# What would be the best way to calculate interest rate using the following notations

I need to calculate the interest rate automatically, but my maths is not very strong and i couldnt figure it out. I am pretty sure its just a game of seconds for the experts. Thank you very much for the help.

Minimum loan = 100,000 Maximum loan = 1,000,000

At 100k the loan interest is 7% whereas at 1000k the interest rate is 5%.

Now, using that data, how can a computer automatically calculate the interest rate based on that scale? Like interest for 500k or 437,698$Against thanks alot - ## 1 Answer There is no automatic answer, the appropriate interest rate is a business decision, which might be influenced by various factors. But a mathematically simple possibility is to use linear interpolation. For a sum$S$between$100000$and$1000000$we would then set the interest rate at the following percentage: $$7 -(7-5)\frac{S-100000}{1000000-100000}.$$ We have given the answer in "raw" form in order to make it easy to adjust if the numbers change. Edit: This is in answer to a request for a derivation. We will lie a bit about how we actually did it. Linear interpolation is such a common tool that after a while the answer becomes almost automatic. Instead we will use an unpleasant amount of algebra. Let$A$be the lower amount, here$100000$, and$B$the higher amount, here$1000000$. Let$a$be the interest rate at the lower amount, here$7$(percent), and let$b$be the interest rate at the higher amount. Let$x$be the interest rate when the loan size is$S$. (If the symbols$A$,$B$,$a$,$B$get in the way of understanding, substitute their actual values in your problem in the algebra below.) Assume that$x$is a linear function of$S$. More precisely, assume that$x=a-(pS+q)$, where$p$and$q$are numbers to be determined. When$S=A$, we have$x=a$, so$0=pA+q$. Thus$q=-pA$. When$S=B$, we have$x=b$, so$b=a-(pB+q)=a-(pB-pA)$. It follows that$a-b=p(B-A)$, and therefore$p=\frac{a-b}{B-A}\$.

We find that $$x=a-(pS-pA)=a-p(S-A)=a-\frac{a-b}{B-A}(S-A)=a-(a-b)\frac{B-A}{S-A}.$$ That is the general formula.

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Thanks for the help, but if possible could you please also tell me the derivation? Im curious. – Aayush Agrawal Jan 26 '13 at 5:25