Question 310674: This question was answered but I'm still confused. Can I please have some more help? High school was ten years ago, what little math ability I had is gone.
Thanks Again
My question is why is the quanity equation multiplied by 20 and not.2. Also when solving the two equations why does the 20a cancel, but not the 550? I know these are probably very simple, but I just don't understand.
How many ml of medication A which has a 20% concentration, and medication B which has a 65% concentration should be mixed to form a 550 ml solution that is 30.9%
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Equation:
Quantity Equation:::::::: a + b = 550 ml
Active Ingredient Eq: 0.20a + 0.65b = 0.309*550
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Solving using 2 equations and elimination:
Multiply thru the 1st equation by 20
Multiply thru the 2nd equation by 100
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20a + 20b = 20*550
20a + 65b = 30.9*550
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Subtract 1st from 2nd and solve for "b":
45b = 10.9*550
b = 133.22 ml (amt. of 65% solution needed in the mixture)
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Since a + b = 550, a = 416.78 ml (amt. of 20% solution needed in the mix)
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Actually, you could multiply the Quantity equation by 2, but then you would have to multiply the Active Ingredient equation by 10 instead of 100.
As to your other question, let me illustrate it this way. In the first place, I would have used -20 as a multiplier for the Q equation and 100 for the AI equation, thus:
Now, you could multiply out the constant terms in the right-hand sides, and that would actually answer your question about "why the 550 doesn't cancel". But let's proceed without that step. Since I used a -20 multiplier for the Q equation, we can actually add the two equations rather than subtracting them. This technique gives you much less chance for a sign error. But rather than just showing the sum of the two, let's use the idea that if  and  , then  to write:
\ -\ (20\,\cdot\,550))
Now,  and that is why the  goes away.  , just like you wrote. For the constant term, let's do a little factoring. Factor out the 550:
\ -\ (20\,\cdot\,550)\ =\ 550(30.9-20)\ =\ 550\,\cdot\,10.9)
Putting it all back together:
And finally, as you stated,
John
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