A chemist is mixing a 20% NaCl solution with a 35% NaCl solution. The chemist wishes to make a mixture that is at least 30% NaCl
but no more than 4 liters from the two types of solutions.
Let a be the volume of 20% NaCl, in liters, and b be the volume of 35% NaCl, in liters, in the mixture. Which of the following systems represents all of the constraints
that a and b must satisfy?
Approach
Let's break apart the meaning of the correct option:
⎩⎨⎧a>0b>0a+b≤420a+35b≥30(a+b)we need some amount of the 20% NaCl solution; it cannot be negativewe need some amount of the 35% NaCl solution; it cannot be negativethe combined volume of solution is no more than 4 litersthe combined volume of NaCl needs to be greater than 30%
Perhaps the most confusing relationship involves the volume of NaCl:
20a+35b≥30(a+b)
In cases like this, we need to make sure the corresponding parts make sense as a whole.
20a+35b≥30(a+b)✓
Looking at structure is very important in mathematics. You've done it plenty of times, though it may be done without you expliciting thinking it. For example, would this be reasonable as written?
12 inches+5 inches≥10 feet
Similarly, the first two choices do not seem reasonable by the same logic.
20a+35b≤30 ✖
Sidenote
Mixture problems show up rarely on the SAT. It is not entirely necessary to memorize a seperate formula for it. Instead, treat it like a system of equations.
Make one equation for the total volume of solution and one equation for the total volume of the solute or identifiable substance (NaCl in our case).
a+b≤4volume of solution
The volume of NaCl is taken by multiplying the percent concentration with the volume of the solution. For example, 20% of solution a is NaCl. Therefore,
0.20⋅a=volume of NaCl in the 20% solution
Accordingly, the volume of NaCl equation:
0.20a+0.35b≥.30(a+b)volume of NaCl only
We can multiply each term by 100 to avoid working with decimals.