Algebra Story Problem: Find the speed of the current
Two miles upstream from his starting point, a canoeist passed a log floating in the river's current. After paddling upstream for one more hour, he paddled back and reached his starting point just as the log arrived. Find the speed of the current.
Answer STEP 1:
You are asked to find the speed of the current. Assume a variable for this quantity, say, c
. We come across some other quantities in the given information which can also be denoted by variables
for ease in reference.
See how the parentheses get cleared when each term inside the parentheses is multiplied by the factor outside.
- t – total time, right from when the canoeist sees the log till he reaches back at the starting point.
- r – rate of the canoeist
- x – distance paddled upstream by the canoeist after he sees the log.
Recall that the distance traveled is the product of rate and time. Since the rate of the log is the same as that of the current, the log also has covered 2 miles.
This will help you to frame an equation, 2 = ct
. You can also rewrite it as t = 2/c
It is to be noted that t is the total time taken by the canoeist to travel the upstream distance x
, then downstream and finally 2 miles.
Since the direction of current is against that of the canoeist, the time taken to cover his upstream journey will be:
Similarly, the time taken to cover downstream x
distance is equal to:
Also, the time it takes for the canoeist to cover the final 2 miles downstream is:
The sum of these time periods equal t
It takes 1 hour for the person to travel the upstream distance x
. So, you can replace x/r-c
with 1 and x
with r – c. To make the denominators of the fractions
same, you can also find an equivalent fraction for 1.
Now, add the numerators of the fractions keeping the denominator.STEP 5:
We have already obtained an expression for t
at the start. Equate the two expressions.
Cross multiply to clear the fractions give:
See how we can remove the parentheses
from each side. Multiply the terms inside by the factor outside.STEP 6:
The final step is to solve the equation for the variable c
The speed of the current is 1 mile per hour