Choose the sentence that correctly uses prepositions to describe moving to the back of a booth and waiting there.

Introduction / Context:
This question examines your sense of natural preposition choice and phrase structure in a narrative sentence. The characters are changing their physical position relative to a booth, and the sentence must clearly express that movement in idiomatic English.

Given Data / Assumptions:
The core situation is that some people hurry to a safer or more hidden position at the back of a booth and then wait until a malformed youth has gone away. The options change the prepositions to, for, and towards, and also change the noun phrase that describes the rear part of the booth.

Concept / Approach:
When we describe movement in English, we usually use verbs like hurry, go, or run followed by the preposition to or towards plus a clear place phrase. The phrase the back of the booth is more natural than the booth’s back in modern prose, although both can be used. The preposition for is generally not used directly with a destination in this kind of sentence. We are looking for a version that sounds like standard narrative English.

Step-by-Step Solution:
First, read option A: They hurried to the booth’s back. This is understandable, but the possessive form the booth’s back is slightly stiff compared to the more usual back of the booth.Second, read option B: They hurried for towards the booth’s back. This combines for and towards in an ungrammatical way, so it must be rejected.Third, read option C: They hurried for the booth’s backing. Here backing is used incorrectly for back, and for remains a wrong preposition.Fourth, read option D: They hurried towards the back of the booth and waited for the malformed youth to disperse. This combines a natural motion verb with towards and the phrase the back of the booth, which is clear and idiomatic.Finally, compare the options. Option D gives the most natural and correct description in modern English.

Verification / Alternative check:
We can test common expressions in daily English: people hurry to the back of the hall, move to the back of the line, or go towards the back of the bus. All these patterns use to or towards plus the back of followed by a noun. The phrase the booth’s back is not wrong, but back of the booth aligns better with these standard patterns and therefore reads more smoothly to most readers.

Why Other Options Are Wrong:
Option A is grammatically acceptable but slightly awkward in style. In a strict exam context, however, the examiner usually wants the most natural wording, which is given in option D. Option B is clearly ungrammatical, since hurry for towards does not form a standard prepositional combination. Option C misuses backing, which refers to support or endorsement, not the rear of a physical object, and also uses the wrong preposition for. These flaws make B and C unsuitable choices.

Common Pitfalls:
Learners sometimes overuse the preposition for because they associate it with purpose or destination, but in motion contexts to and towards are usually correct. Another pitfall is to rely too heavily on possessive apostrophe forms like booth’s back, which are more common in older or more literary writing. For clear modern English, it is often better to choose of phrases such as the back of the booth.

Final Answer:
The best sentence is They hurried towards the back of the booth and waited for the malformed youth to disperse.