Teacher needs maths assistance please

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Laser Eyes
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I teach English at a university in China. I don't want any of my students to fail so I need to adjust exam marks. The exam marks range from 20 to 100. I want to adjust them so that the marks will range from 60 to 100, but I want to retain the proportional relationship between the marks. So for example 20 out of 100 will become 60 and 60 out of 100 will become 80. I haven't had to do any maths for many years so can someone please let me know how to do this.
 
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Usually we require some effort and an attempt to solve the problem, because we cannot know whether your story is correct or only a nice trick to get homework solved. So my answer now isn't really allowed. However, I think I can make an exception, as the solution isn't really very complicated. You can use ##new = \dfrac{1}{2}\cdot \, old +50##
You can also find this solution graphically.
 
Thank you. I knew the answer was fairly simple and I could have proven what I said if necessary.