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 Alligation or Mixture :-
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Direct Formula / Rule 8 :

• Theorem: A person travels D km in T hours in two stages. In the first part of the journey, he travels by bus at the speed of x km/hr. In the second part of the journey, he travels by train at the speed of y km/hr.  Then the distance travelled by bus is = ( yT - D ) x Km y - x
 and the distance travelled by train is = ( D - xT ) y Km y - x

• Example :

• A person travels 285 km in 6 hours in two stages. In the first part of the journey, he travels by bus at the speed of 40 km per hr. In the second part of the journey, he travels by train at the speed of 55 km per hr. How much distance did he travel by train?

• Detail Method :
Let the person travels for x hours by the train.
∴ Time for which he travels by bus = (6 - x) hours.
The distance travelled by train = 55 X x km and
the distance travelled by bus = (6 - x) 40 km
According to the question,
40 (6 - x) + 55x = 285
40 X 6 - 40x + 55x = 285
240 - 40x + 55x = 285
15x = 285 - 240
15x = 45 ∴ x = 3 hours.
Distance travelled by train = 55 X 3 = 165 km.

• Ailigation Method :
In this question, the alligation method is applicable for the speed.
 Speed of bus Speed of train
 40 Average Speed 285/6 55 45/6 45/6
∴ time spent in bus : time spent in train = 45/6 : 45/6 = 1 : 1
∴ distance travelled by train = 55 X 3 = 165 km.

• Quicker Method : Here you can use direct formula :
 reqd distance = 285 - 40 X 6 X 55 = 3 X 55 = 165 km. 55 - 40

Exercise :
1. A person travels 255 km in 7 hours in two stages. In the first part of the journey, he travels by bus at the speed of 30 km per hr. In the second part of the journey, he travels by train at the speed of 45 km per hr. How much distance did he travel by bus?
2. A person travels 245 km in 6 hours in two stages. In the first part of the journey, he travels by bus at the speed of 30 km per hr. In the second part of the journey, he travels by train at the speed of 50 km per hr. How much distance did he travel by train?
3. A person travels 490 km in 6 hours in two stages. In the first part of the journey, he travels by bus at the speed of 60 km per hr. In the second part of the journey, he travels by train at the speed of 100 km per hr. What is the ratio between distances travelled by bus and train?
4. Answers : 1 = 120 km,   2 = 162.5 km,   3 = 33 : 65

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