A -liter tank is initially full of a solution in which the ratio of dye to water is . The solution is thoroughly mixed, and liters of the solution are removed. Then liters of pure water are added to the tank. The ratio of dye to water in the tank is now .
Which choice is the value of ?
For mixture-replacement questions, first convert the ratio into actual starting amounts. Then track each substance separately: removing a mixture subtracts both components in their original proportions, while adding a pure substance changes only that one component. Finally, translate the new ratio into an equation between the final amounts.
Hints
Use the initial ratio
Split the liters into equal ratio parts to find the initial amounts of dye and water.
Track both substances
Because the removed liquid is the original mixture, each liter removed takes away both dye and water in the ratio . The replacement adds only water.
Interpret the new ratio
For a dye-to-water ratio of , write an equation relating the final water amount to the final dye amount.
Desmos Guide
Set up the two quantities
Algebra is faster for an exact answer, but Desmos can verify it. Enter for the final dye amount.
Graph the ratio condition
For a dye-to-water ratio, the dye amount must equal one-third of the water amount. Enter .
Find the intersection
Click the intersection of the two lines. Its -coordinate is the number of liters removed and replaced; use the displayed decimal to confirm the exact fractional choice.
Step-by-step Explanation
Find the initial amounts
The total of the ratio parts is . Therefore, the initial amounts are:
- Dye: liters
- Water: liters
Write expressions for the final amounts
Each liter removed contains liter of dye and liter of water. After removing liters and adding liters of pure water, the amounts are:
Use the final ratio
A dye-to-water ratio of means the water amount is times the dye amount.
Solving gives:
So choice A is correct.