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?

Personal

1 Answer
Dec 21, 2016

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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#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)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#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)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#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)#

#color(magenta)("///////////////////////////////////////////////////////")#
#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.