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

Question

I keep coming across such questions during my GRE preparation:

• A is how many times of B versus A is how many times greater than B
• A is what percentage of B versus A is what percentage greater than B

Some websites treat both phrasings in each pair as meaning the same, however I believe that "$A$ is __times greater than $B$" is asking for $$\frac AB-1.$$ 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 to interpret these 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 nonzero real $B,$ $“A$ is $x$ times more than $B”$ means that $$x=\frac{A-B}{|B|}.$$

   So, filling in the blanks with 15, these assertions are all correct:

   • __ is more than 3 by $\boldsymbol{12}.$
   • __ is more than 3 by $\boldsymbol4$ times.
   • __ is $\boldsymbol4$ times more than 3.
   • 3 increases by $\boldsymbol{400\%}$ to __.
   • From 3 to __, there is a $\frac{\text{__}-3}3=\boldsymbol{400\%}$ increase.  (Wikipedia)

   A tall child is not necessarily a tall person, a closed set (e.g., $(−∞,∞)$), is not necessarily a closed interval, and $“4$ times more than” does not actually mean $“4$ times as many as”.

2. However, in practice, the phrasing $“A$ is $x$ times more than $B”$ is at best ambiguous: I fully expect most folks to fill in “__ is $\boldsymbol4$ times more than 3” 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 many dictionaries have legitimised the word “irregardless”.

3. I keep coming across such questions during my GRE preparation:

   In the first place, a major exam should not contain such a question phrasing. I'd go for the technically incorrect reading, since—notwithstanding the fact that this is a mathematics/quantitative exam—that reading is likelier to be the intended one.

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 correct and dealing with real $A$ and nonzero real $B,$ $“A$ is $x$ times bigger/greater/larger than $B”$ might mean the same as the first reading above, or it might mean that $$x=\frac{|A|-|B|}{|B|}$$ (position on the number line versus size/magnitude).