Less weight going up a slope means faster times. How much faster?

Suppose one is considering buying a new, light-weight frame or going on a diet. What is the benefit from having less weight? How much time would be saved over a given distance on a specified slope? How much more distance would be covered in a given time?  

Two riders, identical except that one has less weight than the other, ride a given distance up a hill. The the calculation gives the distance and time between the riders as the lightest rider reaches the given distance.  

The plot shows the benefit from less weight when riding up a hill. The range of the plot is from -10% to +10% grade and from 0 to 5 kg less weight. The table gives specific values for the given speed.  

Note that large changes may produce mathematical results, but may not have real-world meaning. Keep changes small.  

"Negative values for improvements" show that weight is an advantage going down a hill.  The plot shows that weight is more of a penalty going up hill than an advantage going down. 

Example: 

Benefit From Less Weight 

This Much Less Weight  5  kg Over This Distance  2000  meters On Hill of Slope  0.03  Decimal Faster by  8.69  s Ahead by  64.82  m Frontal Area  0.5  m^2 Coefficient Wind Drag  0.5  Dimensionless Air Density  1.226  kg/m^3 Weight Rider & Bike  75  kg Coefficient of Rolling  0.004  Dimensionless Power  250  watts