Rate, Proportion and Ratios

Rate, Proportion and Ratios

In science we often want to know how a quantity relates to another quantity or how something changes over a period of time. To do this we need to know about rate, proportion and ratios.

Rate:

The rate at which something happens is the number of times that it happens over a period of time. The rate is always a change per time unit. So we can get rate of change of velocity per unit time \(\left(\frac{\Delta \vec{v}}{t}\right)\) or the rate of change in concentration per unit time (or \(\frac{\Delta C}{t}\)). (Note that \(\Delta\) represents a change in.)

Ratios and fractions:

A fraction is a number which represents a part of a whole and is written as \(\dfrac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator. A ratio tells us the relative size of one quantity (e.g. the number of moles of reactants) compared to another quantity (e.g. the number of moles of product). \(\text{2}:\text{1}\) ; \(\text{4}:\text{3}\), etc. Ratios can also be written as fractions as percentages (fractions with a denominator of \(\text{100}\)).

Proportion:

Proportion is a way of describing relationships between values or between constants. We can say that \(x\) is directly proportional to \(y\) (\(x\propto y\)) or that \(a\) is inversely proportional to \(b\) (\(a\propto \frac{1}{b}\)). It is important to understand the difference between directly and inversely proportional.

Directly proportional

Two values or constants are directly proportional when a change in one leads to the same change in the other. This is a more-more relationship. We can represent this as \(y\propto x\) or \(y=kx\) where \(k\) is the proportionality constant. We have to include \(k\) since we do not know by how much \(x\) changes when \(y\) changes. \(x\) could change by \(\text{2}\) for every change of \(\text{1}\) in \(y\). If we plot two directly proportional variables on a graph, then we get a straight line graph that goes through the origin \(\left(0;0\right)\):

Inversely proportional

Two values or constants are inversely proportional when a change in one leads to the opposite change in the other. We can represent this as \(y=\frac{k}{x}\). This is a more-less relationship. If we plot two inversely proportional variables we get a curve that never cuts the axis: