Proportional reasoning involves the comparison of ratios between objects or within objects along with the understanding of the fact that scaling of one object with the maintenance of the ratios needs involvement of four variables.
$\frac{a}{b}$ = $\frac{c}{d}$
Whenever there is an increase or decrease in one variable supposing ‘c’, then it has to be compensated by a decrease or increase in ‘a’ or ‘d’. Same is the case with other variables. The changes are in correspondence to maintain the relationship and which quantity to change, how much to change, what to multiply or divide to achieve and maintain the change. It all requires a good knowledge of arithmetic along with the scaling tact and multiplying/dividing methods.

In general, we say that proportional reasoning is a type of mathematical reasoning which involves a sense of multiple comparisons, co-variation and the ability to store and process pieces of information mentally. This is also concerned with prediction and inference and also involves both quantitative and qualitative thought methods.

Events like exchanging money, making scaled diagrams, speed, decay rates, growth rates calculation, changing units of measurement etc. are all common real life examples of the use of methods of proportional reasoning.

We can compare quantities or objects in multiplicative way rather than in an additive way. The comparison is explained by making use of terms like double, half, two thirds greater, by three times, smaller than etc.

We can also say that, proportions are same as talking about equivalent fractions, that is, the fractions that are same once they are reduced to their simplest forms or the fractions that comes out to be same by either means of multiplying or dividing.

The various proportions we deal with are:

Continuous proportion is the one similar to geometric mean. We have three quantities here such that $\frac{a}{b}$ = $\frac{b}{c}$. Here ‘b’ is the geometric mean of the given continuous proportion.

Direct proportion is the one in which two quantities or objects are related to each other directly. For example distance is directly related to speed as well as time;work is directly related to wages as in more work will get you more wages etc.

Inverse proportion is the one in which two quantities or objects are related to each other inversely. For example, the increase in number of persons employed for doing a work will decrease the time taken to complete the work; speed and time are also inversely proportional as when speed is more time taken is less and vice versa.
For making reasoning in a proportional way proper knowledge of the given data has to be undergone as of its units, type, scaling ways, magnitudes, multiplicative scale etc. The right understanding of the given data is also important as without proper understanding the ratio and thus the proportion will vary and may not even be accurate. One has to keep in mind about the relations being direct and inverse between the given objects or quantities. Like, speed and time are inversely related to each other as in more speed will imply less time taken and vice versa. Similarly, time and work are directly related as more work will imply more time. But there are cases where work and time gets inversely related as well but that all depends on the conditions given. Like if more men are employed for the same work less time will be taken. At times, when work is increased with time, time still increases but the increase rate of time might be less as well.
The ratio and proportion are the concepts that complement each other. The ratio said to the way of comparing two numbers. In ratio, two numbers

that are to be compared are separate by a colon sign (:). The ratio of 8 and 12 can be written as 8:12 or $\frac{8}{12}$ or $\frac{2}{3}$.

On the other hand, the proportion is defined as an equation expressing two ratios which are equal. When two ratios are equivalent, they can be expressed as proportion.

For example : $\frac{3}{4}$ = $\frac{6}{8}$ is a proportion.
Let us see certain examples for a better understanding of proportional reasoning concept.
Example 1:

If 200 pens cost Rs. 1000, then how much 10 pens cost?


Then we can form the proportion as

$\frac{200}{1000}$ = $\frac{10}{x}$

$\rightarrow$  X = $\frac{10000}{200}$ = 50

Thus, 10 pens will cost Rs.50

Example 2:

Find the mean of 4 and 9.


Let ‘x’ be the mean of 4 and 9

Then we have,

$\frac{4}{x}$ = $\frac{x}{9}$

$\rightarrow$ x$^{2}$ = 36

$\rightarrow$ x = 6

Hence, the mean of 4 and 9 is 6


$\frac{4}{6}$ = $\frac{2}{3}$


Thus both turn out to be in same fractions. Hence our solution is correct.