Word problems w/ substitution/elimination

Question

Brittany and Hanna can paddle 12 mi upstream in the same time it takes to paddle 28 mi downstream. Determine their paddling rate in still water if the stream flows at 2 mi/hr.

a jetliner can travel 1700 mi from Fargo, ND, Phoenix, AZ in 4 hr with a tailwind, but it takes 3 hr to travel 975 mi from Phoenix to Boise, ID into the same wind. Determine both wind speed and air speed of the plane

Answer 1

Brittany and Hanna can paddle 12 mi upstream in the same time it takes to paddle 28 mi downstream. Determine their paddling rate in still water if the stream flows at 2 mi/hr.

Let's call their paddling speed vp. To find their effective speed going upstream, we need to subtract the stream speed, since it is working against them.

v_up = vp - 2 mi/hr

To find their effective speed going downstream, we need to add the stream speed, since it is working for them.

v_down = vp + 2 mi/hr

The time it takes to paddle 12 miles upstream is

12 mi / (vp - 2 mi/hr) = 12 / (vp-2) hr (If you get confused about how to get time from speed and distance, just check the units. We need time (hr) as the result. We have distance (mi) and speed (mi/hr). If we divide mi by mi/hr, we get mi / (mi/hr) = mi * hr/mi = hr)

The time it takes to paddle 28 miles downstream is

28 / (vp + 2). And these should be equal:

12 / (vp-2) = 28 / (vp+2) (crossmultiplication)

28 * (vp-2) = 12 * (vp+2)

28 vp - 56 = 12 vp + 24 (collect vp terms on one the left side and constants on the right side)

(28-12) vp = 24 + 56

vp = 80/16

Same approach can solve the second problem.