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?

Question

Benjamin Achtenberg · Accepted Answer

Hope this helps!

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.

Madelyn Anderson · Answer

undefined Dividing the interest rate of 0.08% over 4 is typically done when calculating compound interest, especially when interest is compounded quarterly. It means that you're dividing the annual interest rate (0.08%) by the number of compounding periods per year (4 in this case). This is because the interest is being applied more frequently than annually, so you need to adjust the rate accordingly to reflect the smaller compounding periods.

If you were to multiply instead of dividing, it would artificially inflate the interest rate, assuming that the interest is being compounded quarterly. This would lead to an incorrect calculation of the final amount of money accrued through interest over time.