In a mixture problem, each percent describes an ingredient's share of the total volume. Here, equal volumes leave and enter, so the total volume stays fixed even though the vinegar amount changes. Track the ingredient, not just the percent. Write an equation for the vinegar before and after the exchange, then solve it with a one-variable Desmos regression.
Hints
- Hint 1
Let be the original volume in milliliters. A concentration of vinegar means out of every milliliters is vinegar. How would you write the starting vinegar amount using ?
- Hint 2
The container loses milliliters and gains milliliters, so its total volume is still . Only vinegar is removed; the water added brings no vinegar back. What vinegar amount is left?
Step-by-step
Track the vinegar and solve
Step 1Write the starting vinegar amount
Let be the mixture's original volume in milliliters. Since means of the whole, the container starts with milliliters of vinegar.
- Step 2
Track what changes in the exchange
Removing only vinegar leaves milliliters of vinegar. The milliliters of water added contain no vinegar, and they replace exactly the volume removed, so the mixture's total volume is still .
- Step 3
Match the final vinegar amounts
The final mixture is vinegar, so it contains milliliters of vinegar. That's the amount left after the exchange, giving the mixture equation . Don't subtract from and call the result milliliters: the percentages are shares of , not volumes.
- Step 4
Solve for the original volume
Type in Desmos, then type on the next line. The tells Desmos to find the value of that makes both sides equal. Under PARAMETERS, it shows ; tap the fraction button on the line to see . Since represents , the container originally held milliliters. Choice C.