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Question Video: Applications on Systems of Linear Equations

Jacob has $20,000 to invest. His intent is to earn 11% interest on his investment. He can invest part of his money at 8% interest and part at 12% interest. How much does Jacob need to invest in each option to get a total 11% return on his $20,000?


Video Transcript

Jacob has 20,000 dollars to invest. His intent is to earn 11 percent interest on his investment. He can invest part of his money at eight percent interest and part at 12 percent interest. How much does Jacob need to invest in each option to get a total 11 percent return on his 20,000 dollars?

So the way that we’re gonna solve this problem is have a look at the values that are involved. So the first thing we’re going to do is work out how much Jacob is gonna have if he gets his intended 11 percent interest. Well, to work out what would happen if we increase 20,000 dollars by 11 percent, what we’re gonna do is multiply 20,000 by 1.11. And we get the 1.11 multiplier because if we think about 100 percent, well, 100 percent is equal to one. Well, if we add on 11 percent, then it’s gonna be 111 percent. And 111 percent is equal to 1.11. And we get this because percent means by the 100 or out of 100. So what we do is we divide our percentage by 100 to get the decimal. So that would give us our 1.11, which is gonna give us 22,200. So this is gonna be our final amount after investment that jacob is looking for.

So now what we’re gonna do is work out what the two investments are. But the two investments are gonna be 𝑥 and 𝑦 cause that’s the variables we’ve decided to use, with 𝑥 representing the investment at eight percent interest and 𝑦 representing the investment at 12 percent interest. So then, to set up our first equation, we’re gonna get 1.08𝑥 plus 1.12𝑦 equals 22,200. And that’s because after the 11 percent interest, we get the 22,200. And then we get the 1.08𝑥 because we have an increase of eight percent. It’s like if it’s a multiplier of 1.08. And then we’ve multiplied that by the investment amount, which is 𝑥. Then, we get the 1.12 again, similarly, because we’re increasing by 12 percent. And we’ve called this equation one cause this is gonna help us when we identify what the next steps are.

So then, for equation two, what we’re gonna get is 𝑥 plus 𝑦 equals 20,000. And that’s because we know there are two investments and the total that jacob has to invest is 20,000 dollars. So therefore, 𝑥 plus 𝑦 is gonna be equal to 20,000. So now, the best way to solve these simultaneous questions is to use the substitution method. And that’s because we’ve got a simple second equation, which we can rearrange easily to make 𝑥 or 𝑦 the subject. So what we’ve chosen to do is make 𝑥 the subject. So to do this, we’re gonna subtract 𝑦 from each side of the equation. So when we do that, we get 𝑥 equals 20,000 minus 𝑦, which we’re gonna call equation three.

So then, in order to use our substitution method, we’re gonna substitute our new equation for 𝑥 into equation one, so substitute equation three into equation one. So once we’ve done this, the next step is to distribute across our parentheses. So what this is gonna give us is 21,600 minus 1.08𝑦 plus 1.12𝑦 equals 22,200. So now, we can simplify this by subtracting 21,600 from each side of the equation, which is gonna give us 0.04𝑦 equals 600. And then, if you divide each side of the equation by 0.04, you get 𝑦 is equal to 15,000.

So now, we found 𝑦, what we can do is substitute this into equation three to find 𝑥. So when we do this, we get 𝑥 is equal to 20,000 minus 15,000, which is gonna give us an 𝑥-value of 5,000. So therefore, we can say that the investment that’s gonna be needed is 5,000 dollars at eight percent and 15,000 dollars at 12 percent.

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