Mixture and Alligation - Important Points
Alligation Rule
Alligation is method to find the proportion in which ingredients at two different prices must be mixed to produce a mixture at a given price.
If gradients are mixed in a ratio then we can write-
(Quantity of cheaper item)/(Quantity of dearer item) = (Price of dearer – Average Price)/(Average Price – Price of cheaper)
Price Price
of cheaper (c) of dearer (d)
\ /
Average Price (m)
/ \
(d-m) : (m-c)
Concept of Replacement
Suppose a container has a solution from which some quantity of solution is taken out and replaced with one of the ingredients. This process is repeated n times then,
Final Amount of ingredient that is not replaced =
Initial Amount × (Volume after removal/Volume after replacing)
Above formula is not only true for absolute amounts but for ratios as well.
Final ratio of ingredient not replaced to total=
Initial ratio x (Volume after removal/Volume after replacing)
Example 1: In what proportion must rice at Price 82 per kg must be mixed with rice at Price 92 per kg, so that the mixture be worth Rs 86 a Kg?
Solution: CP of unit quantity of Dearer item = 92/kg
CP of unit quantity of Cheaper item = 82/kg
So we can use above formula:
(Quantity of cheaper item)/(Quantity of dearer item) = (Price of dearer – Average Price)/(Average Price – Price of cheaper)
=> 92 – 86 / 86 – 82 = 6 / 4 = 3 / 2
So, the required ratio is 3 : 2
Example 2: How many kg of rice at Rs. 60 per kg, must be be mixed with 30 kg of rice at Rs 25 per kg, so that on selling the mixture at Rs 50 per kg a gain of 25 % on the outlay is obtained?
Solution: As seller is gaining 25 % profit on mixture, so-
CP x 125/100 = 50 => CP = 40
Now use the formula of alligation to find quantity of dearer rice,
(Quantity of cheaper item)/(Quantity of dearer item) = (Price of dearer – Average Price)/(Average Price – Price of cheaper)
=> 60 – 40 / 40 – 25 = 20 / 15 = 30 / x
=> x = 22.5 kg
Example 3: In what proportion must water be mixed with milk to gain 20 % by selling it at cost price?
Solution: Let cost price of milk be Rs 1 per litre, then SP of mixture is also Rs 1 per litre
Now CP of mixture be = 1 - (20 % of Rs 1) = 80 / 100 = Rs 4 / 5
1 0
\ /
4/5
/ \
(4/5 - 0) : (1 – 4/5)
Required ratio = (4/5) / (1/5) = 4 : 1
Example 4: Milk and water are mixed in a vessel A in the ratio 5 : 3 and in vessel B, in ratio 9 : 7. In what ratio should quantities be taken from the two vessels so as to form a mixture in which milk and water will be in the proportion of 7 : 5 ?
Solution: In vessel A, milk = 5 / 5 + 3 = 5 / 8 of the weight of mixture
In vessel B milk = 9 / 9 + 7 = 9 / 16 of the weight of mixture
Now, we have to form a mixture in which milk be 7 / 12 of the weight of the mixture
Now according to rule of alligation:
5/8 9/16
\ /
7/12
/ \
(7/12 – 9/16) : (5/8 – 7/12)
They should be mixed in the ratio = (7/12) – (9/16) : (5/8) – (7/12)
= 1/24 : 1/48 = 2 : 1
Example 6: 800 gm alcohol solution has 40% alcohol in it, How many grams of alcohol should be added to make it 60 % in the solution?
Solution: By applying alligation rule, alcohol and water can be shown in the following way-
40% 100%
\ /
60%
/ \
(40%) : (20%)
So, the two mixtures should be added in ratio 2 : 1
=> 400 gm alcohol should be added.