Investing at Two Different Interest Rates
Date: 11/24/2003 at 00:47:23 From: Ryan Subject: (no subject) I am having problems with simple interest. If Pedro invested $5000 for one year, part at 12% annual interest and the balance at 10% annual interest. His total interest for the year was $568. How much money did he invest at each rate? I know the formula i = prt but I'm lost.
Date: 11/24/2003 at 05:21:59 From: Doctor Jeremiah Subject: Re: (no subject) Hi Ryan, Your formula i = prt is correct, but in this question there are two values of p because some money was done at 10% and some was done at 12%, so we need to treat them as completely separate. That also means that there are two values of r and i (one for each value of p). So we can write four equations that represent the problem: i1 + i2 = 568 (total interest is $568) p1 + p2 = 5000 (total principal is $5000) i1 = p1 * r1 * t (interest earned at rate 1) i2 = p2 * r2 * t (interest earned at rate 2) In your specific problem, t = 1, r1 = 0.10 and r2 = 0.12 so: i1 + i2 = 568 p1 + p2 = 5000 i1 = p1(0.10)(1) i2 = p2(0.12)(1) Now we have four equations and four unknowns. The last two equations tell us what i1 and i2 are equal to, so let's replace the i1 and i2 in the first equation with those two expressions: i1 + i2 = 568 where i1 = p1(0.10)(1) and i2 = p2(0.12)(1) p1(0.10)(1) + p2(0.12)(1) = 568 Now we have these two equations: p1(0.10) + p2(0.12) = 568 p1 + p2 = 5000 If you know how to solve a system of two linear equation, you can solve these two for p1 and p2, which are the amounts that he invested at each rate. If you aren't sure how to solve a system of equations, and this method seems a little complicated, here's another way that uses only one variable, even though the math behind it is the same as what we just did. Let x = the amount Pedro invested at 12% interest. Since his total investment was $5000, he must have 5000 - x left to invest at 10% interest. Do you see why? That's the key step to this method! Now we can make a chart using i = prt and filling in what we know for each of the two investments: Principal Rate Time Interest --------- ---- ---- ------------------- At 12% | x .12 1 (x)(.12)(1) | At 10% |5000 - x .10 1 (5000 - x)(.10)(1) | Since we know that the total interest he earned was $568, we can write an equation showing that adding the two interest amounts from the chart must make 568: (x)(.12)(1) + (5000 - x)(.10)(1) = 568 Distributing we get: .12x + 500 - .10x = 568 You can solve this equation for x and then use x to calculate the principals, or how much he invested at each rate, in the chart above. Let me know if you get stuck or if you have other questions. - Doctor Jeremiah, The Math Forum http://mathforum.org/dr.math/
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