SOLUTION: Student Loans. Dimitri�s two student loans total $9000. One loan is at 4.5% simple interest and the other is at 5.5% simple interest. At the end of 1 year, Dimitri owes $447 in

Question 1092440: Student Loans. Dimitri�s two student loans total $9000. One loan is at 4.5% simple interest and the other is at
5.5% simple interest. At the end of 1 year, Dimitri owes $447 in interest. What is the amount of each loan?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.

It is a standard problem on investments. If x is the amount at 4.5% and y is amount at 5.5%, then you have this system of two equations for two unknowns x + y = 9000, (1) (total amount) 0.045x + 0.055y = 447. (2) (owes) To solve it, first multiply eq(2) by 10000. Then the system takes the form x + y = 9000, (1) 45x + 55y = 447000. (2) Now from eq(1) express y = 9000 - x and substitute into eq(2). You will get a single equation for x 45x + 55(9000-x) = 447000. Simplify it step by step and solve for x: 45x + 495000 - 55x = 447000, -10x = 447000 - 495000 = -48000 ====> x = = 4800. Thus the amount at 4.5% is just found. It is $4800. Then the second amount is the rest of 9000: y = 9000 - 4800 = 4200. Answer. $4800 at 4.5% and $4200 at 5.5%.

Solved.

See the lesson
    - Using systems of equations to solve problems on investment
in this site.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
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The referred lesson is the part of this online textbook under the topic "Systems of two linear equations in two unknowns".

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Free of charge online textbook in ALGEBRA-I
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to your archive and use it when it is needed.

Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!

Wow! That first solution was a lot of work. There are a lot of different ways you could solve the problem using the traditional algebraic approach she used. If I had solved the problem using algebra, my path would have been very different than hers....

But I use what to me is a much faster and easier way of solving mixture problems. This might not seem like a mixture problem; but you can view it that way. You are mixing a 4.5% loan and a 5.5% loan and you are getting a "mixture" of the two loans that is somewhere between 4.5% and 5.5%.

The key to my method for solving the problem is that where the mixture percentage lies between the 4.5% and 5.5% exactly determines the ratio of -- in this problem -- how much of the loan is at each rate.

Equivalently, how much of the loan is at each rate depends exactly on where the total interest lies between what the interest would have been if all of the loan had been at the lower rate and what it would have been if it had all been at the higher rate.

So here is what I do to solve the problem. (Compare the amount of work required in my method to the amount shown in the algebraic solution you received.)

If all of the $9000 loan was at the 4.5% rate, the amount of interest would be
0.045(9000) = 405.

If all of it was at the 5.5% rate, the amount of interest would be
.055(9000) = 495,

The actual amount of interest was 447. Now look where that number 447 lies between 405 and 495.

I like to think of walking along a number line from 405 to 495, passing 447 on the way. The two calculations shown above tell me that 447 is 42/90 of the way from 405 to 495. But the key to this method is that 447 being 42/90 of the way from 405 to 495 tells me that 42/90 of the loan must be at the higher rate.

I didn't bother to simplify the fraction, because I know the total amount of the loans is $9000, so 42/90 is a "nice" fraction to use. I can see immediately that $4200 of the loan must be at 5.5%, leaving $4800 at 4.5%.

Without all the words of explanation, here is all the work that is required to solve the problem by this method.

0.045(9000) = 405
0.055(9000) = 495
495-405 = 90
495-447 = 48
447-405 = 42
amount of loan at 4.5% = (48/90)(9000) = 4800
amount of loan at 5.5% = (42/90)(9000) = 4200

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