Fiona invested $1100 at 8% compound comtinously. At the same time, Maria ivested $1200 at 8% compounded daily. How long will it take for their investments to be equal in value?
after x years, the two amounts will be equal:
1100*e^.08x = 1200*(1+.08/365)^(365x)
x = 9926.18535
To find out how long it will take for Fiona and Maria's investments to be equal, we can set up an equation using the compound interest formula. The formula for compound interest is given by:
A = P * e^(rt),
where:
A is the final amount,
P is the principal amount (initial investment),
e is the base of the natural logarithm (approximately 2.71828, a mathematical constant),
r is the interest rate, and
t is the time in years.
For Fiona's investment:
P1 = $1100
r1 = 8% = 0.08
For Maria's investment:
P2 = $1200
r2 = 8% = 0.08
Setting up the equation:
P1 * e^(r1 * t) = P2 * e^(r2 * t)
Substituting the given values:
$1100 * e^(0.08 * t) = $1200 * e^(0.08 * t)
To solve this equation, we need to isolate the exponential term on one side:
e^(0.08 * t) / e^(0.08 * t) = ($1200 / $1100)
Simplifying:
1 = ($1200 / $1100)
To find the value of t, we can take the natural logarithm (ln) of both sides:
ln(1) = ln($1200 / $1100)
ln(1) = ln(1.0909)
Solving for t:
t = ln(1.0909) / 0.08
Calculating this using a calculator, we get:
t ≈ 0.035 years.
Converting this to months or days would depend on the context of the problem, but the solution above represents the time in years.