ALLIGATIONS
In the chapter on Averages, we have seen the use of weighted average formula. To recollect, the weighted average is
used when a number of smaller groups are mixed together to form one larger group.
If the average of the measured quantity was
A1 for group 1 containing n1 elements
A2 for group 2 containing n2 elements
A3 for group 3 containing n3 elements
A4 for group 4 containing nk elements
We say that the weighted average, Aw is given by :
In the case of the situation where just two
groups are being mixed, we can write this as:
Ex. 1 4 kg of rice at Rs. 5 per kg is mixed with 8 kg of rice at Rs. 6 per kg. find the average price of the mixture.
Sol. 
is the same as
Note : in this case, teh problem associated with the cross method is overcome and the solution becomes graphical.
Case 3 : A1, Aw, n1 and n2 are known; A2 is a unknown.
Ex. 2 5 kg of rice at Rs. 6 per kg is mixed with 4 kg of rice to get a mixture costing Rs. 7 per kg. Find the price of the costlier rice.
Using straight line method :
4 corresponds to 7 – 6 and 5 corresponds to x – 7.
The through process should go like :
4 ® 1
\ 5 ® 1.25
Hence, x – 7 = 1.25
and x = 8.25
ASSIGNMENT
1. If 5 kg of salt costing Rs. 5/kg and 3 kg of salt costing Rs/ 4/kg are mixed, find the average cost of the mixture per kilogram
(A) Rs. 4.5 (B) Rs. 4.625 (C) Rs. 4.75 (D) Rs. 4.125
(E) Rs. 4.25
2. Two types of oils having the rates of Rs. 4/kg and Rs. 5/kg respectively are mixed in order to produce a mixture having the
rate of Rs. 4.60/kg. What should be the amount of the second type of oil if the amount of the first type of oil in the mixture is 40 kg ?
(A) 75 kg (B) 50 kg
(C) 60 kg (D) 40 kg
(E) None of these
3. How many kilograms of sugar worth Rs. 3.60 per kg should be mixed with 8 kg of sugar worth Rs. 4.20 per kg,
such that by selling the mixture at Rs. 4.40 per kg. there may be a gain of 10%?
(A) 6 kg (B) 3 kg
(C) 2 kg (D) 5 kg
(E) 4 kg
4. A mixture of 125 gallons of wine and water contains 20% water. How much water must be added to the mixture in order to
increase the percentage of water to 25% of the new mixture?
(A) 10 gals (B) 8.5 gals
(C) 8 gals (D) 6.66 gals
(E) 8.33 gals
5. Ravi lends Rs. 3600 on simple interest to Harsh for a period of 5 years. He lends a part of the amount at 4% interest and
the rest at 6% and receives Rs. 960 as the amount of interest. How much money did he lend on 4% interest rate ?
(A) Rs. 2800 (B) Rs. 2100
(C) Rs. 2400 (D) Rs. 1200
(E) 1000
6. 400 students took the AMS Mock Cat in Lucknow. 60% of the boys and 80% of the girls cleared the cut off in the
eamination. If the total percentage of students qualifying is 65%, how many girls appeared in the examination ?
(A) 100 (B) 120
(C) 150 (D) 300
(E) 175
7. A man purchased a cow and a calf for Rs. 1300. He sold the calf at a profit of 20% and the cow at a profit of 25%. In this way,
e of the cow.
(A) Rs. 1100 (B) Rs. 600
(C) Rs. 500 (D) Rs. 400
(E) Rs. 800
8. The average salary per head of all employees of a company is Rs. 600. The average salary of 120 officers is Rs. 4000.
If the average salary per head of the rest of the employees is Rs. 560, find the total number of workers in the company ?
(A) 10200 (B) 10320
(C) 10500 (D) 10680
(E) 10800
9. A dishonest milkman purchased milk at Rs. 100 per litre and mixed 5 litres of water in it. By selling the mixture at the
rate of Rs. 10 per litre he earned a profit of 25%. The quantity of the amount of the mixture that he had was ?
(A) 15 litres (B) 20 litres
(C) 25 litres (D) 30 litres
(E) None of these
10. A cistern contains 50 litres of water. 5 litres of water is taken out of it and replaced by wine. The process
is repeated again. Find the proportion of wine and water in the resulting mixture.
(A) 1 : 4 (B) 41 : 50
(C) 19 : 81 (D) 81 : 19
(E) 50 : 41
11. A bartender stole champagne from a bottle that contained 50% of spirit and he replaced what he had stolen with champagne
having 20% spirit. The bottle then contained only 25% spirit. How much of the bottle did he steal ?
(A) 80% (B) 83.33%
(C) 85.71% (D) 88.88%
12. A bag contains a total of 105 coins of Rs. 1,50 p and 25 p denominations. Find the total number of coins of Re 1 if there are
a total of 50.5 rupees in the bag and it is known that the number of 25 paise coins are 133.33% more than the number of 1 rupee coins.
(A) 56 (B) 25 (C) 24 (D) None of these
13. A man possessing Rs. 6800, lent a part of it at 10% simple interset and the remaining at 7.5%
was Rs. 1904. Find the sum lent at 10% rates.
(A) Rs. 1260 (B) Rs. 1700
(C) Rs. 1360 (D) None of these
14. If a man decides to travel 80 kilometres in 8 hours partly by foot and partly on a bycycle, his speed on foot being 8 km/h
and that on bycycle being 16 km/h, what distance would he travel on foot ?
(A) 20 km (B) 30 km
(C) 48 km (D) 60 km
15. Two vessels contain a mixture of spirit and water. In the first vessel the ratio of spirit to water is 8 : 3 and in the second vessel the ratio is
5 : 1. A 35 litre cask is filled from these vessels so as to contain a mixture of spirit and water in the ratio of 4 : 1. How many litres are taken from the first vessel ?
(A) 11 litres (B) 22 litres
(C) 16.5 litres (D) 17.5 litres
ANSWER KEY
1). B 2). C 3) E 4) E 5) D 6) A 7) E 8) B
9) C 10) C 11) B 12) C 13) C 14) C 15) A