[Please watch the video attached at the end of this blog for a visual explanation of Mastering Direct and Indirect Variations]
Direct Variation/ Proportion
Direct variation is when one value changes in direct proportion to another. Quite simply put, when one value increases, the other value increases, or one value decreases, the other value decreases.
In questions, the easiest way to identify direct variation is when the question mentions “x varies as y varies”, “x varies directly as y”, or “x is proportional to y”. All of these mean one thing: the question deals with direct variation.
Variation is denoted by the symbol ∝, which is the ‘proportional sign’. Whenever ‘x’ and ‘y’ or any other two values are on different sides, this means that they are directly proportional to one another:
This notation can be replaced and made into an equation in the form x = ky, where k is a constant.
Example: If z ∝ m. z = 20, m = 4. Find out the value of z when m = 7.
This question is extremely easy to solve.
1) We can see that z is directly proportional to m. We can therefore replace this notation with the equation for direct variation which in this case will be z = k m.
2) By substituting the values of z and m given, you will see that the value of k is 5.
3) Now that the value of k has been found, substitute it in the second situation where the value of m is given but the value of z is not.
- z = k ✕ m
- z = 5 ✕ 7
- z = 35.
Hence when m = 7, the value of z will be 35.
This is the basic structure for all variation questions, and if you practise it well enough, it is one of the easiest areas for you to score marks.
Inverse Variation/ Proportion
Inverse variation is the opposite of direct variation. Inverse variation is when one value changes inversely to another. In inverse variation, when one value increases, the other value will decrease, and when one value decreases, the other value will increase.
The way inverse variation questions are worded is also different to those of direct variation. They will have the words “varies inversely” or “inversely proportional”. The way the proportion is denoted is also different. In this, x is not proportional to y, rather, x is proportional to 1⁄y, and in the same way, when denoting it as an equation, it will be x = k/y, where k is a constant.
Example: b ∝ 1⁄e. If b = 6 and e = 2, what is the value of b when e = 12.
1)We can see that b is inversely proportional to e. We can therefore replace this notation with the equation for inverse variation which in this case will be b = k/e.
2) By substituting the values of b and e given, you will see that the value of k is 12.
3) Now that the value of the constant k has been found, substitute it in the second situation where the value of e is given but the value of b is not.
- b = k / e
- b = 12 / 12
- b = 1
Hence when e = 12, the value of b will be 1.
Direct Variation and Inverse Variation is another one of those lessons you will be able to catch up on in no time. With this being a frequently tested area, this is also an area students score marks well in. Pay close attention to whether the question focuses on direct proportion or inverse proportion.
If you want to double check, once you get the answers, compare them and see. If one value has increased while the other value has decreased, it is inverse proportion, but if both values have increased, then it is direct proportion.
If you are struggling with IGCSE revision or the Mathematics subject in particular, you can reach out to us at Tutopiya to join revision sessions or find yourself the right tutor for you.
Watch the video below for a visual explanation of the lesson on mastering direct and indirect variations. Don’t forget that there is a timed quiz to test whether you have understood the lesson!