Question

Bottle 1 contains a mixture of milk and water in 7 : 2 ratio and Bottle 2 contains a mixture of milk and water in 9 : 4 ratio. In what ratio of volumes should the liquids in Bottle 1 and Bottle 2 be combined to obtain a mixture of milk and water in 3 : 1 ratio?

• A. 27 : 14
• B. 27 : 13
• C. 27 : 16
• D. 27 : 18

Answer 1

The Correct Option is B

To solve this problem, we need to set up equations based on the given ratios and find the required ratio for combining the volumes of Bottle 1 and Bottle 2 to achieve the desired ratio in the resultant mixture.

Let the volume of the liquid taken from Bottle 1 be \( x \) and from Bottle 2 be \( y \). We aim to determine the values of \( x \) and \( y \) such that when these two solutions are mixed, the ratio of milk to water in the final mixture is 3:1. Below are the step-by-step calculations:

1. Bottle 1 has a milk to water ratio of 7:2. So, the fraction of milk in Bottle 1 is \( \frac{7}{9} \) and water is \( \frac{2}{9} \).
2. Bottle 2 has a milk to water ratio of 9:4. So, the fraction of milk in Bottle 2 is \( \frac{9}{13} \) and water is \( \frac{4}{13} \).
3. The desired mixture should have a milk to water ratio of 3:1. Thus, the fraction of milk must be \( \frac{3}{4} \) and the fraction of water must be \( \frac{1}{4} \).
4. Set up the equation for milk:
   \[ \frac{7}{9}x + \frac{9}{13}y = \frac{3}{4}(x + y) \]
5. Solving the equation:
   Multiply throughout by 468 (LCM of 9, 13, and 4) to clear the fractions:
   \[ 364x + 324y = 351(x + y) \]
   Simplify:
   \[ 364x + 324y = 351x + 351y \]
   \[ 13x = 27y \]
   So, the ratio of the volumes is:
   \[ x : y = 27 : 13 \]

Therefore, the liquids in Bottle 1 and Bottle 2 should be combined in the ratio of 27:13 to obtain the desired mixture.