document.write( "Question 1203599: The formula for an investment worth with interest compounded annually is A=P(1+i)^n, where P represents the initial investment, i is the interest rate, and A is the worth of the investment after n years.\r
\n" ); document.write( "\n" ); document.write( "a) Rearrange the formula for P. What was the initial investment of an investment worth $1000 that compounded 10% interest for 10 years?
\n" ); document.write( "b) Rearrange the formula for i. What is the interest rate of an investment whose worth went from $1000 to $1200 in 2 years?
\n" ); document.write( "c) Explain a method with which you could estimate how many years it would take for an investment to reach a certain worth at a certain interest rate.
\n" ); document.write( "d) Estimate how many years would it take an investment of $2100 at 20% interest to reach a worth of $5225?\r
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Algebra.Com's Answer #839294 by Theo(13342)\"\" \"About 
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The formula for an investment worth with interest compounded annually is A=P * (1 + i)^n, where P represents the initial investment, i is the interest rate, and A is the worth of the investment after n years.\r
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\n" ); document.write( "\n" ); document.write( "a) Rearrange the formula for P. What was the initial investment of an investment worth $1000 that compounded 10% interest for 10 years?\r
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\n" ); document.write( "\n" ); document.write( "start with A = P * (1 + i) ^ n.
\n" ); document.write( "divide both sides of the equation by (1 + i) ^ n to get:
\n" ); document.write( "A / ((1 + i) ^ n) = P
\n" ); document.write( "switch sides to get:
\n" ); document.write( "P = A / ((1 + i) ^ n)
\n" ); document.write( "when A = 1000 and i = 10% interest copounded for 10 years, the formula becomes:
\n" ); document.write( "P = 1000 / ((1 + .10) ^ 10 = 385.5432894.
\n" ); document.write( "that's your solution.
\n" ); document.write( "by replacing P with that in the original equation and appling 10% compounded yearly to it for 10 years to get:
\n" ); document.write( "A = 385.5432894 * (1 + .10) ^ 10 = 1000.
\n" ); document.write( "this confirms that P was calculated correctly.\r
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\n" ); document.write( "\n" ); document.write( "b) Rearrange the formula for i. What is the interest rate of an investment whose worth went from $1000 to $1200 in 2 years?\r
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\n" ); document.write( "\n" ); document.write( "A = P * (1 + i) ^ n becomes:
\n" ); document.write( "1200 = 1000 * (1 + i) ^ 2
\n" ); document.write( "divide both sides of the equation by 1000 to get:
\n" ); document.write( "1.2 = (1 + i) ^ 2
\n" ); document.write( "take the square root of both sides of the equation to get:
\n" ); document.write( "1.2 ^ (1/2) = 1 + i
\n" ); document.write( "subtract 1 from both sides of the equation to get:
\n" ); document.write( "1.2 ^ (1/2) - 1 = i
\n" ); document.write( "simplify to get:
\n" ); document.write( ".095445115
\n" ); document.write( "that's your interest rate.
\n" ); document.write( "confirm by replacing i in the original equation and solving for A to get:
\n" ); document.write( "A = 1000 * (1 + .095445115) ^ 2 = 1200.
\n" ); document.write( "this confirms that the value of i is correct.\r
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\n" ); document.write( "\n" ); document.write( "c) Explain a method with which you could estimate how many years it would take for an investment to reach a certain worth at a certain interest rate.\r
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\n" ); document.write( "\n" ); document.write( "the general formula is A = P * (1 + i) ^ n
\n" ); document.write( "you know A and you know P and you know i, but you don't know n.
\n" ); document.write( "you would use logarithms to find the value of n as follows.
\n" ); document.write( "you would divide both sides of the equation by P to get:
\n" ); document.write( "A/P = (1 + i) ^ n
\n" ); document.write( "you would take the log of both sides of this equation to get:
\n" ); document.write( "log(A/P) = log((1+i)^n)
\n" ); document.write( "by rule of logs, this becomes:
\n" ); document.write( "log(A/P) = n * log(1 + i)
\n" ); document.write( "you would divide both sides of the equation by log(1 + i) to get:
\n" ); document.write( "log(A/P) / log(1 + i) = n\r
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\n" ); document.write( "\n" ); document.write( "d) Estimate how many years would it take an investment of $2100 at 20% interest to reach a worth of $5225?
\n" ); document.write( "A = P * (1 + i) ^ n becomes:
\n" ); document.write( "5225 = 2100 * (1 + .2) ^ n.
\n" ); document.write( "simplify to get:
\n" ); document.write( "5225 = 2100 * 1.2 ^ n
\n" ); document.write( "divie both sides of the equation by 2100 to get:
\n" ); document.write( "5225/2100 = 1.2 ^ n
\n" ); document.write( "take the log of both sides of the equation to get:
\n" ); document.write( "log(5225/2100) = log(1.2^n)
\n" ); document.write( "by rule of logs, this becomes:
\n" ); document.write( "log(5225/2100) = n * log(1.2)
\n" ); document.write( "divide both sides of the equation by log(1.2) to get:
\n" ); document.write( "log(5225/2100) / log(1.2) = n
\n" ); document.write( "solve for n to get:
\n" ); document.write( "n = 4.999504552.
\n" ); document.write( "replace n in the oiginal equation with that and solve for A to get:
\n" ); document.write( "A = 2100 * 1.2 ^ 4.999504552 = 5225, confirming the value of n is good.\r
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\n" ); document.write( "\n" ); document.write( "note that the log rule that allows this is log(b^a) = a * log(b).
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