Alligations or Mixtures
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Q.

In a 729 litres mixture of milk and water, the ratio of milk to water is 7:2.

To get a new mixture containing milk and water in the ratio 7:3, the amount of water to be added is:

 A.

51 litres

 B.

61 litres

 C.

71 litres

 D.

81 litres

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Solution:
Option(D) is correct

Quantity of milk in 729 litre of mixture,

$= \dfrac{7}{9}\times 729 = 567$ litre

Quantity of water,

$= 729-567 = 162$ litre.

Let $x$ litre of water be added to make ratio $7:3$.

Milk                water
$567$              $(162+x)$
        \           /
        mixture
      $(729+x)$
      /           \
$567$     :     $(162+x)$
  $7$        :           $3$

$\Rightarrow \dfrac{7}{3} = \dfrac{567}{162+x}$

$\Rightarrow 162\times 7 + 7x = 567\times 3$

$\Rightarrow 7x = 1701-1134 = 567$

$\Rightarrow x = \dfrac{567}{7}$

$= \textbf{81 litre}$ water is to be added.

Edit: For an alternate solution ( a quick one), do check KARTIK's comment.

Edit 2: For yet another shortcut alternate solution, check comment by Rajat Srivastava.


(7) Comment(s)


Deepak
 ()

Ratio 7:2 gives 567 part milk ie 7 part.

So 1 part milk=567/8=81.

Increse in water is 1 part ..so amount of water added is 81.



Indra Kiram
 ()

milk:water :- 7: 2; Milk:-567, water:-162,let water added be X. In new solution Milk:- 567, Water: 162+x, Total solu:-729+x and ratio is 7:3

simply $7/10(729+x)=567$, upon calculation we get X =81.



Himal Hazra
 ()

7:2 ratio hence 7/9 * 729 = 567 rest is water 729-567=162 what must be added to water to have ratio 7:3

let x be water added

567/(162+x) = 7/3

x = 81 ltrs ans



Rajat Srivastava
 ()

2 units out of a total 9 constitute water with a total of 729 litres.

Therefore 1 unit of water $= (\frac{2}{9}) \times 729 = 81$.

In the new solution, water is 1 unit more and the amount of milk remaining same.

Units of water added $(3-2) = 1$ unit

$= \textbf{81 litres}.$



Sandeep
 ()

What if the new ratio to be obtained is 8:5?


Math
 ()

Since only water is added, milk quantity remains same, 567ltrs. Now final mixture ratio 8:5, do 8parts=567;

therefore

5parts= 567*5/8s


KARTIK
 ()

milk:water $= 7:2$

Amount of milk $= 7x$

Amount of water $= 2x$

Total amount of mixture, $7x + 2x = 9x=729$

$\Rightarrow x=\dfrac{729}{9} = 81$

So, for every increase in ratio there is increase of 81

Now Ratio has been changed from $7:2$ to $7:3$ which is an increase in the ratio by 1, so 81 litres of water is to be added.

Now taking it forward,

If final ratio were $7:5$, (increase of ratio by 3)

Then increase would be $81*3=243$