How much of a 45% solution must be added to 1 liter of a 60% solution in order to make a 50% solution?

0.45x+0.60*1=(x+1)*0.5
0.6-0.5=0.5x-0.45x
1.0=0.05x 
x=20 liters

length of a dog run is 9 feet more than twice the width and the perimeter is 34 feet, what is the length and width

1) l=2w+9
2) 34=2(l+w) 


from 2)...
 17=l+w
17=(2w+9)+w
17-9=3w
w=8/3 

so 
l=
2*8/3+9=
16/3+27/3=
43/3

Four times a number added to 5 is divided by 6. The result is 7/2. Find the number.

(4x+5)/6=7/2
6(4x+5)/6=6*7/2
4x+5=21
4x=16
x=4

There is a certain number less than 100. The sum of the digits is 7 and the product of the digits is 6. What is the number?

we describe the number as 100=xy so that x is in the 10's place and y in the 1's place, then: 
1) x+y=7
2) xy=6 

x=6/y 
6/y+y=7
6+y^2=7y
y^2-7y+6=0
(y-6)(y-1)=0 
y=6 or y=1 

so we see that there are two numbers possible: 16 and 61

The graph is a parabola with vertex (3,-8) and passing through the origin. Write the quadratic equation of this graph.

We use the Standard Form: 

from the vertex we get:
 
since the origin is on the graph, 
 

 
 

so we can write

the perimeter of a rectangle is 112 cm . the length is 30 cm . calculate the width .

P=2(l+w) 
w=P/2-l
w=112/2-30
w=26 cm

The sum of two numbers is 78. The second number is 14 more than 3 times the first number. Find the two numbers.

x+y=78
y=3x+14 


so...
x+3x+14=78
4x+14=78
4x=64
x=16 

therefore...
y=78-16
y=62

a number added to eighteen times its reciprocal is eleven

x+18/x=11
x^2+18=11x
x^2-11x+18=0
(x-9)(x-2)=0 
x=2 or x=9