A chemist mixes a 10% saline solution with a 20% saline solution to make 500 millimiters of a 16% saline solution. How many millimeters of each solution does the chemist mix together?

Question

How do I write an equation for this problem?

Answer 1

Hi Megan,

This word problem is asking us to find two quantities, the amount of the 10% solution and the amount of the 20% solution. Each of these things will be a variable, like x or y. Two determine two variables, we need two formulas.

So lets set up two formulas based off what we know. We know that the amount of the 10% solution and the 20% solution will add to a total of 500 millimeters. Lets call the amount of the 10% solution "x" and the amount of the 20% solution "y" and so we know that "x" and "y" add together to make 500 millimeters.

x + y = 500

Thats one formula down, and one to go!

We also know that the 10% solution and 20% solution combine to make an average salinity of 16%. The amount of each solution (our variables) will determine how much "influence" each solution has on the average.

Average = (sum of the numbers) / (the number of numbers)

16% = (sum of the numbers) / 500

16%(500) = sum of the numbers.

When I say "sum of the numbers" we're used to that number set looking like "1, 2, 3, 3, 5, 5, 7, 10, 11, 11" where we have a simple list of numbers. But another way we could have the same information would be to have just the list of possible numbers, and the frequency with which they appear in the set. So to follow this example, the number "0" appears in the set 0% of the time and the number "5" appears 2 times, and so on. We can multiply the number of occurrences by the number to determine the "sum" of all the numbers, or in other words how heavily the number "influences" the average.

16%(500) = "sum of the numbers" = "each number multiplied by its occurrences" = (10%)(x) + (20%)(y)

Note that the percent (%) doesn't really matter. If we multiplied both sides of the formula by 100, the % would go away without anything else changing.

10x + 20y = 16*500 = 8000

So now we have our two formulas.

10x + 20y = 8000

x + y = 500

Let's solve for one of the variables in the simpler formula.

x + y = 500

-y -y

x = 500 - y

Now we know that x is the same as "500-y" and we can combine that knowledge with the other formula.

10x + 20y = 8000

x = 500 - y

10(500-y) + 20y = 8000

Now we have one variable and one formula, which we can solve.

10(500-y) + 20y = 8000

distribute

5000 - 10y + 20y = 8000

simplify terms

5000 + 10y = 8000

subtract 5000 from each side

10y = 3000

divide each side by 10

y = 300

Great! Remember that Y = the amount of the 20% solution. But we still need to find "x" which is the amount of the 10% solution.

x + y = 500

substitute

x + 300 = 500

subtract 300 from each side

x = 200

Final answer: 300 millimeters of the 20% solution and 200 millimeters of the 10% solution.

This makes sense logically too, which is a great way to check our work. 16% is roughly half way between 10% and 20% but slightly closer to 20%. Half of 500 is 250. So it makes sense that a little over half of the amount is the 20%.

Hope this helped! Feel free to message me any time you have other questions!

Answer 2

10X + 20(5-X) = 16(5)

10X + 100 -20X = 80

10X = 20

X = 200 ml of 10% solution mixed with

5-2 = 300 ml of 20% solution to get 500 ml of 16%