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In IELTS writing tasks, I often come across examples like these:

For part-time work the opposite is true, with women earning an average of $5 per hour.

and

Canada is the reverse of the world average, with 8% of the water consumed by agriculture.

What kind of structure is the with + number/somebody + gerund?

  • "With" is a preposition, so they are PPs whose head has a non-finite subordinate clause as complement. Their function is that of adjunct. "With" typically has the meaning of accompaniment, so we could say that the proposition expressed in the subordinate clause accompanies the one expressed in the main clause. – BillJ Jun 6 '18 at 6:46
  • RHK Webster's gives an example of this common usage (though AHD and Collins seem to lack one): with prep. ... << 17. (used as a function word to specify an additional circumstance or condition): We climbed the hill, with Jeff following behind. >> A paraphrase might include 'The situation here is ...' used either contrastively ('For part-time work the opposite is true: the situation here is that women earn an average of $5 per hour') or to expand on / add a particularisation to a statement ('Solubilities of salts in water vary ... – Edwin Ashworth Jun 6 '18 at 9:02
  • enormously, with that of silver perchlorate (up to 500 g per 100 mL of water) being astonishingly high' <==> 'Solubilities of salts in water vary enormously: the situation with silver perchlorate, which has a solubility of up to 500 g per 100 mL of water, is remarkable'). – Edwin Ashworth Jun 6 '18 at 9:08
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Only the OP's first example uses the gerund, (preposition + gerund). The second sentence is an example of the passive voice.

In sentences that are in the active voice, the verb that follows a preposition, e.g. with, for, of, at,… etc. takes the gerund form.

  1. In 2018, nearly a third of Irish consumers used their mobiles for making purchases compared to 41% in the UK.
  2. The survey shows that purchasing online is increasing sharply, with 51% of British consumers shopping online at least once a week.

In sentences that use a passive voice construction, the main verb is in the past participle. Here mobile phone is the agent, and “via” is used instead of “by”.

  1. In Ireland, 29% of all purchases was (done) via mobile phone compared to 41% in the United Kingdom.
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The structure in question is, interestingly, a phrase which contains a phrase which contains a phrase. Let's start at a lower level and build our way up.


Participial Phrases

'Earning' and 'consumed' are both participles (note: they are not gerunds, which act as nouns; instead they modify nouns like adjectives do). 'Earning' is an active participle, and the participial phrase "earning an average of $5 per hour" modifies the noun 'women'. 'Consumed'is a passive participle, and the participial phrase "consumed by agriculture" modifies the noun phrase "8% of the water".

Bonus notes:

  1. These are sometimes called 'participial clauses', although they are technically not clauses since they have no word acting as a proper verb. However, they can take the place of clauses.
  2. The phrases are equivalent to the relative clauses: [women] "who earn an average of $5 per hour"; and [8% of the water] "which is consumed by agriculture". Because of this, the participial phrases can be referred to as 'reduced relatives'.

Noun Phrases

If we combine the nouns with their respective modifiers, we get noun phrases:

  • "women earning an average of $5 per hour"
  • "8% of the water consumed by agriculture"

Prepositional Phrases

When we have a preposition followed by a noun or noun phrase, that gives you a prepositional phrase. In this case, the bold words of your original sentences form prepositional phrases, which act like adverbs to modify the entire clauses that come before them (the collection of noun-bold words in the original sentences).


Summary

Each of your sentences is a clause followed by a prepositional phrase which includes a noun phrase which includes a participial phrase. For visual reference:

clause {prepositional phrase [noun phrase (PARTICIPIAL PHRASE) ] }

  • For part-time work the opposite is true, with women EARNING AN AVERAGE OF $5 PER HOUR.
  • Canada is the reverse of the world average, with 8% of the water CONSUMED BY AGRICULTURE.
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I'm not entirely sure what you mean by "structure." Structurally, both of those sentences are simple sentences, but it's clear that's not what you're asking.

Also, your description, "with + number/somebody + gerund," doesn't actually fit both of those examples. What you have in both is a prepositional phrase: a preposition, followed by a noun, followed by a participle that modifies that noun.

In both sentences, the prepositional phrase itself is a modifier.

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Traditional grammar leads us to analyse a structure in completely different ways according to one small thing such as what part of speech it is introduced with. However this structure is interesting because what is essentially the same structure is found in different languages, but introduced in different ways.

In Latin we have:

The ablatives of a participle and a noun (or pronoun) are used to form a substitute for a subordinate clause defining the circumstances or situation in which the action of the main verb occurs. The ablatives are only loosely connected grammatically to the remainder of the sentence, hence its name absolute (absolūtus = free or unconnected).

An Ablative Absolute with a perfect passive participle is widely used in classical Latin to express the cause or time of an action:

Hīs verbīs dictīs, Caesar discēdit. With these word having been said, Caesar departs.

Acceptīs litterīs, Caesar discēdit. With the letter having been received, Caesar departs.

Leōne vīsō, fēminae discessērunt. With the lion having been seen, the women departed.

Equally common is an Ablative Absolute with a present active participle:

Leōne adveniente, fēmina discēssit. With the lion approaching, the woman left. "Ablative Absolute" in Latin Library

In each of these examples the verbal form is definitely a participle as it agrees with the noun. Note that although we do not have ablatives, with+noun phrase is a valid translation. In Ancient Greek they do not have ablatives, but they use a genitive instead.

In Scots Gaelic the same structure is found, but introduced by agus "and". The semantic relationship between the main clause and the absolute clause is a bit variable as in the English structure. Here it is used to describe circumstances. Most commonly it is used to give a consequence:

Thachair seo air Là na Bliadhn' Ùire ann an 1919 agus seòladairean a' tilleadh bhon Chiad Chogadh. This happened on New Year's Day 1919, and sailors at returning from the First War. BBC News Website 5/9/2018

A' tilleadh translates literally as "at returning" (so tilleadh is technically a gerund) but usually corresponds to the use of the present participle in English.

Bha iad a' tilleadh. They were returning.

Just as in English it can be difficult to tell the difference between a gerund and a present participle.

Thus we have the same structure introduced in different ways:

  • Ablative + noun + participle in Latin
  • Genitive + noun + participle in Ancient Greek
  • "And" + noun + "at" + gerund in Gaelic (where "at" + gerund is the equivalent of present participle in English.

This shows that looking at the introductory word is not necessarily the best way to analyse the structure.

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