Question

On a particular day, the wind added 2 miles per hour to Alfonso's rate when he was cycling with the wind and subtracted 2 miles per hour from his rate on his return trip. Alfonso found that in the same amount of time he could cycle 63 miles with the wind, he could go only 51 miles against the wind.What is his normal bicycling speed with no wind?

Answer 1

I set this up in a table, keeping in mind the distance = rate * time formula in mind.

Answer 2

distance rate time

with wind

without wind

Answer 3

Now let's fill it in.

Answer 4

With the wind, his rate is "whatever his speed is" plus 2 miles per hour, so that goes into the rate column:

distance rate time

with wind r + 2

Answer 5

Against the wind, he goes 2 miles slower, right? So let's put that in the table too.

Answer 6

distance rate time
with wind r + 2

against wind r - 2

Answer 7

The key thing here is to read the problem closely enough to catch the "in the same amount of time" part. THAT tells us that the times are equal to one another, so "t" goes in to both the "time" columns.

Answer 8

distance rate time
with wind r + 2 t
against wind r - 2 t

Answer 9

With the wind he traveled 63 miles, and against it he traveled 51:

Answer 10

distance rate time
with wind 63 r + 2 t
against wind 51 r - 2 t

Answer 11

The t variables are equal to one another. That means that in an equation situation, we can say that t = t. Solving for t in the distance formula, we get:
\[t=\frac{ d }{ r }\], and we can set the distance over rate in both rows equal to one anotheer, because, remember, that t = t.

Answer 12

The distance in the "with wind" is 63, and the rate is r + 2. Remember that it is rate we are solving for. By setting the equations equal to one another, because there times are equal to one another, eliminates the extra variable, and now we have equations in terms of only one variable, the r, which is what we are looking for.

Answer 13

\[\frac{ 63 }{ (r+2) }=\frac{ 51 }{ (r-2) }\]

Answer 14

Cross multiply to get:
63(r-2)=51(r+2)

Answer 15

63r - 126 = 51r + 102

Answer 16

Can you take it from there? What you find for "r" is the rate he goes with no wind at all.

Answer 17

Yes, I can and thanks for the help :)

Answer 18

Any time!