Both the Gleim and the ASA Knowledge Test Study Guides give a formula for for calculating the distance travelled along a DME arc that just didn’t look right to me.
The formula is (# of degrees x DME arc) ÷ 60. But the true formula should be based on the percentage of the circle that the airplane travels.
The formula for the circumference of a circle (the distance around it) is 2πr where r is the radius of the circle (DME arc in their notation). So the distance travelled should be the percentage of the circle. The formula is (2π x DME arc) x (# of degrees / 360). You can rewrite it as (# of degrees x DME arc) ÷ 57.3. Which is close to 60.
In the example with the GNATS.ONE departure, their method gives (15*(333-251))/60 = 20.5 miles. The correct method gives (2π * 15) * (82/360) = 21.5. About 5% more.
I don”t know which method the people who wrote the test use, but if they use the incorrect method, it helps explain why none of the answers is ever the right one.