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27 October 2014, 16:52 | Updated: 27 October 2014, 16:55
At the moment getting from Manchester to Leeds by train takes around 55 minutes. This proposal from HS2 Ltd chairman Sir David Higgins would reduce that to between 25 and 35 minutes.
David Cameron said: ``I am passionate for high-speed rail. It is important for our country that high-speed rail works for Britain.
``These sort of decisions - decisions about our country and the future of rail - matter. They are not always popular. HS2 is not always popular. But I profoundly believe they are right.''
Earlier Sir David had said his plans for HS3 were ``a start of a conversation''.
He said: ``This is not just a single project. It's more than the eye-catching journey-time reductions. HS3 will give northern England rail services twice the capacity and much more reliability.''
There has been criticism too, the Institute of Economic Affairs has described it as ``little more than a costly vanity project'' Its head of transport Dr Richard Wellings said It would be better to spend money on smaller schemes rather than ``creating headline-grabbing policies''.