The ratio of glycerin and water in a mixture is 3 ∶ 1. How much fraction of the mixture must be drawn off and substituted by water so that the ratio of glycerin and water becomes 1 ∶ 1?

Shortcut Trick

Initial quantity of glycerin = 3/4

Remaining quantity of glycerin = 1/2 (as glycerin : water = 1 : 1)

Now,

Remaining quantity = Initial quantity (1 - fraction removed)

⇒ 1/2 = 3/4 (1 - fraction removed)

⇒ 2/3 = (1 - fraction removed)

⇒ fraction removed = 1 - 2/3 = 1/3

Hence, 1/3 is the correct answer.

Alternate Method 

Given:

Initial ratio of glycerin and water = 3:1

Final ratio of glycerin and water = 1:1

Formula Used:

Let the total mixture be x liters.

Amount of glycerin initially = 3/4x

Amount of water initially = 1/4x

Let the fraction of the mixture drawn off and substituted by water be y/x.

After drawing off the mixture and substituting by water, the amount of glycerin left = 3/4x - 3/4y

Amount of water after substitution = 1/4x - 1/4y + y

Final ratio of glycerin to water = 1:1

Calculation:

Given final ratio 1:1, we equate the amounts of glycerin and water:

3/4x - 3/4y = 1/4x - 1/4y + y

⇒ 3/4x - 1/4x = 3/4y - 1/4y + y

⇒ 1/2x = 3/4y + 3/4y

⇒ 1/2x = 6/4 y

⇒ y = 1/2x × 4/6 = 4/12x = 1/3x

So, the fraction of the mixture to be drawn off and substituted by water is 1/3.

The correct answer is option (1).