Given the equation A=250(1.1)t, you can determine that the interest is compounded annually and the interest rate is 10%. Suppose the interest rate were to change to being compounded quarterly. Rewrite the equation to find the new interest rate that would keep A and P the same.
What is the approximate new interest rate?
Convert your answer to a percentage, round it to the nearest tenth, and enter it in the space provided, like this: 42.53%
The answer to this problem is 9.6
Can someone tell me the answer already. Why give the equation but not the answer.
A = 250(1 + 0.1/4)^(4t)
i agree anon
To rewrite the equation to find the new interest rate, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal (initial amount)
r = the interest rate
n = the number of times interest is compounded per year
t = the number of years
Given the original equation A = 250(1.1)t, we can see that the interest is compounded annually (n = 1) and the interest rate is 10% (r = 0.10).
To find the new interest rate that would keep A and P the same but with quarterly compounding, we need to adjust the equation to reflect the new values. Since interest is compounded quarterly, n = 4 (quarterly compounding means 4 times per year).
The formula now becomes:
A = P(1 + r/4)^(4t)
Since we want to keep A and P the same, we can equate the two equations:
250(1.1)t = P(1 + r/4)^(4t)
Divide both sides of the equation by P:
250(1.1)t / P = (1 + r/4)^(4t)
Simplify:
1.1t = (1 + r/4)^(4t)
Now, we need to isolate the new interest rate variable (r). Taking the natural logarithm (ln) of both sides of the equation will help us achieve this:
ln(1.1t) = ln((1 + r/4)^(4t))
Using the logarithmic property of exponents, we can simplify further:
ln(1.1t) = 4t × ln(1 + r/4)
Now, we can isolate r by dividing both sides of the equation by 4t and then multiplying the entire equation by 4:
4 × ln(1.1t) / (4t) = ln(1 + r/4)
ln(1.1t) / t = ln(1 + r/4)
Now, we solve for r by multiplying both sides by 4:
4 × ln(1.1t) / t = r/4
Multiply both sides by 4 to isolate r:
r = 4 × 4 × ln(1.1t) / t
Now, we can substitute the given values into this equation to find the new interest rate.
Using t = 1 (as it was given that interest is compounded annually), we get:
r ≈ 4 × 4 × ln(1.1) / 1
Calculating this value gives:
r ≈ 7.77
To convert this to a percentage, we multiply by 100:
r ≈ 7.77 × 100 = 777%
Therefore, the approximate new interest rate that would keep A and P the same with quarterly compounding is 777%. Rounding this to the nearest tenth gives us 777.0%.