document.write( "Question 641683: I'd really appreciate your help sir/madam.
\n" ); document.write( "I have tried solving it many times but I am really confused about the problem.
\n" ); document.write( "A woman has a certain amount of money invested. If she had $6000 more invested at a rate 1% lower, she would have the same yearly income from the investment. Furthermore, if she had $4500 less invested at a rate 1% higher, her yearly income from the investment would also be the same. How much does she have invested, and what rate is it invested?
\n" ); document.write( "Thanks a lot. \n" ); document.write( "

Algebra.Com's Answer #403760 by KMST(5328) $\"\"$ $\"About$

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Let $\"x\"$ be the amount (in $) of money the woman has invested.
\n" ); document.write( "Let $\"y\"$ be the rate at which she invested her money, written as a decimal (so 1% would be 0.01, 2% would be 0.02, and so on).
\n" ); document.write( "The interest she gets every year (her yearly income) is calculated as
\n" ); document.write( "$ $\"x%2Ay\"$
\n" ); document.write( "An interest rate 1% lower than $\"y\"$ would be $\"y-0.01\"$ .
\n" ); document.write( "If the woman had invested $6000 at that lower rate, her yearly income would be
\n" ); document.write( "$ $\"6000%28y-0.01%29\"$ , and that is the same as $ $\"x%2Ay\"$ , so
\n" ); document.write( " $\"6000%28y-0.01%29=xy\"$ .
\n" ); document.write( "An interest rate 1% higher than $\"y\"$ would be $\"y%2B0.01\"$ .
\n" ); document.write( "If the woman had invested $4500 at that lower rate, her yearly income would be
\n" ); document.write( "$ $\"4500%28y%2B0.01%29\"$ , and that is the same as $ $\"x%2Ay\"$ , so
\n" ); document.write( " $\"4500%28y%2B0.01%29=xy\"$ .
\n" ); document.write( "We have a system of non-linear equations:
\n" ); document.write( " $\"system%286000%28y-0.01%29=xy%2C4500%28y%2B0.01%29=xy%29\"$
\n" ); document.write( "From there we get
\n" ); document.write( " $\"6000%28y-0.01%29=4500%28y%2B0.01%29\"$ --> $\"6000y-60=4500y%2B45\"$ --> $\"6000y-60-4500y=4500y%2B45-4500y\"$ --> $\"1500y-60=45\"$ --> $\"1500y-60%2B60=45%2B60\"$ --> $\"1500y=105\"$ --> $\"1500y%2F1500=105%2F1500\"$ --> $\"highlight%28y=0.07%29\"$ ,
\n" ); document.write( "The interest rate is 7%.
\n" ); document.write( "Substituting into $\"6000%28y-0.01%29=xy\"$ ,
\n" ); document.write( " $\"6000%280.07-0.01%29=0.07x\"$ --> $\"6000%280.06%29=0.07x\"$ --> $\"360=0.07x\"$ --> $\"360%2F0.07=0.07x%2F0.07\"$ --> $\"360%2F0.07=x\"$ --> $\"highlight%28x=5142.86%29\"$ (rounded).
\n" ); document.write( "The woman had $5142.86 invested.\r
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