Respuesta :
Answer:
amount deposit each quarter is $3413.96
Step-by-step explanation:
Given data
amount required = $4000
time period (t) = 2 years
rate (r) = 8% = 0.08
to find out
How much amount deposit each quarter principal, principal
solution
we know compound interest formula and we know 4 quarters in 1 year so n = 4
amount = principal [tex](1 + r/n)^{nt}[/tex]
here put all these value amount rate time period and find the principal amount
principal = amount / [tex](1 + r/n)^{nt}[/tex]
principal = 4000 / [tex](1 + 0.08/4)^{8}[/tex]
principal = 3413.96
amount deposit each quarter is $3413.96
The person should deposit [tex]\boxed{\$\bf 342}[/tex].
Further explanation:
Annuity can be defined as the series of payment made in equal interval of time.
There are mainly two types of annuity.
1. Annuity advance.
2. Annuity arrear
Given:
The person wishes to have [tex]\$\ 3000[/tex] in [tex]2[/tex] years to buy a fancy new stereo system.
Formula used:
The formula for the future value of annuity is given as,
[tex]\boxed{FV=C\left[\dfrac{(1+i)^{n}-1}{i}\right](1+i)}[/tex] …… (1)
Here, [tex]i[/tex] is the interest rate, [tex]FV[/tex] is the future value, [tex]C[/tex] is the payment size and [tex]n[/tex] is the number of cash flow.
Calculation:
Consider the quarterly payment as [tex]C[/tex].
The amount after [tex]2[/tex] year is [tex]\$ 3000[/tex]. Therefore, the future value is [tex]\$ 3000[/tex].
Interest rate for [tex]8%[/tex] compounded quarterly is calculated as shown below:
[tex]\begin{aligned}i&=\dfrac{0.08}{4}\\&=0.02\end{aligned}[/tex]
To calculate the value of [tex]C[/tex] substitute [tex]0.02[/tex] for [tex]i[/tex], [tex]8[/tex] for [tex]n[/tex] and [tex]3000[/tex] for [tex]FV[/tex] in equation (1) as shown below:
[tex]\begin{aligned}3000&=C\left[\dfrac{(1+0.02)^{8}-1}{0.02}\right](1+0.02)\\C\left[\dfrac{(1+0.02)^{8}-1}{0.02}\right]&=\dfrac{3000}{1.02}\\C\left[\dfrac{1.172-2}{0.02}\right]&=2941.17\\C\cdot (0.172)&=2941.17\times 0.02\\C&=\dfrac{58.823}{0.172}\\C&\approx342\end{aligned}[/tex]
Therefore, the person should deposit [tex]\boxed{\bf \$342}[/tex] each quarter in to an account paying [tex]8\%[/tex] compounded quarterly.
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Answer details:
Grade: Middle school
Subject: Mathematics
Chapter: Compound interest
Keywords: Compounded quarterly, account, deposit, future value, interest rate, fancy, stereo system, annuity, level annuity, arrear, annuity advance, annuity arrear.
