Mrs. Danforth packed $5$ bags of fun-sized candies for her family's picnic. Each bag contains $16$ pieces. She had intended to pass out the same number of candies to each of her $4$ children. But Cameron, her second-youngest, argues that she should instead distribute the candies in proportion to the children's ages. All of the children have different ages. If Cameron can convince his mother to give out the candies his way, he will gain two candies while his four-year-old sister will lose twelve. How old is Mrs. Danforth's oldest child?
