Problem 40 - Algebra and Trigonometry

Question

What does it mean if two quantities vary directly?

Step-by-Step Solution

Answer

If two quantities vary directly, it means they change at a constant rate relative to each other. For example, if a car moves at a constant speed, the distance it covers varies directly with the time it travels, since they change at the same ratio.

Step 1: Direct Variation Definition

Direct variation describes a simple relationship between two variables. If two quantities vary directly, it means that when one quantity increases, the other quantity will increase as well, and if one quantity decreases, the other one will decrease too. We can write it mathematically as \(y = kx\), where \(y\) and \(x\) are the two quantities, and \(k\) is the constant of variation.

Step 2: Understanding the Constant of Variation

The constant of variation \(k\) is the factor by which \(x\) is multiplied to get \(y\). When \(x\) changes, \(y\) changes in direct proportion to \(x\), which means they will increase or decrease together. The constant \(k\) is the ratio of \(y\) and \(x\) (i.e., \(k = \frac{y}{x}\)), and this ratio is the same for all values of \(y\) and \(x\).

Step 3: Example of Direct Variation

For example, if a car travels at a constant speed, the distance it covers (d) and the time (t) it travels are in direct variation. If the car travels for 2 hours (t=2), and covers a distance of 120 kilometres (d=120), then the constant speed (k) would be \(\frac{120}{2} = 60\) kilometres per hour. This means that for every hour (1 unit of time), the car covers 60 kilometres (60 units of distance). Hence, the distance varies directly with time, with a constant ratio of 60.