Direct Proportion

What is direct proportion?

Direct proportionality indicates a situation in which, when one term is multiplied by a certain amount, the same exact thing happens to the second term.

In the same way, when one term is divided by a certain amount, the same exact thing happens to the second term.

The ratio between both magnitudes remains constant.

Let's look at an example from everyday life

Imagine traveling in some vehicle while the roads are quite empty, without any traffic jams.

As you travel more time, you will pass more and more kilometers.

It can be said that, as time goes by, the distance also increases.

Let's look at a graphical representation of direct proportionality

$Y=aX$

Represents direct proportionality.

As $X$ increases, so does $Y$ .

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Test your knowledge

Question 1

Find the part of the whole:

In a box there are 28 candies, $\frac{1}{4}$ of them are orange.

How many orange candies are there in the box?

Question 2

Given two circles.

The length of the diameter of circle 1 is 4 cm.

The length of the diameter of circle 2 is 10 cm.

How many times larger is the area of the circle in drawing 2 than the area of the circle in drawing 1?

Question 3

Given two circles

The length of the radius of circle 1 is 4cm.

The length of the radius of circle 2 is 10cm.

How many times greater is the area of the circle in drawing 2 than the area of the circle in drawing 1?

How can we check if there is direct proportionality?

To see if there is direct proportionality, we must find out if both terms increase or decrease by the same amount of times.

Let's look at an example

Given the following table:

We will see if every time $X$ increases by a specific amount, $Y$ also increases in the same proportion.

If this occurs, it means there is direct proportionality. If not, then there isn't.

Let's ask:

By how much does $X$ increase from $2$ to $4$ ?

The answer is it multiplies by $2$ .

And by how much does $Y$ increase from $5$ to $10$ ?

The answer is it multiplies by $2$ .

Let's continue,

By how much does $X$ increase from $2$ to $6$ ? The answer is it multiplies by $3$ .

And by how much does $Y$ increase from $5$ to $15$ ?

The answer is it multiplies by $3$ .

We will continue examining and discover that indeed every time $X$ is multiplied by a certain number, $Y$ also increases, multiplied by the same number.

We will see it in the following way:

Let's look at a verbal example

Diana's credit card company charges a monthly fee of $2$ $, plus $1$ $ for each bank transaction.

Is the ratio of the amount Diana has to pay to the number of transactions she made during the month directly proportional?

Solution:

To answer this kind of question, it is convenient to draw a table:

$X$ represents the number of transactions Diana made

$Y$ represents the amount Diana has to pay

Notice, the question says that the credit card company applies a cost of $2$ $ each month, that is, even if Diana does not make any transactions, she will have to pay $2$ $.

Let's draw a table:

Now let's see:

Does the $X$ multiply by a certain number and also the $Y$ increase multiplied by the same number?

The answer is no.

We can see that when the $X$ doubles and goes from $1$ to $2$

the $Y$ does not double! From $3$ to $4$ what it does is $\frac{4}{3}$ .

Therefore, we can determine that the ratio of the amount Diana has to pay to the number of transactions she made during the month is not directly proportional.