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 Alligation or Mixture :-
 ::Important Formulas ::Examples

Direct Formula / Rule 20 :

• Theorem: A man mixes M1 litres of milk at Rs x per litre with M2 litres at Rs y per litre. Amount of water, that should be added to make the average value of the mixture Rs z per litre, is given by  = [ M1(x - z) + M2(y - z) ] litres. z

• Example :

• A man mixes 5 kilolitres of milk at Rs 600 per kilolitre with 6 kilolitres at Rs 540 per kilolitre. How many kilolitres of water should be added to make the average value of the mixture Rs 480 per kilolitre?

• Detail Method :
According to the question,
Cost of 5 kilolitres of milk = 600 X 5 = Rs 3000
Cost of 6 kilolitres of milk = 540 X 6 = Rs 3240
Now, we suppose that x kilolitres of water is added.
Total amount of the mixture = 5 + 6 + x = (11 + x) kilolitres.
Total cost of the mixture = Rs 3000 + Rs 3240= Rs 6240
From the question.  = 6240 = 480 11 + x
 or, 11 + x 6240 = 13 480
∴ x = 2 kilolitres

• Ailigation Method :
Cost of milk when two qualities are mixed  = 5 X 600 + 6 X 540 = Rs 6240 per kilolitre. 5 + 6 11
Cost of water = Rs 0/ kilolitre.  So, First mixture (milk) Second mixture (water)
 6240/11 Average Speed 480 0 480 960/11
∴ Ratio of milk and water = 480 : 960/11 = 1 : 2/11 = 11 : 2
Which implies that 11 kilolitres of milk should be mixed with 2 kilolitres of water. Thus 2 kilolitres of water should be added.

• Quicker Method : Here you can use direct formula :
The required amount of water  = 5 X (600 - 480) + 6 X (540 - 480) 480
 = 5 X 120 + 6 X 60 = 960 = 2 kilolitres. 480 480

Exercise :
1. A man mixes 5 kilolitres of milk at Rs 6000 per kilolitre with 6 kilolitres at Rs 5400, and with sufficient water to make the average value Rs 4800 per kilolitre. How many kilolitres of water has he added?
2. A man mixes 6 kilolitres of milk at Rs 650 per kilolitre with 7 kilolitres at Rs 600, and with sufficient water to make the average value Rs 540 per kilolitre. How many kilolitres of water has he added?
3. A man mixes 6 kilolitres of milk at Rs 325 per kilolitre with 9 kilolitres at Rs 300, and with sufficient water to make the average value Rs 270 per kilolitre. How many kilolitres of water has he added?
4. Answers : 1 = 2 kilolitres,   2 = 2 kilolitres,   3 = 2 2/9 kilolitres

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