distance = rate*time = speed*time = velocity*time.

Let x = distance down river that Duncan paddled to, in miles.

And c = rate of current or stream, in mi/hr.

So, Duncan's best rate in still water is 10c mi/hr.

Calum:

When Calum and Duncan met at Duncan's starting/finishing point, Calum travelled half a mile, or (1/2)mi.

The time he spent doing that is

distance = rate*time

1/2 = c *T

T = (1/2)/c = 1/(2c) hrs. -----------**

Duncan:

Total time of Duncan paddling downstream to x and then paddling upstream to his starting point is the same as Calum's T.

Downstream, Duncan's net rate is 10c +c = 11c

So,

distance = rate*time,

x = (11c)(t1)

t1 = x/(11c) hr. --------time spent paddling x miles downstream.

Upstream, Duncan's net rate is 10c -c = 9c

So,

distance = rate*time,

x = (9c)(t2)

t2 = x/(9c) hr. --------time spent paddling x miles upstream.

------------------------------

Now,

t1 +t2 = T

x/(11c) +x/(9c) = 1/(2c)

Clear the fractions, multiply both sides by (11*9*2*c),

x(9*2) +x(11*2) = 1(11*9)

18x +22x = 99

40x = 99

x = 99/40 = 2.475 mi. ----------------answer.

[In feet, that is (2.475mi)(5280ft/1mi) = 13,068 ft.]