Santiago would need to invest approximately $260 to the nearest dollar for the value of the account to reach $320 in 14 years when the interest rate is 1.5% compounded quarterly.
To find out how much Santiago needs to invest in an account to reach a value of $320 in 14 years with an interest rate of 1.5% compounded quarterly, you can use the compound interest formula:
![\[A = P \left(1 + (r)/(n)\right)^(nt)\]](https://img.qammunity.org/2024/formulas/business/high-school/qbt64n0zcusl0cr29qewk9ldkubuogsf2v.png)
Where:
-
is the future value of the investment ($320 in this case).
-
is the initial principal (the amount Santiago needs to invest).
-
is the annual interest rate (1.5% or 0.015 as a decimal).
-
is the number of times the interest is mixed per year (quarterly, so
).
-
is the number of years (14 years in this case).
We need to solve for
, so let's rearrange the formula:
![\[P = (A)/(\left(1 + (r)/(n)\right)^(nt))\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/8tz0ck2b2ecnk55z09fqi6fm9tw35gg4r7.png)
Now, plug in the values:
![\[P = (320)/(\left(1 + (0.015)/(4)\right)^(4 * 14))\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/lnm1zsfv8vvzh2n466mm2fe7v9hagkznbs.png)
Calculate the exponent:
![\[P = (320)/(\left(1 + 0.00375\right)^(56))\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/omq6hrtyvks52se2epnuxo11lip9ikmlxn.png)
Now, calculate the expression inside the parentheses:
![\[P = (320)/((1.00375)^(56))\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/j9ya5ttfxyr8s2a72b3w6ulycawyw4fxap.png)
Calculate the value inside the parentheses:
![\[P = (320)/(1.22804768215)\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/3way47rm34v4hdn7inftrhl8mqagzrd4jd.png)
Now, divide to find the value of
:
![\[P \approx (320)/(1.22804768215) \approx 260.19\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/myvndz8d4mw4cnkrte0qun60uvadc5epra.png)
So, Santiago would need to invest approximately $260 (rounded to the nearest dollar) for the value of the account to reach $320 in 14 years with an interest rate of 1.5% compounded quarterly.