Simple/Compound Interest Rates

I have found that after the 5 years both accounts should have the amount 1.298C in them, (I defined C as the initial deposit) however I am not too sure on how I go about finding 'n' from this (when the second account raises to 6%). Any help would be much appreciated!

I'm assuming the simple interest is paid per year?

Call $t_1$ the amount of time that 5% interest is paid and $t_2$ the amount of time 6% is paid $(t_1+t_2=5)$.

Then the interest in the first $t_1$ years is $0.05Ct_1$ and the interest in the next $t_2$ years is $0.06Ct_2$ so the total amount in the

account after 5 years will be $C + 0.05Ct_1 + 0.06Ct_2=C(1+0.05t_1+0.06t_2)$

Can you carry on from here? You're not going to get exactly the same as the compound interest but due to rounding of money, you're looking for $t_1$ and $t_2$ that get you closest to the compound interest.

Sorry I'm still not really sure, I did get the same expressions for the total amount and time as you but not sure where to go from those.

Setting the compound and simple expressions equal to each other:

$C\times 1.0535^5 = C(1+0.05t_1+0.06t_2)\Rightarrow 1.0535^6 = 1+0.05t_1+0.06t_2$

So now you need to look for $t_1$ and $t_2$ which satisfy that equation. Since they need to add up to 5, there aren't many choices e.g. 1,4 or 2,3 etc.

From doing that I know that $t_1$ has to be between 2 - 3, and with further calculations between 2 - 2.05, but the exact answer is n = 2.04 years which I'm not sure how to prove without sort of guessing.

How did you get 2-3? And where did you get the answer of 2.04?

I can't see how the answer can be a decimal.

Sorry made a mistake, in order to get close to the the total compound amount then surely t1 will not even be 1 year with t2 > 4, but 2.04 is apparently the answer on my worksheet. I can only assume that the decimal comes from considering daily interest too, so 0.04 would equate to 14 days. But now I am also wondering where 2.04 comes from when 2.04 would give 1.2796C as the total amount which isn't really close to 1.298C..?

I don't see how the answer could be 2.04. Either the interest will be payed yearly or monthly (I doubt anything else). How can you switch the type of interest after 2.04 years?