A PERSON SAVES #3000 AT 4 1/2% COMPOUND INTEREST. SHE ADDS #800 TO HER AMOUNT AT THE END OF EACH YEAR. FIND HER TOTAL SAVINGS AFTER 2 YEARS
To calculate the total savings after 2 years, we need to apply the compound interest formula for each year and also take into account the additional deposit of #800 at the end of each year.
The compound interest formula for an amount P, compounded annually at an interest rate r for t years, is given by:
\[ A = P(1 + \frac{r}{100})^t \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (in decimal form).
- \( t \) is the time the money is invested for, in years.
We start with an initial savings of \( P = #3000 \), an interest rate of \( r = 4.5\% = 0.045 \), and we'll calculate the savings over \( t = 2 \) years.
### After the first year:
\[ A_1 = P(1 + \frac{r}{100})^1 \]
\[ A_1 = 3000(1 + 0.045)^1 \]
\[ A_1 = 3000(1.045) \]
\[ A_1 = 3135 \]
So after the first year, she will have #3135.
Now she adds #800 to this amount for the next year's investment, so the new principal \( P_2 \) for the second year is:
\[ P_2 = A_1 + 800 \]
\[ P_2 = 3135 + 800 \]
\[ P_2 = 3935 \]
### After the second year:
\[ A_2 = P_2(1 + \frac{r}{100})^1 \]
\[ A_2 = 3935(1 + 0.045)^1 \]
\[ A_2 = 3935(1.045) \]
\[ A_2 = 4111.175 \]
Now you just add the #800 end-of-the-year deposit again to the amount at the end of the second year:
\[ Total = A_2 + 800 \]
\[ Total = 4111.175 + 800 \]
\[ Total = 4911.175 \]
Thus, after 2 years with the given interest and yearly deposits, her total savings would be approximately #4911.18 (rounded to the nearest penny). Please note that financial institutions may round figures differently, and this calculation would also assume the interest is compounded exactly once per year, no withdrawals are made, and no taxes or fees are deducted from the savings.