Mixture and Alligation
(i) To find the mean or average value of mixture when the prices of two or more ingredients which may be mixed together and the proportion in which they mixed are given.
(ii) To find the proportion in which the ingredients at given prices must be mixed to produce a mixture at a given price.
(1) The word Alligation literally means linking. The rule takes its name from the lines or links used in working out questions on the mixture.
(2) Alligation method is applied to the percentage value, ratio, rate, prices, speed etc and not for absolute values. That is, whenever per cent, per hour, per kg, per km etc, are being compared, we can use Alligation.
Rules of Alligation
if the gradients are mixed in a ratio, then
We represent it as under:
Then, (cheaper quantity) : (dearer quantity) = (d – m ) : (m – c)
Ex1: In what proportion must rice at Rs3.10 per kg be mixed with rice at Rs3.60 per kg, so that the mixture be worth Rs3.25 a kg?
By the alligation rule :
they must be mixed in the ratio 7 :3.
Milk and Water
Ex2: A mixture of certain quantity of milk with 16 liters of water is worth 90 P per liter. If pure milk be worth Rs1.08 per liter, how much milk is there in the mixture?
The mean value is 90 P and price of water is 0 P.
By the Alligation Rule, milk and water are in the ratio of 5 :1.
quantity of milk in the mixture = 5 ×16 = 80 liters.
Ex3: 300 gm of sugar solution has 40% sugar in it. How much sugar should be added to make it 50% in the solution?
The existing solution has 40% sugar, and sugar is to be mixed ; so the other solution has 100% sugar. So, by alligation method;
the two mixture should be added in the ratio 5 :1
Therefore, required sugar =
Direct formula :
Quantity of sugar added =
In this case,
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