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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 : 3 B. 4 : 3 C. 3 : 2 D. 3 : 4

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 1kg1st kind rice CP of 1kg2nd kind rice 7.2 5.7 Mean Price 6.3 6.3 - 5.7 = .6 7.2 - 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 litres B. 24 litres C. 30 litres D. 32 litres

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. $\dfrac{4}{3}$

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 Jar Concentration of alcohol in 2nd Jar 40% 19% Mean 26% 26-19=7 40-26=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 kg B. 58 kg C. 63 kg D. 60 kg

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 kind CP of 1 kg sugar of 2nd kind Rs. 9 Rs. 7 Mean Price Rs.8.4 8.4 - 7 = 1.4 9 - 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 : 7 B. 7 ; 6 C. 7 : 8 D. 8 : 7

Explanation:

By rule of alligation,

 Cost of 1 kg rice of 1st kind Cost of 1 kg rice of 2nd kind 9.3 10.80 Mean Price 10 10.8-10 = .8 10 - 9.3 = .7

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