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 observe an example that illustrates this concept.

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:

Do you know what the answer is?

Question 1

There are two circles.

One circle has a radius of 4 cm, while the other circle has a radius of 10 cm.

How many times greater is the area of the second circle than the area of the first circle?

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.

Examples and exercises with solutions on direct proportion

Exercise #1

Given the rectangle ABCD

AB=X

The ratio between AB and BC is $\sqrt{\frac{x}{2}}$

We mark the length of the diagonal A the rectangle in m