Modes of comparing quantities

Comparing quantities

If we look two quantities A and B, the following are equivalent:

(1)   A is greater than B

(2)   A > B

(3)   B is less than A

(4)   B < A.

Example: “3.6 kg is greater than 900 g” is equivalent to “900 g is less than 3.6 kg”

In order to tell more about the situation - to characterize the way how A is greater than B  - we can use some comparing conventions. I mention here the most common four conventions.

Purely additive comparison (= comparing by the difference)

In this convention the extra information is given by the positive difference (D) between the quantities compared. Then the following are equivalent:

(1)   A is (the amount) D greater than B

(2)   A = B + D

(3)   B is (the amount) D less than A

(4)   B = A – D.

Example: “A is 25 km greater than B.” is equivalent to “B is 25 km less than A”.

This convention uses only addition and its inverse operation, subtraction. The greater quantity is obtained by adding and the lesser by subtracting.

Purely multiplicative comparison (= comparing by the ratio)

We use the additive mode of comparing mostly when the quantities we compare have about the same magnitude. In most other cases the multiplicative mode is more natural.

The multiplicative comparing mode requires that both quantities are positive.

Here we have a couple of different ways to formulate the sentences. I take at first the way which is completely analogical to the additive comparison.

Now the key information is the ratio (R), greater than 1, between the quantities compared. The following are equivalent:

(1)   A is R times greater than B

(2)   A = R x B

(3)   B is R times less than A

(4)   B = A / R.

Example: “A is 3.6 times greater than B.” is equivalent to “B is 3.6 times less than A.”

This convention uses only multiplication and its inverse operation, division.  The greater quantity is obtained by multiplying and the less by dividing.

Because schools generally don’t teach this formulation (at least properly), many people mix it up with the conventions used in comparing with percentage or fractions, which we discuss later. For this reason, it may be “safer” to say something like “A equals 3.6 times B” and “B equals A divided by 3.6”.

Nevertheless, saying “times greater than” or “times less than” is practical and symmetrical; it has a long tradition among people, and its equivalent occurs in most languages. And, which is the most important: it collides with no other convention within mathematics! If somebody of us don’t want to use this kind of saying, (s)he has the right to avoid. BUT: We ought to understand those who use it!
I have seen many posts in FB, where somebody tries to “prove” for example that:

“If A = 5 m and B is 10 times less than A, then B = -45 m.”
 This mad result is obtained by using a wrong formula! (The right answer: B = 0.5 m)

The “mixed modes” use both multiplicative and additive calculations

A)     We can tell how big the difference is in percentages of the other quantity:

“A is P% greater than B” means: “(A – B) is P% of B”
“B is Q% less than A” means: “(A - B) is Q% of A”

Example: “100 g is 25% greater than 80 g.”, and “80 g is 20% less than 100 g.”

If you ask, what is 1000% less than 5 m, the answer is really -45 m,
because the difference 5 m – (-45 m) = 50 m, which  is 1000% of 5 m.
However, the conventions “1000% less than” and “10 times less than” are not the same! They are two totally independent modes.
You could think that 1000% and 10 would be replaceable,
but 1000% and 10 times are not! Both have their key symbols: “%” and “times”, which refer to different modes.

B)     Another frequently used mixed mode is comparing with fractions. The idea is the same as in the preceding percentual mode, but here we use fractions instead of percentages.

“A is P/Q greater than B” means: “(A – B )/B = P/Q”
“B is P/Q less than A” means: “(A – B )/A = P/Q”

Example: “Number 4 is 1/3 greater than number 3.”, and “Number 3 is 1/4 less than number 4.”

This mode can naturally be used also when the fraction reduces to an integer. We can say that
“18 km is 10/2 greater than 3 km.”   
Again, some people are lured to think that this collides with the fact that
“15 km is 5 times greater than 3 km.”
Not so! We cannot replace “5 times greater” by “10/2 greater”. The former one should be identified as a purely multiplicative way of comparing. Thus, there we don’t add anything to anything!

When comparing quantities, we must be alert in order to choose the right convention. One of the most common errors can occur in the “political mathematics”:

If a party A wins 24% of votes and another party B wins 20%, one could be lured to say that the support of A is 4% greater than the support of B.  However, the formula “24% - 20%” is not correct, because “4% greater than” does not refer to the purely additive mode of comparing. Thus, as you know: The support of A is even 20% greater than the support of B.
But: the support of A is 4 percentage units greater than the support of B.
(The problem was hidden in the fact that the support is measured using percentage as unit. That is why we must use two different way of saying.)