Which of the following is not a direct proportion?
A. y=3x
B. y=45x
C. y=72x
D. y=2

Variation, in general, will concern two variables, say height and

weight of a person, and how when one of these changes, the other might

be expected to change.  We have direct variation if the two variables

change in the same sense, i.e. if one increases, so does the other.  

We have indirect variation if one going up causes the other to go

down.  An example of this might be speed and time to do a particular

journey, so the higher the speed the shorter the time.

Normally we let x be the independent variable and y the dependent

variable, so that a change in x produces a change in y.  For example,

if x is number of motor cars on the road and y the number of

accidents, then we expect an increase in x to cause an increase in y.  

(This obviously ceases to apply if number of cars is so large that

they are all stationary in a traffic jam.)

When x and y are directly proportional then doubling x will double the

value of y, and if we divide these variables we get a constant result.

Since if y/x = k  then (2y/2x) = k  where k is called the constant of

proportionality

We could also write this  y = kx

Thus if I am given the value of x, I multiply this number by k to find

the value of y.

Example: Given that y and x are directly proportional and y = 2 when

x = 5, find the value of y when x = 15.

We first find value of k, using y/x = k

                               2/5 = k

Now use this constant value in the equation y = kx for situation when

x = 15

     y = (2/5)*15

       = 30/5 = 6

If you want to do this quickly in your head, you could say, x has been

multiplied by a factor 3 (going from 5 to 15), so y must also go up by

a factor of 3.  That means y goes from 2 to 6.

Indirect Variation.

We gave an example of inverse proportion above, namely speed and time

for a particular journey.  In this case if you double the speed you

halve the time. So the product speed x time = constant

In general, if x and y are inversely proportional then the product xy

will be constant.

              xy = k

  or          y = k/x

Example:  If it takes 4 hours at an average speed of 90 km/hr to do a

certain journey, how long would it take at 120 km/hr

   k = speed*time = 90*4 = 360 (k in this case is the distance)

Then time = k/speed

         = 360/120

         = 3 hours.

To do this in your head, you could say that speed has changed by a

factor 4/3, so time must change by a factor 3/4.  However, for the

usual type of problem, go through the steps I outlined above.

Step-by-step explanation:

I hope these examples have made the idea of variation (both direct and

inverse) reasonably clear.

Variation, in general, will concern two variables, say height and

weight of a person, and how when one of these changes, the other might

be expected to change.  We have direct variation if the two variables

change in the same sense, i.e. if one increases, so does the other.  

We have indirect variation if one going up causes the other to go

down.  An example of this might be speed and time to do a particular

journey, so the higher the speed the shorter the time.

Normally we let x be the independent variable and y the dependent

variable, so that a change in x produces a change in y.  For example,

if x is number of motor cars on the road and y the number of

accidents, then we expect an increase in x to cause an increase in y.  

(This obviously ceases to apply if number of cars is so large that

they are all stationary in a traffic jam.)

When x and y are directly proportional then doubling x will double the

value of y, and if we divide these variables we get a constant result.

Since if y/x = k  then (2y/2x) = k  where k is called the constant of

proportionality

We could also write this  y = kx

Thus if I am given the value of x, I multiply this number by k to find

the value of y.

Example: Given that y and x are directly proportional and y = 2 when

x = 5, find the value of y when x = 15.

We first find value of k, using y/x = k

                              2/5 = k

Now use this constant value in the equation y = kx for situation when

x = 15

    y = (2/5)*15

      = 30/5 = 6

If you want to do this quickly in your head, you could say, x has been

multiplied by a factor 3 (going from 5 to 15), so y must also go up by

a factor of 3.  That means y goes from 2 to 6.

Indirect Variation.

We gave an example of inverse proportion above, namely speed and time

for a particular journey.  In this case if you double the speed you

halve the time. So the product speed x time = constant

In general, if x and y are inversely proportional then the product xy

will be constant.

             xy = k

 or          y = k/x

Example:  If it takes 4 hours at an average speed of 90 km/hr to do a

certain journey, how long would it take at 120 km/hr

  k = speed*time = 90*4 = 360 (k in this case is the distance)

Then time = k/speed

        = 360/120

        = 3 hours.

To do this in your head, you could say that speed has changed by a

factor 4/3, so time must change by a factor 3/4.  However, for the

usual type of problem, go through the steps I outlined above.

Step-by-step explanation:

I hope these examples have made the idea of variation (both direct and

inverse) reasonably clear.

definitely a. the others have almost all data points on one side of the line.

step-by-step explanation:

x is the total time in minutes for the free response

you are given number of questions as 15 - m and time per item as 8 minutes

to find x multiply the two givens together:

x = 8 * (15-m) = 8(15-m)