Learn Russian: Без четверти десять пассажиры будут проходить через турникет. - At a quarter to ten, passengers will be passing through the turnstile.

It literally means "without a quarter, ten." To express minutes to the upcoming hour, Russian uses the preposition без (without) followed by the minutes in the genitive case (четверти) and the upcoming hour in the nominative case (десять).

Why isn't there a word for "at" before Без четверти десять?

When you use the без + [minutes] + [hour] structure to say "at [a certain time]", you do not need the preposition в. The phrase itself serves as the time marker indicating when the action happens.

What is the dictionary form of четверти and why does it end in -и here?

The dictionary form is четверть (a quarter), which is a feminine noun ending in a soft sign (-ь). The preposition без always requires the genitive case. For feminine nouns ending in -ь, the genitive ending is -и.

Why does the sentence use the imperfective future будут проходить instead of a perfective verb?

The imperfective future emphasizes an ongoing process at a specific moment in time. Translating to "will be passing," it paints a picture of the action happening continuously at 9:45. A perfective verb (like пройдут) would imply the action is completely finished right at that moment.