This randomized phase II trial studies how well abbreviated breast magnetic resonance imaging (MRI) and digital tomosynthesis mammography work in detecting cancer in women with dense breasts. Abbreviated breast MRI is a low cost procedure in which radio waves and a powerful magnet linked to a computer and used to create detailed pictures of the breast in less than 10 minutes. These pictures can show the difference between normal and diseased tissue. Digital tomosynthesis mammography is a procedure that uses multiple x-rays pictures of each breast to produce a 3-dimensional rendering of the entire breast. Combined screening with abbreviated breast MRI and digital tomosynthesis mammography may be a better method to screen women with dense breasts.