In everyday life we observe that when one quantity changes, another quantity also changes in a related way. If we buy more pencils, we have to pay more money; if fewer workers are employed, it takes more days to finish a job; and if a car moves faster, it covers a given distance in less time. These relationships between changing quantities are of great practical importance, and mathematics helps us describe them precisely.
In this chapter we will study two important kinds of relationships between quantities: direct proportion and inverse proportion. When two quantities increase together or decrease together in the same ratio, they are said to be in direct proportion. When one quantity increases while the other decreases in such a way that their product remains constant, they are said to be in inverse proportion. We will learn how to identify these relationships and how to solve problems using them.
Two quantities x and y are said to be in direct proportion if they increase together or decrease together, and the ratio of the corresponding values remains constant. That is: $$\frac{x}{y} = k \quad \text{or} \quad y = kx$$ where k is a constant called the constant of proportionality.
In direct proportion, if x increases, y increases in the same ratio; if x decreases, y decreases in the same ratio.
If 2 kg of sugar costs 80 rupees, then 5 kg of sugar costs 200 rupees. The ratio of quantity to cost is always 1:40, so the quantities are directly proportional.
In direct proportion problems we can find the value of one unit first and then multiply. Alternatively, we set up the equation x1/y1 = x2/y2.
Two quantities x and y are said to be in inverse proportion if an increase in one causes a proportional decrease in the other, such that their product remains constant. That is: $$x \times y = k \quad \text{or} \quad y = \frac{k}{x}$$ where k is a constant.
In inverse proportion, when x increases, y decreases; when x decreases, y increases.
If 6 workers can finish a job in 10 days, then 12 workers can finish it in 5 days. The product of workers and days is always 60, so the quantities are inversely proportional.
To identify whether two quantities are in direct or inverse proportion, think about what happens when one quantity increases. If the other quantity also increases, it is direct; if the other quantity decreases, it is inverse.
| Feature | Direct proportion | Inverse proportion |
|---|---|---|
| When one quantity increases | Other increases | Other decreases |
| Relationship of ratio | x/y is constant | x x y is constant |
| Equation | y = kx | y = k/x |
| Graph | Straight line through origin | Curve (hyperbola) |
| Example | Cost and quantity of goods | Workers and time for a job |
| Situation | Type of proportion |
|---|---|
| More items, more cost | Direct |
| More speed, less time | Inverse |
| More workers, less days | Inverse |
| More petrol, more distance | Direct |
| More people, more food needed | Direct |
| Fixed amount, more shares, less each | Inverse |
Let us work through a typical direct proportion problem in detail. Suppose a train covers 240 kilometres in 3 hours at a constant speed. How far will it travel in 5 hours? Since the speed is constant, distance and time are directly proportional, so the ratio distance/time remains fixed. We can write x1/y1 = x2/y2, where x1 = 240 km, y1 = 3 hours and y2 = 5 hours. Substituting, we get 240/3 = x2/5. Cross-multiplying gives 3 x x2 = 240 x 5 = 1200, so x2 = 1200/3 = 400 kilometres. Alternatively, the unitary method tells us that the train covers 240/3 = 80 kilometres in one hour, so in 5 hours it covers 80 x 5 = 400 kilometres. Both methods give the same answer, and the fact that they agree is a useful way to check our work.
Now consider an inverse proportion problem. If 8 pumps can empty a tank in 6 hours, how long will 12 pumps take, assuming every pump works at the same rate? Here more pumps mean less time, so the product of the number of pumps and the time taken is constant. Using x1 x y1 = x2 x y2 with x1 = 8 pumps, y1 = 6 hours and x2 = 12 pumps, we get 8 x 6 = 12 x y2, that is, 48 = 12 x y2. Dividing both sides by 12 gives y2 = 4 hours. The answer makes sense because increasing the pumps from 8 to 12, a one-and-a-half times increase, should reduce the time from 6 hours to 4 hours, which is exactly two-thirds of the original time. This kind of common-sense check helps students detect careless mistakes, because an answer that says 12 pumps take longer than 8 pumps would clearly be wrong.
Direct and inverse proportions help us understand and predict how quantities change in relation to each other. In this chapter we learnt that two quantities are in direct proportion when their ratio remains constant, and in inverse proportion when their product remains constant. We studied how to solve problems using the equations x1/y1 = x2/y2 for direct proportion and x1 x y1 = x2 x y2 for inverse proportion, and how to distinguish between the two using simple reasoning. These concepts have applications in commerce, science, engineering and daily life, from calculating bills and discounts to planning work and travel. Mastering proportions gives students a strong foundation for ratios, percentages, variations and graphs in higher classes.