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 observe an example that illustrates this concept.

Detailed explanation

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Test yourself on ratio!

How many times longer is the radius of the red circle than the radius of the blue circle?

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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

How many times longer is the radius of the red circle, which has a diameter of 24, than the radius of the blue circle, which has a diameter of 12?

Question 2

How many times longer is the radius of the red circle than the radius of the blue circle?

Question 3

There are 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 circle 2 than the area of circle 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:

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?

Question 2

There are two circles.

The length of the radius of circle 1 is 6 cm.

The length of the diameter of circle 2 is 12 cm.

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

Question 3

How many times longer is the radius of the red circle (14 cm) than the radius of the blue circle, which has a diameter of 7?

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.

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

Check the correct argument:

Video Solution

Step-by-Step Solution

Given that:

$\frac{AB}{BC}=\sqrt{\frac{x}{2}}$

Given that AB equals X

We will substitute accordingly in the formula:

$\frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}$

$x\sqrt{2}=BC\sqrt{x}$

$\frac{x\sqrt{2}}{\sqrt{x}}=BC$

$\frac{\sqrt{x}\times\sqrt{x}\times\sqrt{2}}{\sqrt{x}}=BC$

$\sqrt{x}\times\sqrt{2}=BC$

Now let's focus on triangle ABC and use the Pythagorean theorem:

$AB^2+BC^2=AC^2$

Let's substitute the known values:

$x^2+(\sqrt{x}\times\sqrt{2})^2=m^2$

$x^2+x\times2=m^2$

We'll add 1 to both sides:

$x^2+2x+1=m^2+1$

$(x+1)^2=m^2+1$

Answer

$m^2+1=(x+1)^2$

Exercise #2

Given the rectangle ABCD

AB=X the ratio between AB and BC is equal to $\sqrt{\frac{x}{2}}$

We mark the length of the diagonal $A$ with $m$

Check the correct argument:

Video Solution

Step-by-Step Solution

Let's find side BC

Based on what we're given:

$\frac{AB}{BC}=\frac{x}{BC}=\sqrt{\frac{x}{2}}$

$\frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}$

$\sqrt{2}x=\sqrt{x}BC$

Let's divide by square root x:

$\frac{\sqrt{2}\times x}{\sqrt{x}}=BC$

$\frac{\sqrt{2}\times\sqrt{x}\times\sqrt{x}}{\sqrt{x}}=BC$

Let's reduce the numerator and denominator by square root x:

$\sqrt{2}\sqrt{x}=BC$

We'll use the Pythagorean theorem to calculate the area of triangle ABC:

$AB^2+BC^2=AC^2$

Let's substitute what we're given:

$x^2+(\sqrt{2}\sqrt{x})^2=m^2$

$x^2+2x=m^2$

Answer

$x^2+2x=m^2$

Exercise #3

How many times longer is the radius of the red circle than the radius of the blue circle?

Video Solution

Answer

Exercise #4

How many times longer is the radius of the red circle, which has a diameter of 24, than the radius of the blue circle, which has a diameter of 12?

Video Solution

Answer

Exercise #5

How many times longer is the radius of the red circle than the radius of the blue circle?

Video Solution

Answer

$2$

Check your understanding

Question 1

How many times longer is the radius of the red circle than the radius of the blue circle?

Question 2

There are 18 balls in a box, $\frac{2}{3}$ of which are white.

How many white balls are there in the box?

Question 3

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

How many orange balls are there in the box?

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