Know the important mathematical properties of direct proportion!

Know the important mathematical properties of direct proportion!

Didn't we see yesterday that all equivalent fractions are direct proportional!

Let us tabulate equivalent fractions of ¼ as numerator and denominator as follows?

Numerator	1	2	3	4	5	6	7	8	9	10
Denominator	4	8	12	16	20	24	28	32	36	40

 

Looking at the above table, what do you think?

Look from left to right. That is, look from here to there. Numerators go incrementally. Not only numerators? Denominators are also increasing.

Now look from right to left. I mean look from there to here. Numerators go lower. Not only numerators? Denominators also go low in values.

Now consider one more thing. As the numerator increases, the denominator also increases, doesn't it? Or if the numerator decreases when viewed from right to left, the denominator also decreases accordingly!

It is this mathematical property that you should observe.

As the numerator increases the denominator also increases. As the numerator decreases, the denominator also decreases. This is the mathematical property of direct proportionality. If it increases, both the elements numerator and denominator will increase. If it decreases, both elements, the numerator and denominator, will decrease.

And if the opposite happens, should I say what it is? That is the inverse proportion. That means numerator decreases as denominator increases or numerator decreases as denominator increases.

I think you understand better now.

Let's focus on the multiplication tables anyway shall we? If so, what we saw above is the fourth table? Are the numbers that are multiplied by the order of four in the numerator and the product that is multiplication value in denominator?

Is that all? In terms of geometry and measurements, all that is numerator is the sides of a square. Aren't all denominators the perimeters of a square with corresponding sides?

See where we start and where we end up. See how each of the math topics relate to each other.

Now you should know an important mathematical property of equivalent fractions.

Take any two equal fractions from the above table and equate them.

For example

2 / 8 = 3 / 12

Let's simplify both of these on both sides.

Does dividing the fraction on the left up and down by 2 and come to ¼?

Similarly, if the fraction on the right is rounded up and down by three i.e. divided, does it come to ¼?

Not only these two equivalent fractions are equal, but if we take the other equivalent fractions in the table and simplify them, we get ¼.

Thus, no matter how you simplify equivalent fractions, they will return to their simplest form. Do I need to tell you how we can represent this with variables? Does x / y = a  constant?

This is an important mathematical property of equivalent fractions. An important mathematical property not only for equivalent fractions but also for direct proportion.

Also cross multiply the equal fractions 2 / 8 = 3 / 12 that we equated above. That is, multiply the numerator of ​​the fraction on the right by the denominator of the fraction on the left. Similarly, multiply the numerator of the fraction on the right by the denominator of ​​the fraction on the left. Equate the two multiplied products.

Equally means,

2 × 12 = 3 × 8

Does 24 = 24 come up?

Similarly, if we take any two equal fractions mentioned in the table and do cross multiplication, we get the same answer on both sides of the equation.

This is also an important mathematical property of equivalent fractions. It is also an important mathematical property for direct proportion. It is by using this property that we are going to easily find the answer to proportionality calculations. So remember this mathematical property well.

I told you that today I will tell you about the application of proportionality in life. But look, I've said a lot, so I'll say it tomorrow!

Wait a day. Let us know the life application of direct proportion tomorrow.

Before that, let's summarize and remember the important concepts we have learned in direct proportion.

1. If one element increases in direct proportion, the other element also increases. Or if one element decreases, the other element also decreases.

2. Taking and reducing any ratio of a direct proportion gives a constant.

3. If two ratios of a direct proportion are cross-multiplied, the products obtained are equal.

What we compiled is correct? Remember this well. We will see about the rest tomorrow.

*****