# Alligation or Mixture Questions

Q:

Tea worth Rs. 126 per kg are mixed with a third variety in the ratio 1: 1 : 2. If the mixture is worth Rs. 153 per kg, the price of the third variety per kg will be

 A) Rs. 169.50 B) Rs.1700 C) Rs. 175.50 D) Rs. 180

Answer & Explanation Answer: C) Rs. 175.50

Explanation:

Since first second varieties are mixed in equal proportions, so their average price = Rs.(126+135/2) = Rs.130.50
So, the mixture is formed by mixing two varieties, one at Rs. 130.50 per kg and the other at say, Rs. x per kg in the ratio 2 : 2, i.e., 1 : 1. We have to find x.

Cost of 1 kg tea of 1st kind         Cost of 1 kg tea of 2nd kind

x-153/22.50 = 1  => x - 153 = 22.50  => x=175.50.
Hence, price of the third variety = Rs.175.50 per kg.

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Q:

A can contains a mixture of two liquids A and B in the ratio  7 : 5. When 9 litres of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 7 : 9. How many litres of liquid A was contained by the can initially?

 A) 10 B) 20 C) 21 D) 25

Explanation:

Suppose the can initially contains 7x and 5x litres of mixtures A and B respectively

Quantity of A in mixture left =$\inline (7x-\frac{7}{12}\times 9)=(7x-\frac{21}{4})$ litres.

Quantity of B in mixture left =  $\inline (5x-\frac{5}{12}\times 9)=(5x-\frac{15}{4})$litres.

$\inline \frac{(7x-\frac{21}{4})}{(5x-\frac{15}{4}+9)}=\frac{7}{9}$

$\inline \Rightarrow \frac{28x-21}{20x+21}=\frac{7}{9}$

$\inline \Rightarrow 252x-189=140x+147$

$\inline \Rightarrow x=3$

So, the can contained 21 litres of A.

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Q:

8 litres are drawn from a cask full of wine and is then filled with water. This operation is performed three more times. The ratio of the quantity of wine now left in cask to that of the water is 16 : 65. How much wine the cask hold originally?

 A) 18 litres B) 24 litres C) 32 litres D) 42 litres

Answer & Explanation Answer: B) 24 litres

Explanation:

Let the quantity of the wine in the cask originally be x litres
Then, quantity of wine left in cask after 4 operations =$\inline \fn_cm \left [ x(1-\frac{8}{x})^4 \right ]$litres

$\inline \fn_cm \therefore \left [ \frac{x(1-\frac{8}{x})^4}{x} \right ]=\frac{16}{81}$

$\inline \fn_cm \Rightarrow \left [ 1-\frac{8}{x} \right ]^4=(\frac{2}{3})^4$

$\inline \fn_cm \Rightarrow \left ( \frac{x-8}{x} \right )=\frac{2}{3}$

${\color{Black}&space;\Rightarrow&space;3x-24=2x}$

${\color{Black}&space;\Rightarrow&space;x=24}$

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Q:

A vessel is filled with liquid, 3 parts of which are water and 5 parts of 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?

 A) 1/3 B) 1/4 C) 1/5 D) 1/7

Explanation:

Suppose the vessel initially contains 8 litres of liquid.

Let x litres of this liquid be replaced with water.
Quantity of water in new mixture =  $\inline (3-\frac{3x}{8}+x)$litres.

Quantity of syrup in new mixture =  $\inline (5-\frac{5x}{8})$ litres.

${\color{Blue}&space;\therefore&space;}$ $\inline (3-\frac{3x}{8}+x)=(5-\frac{5x}{8})$

=>   5x + 24 = 40 - 5x

=>   10x = 16    => x = 8/5

So, part of the mixture replaced = $\inline \left ( \frac{8}{5} \times \frac{1}{8}\right )$ = 1/5.

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Q:

A merchant has 1000 kg of sugar part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. The Quantity sold at 18% profit is

 A) 400 kg B) 560 kg C) 600 kg D) 640 kg

Answer & Explanation Answer: C) 600 kg

Explanation:

By the rule of alligation:

Profit of first part                         Profit of second part

So, ratio of 1st and 2nd parts = 4 : 6 = 2 : 3.

${\color{Blue}&space;\therefore&space;}$ Quantity of 2nd kind = (3/5 x 1000)kg = 600 kg.