Please help its due tonight I'm so confused!!

Question

Flying against the wind, an airplane travels 7560

 kilometers in 9

 hours. Flying with the wind, the same plane travels 9760

 kilometers in 8

 hours. What is the rate of the plane in still air and what is the rate of the wind? 

Answer 1

These used to confuse me, too. Remember that distance = rate x time, right?

Let p = the speed of the plane and w = the speed of the wind.

So, let's say I row 4 kph but there's a 2 kph current. If I row upstream, I'm travelling 4 - 2 = 2 kph. If I row downstream, I travel 4 + 2 = 6 kph. Get it?

Setting up a system of equations...

9(p - w) = 7560

8(p + w) = 9760

p - w = 840

p + w = 1220

2p = 2060

p = 1030 kph

1030 - w = 840

w = 190 kph

Answer 2

Without any other info, we can assume that the wind is slowing down the plane by the wind's speed. Speed, or velocity, is a measurement of distance/time, so we can write an equation like this:

v = d / t, where v = velocity (or speed)

d = distance

t = time

First let's calculate the speed the plane is actually travelling against wind and with wind:

Against wind:

v = d / t

= 7560 km / 9 hours

= 840 km/hr

With wind:

v = d / t

= 9760 km / 8 hours

= 1220 km/hr

When the plane is travelling against the wind, we can say that the velocity the plane is travelling is equal to the speed the plane would travelling without wind and subtracting the velocity of the wind. To write that out in math terms, we could say:

v_T = v_a - v_w, where v_T = total velocity, or the actual velocity the plane is travelling

v_a = the velocity the plane is trying to go without wind

v_w = the velocity of the wind

We know that this speed is equal to 840 km/hr from earlier, so let's replace v_T with 840:

v_T = v_a - v_w

840 km/hr = v_a - v_w

When the plane is travelling with the wind, we can say that the velocity the plane is travelling is equal to the speed the plane would travelling without wind and adding the velocity of the wind. To write this out using the same variables from earlier, we could say:

v_T = v_a + v_w

We know this speed is equal to 1220 km/hr, so let's replace v_T with 1220 km/hr:

v_T = v_a + v_w

1220 km/hr = v_a + v_w

Now we have written two equations with the same two variables. If we solve for one of the variables in the first equation, we can rewrite the second equation to be in terms of only one variable and solve for that variable. Let's solve for v_a in our first equation for when the plane is travelling against the wind:

840 km/hr = v_a - v_w

v_a = 840 km/hr + v_w

Now we can use this to replace v_a in our second equation and solve for v_w:

1220 km/hr = v_a + v_w

= (840 km/hr + v_w) + v_w

= 840 km/hr + 2v_w

380 km/hr = 2v_w

v_w = 190 km/hr

We just solved for the speed of the wind. Now we can use this in any of our equations to solve for v_a:

v_a = 840 km/hr + v_w

= 840 km/hr + 190 km/hr

= 1030 km/hr

So there you have it, the speed of the plane v_a = 1030 km/hr, and the speed of the wind v_w = 190 km/hr