jordan invested $97,000 in an account paying an interest rate of 1.6% compounded monthly. assuming no…

jordan invested $97,000 in an account paying an interest rate of 1.6% compounded monthly. assuming no deposits or withdrawals are made, how long would it take, to the nearest year, for the value of the account to reach $109,700?\nanswer

Answer

Explanation:

Step1: Recall compound interest formula

The compound interest formula for compounded monthly is: $$A = P\left(1+\frac{r}{12}\right)^{12t}$$ Where: $A = 109700$ (final amount), $P = 97000$ (principal), $r = 0.016$ (annual interest rate), $t$ = time in years.

Step2: Substitute values into formula

$$109700 = 97000\left(1+\frac{0.016}{12}\right)^{12t}$$

Step3: Isolate the exponential term

Divide both sides by 97000: $$\frac{109700}{97000} = \left(1+\frac{0.016}{12}\right)^{12t}$$ $$1.130928 = \left(1.001333\right)^{12t}$$

Step4: Apply natural logarithm to both sides

Take $\ln$ of both sides to bring down the exponent: $$\ln(1.130928) = 12t \cdot \ln(1.001333)$$

Step5: Solve for t

Calculate the logarithms and rearrange: $$t = \frac{\ln(1.130928)}{12 \cdot \ln(1.001333)}$$ $$\ln(1.130928) \approx 0.1227$$ $$\ln(1.001333) \approx 0.001332$$ $$t = \frac{0.1227}{12 \cdot 0.001332} = \frac{0.1227}{0.015984} \approx 7.68$$

Step6: Round to nearest year

Round 7.68 up to the nearest whole number.