a) 1150 dollars invested at 9% annual interest rate (compounded yearly) or
b) 1475 invested at 6% annual interest (compounded yearly) after
When would the two investments have equal value?
let the time be t
1150(1.09)^t = 1475(1.06)^t
46(1.09)^t = 59(1.06)^t
take log of both sides , and using simple log rules
log46 + tlog1.09 = log59 + tlog1.06
tlog1.09 - tlog1.06 = log 59 - lot 46
t(log 1.09 - log 1.06) = log 59 - log 46
t = (log59 -log46)/(log1.09 - log1.06) = 8.918
it would take appr 9 years
check:
1150(1.09^9 = 2497.68
1475(1.06)9 = 2491.98
using t = 8.918
1150(1.09)^8.918 = 2480.09
1475(1.06)^8.918 = 2480.13 , not bad
To determine when the two investments will have equal value, we need to set up an equation where the value of investment a) is equal to the value of investment b). We'll use the compound interest formula for both investments and solve for the time (t) when the two amounts are equal.
For investment a):
A = P * (1 + r)^t
Where:
A is the future value of the investment
P is the principal amount (initial investment)
r is the annual interest rate (expressed as a decimal)
t is the number of years
For investment b):
A = P * (1 + r)^t
Let's substitute the given values into the formulas:
For investment a):
Aa = 1150 * (1 + 0.09)^t
For investment b):
Ab = 1475 * (1 + 0.06)^t
Now, we can set the two equations equal to each other and solve for t:
1150 * (1 + 0.09)^t = 1475 * (1 + 0.06)^t
Next, we divide both sides by (1 + 0.09)^t:
1150 = 1475 * [(1 + 0.06) / (1 + 0.09)]^t
Simplifying further:
(1 + 0.06) / (1 + 0.09) = 106/109
1150 = 1475 * (106/109)^t
Finally, we can solve the equation using logarithms. Taking the logarithm of both sides, we can isolate t:
log(1150) = log(1475 * (106/109)^t)
Using logarithm properties:
log(1150) = log(1475) + t * log(106/109)
Simplifying further:
t = [log(1150) - log(1475)] / log(106/109)
Using a calculator or software, evaluate the right side of the equation to find the value of t when the investments will have equal value.