Expert: Josh - 3/26/2009

Question
QUESTION: Hello:

If I want to express 13 mills as a decimal part of $1.00, the answer is $0.013. This amount is found by dividing 13 mills by 1000 mills/$1.00.

Can 13 mills be expressed as a decimal part of another amount instead of $1.00?  For example, Express 13 mills as a decimal part of $1.50 or express 13 mills as a decimal part of $0.10. What would these amounts be?

I thank you for your reply.



ANSWER: Hi Kenneth,

I think we need to draw a line between mathematics and the terminologies used outside of mathematics. In the financial world, people identify 1000 mills as equivalent to a dollar. In other words, 1 mill corresponds to 0.001 dollar. This ratio is a constant and it never change. It is kind of strange to consider "expressing 13 mills as a decimal part of 0.10". On its own, without any need of clarification on what the "unit" represents, 13 mills is always 13 x (1/1000) of a unit.

To avoid any confusion, think of "milli" as a prefix. In mathematics, "milli" may be regarded as a multiplier of 1 one thousandth.
So, 1 milli unit is equivalent to 1 x (1/1000) of a unit.
13 milli-unit is equivalent to 13 x (1/1000) of a unit (or 0.013).
100 milli-unit is equivalent to 100 x (1/1000) of a unit (or 0.100).
This is consistently how the word "milli" is understood. Again, we don't care what the unit is.

Now, if we want to express a quantity (A) as a proportion of any number (B). We can always divide A by B to get the ratio, then multiply it by 100 to get a percentage.

If A equals thirteen mills, then A=0.013. This number does not magically appear as something else. In decimal form, it can only be expressed as 0.013. Now, it is a separate matter to compare this relative to something else. If we pose the question, how much is thirteen mills in terms of a dollar thirty, then we divide A=0.013 by B=1.30. We see that thirteen mills represents 0.01 or 1% of 1.30, whether the unit is in dollar, or meter.

To answer your question regarding "13 mills as a decimal part of $0.10", it is better to consider "13 mills as a proportion of 0.10 (regardless of the measurement unit)" and express the answer in decimal form.

Here, we have B=0.10, A=0.013. Using the concept of multipliers, "thirteen milli" is directly translated as 13 x 0.001. The answer is A/B=0.013/0.10=0.13. i.e., A represents 13% of B.

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

QUESTION: Hello Josh:

I want to thank you for the reply. I'm somewhat confused.

If 13 mills is expressed as a decimal part of $1.50, what is this amount?

I thank you for your follow-up reply.

Answer
Hi Kenneth,

I must say, I find it very awkward to mingle word units in equations. If you don't mind, I like to drop all references to "mills" and "dollars".  I prefer to look at mills (I call it "milli-unit") as 0.001 of a unit. Let us agree on this, regardless of what the unit is about. The unit can be "meter" for length, "litre" for volume etc. The interpretation is still the same.

1 milli-meter equals 0.001 meter.
1 milli-liter equals 0.001 liter.
1 Mills (think of it as "milli-dollar" if you must) corresponds to 0.001 of a dollar.

I think it is rather strange to express "13 mills as a decimal part of $1.50". Let me rephrase this. I think what you really want to do is take "13 mills" as a proportion of 1.50 (regardless of the measurement unit) and express the answer in decimal form.

13 mills (13 x 0.001) equals 0.013.
This, as a proportion of 1.50, is given by 0.013/1.50.
The answer in decimal form is 0.00866666...
It is a recurring decimal figure. Now, you see why I didn't use these numbers in an example.

To summarize, we can avoid any confusion by not mingling word units in any calculation. Treat "mills" as a multiplier of 0.001. Then, 2 milli-unit, is simply 2 x 0.001.

e.g., what is 13 millimeter as a proportion of 1.3 meter in decimal form? Well, 13 [millimeter] is 13 x 0.001 [meter]. Therefore, the ratio is 0.013/1.3 = 0.01.