A couple needs $15,000 as a down payment for a home. If they invest the $10,000 they have at 8% compounded quarterly, how long will it take for the money to grow into $15,000?
10000(1.02)^n = 15000 , where n is the number of quarter years.
1.02^n = 1.5
take log of both sides
log 1.02^n = log 1.5
n log 1.02 = log 1.5
n = log 1.5/log 1.02 = 20.475 quarters
= 5.12 years , 20.475 quarters or 61.4 months
take your pick
To determine how long it will take for the $10,000 investment to grow into $15,000 at an interest rate of 8% compounded quarterly, we can use the compound interest formula:
A = P(1 + r/n)^(nt),
where:
A = the final amount ($15,000),
P = the principal amount ($10,000),
r = the annual interest rate (8% = 0.08),
n = the number of times the interest is compounded per year (quarterly = 4 times),
t = the number of years.
We need to solve the formula for t, which represents the time required. Rearranging the formula, we have:
t = (log(A/P)) / (n * log(1 + r/n)).
Let's substitute the given values into the formula and calculate the result:
t = (log(15000/10000)) / (4 * log(1 + 0.08/4)).
t = (log(1.5)) / (4 * log(1.02)).
t = (0.1761) / (4 * 0.0069).
t = 0.1761 / 0.0276.
t ≈ 6.38.
Therefore, it will take approximately 6.38 years for the $10,000 investment to grow into $15,000 at an interest rate of 8% compounded quarterly.
To calculate how long it will take for the $10,000 to grow into $15,000 at an interest rate of 8% compounded quarterly, we can use the compound interest formula:
A = P(1 + r/n)^(n*t)
Where:
A = the future value (in this case $15,000)
P = the principal amount (in this case $10,000)
r = the annual interest rate (8% or 0.08 in decimal form)
n = the number of times the interest is compounded per year (4 for quarterly)
t = the number of years
Let's solve for t:
$15,000 = $10,000(1 + 0.08/4)^(4*t)
Divide both sides of the equation by $10,000:
1.5 = (1.02)^(4*t)
Take the natural logarithm (ln) of both sides:
ln(1.5) = ln(1.02)^(4*t)
Using the power rule of logarithms:
ln(1.5) = (4*t) * ln(1.02)
Rearrange the equation to solve for t:
t = ln(1.5) / (4 * ln(1.02))
Using a calculator, we can solve for t:
t ≈ 14.17 years
So, it will take approximately 14.17 years for the $10,000 to grow into $15,000 at an interest rate of 8% compounded quarterly.