Kenny wants to invest $250 every three months at 5.2%/a compounded quarterly. He would like to have at least $6500 at the end of his investment. How long will he need to make a regular payments?

Question

An engaging, graphical representation of a financial investment scenario. The image should visually depict Kenny, a Caucasian male investor, placing $250 into a money box every three months. The money box should be designed to indicate a rise in value with time, symbolizing the 5.2% annual interest compounded quarterly. Background elements could include symbols of time passing like clock or calendar. The endpoint should show a visual representation of $6500, Kenny's goal. Note should be taken to not include any textual elements in the image.

Kenny wants to invest $250 every three months at 5.2%/a compounded quarterly. He would like to have at least $6500 at the end of his investment. How long will he need to make a regular payments?

*PLEASE SHOW WORK*

Answer 1

.052/4 = .013 = r

so
multiply by 1.013 every quarter year
n = number of quarter years

Google amount of a sinking fund
S = 250 [(1+r)^n - 1 ] /r
6500 /250 = 26 = [(1+r)^n - 1 ] /r

26 = [ 1.013^n - 1 ]/.013

.338 = 1.013^n - 1
1.338 = 1.013^n
so
n log 1.013 = log 1.338
n (.0056094) = .12646
n = 22.54 quarters
so 5.6 years

Answer 2

for example

http://www.math.utep.edu/Faculty/cmmundy/Math%201320/Worksheets/Sinking%20Funds%20&%20Annuities.pdf

Answer 3

Well, Kenny certainly has some lofty financial goals! Let's break down the problem and solve it step by step.

To find out how long it will take Kenny to reach his goal, we need to use the compound interest formula:

A = P * (1 + r/n)^(n*t)

Where:
A = the future value of the investment ($6500 in this case)
P = the regular payment made every three months ($250 in this case)
r = the annual interest rate (5.2% or 0.052 in decimal form)
n = the number of times interest is compounded per year (quarterly in this case)
t = the number of years that Kenny wants to invest

Since Kenny wants to reach $6500, we can set up the equation as follows:

6500 = 250 * (1 + 0.052/4)^(4*t)

Now, let's solve for t:

6500/250 = (1 + 0.052/4)^(4*t)

26 = (1 + 0.013)^4t

Taking the log of both sides:

log(26) = log((1 + 0.013)^4t)

Using logarithm rules, we can bring down the exponent:

log(26) = 4t * log(1 + 0.013)

Now, divide both sides by 4 * log(1 + 0.013):

t = log(26) / (4 * log(1 + 0.013))

Calculating the values:

t ≈ 6.23 years

So, it will take Kenny approximately 6.23 years to reach his goal of $6500 by making regular payments of $250 every three months.

Remember, investing is no joke! It's always a good idea to consult with a financial advisor before making any investment decisions.

Answer 4

To find out how long Kenny will need to make regular payments, we can use the future value of an annuity formula:

FV = P * (1 + r/n)^(nt) - P * ((1 + r/n)^nt - 1) / (r/n)

Where:
FV = Future Value
P = Payment per period
r = Interest rate per period
n = Number of compounding periods per year
t = Number of years

Given:
P = $250
r = 5.2% = 0.052
n = 4 (compounded quarterly)
FV = $6500

Let's substitute the given values into the formula and solve for t:

$6500 = $250 * (1 + 0.052/4)^(4t) - $250 * ((1 + 0.052/4)^(4t) - 1) / (0.052/4)

Simplifying the formula:

$6500 = $250 * (1 + 0.013)^4t - $250 * ((1 + 0.013)^(4t) - 1) / 0.013

Now, we can solve for t by isolating it:

$6500 - $250 * ((1 + 0.013)^(4t) - 1) / 0.013 = $250 * (1 + 0.013)^4t

$6250 = $250 * ((1.013)^(4t) - 1) / 0.013

$6250 * 0.013 = $250 * ((1.013)^(4t) - 1)

$81.25 = (1.013)^(4t) - 1

$81.25 + 1 = (1.013)^(4t)

82.25 = (1.013)^(4t)

To solve for t, we can take the natural logarithm of both sides:

ln(82.25) = ln((1.013)^(4t))

Using the property ln(a^b) = b * ln(a):

ln(82.25) = 4t * ln(1.013)

Now, solve for t by dividing both sides by 4 * ln(1.013):

t = ln(82.25) / (4 * ln(1.013))

Using a calculator, we can find:

t ≈ 6.47

Therefore, Kenny will need to make regular payments for approximately 6.47 years to have at least $6500 at the end of his investment.

Answer 5

To determine how long Kenny needs to make regular payments to reach the desired amount, we can use the formula for compound interest:

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

Where:
A = the future value of the investment
P = the regular payment amount
r = the annual interest rate (in decimal form)
n = the number of compounding periods per year
t = the number of years

In this case, the regular payment amount is $250, the annual interest rate is 5.2%/a (0.052 in decimal form), and the interest is compounded quarterly (n = 4).

We want to find the value of t when A reaches $6500.

Let's solve for t:

A = P(1 + r/n)^(nt)
6500 = 250(1 + 0.052/4)^(4t)

Divide both sides of the equation by 250:

6500/250 = (1 + 0.052/4)^(4t)

26 = (1 + 0.013)^4t

Now, take the natural logarithm (ln) of both sides:

ln(26) = ln[(1 + 0.013)^4t]

Using the properties of logarithms, we can rewrite the equation as:

ln(26) = 4t * ln(1 + 0.013)

Divide both sides by 4 * ln(1 + 0.013):

ln(26) / [4 * ln(1 + 0.013)] = t

Using a calculator, we can calculate the value of t:

t ≈ 26 / [4 * ln(1 + 0.013)] ≈ 8.21

Therefore, Kenny will need to make regular payments for approximately 8.21 years to reach a minimum of $6500 at the end of his investment.