"$x$ times as many as" versus "$x$ times more than"?

Question

I keep coming across such questions during my GRE preparation:

• A is $x$ times of B versus A is $x$ times greater than B

Some websites treat the two phrasings as synonymous, however I believe that the second phrasing, "A is $x$ times greater than B", means that $$x=\frac AB-1.$$ As such, I think that these four options are all correct:

Let A=9 and B=3.

1: A is 3 times of B.
2: A is 2 times greater than B.
3: A is 300% times of B.
4: A is 200% times greater than B.

Yet the author of the following question would pick only options 1 and 3 above:

What is the right convention for interpreting such phrasings?

Answer 1

The phrase "A is what % of B" should be written as $A=x\cdot B$. And now solve for x, and then multiply by 100.

Example 1a: If A is 100, and B is 50, then $100=x\cdot 50$, means that $x = 2$, and A is 200% of B.

The phrase "A is what % greater than B," should be written as $A=x\cdot B$, just as before. But now, when you solve for x, and multiply by 100, you want to take the additional step of subtracting 100. Notice that this will only work if A is actually greater than B.

Example 1b: In the above example, A would be 100% greater than B.

Example 2: if A is 150, and B is 100, then solving for x in $A=x\cdot B$, would give us $x = 1.5$, and so A is 150% of B. But A is 50% greater than B.

Answer 2

The question that you link to uses the phrase "...how many times larger was...".

You use the phrase greater than in your question, and this is what causes the problem. There is an ambiguity about whether it is the comparative sizes of their differences that you discuss.

The fact is that 2 is two times larger than 1. The fact is that 1.5 is three times larger than 0.5. The fact is that $b$ is $(b \div a)$ times larger than $a$. The page you link to is correct.

Answer 3

1. I wholly agree with your interpretation that for real $A$ and positive real $B,$ $\text“A$ is $x$ times greater than $B\text”$ means that $$A=B+xB.$$

   As such, 15 is the correct answer for all these statements—including the third one:

   • __ is greater than 3 by $\boldsymbol{12}.$
   • __ is greater than 3 by four times.
   • __ is four times greater than 3.
   • __ is $\boldsymbol{400\%}$ greater than 3.
   • 3 increases by $\boldsymbol{400\%}$ to __.
   • From 3 to __, there is a $\frac{\text{__}-3}3=\boldsymbol{400\%}$ increase.  (Wikipedia)

   The lexical chunk “four times greater than” doesn't inherently mean “four times as much as”, just as a tall child isn't necessarily a tall person and a closed set (for instance, $(−∞,∞)\,)$ isn't necessarily a closed interval. The point is that just as “closed set” is describing an object that is closed as a set rather than closed as an interval, and “tall child” is narrowing down the set of children rather than the set of all people, the “four times” in “four times faster/longer” is specifically modifying fastERness/longERness.

2. In practice, however, the phrasing $“A$ is $x$ times greater than $B”$ is (at best needlessly) ambiguous: the majority will fill in the third blank above with 12 rather than 15. This technically incorrect reading is prevalent even in print publication, so it is unfortunately arguably just about idiomatic—analogous to how the word “irregardless” has been legitimised by many dictionaries. (Yet another example of intellectual sloppiness is mindlessly answering from 50% to 75%, the increase is ____ with 25% rather than 50% or 25 percentage points.)

   I keep coming across such questions during my GRE preparation

   A high-stakes exam shouldn't contain such functionally ambiguous phrasings. I'd go for the technically incorrect reading—notwithstanding that it's a mathematics/quantitative exam—since that reading is likelier to be the intended one.

3. This top-voted answer at English Language & Usage Stack Exchange makes the same observation:

   This is indeed a classic. The question has been asked many times around the web, and there appear to be two schools: one that agrees with you, and one that thinks both constructions mean the same.

   I think those people are nuts, but, hey, they might be the majority. I say, why use a construction that is either illogical or ambiguous when you have a perfectly good alternative? But language isn't logical, especially not idiom, so I suppose I cannot call my argument objective.

   I think “3 times more than 5 sweets” means “15 sweets total” to most people, though I'd never use it. You will even see it in newspapers. The exact same problem exists in Dutch, with the same sides to choose between.

4. If still being technically accurate and dealing with real $A$ and nonzero real $B,$ $“A$ is $x$ times bigger/larger than $B”$ might mean the same as the first reading above (referring to position on the number line), or it might mean that $$|A|=|B|+x|B|$$ (referring to size/magnitude).