Linear equation word problem: saline

We're told to make a table and solve. So they tell us that we have 50 ounces of a 25% saline solution, a mixture of water and salt. How many ounces of a 10% saline solution must you add to make a new solution that is 15% saline? So let's make this table that they're talking about. Let's write amount of solution. Let me write total amount of solution or maybe I should say total volume of solution. And then the next column I'll say percent saline. And then we can use this information to figure out total amount of saline. And let's list it for each of the two solutions that they talk about. We're starting with 50 ounces of a 25% saline solution. So this is what we start with. Starting solution, we have 50 ounces of it, so I'll just write 50. We'll assume everything is in ounces. It is 25% saline. So if we wanted to figure out the total ounces of saline, we say, well, we have 50 ounces. Multiply that by 25% and we have the total amount of saline in this solution. So 50 times 25%, that's the same thing as 50 divided by 4, so that's 12.5 ounces of saline in this 50 total ounces. It's 25% saline. Now, let's talk about what we're going to add to it, so solution added. Now, they say how many ounces of a 10% solution? So we don't even know how many ounces we're going to add. That's what we have to actually solve for. So let's call that x for the amount of solution that we have to add. So we don't know how much we're going to add, but we do know that it is a 10% saline solution. And if we know what x is, we know the total amount of saline is going to be 10% of x. If we had 50 here, it would be 10% of 50. If we had 10 here, it would be 10% of 10. So the amount of saline we have in this solution, in x ounces of this solution, is going to be 0.1x, or 10% percent of x. That's what 10% of the solution being saline means. Now, when we add it, what do we end up with? So let me do this in a different color. Resulting solution. Well, if we started with 50 ounces and we add x ounces, we're going to end up with 50 plus x ounces. That's our total volume of the resulting solution. What percent saline is it? Well, our goal is to make a 15% saline solution, so it has to be 15% saline. Now, what's the total amount of saline in it? And this is kind of the main box or cell in this table. There's two ways to get to the total amount of saline in the solution. One, we could just multiply the percent saline times the total value, so we could do that. So let write that. It's 0.15 times 50 plus x. All I did, I multiplied the percent saline times my total volume. That's one way to get the total amount of saline. The other way to get the total amount of saline is to add these two numbers. The 50-ounce solution had 12.5 ounces of saline. We added 0.1x ounces of saline, so if we add these two numbers, it should also be equal to the total amount of saline. So this has to be equal to the sum of these two things. It has to be equal to 12.5 plus 0.1x. And just like that, because 15% of this has to be the same thing as the sum of this, we have one equation and one unknown, and we can solve for x, which is what we need to solve for. How much solution do we need to add? So let's just do that. So 0.15 times 50. Let's see, that's 7.5, if I'm doing that right. Yeah, because 0.15 times 100 would be 15. So this is 7.5 plus 0.15x-- that's the left-hand side-- is equal to 12.5 plus 0.1x. Let me scroll to the right a little bit. Now, we can subtract 7.5 from both sides of this equation. Let me do it in a new color. So if we subtract 7.5 from both sides of this equation, the left-hand side, that cancels out. We're just left with 0.15x is equal to-- 12.5 minus 7.5 is just 5. 5 plus 0.1x. Now, we could subtract 0.1x from both sides of this equation. Let me scroll down a little bit. Let me subtract 0.1, or I could say 0.10x from both sides of this equation. These are the same number. So they cancel out. The left-hand side, we just end up with 0.05x is equal to 5. And now we just divide both sides by 0.05. And we get x is equal to 5 divided by 0.05. That's the same thing as 5 divided by 1/20, or the same thing as 5 times 20. So we get x is equal to 100. So we're done! If you add 100 ounces-- so we figured out that x is equal to 100. If you add 100 ounces of 10% saline solution, you will end up with 150 ounces of 15% saline solution. So if you add 100, 10% of that, we actually added 10 ounces. So you have 12.5 plus 10 ounces is 22.5 ounces of saline in a solution of 150 ounces, which will be 15%. And we're done.