Carothers was born on April 27, 1896 in Burlington, Iowa, to Ira and Mary Evalina Carothers. He was the oldest of four children. He had one brother and two sisters: John, Isobel and Elizabeth. As a youth, Carothers was fascinated by tools and mechanical devices and spent many hours experimenting. He attended public school in Des Moines, Iowa, where he was known as a conscientious student. After graduation, and under pressure from his father, Carothers enrolled in the Capital City Commercial College in Des Moines, where his father was Vice-President, completing the accountancy and secretarial curriculum in July 1915.