Trying to help my 8th grader, but I'm at a loss with this one. Can anyone assist? Thank you.
Gwyneth placed $2,000 in an account that warned 10% interest, compounded annually. How many years did she save if she if she had $2,662 in her account?
you need
2000 * 1.10^t = 2662
1.10^t = 1.331
t log1.10 = log1.331
t = log1.331 / log1.10 = 3 years
2000(1.10)^n = 2662
Kind of hard to believe that this is grade 8 question since it
requires logs to actually solve it, .... anyway ...
1.1^n = 1.331
take logs of both side, and using log rules
n log 1.1 = log 1.331
n = log1.331/log1.1 = 3
ok,
perhaps they were expected to do it this way
after 1 year, amount = 2000(1.1) = 2200
after 2 years, amount = 2200(1.1) = 2420
after 3 years, amount = 2420(1.1) = 2662
well, that was lucky!
I wouldn't have solved it. Thank you!!
Of course! I'd be happy to help you with this problem.
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount in the account
P = the principal amount (the initial amount you start with)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
In this case, Gwyneth placed $2,000 in the account, and the interest rate is 10% (0.10) compounded annually. We need to find the value of t.
We are given that the final amount in her account is $2,662.
Substituting the given values into the formula, we have:
$2,662 = $2,000(1 + 0.10/1)^(1*t)
Now, we need to solve for t. Let's simplify the equation.
$2,662 = $2,000(1.10)^t
Next, divide both sides of the equation by $2,000 to get rid of the coefficient.
1.331 = 1.10^t
Now, we need to solve for t. We can use logarithms to isolate t. Taking the natural logarithm of both sides of the equation:
ln(1.331) = ln(1.10^t)
Using the logarithm rule that ln(a^b) = b * ln(a):
ln(1.331) = t * ln(1.10)
Now, divide both sides by ln(1.10) to solve for t:
t = ln(1.331) / ln(1.10)
Using a calculator, we can evaluate this expression:
t ≈ 3.996
Therefore, Gwyneth saved for approximately 4 years to have $2,662 in her account.
I hope this explanation helps! Let me know if you have any further questions.