Algebra help

Question

Gretchen can paddle 3 miles upstream in twice the time it takes her to paddle 5 miles down stream. If the current is moving at 1 mph, what is gretchen's paddling rate in still water?

Answer 1

The equation that you need to use for this problem is Distance = Rate × Time. 

 

Gretchen's rate of speed going downstream would be her paddling rate in still water plus the current, which is moving in the same direction that she is paddling. Since she can paddle 5 miles downstream in an amount of time when the current is 1 mph, we can make the equation 3 = (G +1) × t. We will note Gretchen's paddling rate with the variable G, and time is represented by the variable t. 

 

Gretchen's rate of speed going upstream would be her paddling rate in still water minus the current, which is moving in the opposite direction from what she is paddling. Since she can travel 3 miles upstream in twice the amount of time as it took her to travel 5 miles downstream, we can make the equation 5 = (G - 1) × 2t. 

 

We have two equations. Let us use a system of equations to solve for t. First, let us expand our two equations by multiplying the t values across the rate quantities. 3 = (G +1) × t becomes 3 = Gt + t. Next, 5 = (G - 1) × 2t becomes 5 = 2Gt - 2t. 

 

Now we have

3 = Gt + t

5 = 2Gt - 2t

 

If we multiply both sides of the first equation by 2, we can get

6 = 2Gt + 2t

5 = 2Gt - 2t

 

Let us now subtract the second equation from the first equation. 

6 - 5 = (2Gt + 2t) - (2Gt - 2t)

Simplify.

1 = 2Gt + 2t - 2Gt + 2t

1 = 4t

0.25 = t

 

Now we know that the amount of time specified by the word problem is 0.25 hours, or 15 minutes. We can input this value to either of the two original equations to solve for G, Gretchen's paddling rate. 

 

5 = (G - 1) × 2t

5 = (G - 1) × 2(0.25)

5 = (G - 1) × 0.5

5 ÷ 0.5 = (G - 1)

10 = G - 1

11 = G

 

Gretchen's paddling rate in still water is 11 mph.

 

 

Answer 2

Let T =Time to go upstream and r = rate in still water

 

3 = (r-1) * 2T

5 = (r+1) * T

 

3 = 2Tr - 2T

5 = Tr + t

 

Multiply the 2 equation by 2 and subtract

 

7 = 4T => T= 7/4

 

3 = (r-1) * 7/2 and r = 13/7

 

I am always uncomfortable when the solution is in messy numbers; nevertheless, I think it is correct.