Ex 3.6, 2
Formulate the following problems as a pair of equations, and hence find their solutions:
(iii) Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.
Let speed of train be x km/hr
& speed of bus be y km/hr
Given that ,
Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train & the remaining by bus.
Train
Distance = 60 km
Speed = x
Time = ?
We know that
Time = (𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 )/𝑆𝑝𝑒𝑒𝑑
Time = 60/𝑥
Bus
Distance = 240 km
Speed = y
Time = ?
We know that
Time = (𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 )/𝑆𝑝𝑒𝑒𝑑
Time = 240/𝑦
Now,
Total time taken = 4 hours
60/𝑥 + 240/𝑦 = 4
Also,
She travels 100 km by train & remaining by bus, she takes 10 min longer.
So, she now takes 4 hours, 10 minutes
Train
Distance = 100 km
Speed = x
Time = ?
We know that
Time = (𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 )/𝑆𝑝𝑒𝑒𝑑
Time = 100/𝑥
Bus
Distance = 200 km
Speed = y
Time = ?
We know that
Time = (𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 )/𝑆𝑝𝑒𝑒𝑑
Time = 200/𝑦
Now,
Total time taken = 4 hours 10 minutes
100/𝑥 + 200/𝑦 = 4 + 10/60 hours
100/𝑥 + 200/𝑦 = 4 + 1/6 hours
100/𝑥 + 200/𝑦 = (4 (6) + 1)/6
100/𝑥 + 200/𝑦 = (24 + 1)/6
100/𝑥 + 200/𝑦 = 25/6
So, our two equations are
60/𝑥 + 240/𝑦 = 4 …(1)
100/𝑥 + 200/𝑦 = 25/6 …(2)
So, our equations become
60u + 240v = 4
100u + 200v = 25/6
6/25 (100u + 200v) = 1
24u + 48v = 1
Now, we have to solve
60u + 240v = 4 …(3)
24u + 48v = 1 …(4)
From (3)
60u + 240v = 4
60u = 4 – 240v
u = (4 − 240𝑣)/60
Putting value of u in (4)
24u + 48v = 1 24 ((4 − 240𝑣)/60) + 48v = 1
2 ((4 − 240𝑣)/5) + 48v = 1
Multiplying both sides by 5
5 × 2 ((4 − 240𝑣)/5) + 5 × 48v = 5 × 1
2(4 – 240v) + 240v = 5
8 – 480v + 240v = 5
– 480v + 240v = 5 – 8
–240v = −3
v = (−3)/(−240)
v = 1/80
Putting v = 1/80 in (3)
60u + 240v = 4
60u + 240 (1/80) = 4
60u + 3 = 4
60u = 4 – 3
60u = 1
u = 1/60
Hence, u = 𝟏/𝟔𝟎 , v = 𝟏/𝟖𝟎
But we have to find x & y
We know that
u = 𝟏/𝒙
1/60 = 1/𝑥
x = 60
v = 𝟏/𝒚
1/80 = 1/𝑦
y = 80
So, x = 60, y = 80 is the solution of the given equation
Therefore
Speed of train = x = 60 km/hr
& Speed of bus = y = 80 km/hr

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.