system of equations

Question

a chemist working on a flu vaccine needs 10% sodium-iodine solution with a 60% sodium-iodine solution to obtain 50 milliliters of 30% sodium-iodine solution. how many milliliters of 10% solution and of the 60% solution should be mixed?

Answer 1

Hi Charley--

 

your first equation is x + y = 50, with x and y representing just the quantities of the 10% and 60% solutions.

 

Now, read carefully about how I use the word "OF" in the next section:

 

the second equation incorporates the percentage strength of the solutions, and if you think of it as "x mL OF 10% solution" and "y mL OF 60% solution," remember that "OF" means "times." (example: "3 groups OF 4" means "3 TIMES 4.")

 

So the second equation is going to be 10x + 60y = 1500, where 10x is the number of mL OF 10% solution, 60y is the number of mL OF 60% solution, and 1500 is 50 mL OF ("times") the 30% solution you want to end up with.

 

SO...

 

x + y = 50

10x + 60y = 1500

 

Time for a substitution! If x + y = 50,

 

you can say that x = (50 - y), and sub that in wherever you see "x"

 

SO, try this:  10(50-y) + 60y = 1500. Remember to distribute when you multiply, so you get:

 

500 - 50y + 60y = 1500. Now combine your "y"s to get

 

500 + 50y = 1500. Now subtract the 500 from each side to get

 

50y = 1000, and divide by 50 to get... y = 20 mL of the 60% solution, and therefore x = 30 mL of the 10% solution.

 

20 + 30 = 50 mL, AND if you ask yourself "do the percentages make sense," the answer is YES, because...

 

The final 30% solution is a bit closer to 10% than it is to 60%, so it makes sense that there would be a bit more of the 10% solution than of the 60% solution.

 

Hope this helps!

 

Matt in New York