# Effective Percentage in Simple Interest

Which simple interest rate over six years is closest to being equivalent to the following: an effective rate of discount of $3\%$ for the first year, an effective rate of discount of $6\%$ for the second year, an effective rate of discount of 9% for the third year, and an effective rate of interest of $5\%$ for the fourth, fifth and sixth years?

A. $6.3\%\quad$ B. $6.4\%\quad$ C. $6.5\%\quad$ D. $6.6\%\quad$ E. $6.7\%\quad$

Answer for this Question is: The effective rate of (simple) interest would be: $$(1−0.03)^{-1}(1−0.06)^{-1}(1−0.09)^{-1}(1+0.05)^3=(1+6i)\implies i\approx 6.6\%$$

My question is how the first three values we are subtracting from 1 and last value is adding 1 why? Then for first three values they are putting power as $-1$ and last value they are putting power as $3$. Please how they solved and logic, please anyone guide me?

• what is the Difference between the Effective Rate of Discount and Effective Rate of Interest please anayone Share I dont Know the Logic – cloud computing in salesforce Aug 22 '16 at 11:25
• Is there any Formula for effective Rate of Discount is 1/1-r/100, if there is any Formula please Explain, I have a Little bit confusing in si and Ci,Mainly the Power Varies, I want Clearcut for this Problem how they solved Guide Me Please – cloud computing in salesforce Aug 22 '16 at 11:41

According to Wikipedia, "the annual effective discount rate expresses the amount of interest paid/earned as a percentage of the balance at the end of the (annual) period". Thus if you have balance $p_{t+1}$ at the end of the period, and $p_t$ at the start of the period, then the effective discount rate, $d$ for that period is

$$d=\frac{p_{t+1}-p_t}{p_{t+1}}=1-\frac{p_t}{p_{t+1}}\iff p_{t+1}=\frac{p_t}{1-d}$$

whereas the effective interest rate, $r$ expresses the amount of interest as a percentage of the balance at the start of the period:

$$r=\frac{p_{t+1}-p_t}{p_{t}}=\frac{p_{t+1}}{p_t}-1\iff p_{t+1}=(1+r)p_t.$$

Thus in your problem, a dollar at the start of the period of 6 years will become $A$ dollars at the end, where

$$A=\left(\frac{1}{1-0.03}\right)\left(\frac{1}{1-0.06}\right)\left(\frac{1}{1-0.09}\right)(1+0.05)^3$$

The simple rate of interest $i$ equivalent to this must satisfy

$$1+6i=A\iff i=\frac{A-1}{6}$$

Doing the calculation gives $i\approx 0.66$.

Note that we can easily show that the effective discount rate, $d$ and effective interest rate $r$ are related as follows:

$$d=\frac{r}{1+r}\iff r=\frac{d}{1-d}$$

• then what about Effective Discount Rate @smcc please Explain – cloud computing in salesforce Aug 22 '16 at 13:54
• What do you mean? Please be more specific. – smcc Aug 22 '16 at 13:55
• Effective Discount Rate Means what, Final step how you got 1+6i=a and then answer i=0.66 – cloud computing in salesforce Aug 22 '16 at 13:56
• To get $0.66$ just calculate $A$ using the formula and then plug into $i=\frac{A-1}{6}$ – smcc Aug 22 '16 at 13:58
• ok@smcc if we are telling si =pnr/100, effective Discount Rate in simple Terms why we need this and Effective interest Rate why we Need this In what it will be useful,simple interest Means calculating the Interest with the Principal like that,i want Explanation in Simple Terms for Effective Interest Rate and Effective Discount Rate – cloud computing in salesforce Aug 22 '16 at 14:00