Emily, Fred, Grace, and Henry work in a book-binding factory. On Tuesday Emily, Fred, and Grace started working at $10:00$ AM and were able to bind $5$ books every $3$ minutes, $3$ books every $2$ minutes, and $2$ books every $1$ minute, respectively. At some later time, Henry joined the group, and Henry was able to bind $7$ books every $4$ minutes. Together, they finished binding $750$ books at exactly $12:15$ PM. At what time did Henry join the group?