Dear Secondary Math students, we will be covering the topic on Direct and Inverse proportion today. This topic is usually tested together with Similar Figures or ** Similar Triangles** in the O Level E Math or N Level Math examinations. Without further ado, let's begin!

__In this note, you will learn:__

__1) What does it mean when the relationship between two variables is directly proportional?__

__2) What does it mean when the relationship between two variables is inversely proportional?__

__3) Graphs involving proportionality__

__1) What does it mean when the relationship between two variables is directly proportional?__

When two variables or quantities are said to be directly proportional, it basically means that the two variables/quantities increase or decrease to keep a ratio constant. The formula for direct proportionality is given by:

y ∝ x

**y = k x **or **y / x = k**, where k is a constant

Some real-life examples of directly proportional relationships are:

1. You work a part-time job that pays you an hourly wage, hence the number of hours you work is directly proportional to the hourly rate, or **number of hours worked ****∝**** hourly rate**.

For example, if the hourly rate is $10, you will receive $10 for working 1 hour, $20 for working 2 hours, $30 for working 3 hours, so on and so forth.

2. You went for a run and ran at a constant pace of 5km/hr. This means that the number of hours you ran is directly proportional to the distance completed, or **number of hours ran ****∝**** distance completed**. You would have completed 5km in the first hour, 10km by the second hour, 15km by the third hour, so on and so forth.

Now, let’s try to solve a question involving direct proportionality:

Jane is trying to bake a cake. She follows a recipe which tells her that: 5 cups of flour are used for every 2 cups of sugar used. However, Jane is trying to bake a cake that is larger than what the recipe suggests, and decided to use 8 cups of flour instead. Hence, how many cups of sugar must Jane use?

__Answer:__

Let y = number of cups of sugar used and x = number of cups of flour used

y = k x

For every 5 cups of flour used, 2 cups of sugar are used, hence:

2 = k (5)

Therefore, k = 0.4 and y = 0.4x

Jane wants to use 8 cups of flour to make a bigger cake, hence:

y = 0.4 (8)

= 3.2 cups

For 8 cups of flour, Jane must use 3 and a 1/5 cups of sugar.

__2) What does it mean when the relationship between two variables is inversely proportional?__

When two variables or quantities are said to be inversely proportional, it basically means that the product between two variables is equal to a constant. When one variable increases, the other must decrease to keep the constant valid. The formula for inverse proportionality is given by:

y ∝ k / x

**y = k / x **or **y x = k**, where k is a constant

Some real-life examples of inversely proportional relationships are:

1. Speed and travelling time: The faster you go, the less time it takes to complete the journey, which means that speed is inversely proportional to travelling time.

2. Under the topic of “Ideal Gas” in Chemistry, Boyle’s Law states that the volume of an ideal gas is inversely proportional to the pressure of the gas. ie. Volume ∝ k / Pressure

Now, let’s try to solve a question involving inverse proportionality:

It takes 5 men 8 hours to repair a road. How long will it take for 9 men to repair a road at the same rate?

__Answer:__

In this question, we understand that the number of men involved in the reparations of the road is inversely proportional to the number of hours taken to repair the road.

Therefore let’s let the time taken to repair the road be t, and since the question also states that the 8 men are working at the same rate as the 5 men, the rate remains a **constant**.

Let y = number of workers involved in the reparations of the road and x = number of hours taken to repair the road

y x = k

(number of workers) (time taken to compete reparations) = k

5 x 8 = k

k = 40

Therefore, 9 x t = 40

t = 40 / 8

= 5 hours

__Graphs involving proportionality__

Drawing graphs involving proportionality is actually really simple, because all you have to do is to base the graphs off the equations. Let’s see how it’s done:

For direct proportionality, the formula is given by: y = kx, which if we recall from the chapter under “Graph of Functions and Graphical Solutions”, it is just simply a straight-line graph!

As for inverse proportionality, the formula is given by: y = k/x, which resembles a reciprocal function graph!

It’s now time to put on your thinking caps and start solving some questions!

1. The voltage, V volts, across an electrical circuit is directly proportional to the current, I amps, flowing through the circuit. When I = 2.4, V = 90

a) Express V in terms of I

b) Find V when I = 5.5

c) Find I when V = 184.5

2. When a fixed volume of water is poured into a cylindrical jar, the depth, D cm, of the water is inversely proportional to the cross-sectional area, A cm², of the cylindrical jar.

When A = 35, D = 140

a) Find the formula for A in terms of D

b) Find A when D = 150

c) Find D when A = 60

3. The mass, M kg, of a solid cube made from lead is proportional to the cube of the length, L cm, of an edge. When L = 0.4, M = 120

a) Find the formula for M in terms of L

b) Find the value of M when L = 0.7

c) Find the value of L when M = 2450, give your answer corrected to 3 significant figures

4. The force of attraction, F newtons, between two spheres is inversely proportional to the square of the distance, d m, between the centers of the spheres.

When d = 1.35, F = 0.08,

a) Express F in terms of d

b) Find F when d = 4.5

c) Find d when F = 0.03, giving your answer corrected to 3 significant figures

And that’s all we have for today, students! ** Math Lobby** hopes that after reading and practicing questions with this article, you are able to have a clear understanding on the concepts of direct proportionality and inverse proportionality.

For any questions, feel free to contact us on Facebook or Instagram! And as always: Work hard, stay motivated and we wish all students a successful and enjoyable journey with Math Lobby!

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