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Alligation is a method of solving arithmetic problems related to mixtures of ingredients. Please note that alligation method is applied for percentage value, ratio, rate, prices, speed, etc. and not for absolute value. That is whenever per cent, per km, per hour, per kg, are being compared, we can use Alligation.
Common trick for Ratio-Proportion and Mixture Alligation : Almost 50% of the questions are solvable just by going through the options. Just go through the questions I have solved in this article and you will know the approach.
Q.1
Note that Rs. 180/kg and Rs. 280/kg are cost prices, while Rs. 320 is the selling price. To apply the alligation formula, all the three prices should be similar. So we will convert SP into CP
Given SP = Rs. 320/kg, Profit = 20%
Hence CP = 320/1.2 = Rs. 800/3
So the Mean price is Rs. 800/3 per kg
Now you can apply the formula-
Type 1 : Type 2 = 280 - 800/3 : 800/3 - 180 = 2 : 13
Answer : (B)
Both the containers have equal capacity. Let us assume that both containers are of 28 litres. Why 28? Because 28 is the LCM of (3 + 1) and (5 + 2) or 4 and 7. So taking the capacity as 28 litres will make your calculations easier.
In Container 1, we have (3/4)*28 = 21 litres of milk and (1/4)*28 = 7 litres of water.
In Container 1, we have (5/7)*28 = 20 litres of milk and (2/7)*28 = 8 litres of water.
Total milk in both the containers = 21 + 20 = 41
Total water in both the containers = 7 + 8 = 15
Milk : Water = 41 : 15
Answer : (D)
Shortcut
Container 1 has 3 times more milk than water
Container 2 has 2.5 times more milk than water
When the contents of the two containers are mixed, the milk will still be more than water. How much more ? Somewhere between 2.5 and 3 times
(D) is the only option where the quantity of milk is around 2.7 times (i.e. between 2.5 and 3) that of the water.
Milk in vessel A = 4/7
Milk in vessel B = 2/5
Milk in vessel C = 1/2 (because in vessel C, milk and water are present in 1:1 ratio)
You have to mix 4/7 and 2/5, to produce 1/2. Hence 1/2 is the Mean Price.
A : B = (1/2 - 2/5)/(4/7 - 1/2) = 14 : 10 = 7 : 5
Shortcut
Final ratio of the three varieties is 5 : 7 : 9
The question asks us the quantity of third variety of tea in the final mixture. From the above ratio, it is clear that the quantity of the third variety is a multiple of 9. So 45 is the only option possible.
Answer : (D)
Method
Let the three quantities be 4x, 5x and 8x
New quantities are 4x + 5, 5x + 10 and 8x + p
Now 4x + 5 : 5x + 10 : 8x + p = 5 : 7 : 9
(4x + 5)/(5x + 10) = 5/7 and (4x + 5)/(8x + p) = 5/9
Solving 1st equation, we get x = 5
Solving 2nd equation, we get p = 5
In the final mixture the quantity of the third variety is 8x + p = 8*5 + 5 = 45
In this question we will use the below formula
So from the above formula
(Quantity of acid left)/(Quantity of acid in the original mixture) = (1 - 4/20)^2 = 16:25
Answer : (A)
Let the original quantities of A and B be 4x and x
In 10 litres, quantity of A = 4/5 * 10 = 8 litres
In 10 litres, quantity of B = (10 - 8) = 2 litres
New quantities of A and B are 2x and 3x
(Original Quantity of A) - (New quantity of A) = 8 litres[Because after taking out 10 litres of the mixture, the quantity of liquid A reduced by 8 litres]
So, 4x - 2x = 8
or x = 4
Hence quantity of liquid A in original mixture = 4*4 = 16 litres
Answer : (C)
Note : In the above question, there were two different ratios 4:1 and 2:3, then too I took the same constant of proportionality for them, i.e. 'x' because the following two conditions were met:
1. The volume of mixture did not change (Like in this question 10 litres were replaced, not removed)
2. The two ratios had same no. of parts (4:1 and 2:3 both have 5 parts)
You can take different constant to solve the question, but that will make the calculations little lengthy.
Since the ratio of alcohol and water is 1:4, hence quantities of alcohol and water in the mixture are 3 litres and 12 litres respectively.
Total volume will become 18 litres after adding 3 litres water
% of alcohol = 3/18 * 100 = 50/3%
Answer : (B)
Q. 9)
Q. 10) A man spends 75% of his income. His income is increased by 20% and he increased his
expenditure by 10%. His savings are increased by
(A) 50% (B) 25% (C) 75/2% (D) 10%
Let the income = Rs. 100
Then expenditure = 75% of 100 = Rs. 75
Savings = Rs. 25
New income = 1.2 * 100 = Rs. 120
New expenditure = 1.1*75 = Rs. 82.5
New savings = 120 – 82.5 = Rs. 32.5
% increase in savings = (37.5 - 25)/25 * 100 = 50%
Answer: (A)
Let us solve some CGL questions that may not seem to belong to Mixture-Alligation category, but are easier to solve via Alligation formula...
This method is equally easy.
Let the distance travelled on foot be X km. Then distance travelled on bicycle = (61 - X)
Now he travelled for 9 hours. Hence
X/4 + (61 - X)/9 = 9
Solve for X
You will get X = 16 km
A : B = 3 : 4 and B: C = 4 : 5
A : B : C = 3 : 4 : 5
Average score of classes A, B and C =
Alternate Method (Apply alligation on interest rates)
Rs. 85 interest is received in 1 year from an investment of Rs. 1500
SI = P*r*t/100
r = 17/3% [Put SI=85, P=1500, t=1]
Now the mean ratio is 17/3
Apply allegation formula-
Money invested at 6% : Money invested at 5% = (17/3)-5 : 6- 17/3 = 2 : 1
#dailyquizadda #BestOfLuck
Alligation is a method of solving arithmetic problems related to mixtures of ingredients. Please note that alligation method is applied for percentage value, ratio, rate, prices, speed, etc. and not for absolute value. That is whenever per cent, per km, per hour, per kg, are being compared, we can use Alligation.
Common trick for Ratio-Proportion and Mixture Alligation : Almost 50% of the questions are solvable just by going through the options. Just go through the questions I have solved in this article and you will know the approach.
Rule of Alligation
Here Mean Price is something which applies on the whole thing. If two
varieties of tea costing Rs. X and Rs. Y respectively are mixed and
sold at Rs. Z, then Z is the mean price because it is price of the
mixture.
Now I will take up some SSC CGL questions of Ratio-proportion and Mixture-Alligation.
Note that Rs. 180/kg and Rs. 280/kg are cost prices, while Rs. 320 is the selling price. To apply the alligation formula, all the three prices should be similar. So we will convert SP into CP
Given SP = Rs. 320/kg, Profit = 20%
Hence CP = 320/1.2 = Rs. 800/3
So the Mean price is Rs. 800/3 per kg
Now you can apply the formula-
Type 1 : Type 2 = 280 - 800/3 : 800/3 - 180 = 2 : 13
Answer : (B)
Q. 2)
Both the containers have equal capacity. Let us assume that both containers are of 28 litres. Why 28? Because 28 is the LCM of (3 + 1) and (5 + 2) or 4 and 7. So taking the capacity as 28 litres will make your calculations easier.
In Container 1, we have (3/4)*28 = 21 litres of milk and (1/4)*28 = 7 litres of water.
In Container 1, we have (5/7)*28 = 20 litres of milk and (2/7)*28 = 8 litres of water.
Total milk in both the containers = 21 + 20 = 41
Total water in both the containers = 7 + 8 = 15
Milk : Water = 41 : 15
Answer : (D)
Shortcut
Container 1 has 3 times more milk than water
Container 2 has 2.5 times more milk than water
When the contents of the two containers are mixed, the milk will still be more than water. How much more ? Somewhere between 2.5 and 3 times
(D) is the only option where the quantity of milk is around 2.7 times (i.e. between 2.5 and 3) that of the water.
Q. 3) 
Milk in vessel B = 2/5
Milk in vessel C = 1/2 (because in vessel C, milk and water are present in 1:1 ratio)
You have to mix 4/7 and 2/5, to produce 1/2. Hence 1/2 is the Mean Price.
A : B = (1/2 - 2/5)/(4/7 - 1/2) = 14 : 10 = 7 : 5
Q. 4)
Shortcut
Final ratio of the three varieties is 5 : 7 : 9
The question asks us the quantity of third variety of tea in the final mixture. From the above ratio, it is clear that the quantity of the third variety is a multiple of 9. So 45 is the only option possible.
Answer : (D)
Method
Let the three quantities be 4x, 5x and 8x
New quantities are 4x + 5, 5x + 10 and 8x + p
Now 4x + 5 : 5x + 10 : 8x + p = 5 : 7 : 9
(4x + 5)/(5x + 10) = 5/7 and (4x + 5)/(8x + p) = 5/9
Solving 1st equation, we get x = 5
Solving 2nd equation, we get p = 5
In the final mixture the quantity of the third variety is 8x + p = 8*5 + 5 = 45
Q. 5)
In this question we will use the below formula
Q. 6)
So from the above formula
(Quantity of acid left)/(Quantity of acid in the original mixture) = (1 - 4/20)^2 = 16:25
Answer : (A)
Q. 7)
Let the original quantities of A and B be 4x and x
In 10 litres, quantity of A = 4/5 * 10 = 8 litres
In 10 litres, quantity of B = (10 - 8) = 2 litres
New quantities of A and B are 2x and 3x
(Original Quantity of A) - (New quantity of A) = 8 litres[Because after taking out 10 litres of the mixture, the quantity of liquid A reduced by 8 litres]
So, 4x - 2x = 8
or x = 4
Hence quantity of liquid A in original mixture = 4*4 = 16 litres
Answer : (C)
Note : In the above question, there were two different ratios 4:1 and 2:3, then too I took the same constant of proportionality for them, i.e. 'x' because the following two conditions were met:
1. The volume of mixture did not change (Like in this question 10 litres were replaced, not removed)
2. The two ratios had same no. of parts (4:1 and 2:3 both have 5 parts)
You can take different constant to solve the question, but that will make the calculations little lengthy.
Q. 8)
Since the ratio of alcohol and water is 1:4, hence quantities of alcohol and water in the mixture are 3 litres and 12 litres respectively.
Total volume will become 18 litres after adding 3 litres water
% of alcohol = 3/18 * 100 = 50/3%
Answer : (B)
Q. 9)
Shortcut
Originally there are 1512 story books
Final ratio of Story books : Others = 15:4
That means the story books are a multiple of 15.
Just look at the options and see which number when added to 1512, will give a multiple of 15
Answer : (C)
Method
Let the no of story and other books be 7x and 2x respectively
Given 7x = 1512
x = 216
Now let the final quantity of story and other books be 15y and 4y respectively.
Since only story books are added to the collection, hence the quantity of other books has remained unchanged.
So 2x = 4y or y = x/2 = 108
We have to find 15y - 7x = 15y - 14y = y [Since x = 2y]
So answer is 108.
Q. 10) A man spends 75% of his income. His income is increased by 20% and he increased his
expenditure by 10%. His savings are increased by
(A) 50% (B) 25% (C) 75/2% (D) 10%
Let the income = Rs. 100
Then expenditure = 75% of 100 = Rs. 75
Savings = Rs. 25
New income = 1.2 * 100 = Rs. 120
New expenditure = 1.1*75 = Rs. 82.5
New savings = 120 – 82.5 = Rs. 32.5
% increase in savings = (37.5 - 25)/25 * 100 = 50%
Answer: (A)
Let us solve some CGL questions that may not seem to belong to Mixture-Alligation category, but are easier to solve via Alligation formula...
Q. 11)
Let the gain% of the remaining land be x%. Then
Given, 2/5 land is sold on loss and hence 3/5 land is sold on profit
So, Profit : Loss = 3:2
Hence 16 : (x - 10) = 3 : 2
16/(x - 10) = 3/2
or x = 62/3%
Answer : (B)
Q. 12)
His average speed for the entire journey = 61/9 km/hour
Time taken on foot : Time taken on bicycle = (9 - 61/9)/(61/9 - 4) = 4:5
Time taken on foot = 4/9 * 9 = 4 hours
Hence distance travelled on foot = (Speed on foot * Time taken on foot) = 4*4 = 16 km
Answer : (C)
Alternate methodThis method is equally easy.
Let the distance travelled on foot be X km. Then distance travelled on bicycle = (61 - X)
Now he travelled for 9 hours. Hence
X/4 + (61 - X)/9 = 9
Solve for X
You will get X = 16 km
Q. 13)
Number of legs per head = 420/180 = 7/3
Now cows have 4 legs while hens have 2 legs
Cows : Hens = 1/3 : 5/3 = 1 : 5
Now total number of cows and hens = total number of heads = 180
Number of cows = 1/6 * 180 = 30
Answer : (B)
Q. 14)
A : B : C = 3 : 4 : 5
Average score of classes A, B and C =
Answer : (B)
Q. 15)
The man spends 75% of his income, that means he saves 25%
Hence Expenditure : Saving = 75 : 25 = 3 : 1
Let the % increase in the savings be x.
I have written 3 : 1 as 30 : 10 so that (% increase in income - %
increase in expenditure) could agree with 3 : 1 ratio. As we know 20 -
10 = 10, not 1.
So now from the figure we can see that x - 20 = 30
Hence x = 50%
Answer : (C)
Q. 16)
If whole money is invested at 6%, then interest = 6% of 1500 = Rs. 90
If whole money is invested at 5%, then interest = 5% of 1500 = Rs. 75
Hence, Money invested at 6% : Money invested at 5% = 10 : 5 = 2 : 1
Money invested at 5% = 1/3 * 1500 = Rs. 500
Answer : Rs. 500
Alternate Method (Apply alligation on interest rates)
Rs. 85 interest is received in 1 year from an investment of Rs. 1500
SI = P*r*t/100
r = 17/3% [Put SI=85, P=1500, t=1]
Now the mean ratio is 17/3
Apply allegation formula-
Money invested at 6% : Money invested at 5% = (17/3)-5 : 6- 17/3 = 2 : 1
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