1. 12/8 +5/6
Consider the LCM of 6 and 8. This is the union, multiplied, of {2,3} and {2,2,2}. We get {2,3,2,2}. Multiplied together this is of course 24. That means, our two factors are:
x/24 + y/24
From here we reason that we should include the 3 of 24/8 and the 4 of 24/6 need be included in the translation of 12/8 and 5/6. We get 12*3/8*3 and 5*4/6*4.
We now have 36/24 and 20/24 totaling to 56/24.
Observe that 56 = 8*7 and 24 = 3*8. We reduce to 7/3. Finally we deduce that 7/3 = 2 and 1/3. This is our final answer.
2. A merry go round goes around 5 times in 15 minutes. How many times does it go around in 2 hours?
We set up an equation using proportions.
5times/15 min. = x times/120 min.
or
x times = 5times*120 min. / 15 min.
x times = 5times 8 = 40 times
x=40
The merry go round goes around 40 times in 2 hours.
3. A car travels 50 miles in 2 hours. How many...
read more

This is another way to find a distance between two parallel lines. This derivation was suggested to me by Andre and I highly recommend him and his answers to any student, who wants to learn math ans physics. This derivation requires the knowledge of trigonometry and some simple trigonometric identities, so this may be suitable for more advanced students.
Once again, we have two lines.
y=mx+b1 (1)--equation for the first line.
y=mx+b2 (2)--equation for the second line.
Now recall that the slope of the line is the tangent of an angle this line forms with the x-axis. Indeed, m=(y2-y1)/(x2-x1), where x1, x2, y1, y2 are the x- and y-coordinates of any two distinct points on the line. If one draws the picture, it will be immediately obvious that m is the tangent of the angle between the line and the x-axis.
The difference b2-b1 gives the relative displacement along the y-axis of two lines....
read more

Suppose, one have two parallel lines given by the equations:
y=mx+b1 and y=mx+b2. Remember, if the lines are parallel, their slopes must be the same, so
m is the same for two lines, hence no subscript for m. How would one approach the problem of finding the distance between those lines?
First, if one draws a picture, he or she shall immediately realize that if a point is A chosen on one of the lines, with coordinates (x1, y1), and a perpendicular line is drawn from that point to the second line, the length of the segment of this new line between two parallel lines give us the sought distance. Let us denote the point of intersection of our perpendicular line with the second line as B(x2,y2).
What do we know of point A and B?
First, since A lies on the first parallel line, its coordinates must satisfy the equation for the first line, that is,
y1=mx1+b1 (1)
Same...
read more