a The amount of time spent traveling is the ratio of the distance traveled to the corresponding rate.
B
b Rearrange the variables by writing u in terms of d.
C
c Are the ways to find the amount of time for each case the same?
A
a 18/5u
B
b 9/2d
C
c Yes, see solution.
Practice makes perfect
a We are told that a rowing team practices rowing 2 miles upstream and 2 miles downstream. The rate of the team while they are rowing downstream is 25 % faster than the rate while they are rowing upstream. If we let u be the rate of the team while they are rowing upstream, we can write expressions to represent their rates.
Upstream & Downstream
u & u + 25u/100= 125u/100
We want to find the total amount of time they spend rowing. To do so let's recall that the amount of time t spent traveling is the ratio of the distance d traveled to the corresponding rate u.
Time=Distance/Rate ⇔ t=d/u
Since we know the corresponding rates in terms of u and the distance is 2 miles, we can write the time expressions in terms of distance and rate for each case.
Distance (mi)
Rate (mi/h)
Distance/Rate
Time (h)
Rowing Upstream
2
u
2/u
t_1=2/u
Rowing Downstream
2
125u/100
. 2 / 125u/100.
t_2=200/125u⇒ 8/5u
Now we will add the expressions t_1 and t_2 to find the total amount of time.
We can express the total time they spend rowing with 185u in terms of u.
b Let's think about the same scenario from Part A. Now we will just represent the team's rate while they are rowing downstream with d rather than 125u100. To get the rate of the team while they are rowing upstream u in terms of d, we need to isolate u from the equation 125u100=d. Lets do it!
Upstream & Downstream
4d/5 & d
One more time, we will find the total amount of time they spend rowing in terms of d. Now let's recall that the amount of time t spent traveling is the ratio of the distance x traveled to the rate d. Notice that we have used different variables from the Part A to avoid confusion.
Time=Distance/Rate ⇔ t=x/d
Having the corresponding rates in terms of d and the distance of 2 miles, we can write the time expressions in terms of distance and rate for each case.
Distance (mi)
Rate (mi/h)
Distance/Rate
Time (h)
Rowing Upstream
2
4d/5
. 2 / 4d/5.
t_1=10/4d⇒ 5/2d
Rowing Downstream
2
d
2/d
t_2=2/d
Now we will add the expressions t_1 and t_2 to find the total amount of time.
We can express the total time they spend rowing with 92d in terms of d.
c Notice that we found the expressions 185u and 92d using the same process. Both of them are equal to the ratio of the distance traveled to the rate. Since the distances are assigned as 2 miles and then the rates are just written with different variables, the total amount of time must be equal for both cases.
