Ten students entered for the competition, the youngest a boy of 12.
十个学生报名参加了这次竞赛，年纪最小的是个12岁的男孩。(the youngest和a boy of 12之间省去了being)
独立主格结构中的being在下列两种情况下一般不能省略，一是在"There being +名词"结构中，二是在逻辑主语是代词的情况下。
There being no bus, we had to walk home.由于没有公共汽车，我们只好走回家。
It being Sunday, all the offices are closed. 因为是星期日，所有办公室都关门。
He turned to me, his eyes sleepy.
= He turned to me, and his eyes were sleepy.
He stood there, his mouth wide open.
他站在那里，嘴张得大大的。(his mouth和wide open之间省去了being)
= He stood there, and his mouth was wide open.
School over, we all went home.
= School was over, and we all went home.
He sat at his desk, his shoes off.
= He sat at his desk and his shoes were off.
He is standing in front of the blackboard, his back towards us.
= He was standing in front of the blackboard, and his back was towards us.
The new teacher came in, a smile on her face.
= The new teacher came in and she had a smile on her face.
The teacher came into the classroom，a rule in his hand.
= The teacher came in and a ruler was in his hand.
The music teacher stood at the door, violin in hand.
音乐老师站在门口,手里拿着一把小提琴。(= The music teacher stood at the door, a violin in his hand.)