Alligation or Mixture

Alligation : It is the rule that enable us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture at a given price.

Mean Price : The cost price of a unit quantity of the mixture is called the mean price.

Rule of Alligation : If two ingredients are mixed, then :

c = Cost price of a unit quantity of cheaper

d= C.P. of a unit quantity of dearer

(Quantity of Cheaper) / (Quantity of dearer) =

{(C.P. of Dearer) - (Mean Price)}

{ (Mean Price) - (C.P. of Cheaper)}

(Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c)

1. In what ratio must tea at Rs. 62 per Kg be mixed with tea at Rs. 72 per Kg so that the mxture must be worth Rs. 64.50 per Kg ?

Sol : By rule of alligation we have

c= C.P of 1 kg of 1st kind (6200 Paise)

d= C.P. of 1 kg tea of 2nd kind

m= Mean Price (6450 Paise)

c : d = (d-m) : (m-c)

(Tea of 1st kind) : (Tea of 2nd Kind) = 750 : 250 = 3: 1

2.In what ratio must water be mixed with milk to gain 20% by telling the mixture at cost price ?

Sol : Let C.P. of Milk be Re 1. per Litre

Then S.P. of 1 Litre of mixture = Re. 1

Gain obtained = 20%

C.P. of 1 litre of mix = (100 x 1\120) = Re. 5/6

c = C.P. of 1kg of water

d= C.P. of 1kg of milk

m =mean price = Re. 5/6

By the rule of alligation we have

water : milk = 1/6 : 5/6 = 1 : 5

3. How many Kgs of rice costing Rs. 800 per kg must be mixed with 36 kg of rice costing Rs. 5.40 per kg so that 20% gain may be obtained by selling the mixture at Rs. 7.20 per kg ?

Sol : S.P. of 1 kg mix = Rs 7.20 , Gain = 20%

C.P. of 1 kg mix = Rs. (7.20 x 100)/120 = Rs. 6

By rule of alligation

Rice of 1st kind : Rice of 2nd kind = 60 : 200 = 3 : 10

Let x kg of rice of 1st kind be mixed with 36kg of rice of 2nd kind. Then

3 : 10 = x : 36 or 10x = 3 x 36 or x=10.8 kg.

4. A man travelled a distance of 90Km in 9 hours partly on foot at 8 kmph and partly on biycle at 17 kmph. Find the distance travelled on foot.

Sol : Average distance covered in 1 hour = 90/9 km = 10km

By rule of alligation we have

Ratio of times taken = 7 : 2

Thus out of 9 hours, he took 7 hours on foot

Distance covered on foot = (8 x 7) Km = 56 km

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