Boyle's law and charle's law  (Page 2/6)

During this chapter, the terms directly proportional and inversely proportional will be used a lot, and it is important that you understand their meaning. Two quantities are said to be proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or if they have a constant ratio. We will look at two examples to show the difference between directly proportional and inversely proportional .

1. Directly proportional A car travels at a constant speed of 120 km/h. The time and the distance covered are shown in the table below.

   Time (mins)	Distance (km)
   10	20
   20	40
   30	60
   40	80

   What you will notice is that the two quantities shown are constant multiples of each other. If you divide each distance value by the time the car has been driving, you will always get 2. This shows that the values are proportional to each other. They are directly proportional because both values are increasing. In other words, as the driving time increases, so does the distance covered. The same is true if the values decrease. The shorter the driving time, the smaller the distance covered. This relationship can be described mathematically as:
   $$y=kx$$
   where y is distance, x is time and k is the proportionality constant , which in this case is 2. Note that this is the equation for a straight line graph! The symbol $\propto$ is also used to show a directly proportional relationship.
2. Inversely proportional Two variables are inversely proportional if one of the variables is directly proportional to the multiplicative inverse of the other. In other words,
   $$y\propto \frac{1}{x}$$
   or
   $$y=\frac{k}{x}$$
   This means that as one value gets bigger, the other value will get smaller. For example, the time taken for a journey is inversely proportional to the speed of travel. Look at the table below to check this for yourself. For this example, assume that the distance of the journey is 100 km.

   Speed (km/h)	Time (mins)
   100	60
   80	75
   60	100
   40	150

   According to our definition, the two variables are inversely proportional if one variable is directly proportional to the inverse of the other. In other words, if we divide one of the variables by the inverse of the other, we should always get the same number. For example,
   $$\frac{100}{1/60}=6000$$
   If you repeat this using the other values, you will find that the answer is always 6000. The variables are inversely proportional to each other.