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Santiago is going to invest in an account paying an interest rate of 1.5% compounded quarterly. how much would santiago need to invest, to the nearest dollar, for the value of the account to reach $320 in 14 years?

2 Answers

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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:


\[A = P \left(1 + (r)/(n)\right)^(nt)\]

Where:

-
\(A\) is the future value of the investment ($320 in this case).

-
\(P\) is the initial principal (the amount Santiago needs to invest).

-
\(r\) is the annual interest rate (1.5% or 0.015 as a decimal).

-
\(n\) is the number of times the interest is mixed per year (quarterly, so
\(n = 4\)).

-
\(t\) is the number of years (14 years in this case).

We need to solve for
\(P\), so let's rearrange the formula:


\[P = (A)/(\left(1 + (r)/(n)\right)^(nt))\]

Now, plug in the values:


\[P = (320)/(\left(1 + (0.015)/(4)\right)^(4 * 14))\]

Calculate the exponent:


\[P = (320)/(\left(1 + 0.00375\right)^(56))\]

Now, calculate the expression inside the parentheses:


\[P = (320)/((1.00375)^(56))\]

Calculate the value inside the parentheses:


\[P = (320)/(1.22804768215)\]

Now, divide to find the value of
\(P\):


\[P \approx (320)/(1.22804768215) \approx 260.19\]

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.

User Mykola Shorobura
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4 votes

The principal investment required to get a total amount of $320 from compound interest at a rate of 1.5% per year compounded 4 times per year over 14 years is $260.

The formula for compound interest is expressed as:


A = P( 1 + (r)/(n))^(nt)

Where A is the accrued amount, P is the principal, r is the interest rate and t is time.

Given that:

Accrued amount A = $320

Interest rate r = 1.5% = 1.5/100 = 0.015

Compounded quarterly n = 4

Time t = 14 years

Principal P =?

Plug the given values into the above formula and solve for the principal:


A = P( 1 + (r)/(n))^(nt)\\\\P = (A)/(( 1 + (r)/(n))^(nt) ) \\\\P = (320)/(( 1 + (0.015)/(4))^(4*14) ) \\\\P = (320)/(( 1 + 0.00375)^(4*14) ) \\\\P = (320)/(( 1.00375)^(4*14) ) \\\\P = \$260

Therefore, Santiago needs to invest approximately $260.

User BhishanPoudel
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