Question

Can someone explain the reasoning behind dividing the interest rate of 0.08% over 4 here? Like, what does it mean in real life if I were to multiply instead of dividing?

Answer 1

Hope this helps!

Explanation:

The equation is based on annual interest being 8% but there is 4 calculations within the year instead of 1 at the end.

The 4 indicates that the name use is 'quarterly'. So each quarter you earn `(8%)/4` interest . The thing is; that each quarterly calculation is assessed on not only the principle sum but also includes the interest previously earned.
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`color(blue)("Condition for calculated annually only")`

Let the initial deposit be `P` (principle sum)

Let principle sum plus any previous total interest be `P+t`

Then for annually calculated we would have

`(P+t)(1+8/100)` at the end of the first year

`(P+t)(1+8/100)^2` at the end of the 2nd year

`(P+t)(1+8/100)^3` at the end of the 3rd year and so on

From that point on, if it continued for 20 years you would have

`(P+t)(1+8/100)^(20)`
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`color(blue)("Condition for calculated quarterly")`

The interest at any 1 calculation would be `1/4xx8/100 = 8/400`

This is the same as: `1/4xx0.08 = (0.08)/4` as in your writings

Not only is the interest modified we would also need to take into account that there are 4 calculation within any 1 year. So instead of the 20 years in my example we would have `4xx20` calculations
So for `n` years we have `4n`

Thus for our example:

`(P+t)(1+8/100)^20" would become "(P+t)(1+8/400)^(4xx20)`
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`color(blue)("Calculated from the beginning when you deposit the value "P)`

Suppose we had `n` years at 8% annual interest compounded quarterly. NOTE THE WORDING!

`P(1+8/400)^(4n)` which is the same as `P(1+0.08/4)^(4n)`

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`color(magenta)("~~~~~~~~~~~~ The second part of your question ~~~~~~~~~~~~~~~~~")`
`color(magenta)("//////////////////////////////////////////////////////")`

It would totally mess up your calculations if you multiplied by 4 instead of divide.