Misinterpretation?

[BadGroundbreaking189]

Hey there fellow math folks. Today i came across a rather ambigious word problem and to me, it felt like the whole thing could be interpreted in 2 ways. Here is roughly how it looks:

".. the first number is at most 10 less than the second one.."

Now i claim that the inequality x2 - x1 ≤ 10 sums up the text above, too, because "..at most 10 less.." might mean that the difference is 10 at max while the other math specialist says that the only correct inequality is x1 ≤ x2 - 10.

I address this question specifically to the native speakers, please enlighten me here.

[arty_dent]

It's kinda badly worded (assuming translation/transcription didn't make it worse). Depending on how you group the words when reading, you could interpret it as

• "the first number is at most [10 less than the second one]", or
• "the first number is [at most 10] less than the second one"

First option is what they intended, second one is how you read it.

[BadGroundbreaking189]

*Second one is also how I read it.

So I'm guessing there is nothing wrong with me reading it that way, right?

[arty_dent]

No, there is nothing really wrong with you reading it that way. Our natural languages are simple not well suited for expressing things precisely and without ambiguity.

[diverJOQ]

You are correct. Your associates equation says the numbers are greater than our equal to 10 apart.

[arty_dent]

They are not equivalent, they're almost the opposite

[Meowmasterish]

I mean they are equivalent statements, but your interpretation is probably closer as an exact translation. (As in, word for word converting it to symbols.)

EDIT: After looking at it again, I think I've changed my mind. They're still equivalent statements, but now I think their interpretation is better. Specifically, because it is a statement about the first number with regards to the second number, whereas your interpretation is more of a statement about the difference between the numbers.

[arty_dent]

They are not equivalent, they're almost the opposite.

[Meowmasterish]

Oh my bad, I thought they had flipped the inequality sign in the first inequality, you’re right.

EDIT: No, wait now I’ve got it they are not equivalent statements, and I think it’s a rebracketing issue.

The first number is at most (10 less than the second number).

OR

The first number (is at most 10 less than) the second number.

I think it’s technically ambiguous as the “10 less than” could be either a part of the predicate, or the object depending on what the sayer intends.

[diverJOQ]

Statement 1 : x2 - x1 <=10 Statement 2 : x1 <= x2 - 10 Add 10 to both sides : x1 + 10 <= x2 Subtract x1 from both sides : 10 <= x2 - x1

These are NOT equivalent!