John P. answered • 11/18/21
PhD Student in Linguistics, I tutor Writing, Math, English, Sci, etc.
The key to solving this is setting up two correct equations right at the start, after which it gets much easier to solve.
Firstly, you need to know that whatever amount (in kilograms) of A and B you have is supposed to equal 80 at the end of this problem. We're ignoring the iron alloy percentage part for now. To that end, the first equation you can set up is simply A + B = 80, where A = amount of A in kilograms, and B = amount of B in kilograms.
Now we have to set up the iron alloy part though, and this is where you want to be careful. I'd start by thinking of the percentages as decimals (though you absolutely don't have to do this), and writing .2A (20% of A) + .6B (60% of B).... and now the question is what you set that equal to. The answer is that you want the weight of the iron part of A plus the weight of the iron part of B to be equal to the weight of the iron part of your answer. This means you want to calculate the iron part of your answer, which would be 52% of 80, or 41.6. So now we have the following two equations:
A + B = 80
.2A + .6B = 41.6
From here, you can solve it however you like. I prefer multiplying everything in the bottom equation by 5, so that I have A + 3B = 208. From there, I solve for A, giving me A = 208 - 3B. From here, we can substitute into the top formula.
208-3B + B = 80.
Solve for B: B = 64.
Then substitute in again: A + 64 = 80.
Solve for A: A = 16. And there you go, you have your answer.
If you have 16 kilograms of A and 64 kilograms of B, you have 80 kilograms total. And also, if you look at the iron portions of those, you have 20% of 16, which is 3.2, and 60% of 64, which is 38.4.
38.4 + 3.2 = 41.6, which is 52% of 80; precisely what we were asked to solve for.
