Question

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58. You invest $200 in an annuity that earns 7% annual interest. After how many years will the value of the annuity double?

Answer 1

The time taken for $200 invested at 7% annual interest compounded yearly:

`\boxed{\mathrm{10.245 \; {years}}}`

which roughly works out to

`\boxed{\mathrm{10\;years\;and\;3\;months}}`

Explanation:

The formula for the accrued value of an amount P deposited at i% interest for a time period of t years compounded annually is given by the formula


`\boxed{A = P(1 + r)^t\;\cdots[1]}`

Here A is the accrued value, P the principal and r = i/100

If you are given A, P and r we can compute t by taking the logarithms on both sides

In Equation [1] we get

`(A)/(P) = (1 + r)^t \;\cdots[2]`

Taking logs on both sides,

`\log A - \log P = \log((1+r)^t)\\\\\log x^a = a \log x\\\\\\`

Therefore the right side becomes

`t \log (1+r)`

Replacing right side of equation [2] with this expression yields


`\log A - \log P = t \log (1+r)\\\\\textrm{Or, }\\\\t =( \log A - \log P)/(\log (1 + r))\\\\`

This is the general equation for determining how long it will take for the principal P to reach the value A if compounded annually at rate r(in decimal)

Since we are interested in seeing our principal P=200 double to A = 400 at r = 7% = 7/100 = 0.07
and
1 + r = 1 + 0.07 = 1.07

`t = (\log 400 - \log 200)/(\log 1.07)\\\\`

`t = \boxed{10.245 \; \textrm{years}}`

or approximately

`\boxed{\mathrm{10\;years\;and\;3\;months}}`

Answer 2

10.24 years (approx. 10 years 3 months)

Explanation:

Most fixed annuities pay annual compound interest.

`\boxed{\begin{minipage}{7 cm}\underline{Annual Compound Interest Formula}\\\\$ A=P\left(1+r\right)^(t)$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}`

Given:

• A = $400 (double the principal)
• P = $200
• r = 7% = 0.07

To calculate after how many years the value of the annuity will double, substitute the given values into the formula and solve for t:

`\implies 400=200(1+0.07)^t`

`\implies 2=(1.07)^t`

`\implies \ln 2=\ln (1.07)^t`

`\implies \ln 2=t \ln 1.07`

`\implies t=(\ln 2)/(\ln 1.07)`

`\implies t=10.2447683...\rm years`

Therefore, the value of the annuity will double after 10.24 years (approximately 10 years 3 months).