First step: calculate the scale at 55ºN

Total length of each parallel = 360º
Total length of the equator = 360º x 60 NM = 21600 NM
Total length of 55ºN parallel = 21600 NM x cos 55º = 12389.25103 NM
Length of 1’ = 12389.25103 NM / 360º / 60’ = 0.5735764364 NM

3.1 mm on the chart = 0.5735764364 NM in reality
1 cm on the chart = 1.850246569 NM in reality => (0.5735764364/0.31)
1 NM = 1.852 km
1.850246569 NM = 3.426656646 km (1.850246569 x 1.852)
3.426656646 km = 342665.6646 cm

• 55ºN scale: 1:342665

Second step: calculate the scale at 40ºN

When the scale at one latitude on a Mercator is known, the scale at any other latitude may be calculate with the following formula:

Scale denominator “A” x cos “B” = Scale denominator “B” x cos “A”
“A” ad “B” are the latitudes
342665 x cos 40 = Scale B x cos 55
Scale B = 457648.8894

Closest answer = 1 : 457650