The following 15.6" Asus K501UB laptop sit right at the limit of your budget. It has a Core i7-6200u CPU, a nVidia 940m graphics chip, 1080p resolution screen, 12GB of RAM and a 1TB hard drive with a 16GB SSD cache.
https/www.amazon.co.uk/ASUS-K501UB-DM021T-Notebook-i7-6500U-Graphics/dp/B01A6XLR2U/ref=sr_1_6?ie=UTF8&qid=1477505130&sr=8-6&keywords=940m
The i7-6200u is a lot more powerful than the i3-5005u CPU which means it has more horse power and technically means the laptop can last longer before she feel it is no longer powerful enough for her. The nVidia is more than capable of playing the Sims 3 and Sims 4, it's much better than the Intel HD 5500 integrated graphics core in i3-5005 CPU. The 1080p screen means that she has enough desktop real estate on the screen so that she can have a word processor on the one side of the screen while having an internet browser on the other half of the screen. That way if she needs to do research for writing a paper she does not have to switch between the two programs if she does not want to.