# Finding ratio of interest rates

I'm reading through Marcel B. Finan's A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 on my own and am unsure how to proceed with a question (page 99, question 11.29). It is:

On January 1, 1980, Jack deposited 1,000 into Bank X to earn interest at the rate of $j$ per annum compounded semi-annually. On January 1, 1985, he transferred his account to Bank Y to earn interest at the rate of $k$ per annum compounded quarterly. On January 1, 1988, the balance at Bank Y is 1,990.76. If Jack could have earned interest at the rate of $k$ per annum compounded quarterly from January 1, 1980 through January 1, 1988, his balance would have been 2,203.76. Calculate the ratio $\frac kj$.

Given are:

• $(1+\frac j2)^{10}(1+\frac k4)^{12}=1.99076$
• $(1+\frac k4)^{32}=2.20376$

I can solve the second of those for $k$ and then (using that value of $k$) the first for $j$, but wonder if there's a more direct way of finding $k/j$.

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