Expert: Sherry Wallin - 1/4/2011

Question
Hello Math Prof, Sherry:

I have to send a new message since only three follow-up questions are allowed. Here is what I have previously sent to you:

QUESTION: Hello:

I read the following in a business mathematics book regarding the percent-day method for calculating simple interest:

"Percent-day method is based on 360 days‚ time and 1% interest. Any interest rate multiplied by any number of days is equal to so many %-days." Example: 82 days at 5% equals 410%-days. "The rest of the calculation to find interest using his method involves multiplying the amount of interest at 1% by the ratio of
%-days to 360."

Here is an example calculation: An investor invested $720 for 92 days at 7%.
What is his interest?

Solution: $720 for 92 days at 7% = 7% X 92 days = 644%-days.
644/360 X $7.20(1% for 360 days) = $12.88

What is the purpose of the (1% for 360 days) in 644/360 X $7.20(1% for 360 days)= $12.88?
The interest is 7% for 92 days.

Here is a clue from the above quotation "...the method involves multiplying the amount of interest at 1% by the ratio of %-days to 360."

The above would appear to be  644/360 X (1% X $720). But I do not know what the purpose of the (1% for 360 days) after the calculation: 644/360 X $7.20, unless it represents  $720 at 1% for 360 days or one business year, which equals $7.20,     
The calculation could be expressed as 644/360 X $720 X 1% X 360/360.                        

1. But how did the % sign and "days" from the 644%-days cancel from the calculation?

2. What is the logic behind this calculation, i.e,  why does it work to produce the correct answer?

I thank you for any helpful reply.

ANSWER: Hi Kenneth~

When you multiply by an interest rate by some number of days, call it n days, you have units of n percent days. As an example:

interest rate 7% = .01(7) = .07 and let n = 92 days, you now have 7%*92days = 644%days. You want to know how much interest one would get for $720 so you multiply the %days by the amount of $: 7%*92days*$720 and then divide by 360 days: 7%*92days*$720/360days. Notice you have days in the numerator and days in the denominator so they cancel and now you have 7%*92*$720/360 = 7%*92*2 = %$1288 (twelve hundred eighty eight percent dollars) But you know that % means n/100 so you have $1288/100 = $12.88

Math Prof

---------- FOLLOW-UP ----------

QUESTION: Hello Math Prof:

I want to thank you for your reply.  I understand your answer, but it is not quite the same as the one I read in the book.

It is as follows: 644/360 X (1% X $720). This equals 644/360 X $7.20.
What has occurred is that $720 was multiplied by 1% and the result, $7.20, is now multiplied by 644/360. The $7.20, I think, equals the interest for one year at 1%. I suppose that the 644/360 is equivalent to 92 days at 7% since these amounts were multiplied to get 644 %-days.

If the $7.20 represents the interest for one year, now it can be multiplied by the 7 and 92 then divided by 360 the get the $12.88. I do not understand how the % sign cancels from the 7%. The calculation involves multiplying 92 by 7 not 92 by 7%. I agree that the "days" cancel from the 92 days/360 days.
The calculation becomes something like the following: $7.20 (interest for one year with 360-day year) multiplied by 7 because this is the number of percents in 7% and then multiplied by 92 days divided by 360.


I thank you for your follow-up reply.

ANSWER: Hi Kenneth~
You made one fatal mistake and that is when the author went from $720 to $7.20 he didn't multiply by 100, he divided by 100. If I take 720% and I want to get to 7.20 then I am moving the decimal point two points to left which is the same as dividing by 100 or multiplying by .01 or multiplying by 1/100. What the author did is he took 7%*92days*720 = 7*.01*92*720 = 12.88. When you multiply a bunch of numbers and then divide by a single number it doesn't matter where you put the 5 sign or for that matter where you put the .01. Remember multiply by .01 is the same as dividing by 100 because .01 = 1/100. You have 7 percent and 92 days and $720, it does NOT matter where that percent sign goes because essentially for 7%*92*720 it is equal to 7*92*720% and remember anywhere you see a '%' sign you can replace it with .01 or 1/100 so what you have is 7*92days*$720*(1/100) or 7*92days*$720*.01 (where % = .01 or 1/100) and when you divide by the 360 you will have $12.88 either way.

Hopefully you see that the % sign is not canceling you are merely taking %n and rewriting it as .01n.

Math Prof

PS What you say in your last paragraph is correct because you have taken the 720 percent and rewritten it as 7.20 by either multiplying by .01, or by multiplying by 1/100, or by dividing by 100 (and in essence moving the decimal point to the left two places).

---------- FOLLOW-UP ----------

QUESTION: Hello Math Prof:

I need additional clarification.

The author indicated: "The rest of the calculation to find interest using his method involves multiplying the amount of interest at 1% by the ratio of %-days to 360."

I am confused by what he indicates by writing "... multiplying the amount of interest at 1% ..."

Is the 1% a different percentage that he used other than the 7% in the above calculation? I get the idea that there are two percentages involved, a 1% one that he multiplied $720 by and a 7% that is used in the calculation, but this is confusing because after multiplying the $720 by 1%, what becomes of the percentage sign in 644%?

I thank you for your follow-up reply.

Answer:  Hi Kenneth~

If I ask you what 10% of 100 is you would tell me 10, correct? If I ask you what 1% of 100 is you would tell me 1. What about if I ask you what 6% of 100 is? Well you could multiply 100 by .06 or you could say I know that 1% of 100 is 1 so 6% of 100 must be 6(1) = 6, right?

Let's go one more step slightly different. Suppose I want to know what 10% of 87 is? You could just multiply 87 by .1 or you could just move the decimal point one place to the left. The important thing here is that you arrive at 8.7. Now if I want 1% I would move the decimal 2 places to the left getting .87. So suppose I want 3% of 87. Then all I have to do is take the .87 times 3 and I will get the amount at 3%. This is all the author is doing. He is finding the interest at 1% and then multiplying by the x%. If this is still not clear I can try to explain in another way.

BTW, Happy New Year

Math Prof

--------New Question------

I understand your last answer. I wanted to explain that the author could not have intended two percentages to be used 1% and the 7%, which later became 644%.  The answer would have been $0.1288 instead of $12.88. I made the calculation as follows:

644/360 X (1% X $720). This equals 644/360 X $7.20 = $12.88. This is correct.

If the second percentage sign is used as in
644%/360 X (1% X $720). This would equal 644 X 1/100 X 1/360 X 1/100 X $720 = $0.1288. As I noticed this is incorrect.

I want to sincerely thank you for your willingness to help with my questions!

Answer
"What is the purpose of the (1% for 360 days) in 644/360 X $7.20(1% for 360 days)= $12.88?
The interest is 7% for 92 days."

The interest is not 7% for 92 days. The 7% is per annum, meaning per year. When they say 7% for 92 days they are really saying 92/360 at 7% which is .07(92/360)*720 = $12.88

Math Prof