Vessel X contains 150 ml of a blend containing apple syrup and water in a 4:1 ratio. Vessel Y contains 100 ml of a blend containing apple syrup and water in a 1:4 ratio. What volume of the mixture must be transferred from Vessel X to Vessel Y so that the proportion of apple syrup to water in Vessel Y becomes 1:1?

The correct answer is 100 ml.

Step 1: Calculate the initial quantities of apple syrup and water in Vessel X.
Vessel X contains 150 ml of blend with apple syrup and water in the ratio 4 : 1.
Total ratio parts in Vessel X =

$4+1=5$

Amount of apple syrup in Vessel X =

$\frac{4}{5}\times 150=120\text{ml}$

Amount of water in Vessel X =

$\frac{1}{5}\times 150=30\text{ml}$

Step 2: Calculate the initial quantities of apple syrup and water in Vessel Y.
Vessel Y contains 100 ml of blend with apple syrup and water in the ratio 1 : 4.
Total ratio parts in Vessel Y =

$1+4=5$

Amount of apple syrup in Vessel Y =

$\frac{1}{5}\times 100=20\text{ml}$

Amount of water in Vessel Y =

$\frac{4}{5}\times 100=80\text{ml}$

Step 3: Determine the composition of the transferred mixture.
Let $v$ ml be the volume of mixture transferred from Vessel X to Vessel Y.
The transferred portion has the same ratio (4 : 1) as Vessel X:
Apple syrup transferred =

$\frac{4}{5}v=0.8v\text{ml}$

Water transferred =

$\frac{1}{5}v=0.2v\text{ml}$

Step 4: Formulate and solve the equation for Vessel Y.
After adding this volume to Vessel Y, the new amounts become:
New amount of apple syrup in Vessel Y =

$20+0.8v$

New amount of water in Vessel Y =

$80+0.2v$

We want the final ratio of apple syrup to water in Vessel Y to be 1 : 1, which means the volume of apple syrup must equal the volume of water:

$20+0.8v=80+0.2v$

Rearranging the equation to solve for $v$:

$0.8v-0.2v=80-20$

$0.6v=60$

$v=\frac{60}{0.6}=100\text{ml}$

Therefore, 100 ml of the mixture must be transferred from Vessel X to Vessel Y.