Need help with speed/time problem?

Question

Debbie traveled by boat 5 miles upstream. Because of the 4-mph current, it took her 20 minutes longer to get there than to return. How fast will her boat go in still water?

Answer 1

Hi,

Let x = speed of the boat in still water in mph

Then the speed upstream is x - 4 and the speed downstream is x + 4. Since distance/rate = time, we can write an equation for time upstream - time downstream = 20 minutes changed to 1/3 of an hour.

....5..........5..........1
------..-..------..=..-----
x.-.4.....x.+.4........3

Multiplying by 3(x - 4)(x + 4) gives the equation:

5(3)(x + 4) -.5(3)(x - 4).=.1(x - 4)(x + 4)
15(x + 4) -.15(x - 4).=.x² - 16
15x + 60 -.15x + 60.=.x² - 16
120 = x² - 16
0 = x² - 136
136 = x²
√136 = x
11.66 = x

The speed of the boat in still water is 11.66 mph. <== answer

To double check this, substitute this into the equation to see if the answer is almost 1/3. (Rounding error will make it slightly different.)

....5..........5..........1
------..-..------..=..-----
x.-.4.....x.+.4........3

....5....................5...............1
-------------..-..-------------..=..--
11.66.-.4.....11.66.+.4.......3

....5...........5..........1
--------..-..-------..=..--
7.66......15.66......3

.6527 - .3193 = .3334

This checks!!

I hope this helps!! :-)


11.66 mph in still water

Answer 2

Well it took her 1/3 of an hour to go 5 miles, so her speed with respect to the ground must be 15 mph. If she was traveling against a 4 mph current, the boat must go 19 mph in still water.

Answer 3

Debbie travels x miles per hour in still water.
The current is 4mph.
5miles/(x-4) mph = time upstream in hours
5miles/(x+4) = time downstream in hours
5/(x-4) = 5/(x+4) +1/3 hr
Multiply each term by 3(x+4)(x-4)
15(x+4) = 15(x-4) + (x+4)(x-4)
15x + 60 = 15x - 60 + x^2 - 16
0 = x^2 - 136
x = sqrt136 = ~11.67 mph

Answer 4

Speed=distance/time...Time=distance/speed
Speed=x...upstream=x-4...downstream=x+4
Distance=5miles
T1(time upstream)=T2(time downstream)+(1/3)
5/(x-4)=5/(x+4)+(1/3)(=(x+19)/(3x+12))
5(3x+12)=(x+19)(x-4)
15x+60=x^2+15x-56
0=x^2-116
x~10.77

I goofed somewhere...