Explain why in order to lose weight you would need to ingest fewer calories per day, or expend enough energy per day, so the net ingested daily calories are less than the calculated dri. how long would it theoretically take to lose the equivalent of one pound of fat if you ingested 100 calories/day less than the dri?

Basically, a person's weight is the result of two antagonistic processes: calorie intake by food and calorie loss by physical activity. Taking more calories than being spent causes them to be stored as body fat. The healthiest diet is the one that follows DRI, with optimal intake of nutrients, vitamins and minerals. That way, the weight should remain constant. Losing weight is associated with making negative net balance of intake and loss of calories. Spending more than taken means that body fat will be transferred in calories and being used to compensate for decreased intake or increased physical activity. It is a general rule of thumb that for losing one pound one has to make a negative net balance of around 3500 calories. So, ingesting 100 calories less then DRI every day means that it will take around 35 days to lose a pound of fat.

Of course, this is only theoretically, since food type, water intake, age, sleep, other conditions etc. contribute to dynamics of losing or gaining weight.