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Question 5: $500 is invested in an account that pays 4.5% per annum, interest compounded monthly. Find how long it takes to reach $5000. 500(1+0.045)^(12 t)(4.5)/(100)=0.045

Question 55: $500 \$ 500 is invested in an account that pays 4.5% 4.5 \% per annum, interest compounded monthly. Find how long it takes to reach $5000 \$ 5000 .\newline500(1+0.045)12t4.5100=0.045 500(1+0.045)^{12 t} \frac{4.5}{100}=0.045

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Q. Question 55: $500 \$ 500 is invested in an account that pays 4.5% 4.5 \% per annum, interest compounded monthly. Find how long it takes to reach $5000 \$ 5000 .\newline500(1+0.045)12t4.5100=0.045 500(1+0.045)^{12 t} \frac{4.5}{100}=0.045
  1. Identify Formula and Variables: Identify the formula for compound interest and the variables involved.\newlineThe formula for compound interest is A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}, where:\newlineAA is the amount of money accumulated after nn years, including interest.\newlinePP is the principal amount (the initial amount of money).\newlinerr is the annual interest rate (decimal).\newlinenn is the number of times that interest is compounded per year.\newlinett is the time the money is invested for, in years.\newlineWe need to solve for tt.
  2. Plug in Given Values: Plug in the given values into the compound interest formula.\newlineWe are given:\newlineP=$500P = \$500\newlineA=$5000A = \$5000\newliner=4.5%r = 4.5\% per annum =0.045= 0.045 (as a decimal)\newlinen=12n = 12 (since the interest is compounded monthly)\newlineWe need to find tt.\newlineThe formula with our values is:\newline$5000=$500(1+0.045/12)12t\$5000 = \$500(1 + 0.045/12)^{12t}
  3. Simplify Equation: Simplify the equation by dividing both sides by $500\$500.$5000/$500=(1+0.045/12)12t\$5000 / \$500 = (1 + 0.045/12)^{12t}10=(1+0.00375)12t10 = (1 + 0.00375)^{12t}
  4. Further Simplify: Simplify the equation further. 10=(1.00375)12t10 = (1.00375)^{12t}
  5. Take Natural Logarithm: Take the natural logarithm of both sides to solve for tt.ln⁡(10)=ln⁡((1.00375)(12t))\ln(10) = \ln((1.00375)^{(12t)})ln⁡(10)=12t⋅ln⁡(1.00375)\ln(10) = 12t \cdot \ln(1.00375)
  6. Isolate and Divide: Isolate tt by dividing both sides by (12⋅ln⁡(1.00375))(12 \cdot \ln(1.00375)).t=ln⁡(10)(12⋅ln⁡(1.00375))t = \frac{\ln(10)}{(12 \cdot \ln(1.00375))}
  7. Calculate Value of t: Calculate the value of t using a calculator.\newlinet≈ln⁡(10)12×ln⁡(1.00375)t \approx \frac{\ln(10)}{12 \times \ln(1.00375)}\newlinet≈2.3025850929912×0.00373766961828t \approx \frac{2.30258509299}{12 \times 0.00373766961828}\newlinet≈2.302585092990.0448516358194t \approx \frac{2.30258509299}{0.0448516358194}\newlinet≈51.3508826129t \approx 51.3508826129
  8. Interpret Result: Interpret the result.\newlineSince tt represents the time in years, it takes approximately 51.3551.35 years for the investment to grow from $500\$500 to $5000\$5000 with an interest rate of 4.5%4.5\% per annum, compounded monthly.

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