Question 18·Hard·Linear Equations in One Variable
A chemist must prepare liters of a solution that is acid by mixing Solution X, which is acid, with Solution Y, which is acid. If represents the number of liters of Solution X the chemist should use, which equation can be used to find ?
For mixture/percent problems, first choose a variable for one component (here, liters of one solution) and express the other component using the total (like ). Convert each percent to a decimal and multiply by the corresponding liters to get the amount of pure substance from each part. Then write an equation where the sum of these amounts equals the amount of pure substance in the final mixture. Finally, match your equation to the answer choices without fully solving it, to save time.
Hints
Relate the two solution amounts
If liters come from Solution X and the total is 20 liters, how can you write the number of liters of Solution Y in terms of ?
Turn percents into amounts of acid
For each solution, multiply its decimal form of the percent concentration (like 40% ) by the liters used to get the amount of pure acid from that solution.
Connect the parts to the whole
The acid from Solution X and the acid from Solution Y combine to make the acid in the final 20-liter mixture. How can you write an equation that shows "acid from X + acid from Y = acid in final mixture"?
Desmos Guide
Model the total acid from the mixture
In Desmos, type f(x) = 0.40x + 0.10(20 - x) to represent the total amount of pure acid (in liters) coming from liters of the 40% solution and liters of the 10% solution.
Model the acid needed in the final solution
Type g(x) = 0.25*20 to represent the constant amount of acid required in 20 liters of 25% solution (this will graph as a horizontal line).
Use graphs to understand the equation form
Look at how (total acid from the two solutions) compares to (acid needed in the final solution). The correct equation is the one where "total acid from X and Y" is set equal to "acid required in the final mixture," which corresponds to setting equal to .
Step-by-step Explanation
Translate the variable into amounts of each solution
You are told that is the number of liters of Solution X (40% acid).
The total volume must be 20 liters. If liters come from Solution X, then the remaining liters must come from Solution Y:
- Liters of Solution X:
- Liters of Solution Y:
Express the amount of pure acid from each solution
To find the amount of acid (not total volume) from each solution, multiply the percent (as a decimal) by the liters used.
- Solution X: 40% acid, so acid from X is .
- Solution Y: 10% acid, and we use liters, so acid from Y is .
These two expressions represent the liters of pure acid contributed by each solution.
Express the amount of acid in the final mixture
The final mixture is 20 liters and 25% acid.
Amount of pure acid in the final solution is
This is the total acid we want after mixing the two solutions.
Set up the equation and match it to a choice
The total acid from Solution X plus the total acid from Solution Y must equal the total acid in the final mixture.
So we add the acid amounts from each solution and set that equal to the acid in the final mixture:
This matches answer choice C.