4x = 252\\ \\ [/tex]
[tex]\sf\implies \sf \: x = 63 \: \\ \\ [/tex]
Hence,
[tex]\qquad \:\boxed{\begin{aligned}& \:\sf \:Speed\:of\:first\:motorcyclist=63 \: km \: per \: hr \: \\ \\& \:\sf \: Speed\:of\:first\:motorcyclist=70 \: km \: per \: hr \end{aligned}} \qquad \: \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
[tex]\large\underline{\sf{Solution-2}}[/tex]
Let assume that present age of younger cousin is x years.
So, present age of elder cousin is x + 10 years.
Fifteen years ago,
Age of younger cousin = x - 15 years
Age of elder cousin = x + 10 - 15 = x - 5 years
According to statement, 15 years ago, if the elder one was twice as old as the younger one.
[tex]\sf \: x - 5 = 2(x - 15) \\ \\ [/tex]
[tex]\sf \: x - 5 = 2x - 30\\ \\ [/tex]
[tex]\sf \: x - 2x = 5 - 30\\ \\ [/tex]
[tex]\sf \: - x = - 25\\ \\ [/tex]
[tex]\sf\implies \sf \: x = 25 \\ \\ [/tex]
Hence,
[tex]\qquad \:\boxed{\begin{aligned}& \:\sf \: Present\:age\:of\:younger\:cousin=25 \: years \: \\ \\& \:\sf \: Present\:age\:of\:elder\:cousin=35 \: years\end{aligned}} \qquad \: \\ \\ [/tex]