Can someone help me out w/ these Problems... I CAN'T FIGURE THEM OUT!!!!?

Question

1.Steve traveled 200 miles at a certain speed. Had he gone 10mph faster, the trip would have taken 1 hour less. Find the speed of his vehicle.

2.The Hudson River flows at a rate of 3 miles per hour. A patrol boat travels 60 miles upriver, and returns in a total time of 9 hours. What is the speed of the boat in still water?


3.A designer attempts to arrange the characters of his artwork in the form of a square grid with equal numbers of rows and columns, but finds that 24 characters are left out. When he tries to add one more row and column, he finds that he has 25 too few characters. Find the number of characters used by the designer.

Answer 1

1.200/x-200/(x+10)=1
200(x+10)-200x=x(x+10)
200x+2000-200x=x^2+10x
x^2+10x-2000=0
(x+50)(x-40)=0
x=-50 or 40
ignoring the negative value x=40
original speed of the car=40mph

2.60/(x+3)+60/(x-3)=9

60(x-3)+60(x+3)=9(x^2-9)
120x=9x^2-81
9x^2-120x-81=0
x=[120+/-rt(14400+2916)]/18
=120+131.6/18=14 approximately
speed of boat in still water =14 mph

3.x^2+24=(x+1)^2-25
(x+!)^2-x^2=49
2x+1=49
2x=48
x=24
no of characters=576+24=600

Answer 2

I think you will find that the difficult part with these problems is setting up the equations. Once you have done that, solving is usually trivial.
Let us look at the first one.
Define the speed he traveled at as v and time he spent traveling as t. We know that distance = rate * time. So, putting in the values we know, 200 = v*t
However, we also know that if he had traveld ten mph faster, he would have covered the same distance in one hour less.
So we can also say that 200 = (v+10)*(t-1). Thus, we have a system of two equations with two variables. I'll leave it to you to solve them.

I'll leave the second one for you to set up yourself. Just really think about what is happening -- the river "helps" the boat when it travels downriver, and slows the boat when it travels up river.

As for the third question, lets call the number of characters c and the number of rows (or columns) n. Thus, there are n*n spots in the initial grid.
We know that c - n*n = 24 -- there are 24 more characters than there are spaces in the grid. We also know that when the number of rows and colums are each increased by one, then:
(n+1)*(n+1) -c = 25 -- there are now 25 more grid spaces than characters. Yet again, we have two variables and two equations, which can be easily solved.

Answer 3

1. let t1,t2 be the time he travelled in 2 cases t2 = t1 - 1
v1,v2 are the velocities v1+ 10 = v2
S = 200 miles
we have : t1 = S/v1 ; t2 = S/v2 = S / (v1 + 10)
=> S/v1 - 1 = S/(v1 + 10)
solve this equation we will have v1 = 40 mph

2. let V be the velocity of the boat in still water
S is the length of the river S = 60
the velocity of the boat when it travels upriver is v1 = V - 3;
the velocity of the boat when it returns is v2 = V + 3
the total time of the trip : S/v1 + S/v2 = 9
=> S/(V-3) + S/(V+3) = 9
solve this equation we will have : V = 13.98 mph