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Solved Examples(Set 2) - Mixture and Alligation

6. Find the ratio in which rice at Rs. 7.20 a kg be mixed with rice at Rs. 5.70 a kg to produce a mixture worth Rs. 6.30 a kg.
A. 2 : 3B. 4 : 3
C. 3 : 2D. 3 : 4
Discuss
answer with explanation

Answer: Option A

Explanation:

CP of 1kg 1st kind rice = Rs.7.20
CP of 1kg 2nd kind rice = Rs.5.70
CP of 1kg mixed rice = Rs.6.30

By rule of alligation,

CP of 1kg
1st kind rice
CP of 1kg
2nd kind rice
7.25.7
Mean Price
6.3
6.3 - 5.7 = .67.2 - 6.3 = .9

Required Ratio = .6 : .9 = 6:9 = 2:3

7. 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 did the cask originally hold?
A. 26 litresB. 24 litres
C. 30 litresD. 32 litres
Discuss
answer with explanation

Answer: Option B

Explanation:

Let initial quantity of wine $=x$ litre

After a total of 4 operations, quantity of wine
$=x\left(1-\dfrac{y}{x}\right)^n\\= x\left(1-\dfrac{8}{x}\right)^4$

Given that after a total of 4 operations, the ratio of the quantity of wine left in cask to that of water = 16 : 65

$\Rightarrow \dfrac{x\left(1-\dfrac{8}{x}\right)^4}{x} = \dfrac{16}{81}\\\Rightarrow \left(1-\dfrac{8}{x}\right)^4 = \left(\dfrac{2}{3}\right)^4\\\Rightarrow \left(1-\dfrac{8}{x}\right) = \dfrac{2}{3}\\\Rightarrow \left(\dfrac{x-8}{x}\right) = \dfrac{2}{3}\\\Rightarrow 3x-24=2x\\\Rightarrow x=24$

8. A jar full of whiskey 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
A. $\dfrac{3}{4}$B. $\dfrac{3}{2}$
C. $\dfrac{2}{3}$D. $\dfrac{4}{3}$
Discuss
answer with explanation

Answer: Option C

Explanation:

Concentration of alcohol in 1st Jar = 40%
Concentration of alcohol in 2nd Jar = 19%
After the mixing, Concentration of alcohol in the mixture = 26%

By rule of alligation,

Concentration of alcohol in 1st JarConcentration of alcohol in 2nd Jar
40%19%
Mean
26%
26-19=740-26=14

Hence ratio of 1st and 2nd quantities = 7 : 14 = 1 : 2

i.e., $\dfrac{2}{1+2}=\dfrac{2}{3}$ part of the whisky is replaced.

9. How many kilograms of sugar costing Rs. 9 per kg must be mixed with 27 kg of sugar costing Rs. 7 per kg so that there may be a gain of 10 % by selling the mixture at Rs. 9.24 per kg?
A. 56 kgB. 58 kg
C. 63 kgD. 60 kg
Discuss
answer with explanation

Answer: Option C

Explanation:

Selling Price(SP) of 1 kg mixture= Rs. 9.24

Profit = 10%

Cost Price(CP) of 1 kg mixture $=\dfrac{100}{(100+\text{Profit}\%)}× \text{SP}$
$=\dfrac{100}{(100+10)}× 9.24\\= \dfrac{100}{110}× 9.24=\dfrac{92.4}{11}=\text{ Rs.} 8.4$

By rule of alligation,

CP of 1 kg sugar of 1st kindCP of 1 kg sugar of 2nd kind
Rs. 9Rs. 7
Mean Price
Rs.8.4
8.4 - 7 = 1.49 - 8.4 = 0.6

i.e., to get a cost price of 8.4, the sugars of kind1 and kind2 should be mixed in the ratio 1.4 : 0.6 = 14 : 6 = 7 : 3

Suppose $x$ kg of kind1 sugar is mixed with 27 kg of kind2 sugar.
then $x$ : 27 = 7 : 3
$\Rightarrow 3x=27×7\\\Rightarrow x=9×7=63$

10. In what ratio should rice at Rs.9.30 per kg be mixed with rice at Rs. 10.80 per kg so that the mixture be worth Rs.10 per kg ?
A. 6 : 7B. 7 ; 6
C. 7 : 8D. 8 : 7
Discuss
answer with explanation

Answer: Option D

Explanation:

By rule of alligation,

Cost of 1 kg rice of 1st kindCost of 1 kg rice of 2nd kind
9.310.80
Mean Price
10
10.8-10 = .810 - 9.3 = .7

Required ratio = .8 : .7 = 8 : 7.

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