Expressing quantitative change: not as clear as it may seem sometimes
First, a small point, but an important one: quantitative versus qualitative properties or traits.
Red versus blue is a qualitative distinction. “Tastes like strawberry, not chocolate” is a qualitative
distinction. Colors, aromas, flavors, textures (rough/smooth) are examples of qualitative features. These
are important in describing things in biology. Quantitative descriptions attach numbers: how many, how
much, how fast, how long.
When we use quantitative descriptions and statements of things, we should be able to be specific, exact or
at least clear in our meaning. Of course, there are times when approximation is appropriate and useful.
But there are commonly used quantitative expressions that can be quite misleading. When a speaker or
writer uses an expression to compare two things, for example, the listener or reader should be able to
understand the intention, what the speaker or writer means. If the intended meaning and the understood
meaning are different, problems arise.
Here are three items taken from internet postings, as examples of a problem. What do they mean? Note
that such expressions are commonplace in newspapers, magazines and other sources, not just on-line.
1. “Influenzavirus B mutates at a rate 2 to 3 times lower than type A.”
2. “BPA Levels in US Foods 1,000 Times Less Than Limits, Study Finds”
2. “New Study: Fluoride Protection 100 Times Less Than Previously Thought”
Case: If I have $100 and you have $200, we could agree that you have twice as much as I do. That
statement of comparison is unambiguous. Saying that “you have 100% more than I do” is also clear,
since we understand that 100% represents the totality of the reference amount, which is $100 here.
Saying that “your amount is one fold greater than mine” is also clear. “Fold” simply means 100% of the
original quantity or reference quantity.
If I earn another $200, then it is correct to say that:
• my amount has tripled
• my amount is now 3-fold what it was before, or 3-fold as much as it was before
• my amount has increased 2-fold (where the original $100 = 1-fold)
• I have 2-fold more than I had before
• my amount has increased 200% (where the original $100 = 100%)
• I have 200% more than I had before
• my amount is now 300% of what it was. (If I still had only the original $100 after some time, then I
would say that my amount now is 100% of what I had before.)
• I have three times as much as I had before
• I have two times more than I had before (“times” means 100% or 1-fold)
You see that some of these comparative statements refer only to the amount of increase (just the change),
not including the original amount. In working problems that is an important distinction.
Now, back to those internet citations. If that $100 was my original amount, then what does it mean to
say that I actually have “100 times less than previously thought?” If the “times” expression in such
statements of comparison means the original amount, 100%, 1-fold, then my amount has decreased from
$100 to -$9900, since 100 x 100 = 10,000.
Such is probably not the intended meaning of “100 times less.” More likely the meaning is that fluoride
protection is actually 1% as effective as previously thought. So, if we take the new value for fluoride
protection (call it X), then previously we thought that the fluoride protection value was 100X. The
difference between then and now is a factor of 100. The “times” comparison in one direction is fine, but
in the other direction it’s wrong and possibly confusing. It would be correct to say that the protection
factor decreased by 99% when new research data became available.
And how about that “1000 times less” in the BPA citation? (BPA is a chemical found in lots of products;
it may be harmful to human health.) That’s an X versus 1000X comparison: a 99,900% increase in one
direction and a 99.9% decrease in the opposite direction. And notice that:
• from X to 2X = 100% increase
• from X to 10X = 900% increase
• from X to 100X = 9900% increase
• from X to 1000X= 99,900% increase
Formula for expressing % change:
Let A1 = reference value (starting value)
Let A2 = new value (changed)
then the % change is ((A2 - A1) ÷ A1) X 100%
Ex. If attendance at a concert was 800 people on Wednesday night and 1000 on Saturday night, then the
% change is ((1000 - 800) ÷ 800) X 100% = (200 ÷ 800) X 100% = 0.25 X 100% = 25 % change The
positive sign means that the change was an increase.
Ex. If 100,000 people watched a TV program last week but only 65,000 watched it tonight, then the %
change is ((65, 000 - 100,000) ÷ 100,000) X 100% = (-35,000 ÷ 100,000) X 100% = -0.35 X 100% =
-35% change. Of course, the negative sign means that the change was a decrease.
Ex. If your resting heart rate is 70 beats/minute and 120 bpm after climbing 4 flights of stairs, the %
change is ((120 bpm-70 bpm) ÷ 70bpm) X 100% = (50 bpm ÷ 70 bpm) X 100% = 0.71 X 100% =
71% change.
Note that you could also say that your heart rate at the top of the stairs is 171 % of the resting value, i.e.
the resting value + the amount of change. (120 ÷ 70) X 100 = 171%