If money is invested at $r$ percent interest, compounded annually, the amount of the investment will double in approximately $\dfrac{70} {r}$ years. If Pat’s parents invested 5,000 in a long term bond that pays 8 percent interest, compounded annually, what will be the approximate total amount of the investment 18 years later, when Pat is ready for college? A. $20,000$ B. $15,000$ C. $12,000$ D. $9,000$
ANSWER
Verified
Hint: Find the value of years in which the amount will get double, by substituting the value of $r=8$ in $\dfrac{70}{r}$ years. Now check the multiple of the years at which it will become 18 years and accordingly increase the given amount.
Complete step-by-step solution - Here we are given that if money is invested at $r$ percent interest, compounded annually, the amount of the investment will double in approximately $\dfrac{70}{r}$ years. If Pat’s parents invested $5,000$ in a long term bond that pays 8 percent interest, compounded annually, we have to find the approximate total amount of the investment after 18 years, when Pat is ready for college. Before proceeding with the question, let us know about compound interest. Compound interest is calculated on the principal and the interest accumulated over the previous period. While calculating the compound interest, the amount for a certain period of time becomes the principal for the further period of time. We can use the following formula to calculate the compound interest: Compound Interest = Amount - principal, where the amount is given by the following formula, $A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}$, where, A = Amount for desired time, P = principal, r = rate of interest for desired time and t = time for which the interest is to be calculated. Now let us consider our question. Here we are given that $r$ is the rate of interest and the amount of investment doubles in $\dfrac{70}{r}$ years. So, we have the initial amount as P, then the rate of interest as $r$, and in the time of $\dfrac{70}{r}$ years, we get the amount as 2P. The initial investment made by Pat’s parents in long term bond = $5,000$ The rate of interest at which they invested the initial amount = 8%. So, the time in which their investment will get double = $\dfrac{70}{r}$ years. By substituting the values of r = 8, we get the time in which the investment gets double $=\dfrac{70}{8}$ years = 8.75 years. So, the amount they will get at the end of 8.75 years is = 2 (initial amount) = 2 (10,000. So, after 8.75 years, they get $10,000$. Now in the next 8.75 years, this amount will become double again. So, after 8.75 years from now or 17.5 years they get an amount = 2($10,000$) = $20,000$. Hence after 17.5 $\approx$ 18 years they will get an approximate amount of $20,000$. Therefore, the correct answer is option A.
Note: Many students use the formula for compound interest here, that is $A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}$, which is not needed here because we have been asked for the approximate and not exact amount after 18 years. The options are also given accordingly if we look at them carefully. Take note that in compound interest, the interest for a specific number of years is not the same as in the case of simple interest, where it remains constant.