Comparing Quantities

Introduction

Physical quantities gives a lot of information about an object. They help us understand the nature and properties of it.
Like : temperature — tell us how hot the object it, length- how long the object is, etc.

Many a times, comparing these physical quantities also give us great insights about the object.
Like : if temperature is increasing as the time increases, we can know that how hot the object will be after a certain time.

Thus, comparing quantities is an essential and an important thing to learn.

Ratio and Proportions

Ratios and Proportions are mathematical tools that can be used to compare quantities. Lets learn about them.

Ratio

Sometimes, comparison of two quantities by division is very efficient, and such comparisons can be done by ratios.
Ratio can be defined as the relation between two quantities such that one is represented as number of times the other.

For Example :
100km in 1 hr is the speed of a train.
This can be understood as, for every 100km travelled, an hour of time has taken.
This comparison can be represented by ratio as — ‘100km : 1hr’(read as 100km ‘is to’ 1hr) or ‘100km/1hr’ (read as 100km per 1hr).

now, we can deduce some insights from the ratio 100km:1hr.
first, we see that as the distance or the kilometers travelled increases-the time also increases.
second, we see that time also increases with the distance.
third, the increase is always linear(?).

Simple Ratios
When the ratios are such that they can’t be divided further on,
like 500:50 can be simplified and be written as 10:1 and thus, 10:1 is the simple ratio.

Proportions

When two ratios has the same simple ratio they are said to be in proportion or simply, when two ratios are equivalent, the are proportional to each other.

consider two ratios, 100:10 and 500:50 — we can clearly see that the second ratio is just 5 times the first ratio and or both of their simple ratios forms are 10:1, thus both the ratios are proportional to each other. 100:10 :: 500:50 (‘::’ is the symbol to represent proportionality).

also we see that:
100/10 = 500/50 = 10, and we can see that any ratio proportion to them will have the same value i.e. 10. This constant number is called the proportionality constant, generally represented by k.

if, a:b :: c:d
then, a/b=c/d=k ; where k is the proportionality constant.

Direct Proportions
When two quantities or ratios increase or decrease with one and other, they are said to be directly proportional.
if, a and b are two ratios and a is directly proportional to b
then, a ∝ b & a/b = k or a=kb, where k is the proportionality constant.

Inverse Proportions
When one ratio or quantity increase and the other decreases, they are said to be inversely proportional.
if, a and b are two ratios and a is inversely proportional to b
then, a ∝ 1/b & ab = k or a = k/b, where k is the proportionality constant.

Conclusion

Comparing quantities is really an important thing and you will encounter it very often in your journey of learning physics. Understanding the simplicity and power of this concept will help you go smooth while you learn physics.

Next Steps

Let’s now learn about the basics of coordinate geometry here…