document.write( "Question 1158466: Ruth purchased some furniture on credit and has secured a loan from her bank at 12% per year compounded monthly. Under the terms of her finance agreement, she his required to make payments of $116.25 at the end of each month for 36 months. What was the purchase price of Ruth's furniture?\r
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Algebra.Com's Answer #854646 by KMST(5396)\"\" \"About 
You can put this solution on YOUR website!
This question was classified as Money_Word_Problems.
\n" ); document.write( "That is correct, and so would be Algebra in Finance.
\n" ); document.write( "It is a Word_Problem, it is about financing a purchase, and the algebra in it needs to be understood.
\n" ); document.write( "Someone in Finance will probably apply a formula thinking \"I never used what they taught me in that Algebra course\".
\n" ); document.write( "However, if you understand algebra and the word problem, you do not need formulas, and you can even write your own valid formula.
\n" ); document.write( "Here is my formula for this word problem:
\n" ); document.write( "\"A=P%28%281%2BI%29%5En-1%29%2F%28I%281%2BI%29%5En%29\" , where
\n" ); document.write( "\"A\"\"%22=%22\" amount financed (in $)
\n" ); document.write( "\"P\"\"%22=%22\" monthly payment (in $)
\n" ); document.write( "\"I\"\"%22=%22\" interest rate per compounding period, as a decimal
\n" ); document.write( "\"n\"\"%22=%22\" number of equal payments
\n" ); document.write( "In this case \"system%28P=116.25%2CI=0.01%2Cn=36%29\" , and \"%281%2BI%29%5En=1.01%5E36=1.43076878\" (rounded).
\n" ); document.write( "So, applying that formula with trust without even trying to understand, we get:
\n" ); document.write( "\"A\"\"%22=%22\"\"116.25%2A%281.01%5E36-1%29%2F%280.01%2A1.01%5E36%29\"\"%22=%22\"\"116.25%2A%281.43076878-1%29%2F%280.01%2A1.43076878%29\"\"%22=%22\"\"116.25%2A0.43076878%2A100%2F1.43076878\"\"%22=%22\"\"3499.99744\"\"%22...%22\"\"%22=%22\"\"highlight%283500%29\"
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\n" ); document.write( "What if we do not like to trust and mindlessly apply formulas?
\n" ); document.write( "Then we think. The interest of \"%2212%25%22%2Fyear=%2212%25%22%2F12months=%221%25%22%2Fmonth\" is applied every month,
\n" ); document.write( "so \"%221%25%22=1%2F100=0.01\" times the balance is added to the debt each month.
\n" ); document.write( "After 1 month, the interest is added to the initial amount \"A\" , and the new balance would be
\n" ); document.write( "\"A%2B0.01A=A%281%2B0.01%29=A%2A1.01\".
\n" ); document.write( "Applying the interest for the month means multiplying times \"1.01\"
\n" ); document.write( "If there were no payments, the balance with interest applied each month would be would be
\n" ); document.write( "\"A%2A1.01%5E2\" , \"A%2A1.01%5E3\" , and \"A%2A1.01%5E4\"
\n" ); document.write( "at the end of the second, third, and fourth months respectively.
\n" ); document.write( "At the end of 36 month, it would be \"A%2A1.01%5E36\" if no payments were ever made, but Ruth will pay $116.25 at the end of each month.
\n" ); document.write( "Those payments add into the balance as negative amounts and the interest applies to anything positive or negative added into the balance.
\n" ); document.write( "After 36 months, the balance will be zero if Ruth always pays on time.
\n" ); document.write( "At the end of each month the payment of 116.25 would be credited,
\n" ); document.write( "so the balance after exactly 1 month would be
\n" ); document.write( "\"A%2A1.01-116.25\".
\n" ); document.write( "After 2 months, it would be
\n" ); document.write( "\"%28A%2A1.01-116.25%29%2A1.01-116.25=A%2A1.01%5E2-116.25%2A1.01-116.25\"
\n" ); document.write( "After 3 months, it would be
\n" ); document.write( " .
\n" ); document.write( "After 36 months, the balance can be calculated as
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\n" ); document.write( "I am sure that the sum in brackets above is \"S=1.01%5E35%2B1.01%5E34%2B1.01%5E33%2B%22...%22%2B1.01%5E2%2B1.01%2B1\"\"%22=%22\"\"%281.01%5E36-1%29%2F%281.01-1%29\" .
\n" ); document.write( "So substituting the nicer expression for that sum, we get:
\n" ); document.write( "\"A%2A1.01%5E36-116.25%2A%281.01%5E36-1%29%2F%281.01-1%29=0\" --> \"A%2A1.01%5E36=116.25%2A%281.01%5E36-1%29%2F%281.01-1%29\" --> \"A=116.25%2A%281.01%5E36-1%29%2F%28%281.01-1%29%281.01%5E36%29%29\"
\n" ); document.write( "Substituting \"1.01%5E36=1.43076878\" (rounded), and simplifying, we calculate an approximated result as
\n" ); document.write( " (rounded).
\n" ); document.write( " So the purchase price of Ruth's furniture was \"highlight%28%243500%29\" .
\n" ); document.write( "
\n" ); document.write( "I can calculate that sum \"S\" as the sum of a geometric sequence, where
\n" ); document.write( "\"1.01%2AS-S=1.01%5E36-1\" -->\"S%281.01-1%29=1.01%5E36-1\" --> \"S=%281.01%5E36-1%29%2F%281.01-1%29\" .
\n" ); document.write( "Or maybe I remember those special polynomial products where
\n" ); document.write( "\"%28x-1%29%28x%5E%28n-1%29%2Bx%5E%28n-2%29%2B%22...%22%2Bx%5E2%2Bx%2B1%29=x%5En-1\" --> \"x%5E%28n-1%29%2Bx%5E%28n-2%29%2B%22...%22%2Bx%5E2%2Bx%2B1=%28x%5En-1%29%2F%28x-1%29\" \n" ); document.write( "
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