QA02 MIXTURES / ALLIGATIONS

Alligation• It is the rule that enables us • to find the ratio in which two or more ingredients

at the given price • must be mixed to produce a mixture• of desired price.

Mean Price• The cost of a unit quantity of the mixture is called

the mean price

Rule of Alligation• If two ingredients are mixed, then • C.P of a unit quantity of cheaper = (c) • C.P of a unit quantity of dearer = (d)

C D

D–M

M–C

Mean Price (m)

• 2.In what ratio must a grocer mix two varieties of pulses costing `.15 and `.20 per kg respectively so as to get a mixture worth `.16.50 kg?

First type Mean Price Second type15 16.50 203.50 1.50

3.501.50=

3515=

73

02

• 4. A jar full of whisky contains 40% alcohol. A part

of this whisky is replaced by another containing 19% alcohol and now the percentage of alcohol was found to be 26%. The quantity of whisky replaced is:

First Mean Price Second type40% 26% 19%7 14

The ratio is 7:14 = 1:2. Hence quantity replaced =

04

• 5.In what ratio must water be mixed with milk to gain on selling the mixture at cost price?

• Let C.P. of 1 litre milk be Re. 1.• S.P. of 1 litre of mixture = Re.1, • C.P. of 1 litre of mixture =

First type Mean Price Second type0 1

Ratio of water and milk = 1:6.

05

• If n different vessels of equal size are filled with the mixture of P and Q in the ratio p1 : q1, p2 : q2, ……, pn : qn and content of all these vessels are mixed in one large vessel, then

Quantity of PQuantity of Q

=

p1p1+q1

+p2

p2+q2+…+

pnpn+qn

q1p1+q1

+q2p2+q2

+…+q𝑛p𝑛+qn

• Three equal buckets containing the mixture of milk and water are mixed into a bigger bucket.

• If the proportion of milk and water in the glasses are 3:1, 2:3 and 2:1 then find the proportion of milk and water in the bigger bucket.Let’s say P stands for milk and Q stands for water,So, p1:q1 = 3:1p2:q2=2:3p3 : q3=2:1

So in bigger bucket,Milk : Water = 109 : 71

𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦𝑜𝑓 𝑀𝑖𝑙𝑘𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑊𝑎𝑡𝑒𝑟=

33+1 +

22+3 +

22+1

13+1

+ 32+3

+ 12+1

𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦𝑜𝑓 𝑀𝑖𝑙𝑘𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑊𝑎𝑡𝑒𝑟=

10971

• If n different vessels of sizes x1, x2, …, xn are filled with the mixture of P and Q in the ratio p1 : q1, p2 : q2, ……, pn : qn and content of all these vessels are mixed in one large vessel, then

Quantity of PQuantity of Q

=

p1𝑥1p1+q1

+p2𝑥2p2+q2

+…+pn𝑥𝑛

pn+qnq1 𝑥1p1+q1

+q2 𝑥2p2+q2

+…+q𝑛𝑥𝑛

p𝑛+qn

• Three buckets of size 2 litre, 4 litre and 5 litre containing the mixture of milk and water are mixed into a bigger bucket. If the proportion of milk and water in the glasses are 3:1, 2:3 and 2:1 then find the proportion of milk and water in the bigger bucket.Sol:

• Let’s say P stands for milk and Q stands for water,So, p1:q1 = 3:1 , x1 = 2p2:q2=2:3  ,  x2 = 4p3 : q3=2:1  x3 = 5, so

So in bigger bucket,Milk : Water = 193 : 137

𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦𝑜𝑓 𝑀𝑖𝑙𝑘𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑊𝑎𝑡𝑒𝑟=

3 𝑋 23+1 +

2𝑋 42+3 +

2 𝑋 52+1

1𝑋 23+1

+ 3 𝑋 42+3

+ 1 𝑋 52+1

𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦𝑜𝑓 𝑀𝑖𝑙𝑘𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑊𝑎𝑡𝑒𝑟=

193137

Quantity of PQuantity of Q

=

p1𝑥1p1+q1

+p2𝑥2p2+q2

+…+pn𝑥𝑛

pn+qnq1 𝑥1p1+q1

+q2 𝑥2p2+q2

+…+q𝑛𝑥𝑛

p𝑛+qn

• Suppose a container contains x of liquid from which y units are taken out and replaced by water. After n operations, the quantity of pure liquid = units.

• A container contains 40 litres of milk. From this container 4 litres of milk was taken out and replaced by water. This process was repeated further two times. How much milk is now contained by the container?

• Milk =

• p gram of ingredient solution has a% ingredient in it.

• To increase the ingredient content to b% in the solution

• Quantity of ingredient need to be added = • 125 litre of mixture of milk and water

contains 25% of water. How much water must be added to it to make water 30% in the new mixture?Sol:

• Let’s say p = 125, b = 30, a = 25So from the equation

• 8.92 litres.

• If in x litres mixtures of milk and water, the ratio of milk and water is a:b, the quantity of water to be added in order to make this ratio c:d is

1. In a mixture of 60 litres, the ratio of milk and water is 2:1. If this ratio is to be 1:2, then the quantity of water to be further added is a) 20 litres. b) 30 litres c) 40 litres d) 60 litres.

=

Wednesday, May 3, 2023 C.S.VEERARAGAVAN, APTITUDE TRAINER,veeraa1729@gmail.com 15

x ad bcc a b

• 11. A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?

• Old ratio 3:5 new ratio 1:1• Qty of water to be added =

11

1. A milk vendor has 2 cans of milk. The first contains 25% water and the rest milk. The second contains 50% water. How much milk should he mix from each of the containers so as to get 12 litres of milk such that the ratio of water to milk is 3:5.• Let the cost of 1 litre milk be Re. 1• Milk in 1 litre mix. in 1st can =• Milk in 1 litre mix. in 2nd can =• Milk in 1 litre of final mix. =

01

First Mean Price

Second type

Ratio of two mixtures =

quantity of mixture taken from each can =12X= 6 litres.

• 1. A milk vendor has 2 cans of milk. The first contains 25% water and the rest milk. The second contains 50% water. How much milk should he mix from each of the containers so as to get 12 litres of milk such that the ratio of water to milk is 3:5.

• Let the cost of 1 litre milk be Re. 1• Milk in 1 litre mix. in 1st can =.C.P. of 1 litre mix. in 1st can Re. .• Milk in 1 litre mix. in 2nd can = C.P. of 1 litre mix. in 2nd can Re. • Milk in 1 litre of final mix. =. Mean Price = .

First type Mean Price Second type

Ratio of two mixtures = 1:1So, quantity of mixture taken from each can = 6 litres.

01