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 percent saline solution a mixture of water and salt how many ounces of a 10 percent saline solution must you add to make the new solution or to make a new solution that is 15 percent saline so let's make this table that they're talking about so let's talk let's write amount of solution so let me write total total amount total amount of solution or maybe I should say total volume of solution and then the next column I'll say percent saline so percent saline and then we can use this information to figure out total amount of saline total 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 percent saline solution so this is what we start with we start with this starting solution starting solution we have 50 ounces of it so I'll just write 50 we'll assume everything is in ounces it is 25 percent saline it is 25 percent saline so if we wanted to figure out the total ounces of saline we say well we have 50 ounces multiply that by 25 percent and we have the total amount of saline in this solution so 50 times 25 percent that's the same thing as 50 divided by 4 so that's 12.5 so that's twelve point five ounces of saline in this 50 total ounces it's 25 percent saline now let's talk about what we're going to add to it we are going to add we are going to add so solution added solution added now they say how many ounces of a 10 percent 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 percent saline solution we do know that it is a 10 percent saline so Lou and if we know what X is we know the total amount of saline is going to be the percent ten percent of X if we have if we I don't know what this is if we had fifty year would be ten percent of 50 if we had ten year to be ten percent of ten so the amount of saline we have in this solution and X ounces of the solution is going to be is going to be 0.1 X or 10 percent of X that's what 10 percent 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 resulting solution resulting solution well if we started with 50 ounces and we add X ounces we are going to end up with 50 plus X ounces 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 percent saline solution so it has to be 15 percent saline now what's the total amount of saline in it and this is kind of the main the main I guess box or the main 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 volume so we could do that so let me write that so it's 0.15 times 50 plus X all I did I multiply multiply 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 fifty out solution had twelve point five ounces of saline this we added point 1x ounces of saline so if we add these two numbers it should also be equal to the total amount of saline so this is going this has to be equal to the sum of these two things it has to be equal to 12.5 plus plus 0.1 X and just like that because 15 percent of this has to be the same thing as the sum of this we have one equation in 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 if that's seven and a half if I'm doing that right yeah because 0.15 times 100 would be point would be 15 so this is 7 point 5 this is 7 point 5 plus 0.15 X that's the left-hand side is equal to 12 point 5 plus 0.1 X let me scroll to the right a little bit is equal to 0.1 X now we can subtract 7.5 from both sides of this equation I'm doing 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 0.15 X is equal to 12 point 5 minus 7 point 5 is just 5 5 plus 0.1 X now we could subtract 0.1 X from both sides of this equation let me scroll down a little bit so let me subtract 0.1 or I could say 0.1 0 X from both sides of this equation minus 0.1 0 X these are the same number so they cancel out the left-hand side we just end up with 0.05 X is equal to 5 and now we just divide both sides by 0.05 so divide both sides by 0.05 0.05 and we get X is equal to 5 divided by 0.05 that's the same thing as 5 divided by 1 xx or the same thing as 5 times 20 so we get X is equal to 100 so we're done if you add if you add 100 ounces so we figured out that X is equal to 100 if you add 100 ounces of 10 percent saline solution you will end up with 150 ounces of 15 percent saline solution so if you add 110 percent of that we actually ended added 10 ounces so you have 12.5 + 10 ounces is 22 point five ounces of saline in a solution of 150 ounces which will be 50 percent and we're done