Does "less than" really mean "subtracted from", or is it bad English?

I got involved in a discussion about some Math problems provided in the local primary school education:

• 20 more than 543 is 563
• 25 less than 261 is 236
• 155 less than 310 is 155
• 355 more than 1233 is 1588

Some of us (including me) argued that this is unclear or incorrect English, while others said that it is correct or that we are not creative if we do not understand it (huh?).

In my own Math education, I have learnt that "less than" and "more than" (or rather, "greater than") are comparative operators. However, in the above, they are used as manipulative operators (subtraction or addition) - a usage I never learnt.

At first I thought this must be something peculiar with the Singapore Math education, as those who argued that this is incorrect are mostly foreigners, while the other side is made up of mostly locals.

After some searching for similar use on the Internet, the two instances I could find are an exercise paper from Math Activities Resource Center, Mt. San Antonio College, Walnut, CA (USA) [link], and a lecture paper from Professor Weissman's Algebra Classroom [link].

In the exercise paper, the lecturer also mentions that "less than" is different from "is less than", in that "less than" is used to mean subtraction (where the numbers are inverted), whereas "is less than" is used to compare two numbers.

In the lecture paper, it is mentioned that "3 is less than 7" translates to "3<7", "3 less than 7" translates to "7-3" and "3 less 7" translates to "3-7".

I want to hear from some experts regarding how correct and/or widespread this kind of usage of these terms is. No matter how many times I read the examples back to myself I feel it's odd and wrong to use it this way, and that the correct usage for manipulation should be:

• 20 added to 543 is 563
• 25 subtracted from 261 is 236
• 155 subtracted from 310 is 155
• 355 added to 1233 is 1588

Whereas the correct usage for comparison should be:

• 20 more than 543 is False
• 25 less than 261 is True
• 155 less than 310 is True
• 355 more than 1233 is False

The paper you found describing the difference between less than meaning subtraction and is less than meaning comparison is correct. Here are some examples from published English works I found in the Corpus of Contemporary American English. (more than query) (less than query):

For example, children need to learn that numbers later in the count list have larger quantities and that numbers themselves have magnitudes (e.g., 4 is one more than 3 and one less than 5)

   — Jordan, Nancy et al. “Validating a Number Sense Screening Tool for Use in Kindergarten and First Grade: Prediction of Mathematics Proficiency in Third Grade” School Psychology Review, 2010

As the number of killings has crept this year to 95 -- seven more than the total for all of 2004 -- the city has only sporadically increased foot and bicycle patrols, key elements of any community policing plan.    — Jaxon Van Derbeken, Chronicle Staff Writer. San Francisco Chronicle, December 21, 2005

A month before your trip, you see a newspaper ad showing fares for the same trip are $100 less than what you paid.
   — Rosato, Donna. USA Today, October 8, 1996

The four-cam, 32-valve engine produces 280 horsepower, 15 less than the Northstar, and it drives the rear wheels through a new four-speed automatic transmission.

   — Schuon, Marshall. New York Times “ABOUT CARS; With a New V8, Lincoln Introduces an All-New Mark VIII”, November 11, 1992

These are standard English usages, so get used to them.

Yes, if you are talking in strict mathematical terms, less than is a exclusively comparative operator.

However, outside mathematics, in everyday language, it is normal use it both as a comparative operator as well as to state a quantified difference. That means both these phrases are correct:

I have less apples than you.

I have ten apples less than you.

In the first case, I am just comparing the amount of apples we both possess, but in the second case I am quantifying the difference. In mathematics, I could describe that situation as:

My amount of apples is 10 subtracted from your amount of apples

Or

X = Y - 10

Since your examples come from a primary school environment, I do not find it strange that more natural language is used rather than strict mathematical language in order to teach children the basics of calculus.

25 less than 261 is 236 is much closer to what a child would hear in everyday speech (1 apple less than 5 is 4 apples) and as such it leaves the child to ponder the actual numbers rather than a new jargon (I am not sure many primary school kids are familiar with the vocabulary of subtraction, multiplication and addition).

Indeed, for multiplication I would expect lines like

3 times 4 is 12

Instead of the mathematical

3 multiplied by 4 equals 12

As a matter of fact, the former is the exact form in which I was taught to memorize my tables of multiplication (albeit in Dutch, not English).

One the one side, constructs of the form "a is b-a less than b" are common as English statements. Similarly "a is b-a below b", "a is b-a from b", inside, outside, under etc. where you are comparing quantities and can have a sentence which either indicates the truth of the comparison or the magnitude of the difference.

Many such quantified comparisons can include the difference. "the diver is below the surface" and "the diver is six fathoms below the surface"; "the church was outside the city wall", "the church was three miles outside the city wall", "Anne is less than ten years old", "Ann is one day less than ten years old" are all common.

But the examples you give are "b-a less than b is a" are a backwards "six fathoms below the surface is the diver"; "three miles outside the city wall is the church", "one day less than ten years old is Anne" are unusual word orders.

If they were intended as questions, then either "what is 25 less than 261?" or on a page "_ is 25 less than 261" to fill in the blank would be more normal phrasing.

25 subtracted from 261 is 236 is clear. Because we know what subtraction is, we don't give it a second thought.

However, 25 less than 261 is 236 is also clear. If one thinks concretely about what is said, it reads (to me)

25 fewer than/taken away from 261 is 236.

One can also express it as "236 is 25 less than 261"

Arguably, the 'subtract' binary operation in maths covers three different situations. I'll start by covering the easier situation with the 'add' operation:

(a) Combine a number of elements from set A with a number from set B {3 + 4 = 7}

(b) Increase a base figure by a certain amount {3 --- (+4) ---> 7} {3 has been transformed (increased) to 7}

For the corresponding subtraction:

(c) Split a quantity into two sub-quantities, taking off a chosen number, and identify what's left {7 - 4 = 3} {7 has been partitioned into 4 and 3}

(d) Reduce a quantity by a chosen number to leave a reduced number {7 ---(-4) --->3} {7 has been transformed (reduced) to 3}

(e) Identify how much greater one number is than a second {7 - 4 = 3} {7 is 3 more than 4; 4 is 3 less than 7}.

The model for working out 'the answer' to c, d and e is of course identical, usually written 7 - 4 = 3. This convenience masks the type of situation being modelled. (Though often, especially in the teaching of maths, operations are shown over arrows to indicate transformations rather than splittings / finding differences. Ferrers graphs may be used to show partitions.)

The language used (minus, subtract, take, take away, less / less than // add, plus, and / more than) can correspond either to the general model or to the actual situation being modelled.