Alligation or Mixture Questions and Answers

Q: Two solutions S1 and S2 contain whisky and soda in the ratio 2 : 5 and 6 : 7 respectively. In what ratio these solutions be mixed to get a new solution S3, containing whisky and soda in the ratio 5 : 8 ?

Let the amount taken from S1 be 7x And amount taken from S2 be 13y (2x + 6y)/(5x + 7y) = 5/8 16x + 48y = 25x + 35y 9x = 13y x/y = 13/9 Actual ratios of amounts = 7x/13y = (7/13) * (13/9) = 7/9

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111.

Two solutions S1 and S2 contain whisky and soda in the ratio 2 : 5 and 6 : 7 respectively. In what ratio these solutions be mixed to get a new solution S3, containing whisky and soda in the ratio 5 : 8 ?

A.7:9
B.21:5
C.23:6
D.6:23

Answer: A

Let the amount taken from S1 be 7x
And amount taken from S2 be 13y

(2x + 6y)/(5x + 7y) = 5/8
16x + 48y = 25x + 35y
9x = 13y
x/y = 13/9

Actual ratios of amounts 
               = 7x/13y
               = (7/13) * (13/9)
               = 7/9

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112.

In what ratio should a kind of sugar at 8.70 Rs/kg be mixed with another kind of sugar at 9.70 Rs/kg so that the mixture be worth 9 Rs/kg?

A.3:7
B.4:7
C.7:4
D.7:3

Answer: D

The alligation rule gives the ratio in which two ingredients must be mixed:

\[ \frac{\text{cheaper quantity}}{\text{dearer quantity}} = \frac{\text{dearer} - \text{mean}}{\text{mean} - \text{cheaper}} \]

\[ \frac{\text{Rs. 8.70 sugar}}{\text{Rs. 9.70 sugar}} = \frac{9.70 - 9}{9 - 8.70} = \frac{0.70}{0.30} = \frac{7}{3} \]

The mean of 9 lies much nearer 8.70, so the cheaper sugar dominates - consistent with 7 parts to 3.

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113.

How many kg of rice, of cost 9 Rs/kg must be mixed with 27 kg of rice of cost 7 Rs/kg get a gain of 10 % by selling the mixture at 9.24 Rs/kg?

A.60
B.71
C.63
D.65

Answer: C

Work back from the selling price to the cost of the mixture.

\[ \text{CP} = \frac{9.24}{1.10} = \text{Rs. } 8.40 \text{ per kg} \]

Let \( x \) kg of the Rs. 9 sugar be mixed with 27 kg of the Rs. 7 sugar:

\[ \frac{9x + 7(27)}{x + 27} = 8.40 \]

\[ 9x + 189 = 8.4x + 226.8 \quad\Longrightarrow\quad 0.6x = 37.8 \]

\[ x = 63 \text{ kg} \]

The gain is stated on the mixture, so the selling price must be converted to a cost price before any mixing calculation.

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114.

A drum contains forty liters of whisky. Four liters of whisky is taken out and replaced by soda. This process is carried out twice further. How much whisky is now contained by the container?

A.30 Lt
B.29.16 Lt
C.28.70 Lt
D.27.60 Lt

Answer: B

When a fraction of a solution is repeatedly removed and replaced, the original liquid left after \( n \) operations is

\[ A\left(1 - \frac{x}{A}\right)^{n} \]

With \( A = 40 \), \( x = 4 \) and \( n = 3 \):

\[ 40(0.9)^{3} = 40 \times 0.729 = 29.16 \text{ litres} \]

Each operation leaves 90% of the milk then present, and three such reductions compound rather than add.

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115.

Rice of rate Rs. 126 per kg and Rs. 135 per kg and 3rd variety in the ratio 1 : 1 : 2. If the final mixture is worth Rs. 153 per kg, what is the rate of the third variety per kg?

A.165.4
B.170
C.169
D.175.5

Answer: D

Take the quantities as 1, 1 and 2 kg, four kilograms in all.

\[ 126 + 135 + 2p = 4 \times 153 = 612 \]

\[ 2p = 612 - 261 = 351 \quad\Longrightarrow\quad p = \text{Rs. } 175.50 \]

The third variety carries half the weight of the mixture, which is why its price does not have to be far above 153 to lift the average.

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