a grain barge travels on a river from point a to point b loading and unloading grain. the barge travels at a rate of 5mph relative to the water. the rive flows downstream at a rate of 1mph. if the trip upstream takes 2 hours longer than the trip downstream, how far is it from point a to point b?

Caiti,

Mathematically the given information is as follows:

Time up stream: Tu = Td + 2 (1) Tu & Td are up &down stream times

Distance A to B: D = Tu * Su (2) Su is speed upstream

Distance B to A: D = Td * Sd (3) D is the same up or down stream!

Su is speed upstream: Su = 5 -1 = 4 mph (4) barge is slowed upstream

Sd is speed down stream: Sd = 5 +1 = 6 mph (5 ) barge moves faster downstream

From (2) & (3) find:

Tu*Su = Td*Sd

Tu*Su/Sd = Td

Tu*4/6 = Td

Td = 2/3*Tu (6) Makes sense…Td<Tu

Substitute (6) in (1) and isolate Tu:

Tu = Td +2

Tu= 2/3*Tu +2

Tu – 2/3* Tu = 2

1/3*Tu = 2

Tu = 3*2

Tu = 6

Now, substitute Tu=6 into (1) and isolate Td:

Tu =Td + 2

6= Td +2

6-2 = Td

Td = 4

Find D by substituting Sd and Td results into (3)

D=Td* Sd

D = 4*6

**D= 24 miles**

Check this D result by substituting Tu & Su into (3):

D = Tu*Su

D= 6 * 4

**D=24 miles** OK!