The length of an arc is 10cm. find the angle subtending by the arc if the circumference of the circle of which the arc forms part is 60cm.

There are a couple of ways you can do this... you can easily determine the circumference of the circle since you know its radius:  C = 2πr.  Once you determine this value, you can find out what percentage of the circumference of the circle is covered in 10 cm and multiply that by 360 degrees to get the angle. 

The second way is to use the relationship s=rθ, that relates arc length (s) to the circle's radius (r).  Since you know s = 10 cm and r = 60 cm, you can solve for theta:  θ=s/r = 10 cm/60 cm = 1/6.  Note that the angle in this equation is expressed in radians, so if you need to express this in degrees instead, you'll need to apply the conversion that 360 degrees = 2π radians, or that 1 radian is ~57.3 degrees.

The angle ends up being about 9.55 degrees, or 1/6 radians.

An arc is a part of the circumference of a circle and the angles that subtend the arcs are proportional to the length of the arc.

So, you basically have two arcs to compare: a 10cm arc and a 60cm arc. But the 60cm arc also happens to be the circumference in this case, so you know that the angle that subtends that arc is 360 degrees (the whole circle). The 10cm arc is 1/6 of the whole circumference so the angle is going to be 1/6 of 360 degrees, which is 60 degrees.

Recall:   The length of an arc in a circle is a portion of the circumference of the circle.

          That is,     arc length / circumference  =  arc measure / 360°.

By definition, the degree measure of an arc in a circle is equal to the measure of the central angle that intercepts the arc. With this, we can use the above formula to solve for the arc measure:

     arc measure  =  (arc length/circumference)·360°

Given:     arc length = 10 cm

              circumference of circle = 60 cm

     arc measure = (10 cm/60 cm)·360°

                        = (1/6)·360°

                        = 360°/6

                        = 60°

Using circumference = 2*pie*r and given that the circumference is 60cm you are able to solve for your "r" value. Then use arc length = r*theta solve for "theta" using the value you got from the previous step and the arc length you where given.