Direct Proportion

🏆Practice ratio

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.

Direct Proportion in a table

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Test yourself on ratio!

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

How many white balls are there in the box?

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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 Y=aX

Represents direct proportionality.

As X X increases, so does Y Y .

represents direct proportionality. As X increases, so does Y.


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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:

table 5,10,15,20 tutorela

We will see if every time X X increases by a specific amount, Y 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 X increase from 2 2 to 4 4 ?

The answer is it multiplies by 2 2 .

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

The answer is it multiplies by 2 2 .

Let's continue,

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

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

The answer is it multiplies by 3 3 .

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

We will see it in the following way:

grows multiplied by the same number


Do you know what the answer is?

Let's look at a verbal example

Diana's credit card company charges a monthly fee of 22$, plus 11$ 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 X represents the number of transactions Diana made

Y Y represents the amount Diana has to pay

Notice, the question says that the credit card company applies a cost of 2 2 $ each month, that is, even if Diana does not make any transactions, she will have to pay 22$.

Let's draw a table:

table X and Y

Now let's see:

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

The answer is no.

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

the Y Y does not double! From 3 3 to 4 4 what it does is 43 \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

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?

Video Solution

Step-by-Step Solution

The area of a circle is calculated using the following formula:

where r represents the radius.

Using the formula, we calculate the areas of the circles:

Circle 1:

π*4² =

π16

Circle 2:

π*10² =

π100

To calculate how much larger one circle is than the other (in other words - what is the ratio between them)

All we need to do is divide one area by the other.

100/16 =

6.25

Therefore the answer is 6 and a quarter!

Answer

614 6\frac{1}{4}

Exercise #2

The rectangle ABCD is shown below.

AB = X

The ratio between AB and BC is x2 \sqrt{\frac{x}{2}} .


The length of diagonal AC is labelled m.

XXXmmmAAABBBCCCDDD

Choose the correct answer.

Video Solution

Step-by-Step Solution

We know that:

ABBC=x2 \frac{AB}{BC}=\sqrt{\frac{x}{2}}

We also know that AB equals X.

First, we will substitute the given data into the formula accordingly:

xBC=x2 \frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}

x2=BCx x\sqrt{2}=BC\sqrt{x}

x2x=BC \frac{x\sqrt{2}}{\sqrt{x}}=BC

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

x×2=BC \sqrt{x}\times\sqrt{2}=BC

Now let's look at triangle ABC and use the Pythagorean theorem:

AB2+BC2=AC2 AB^2+BC^2=AC^2

We substitute in our known values:

x2+(x×2)2=m2 x^2+(\sqrt{x}\times\sqrt{2})^2=m^2

x2+x×2=m2 x^2+x\times2=m^2

Finally, we will add 1 to both sides:

x2+2x+1=m2+1 x^2+2x+1=m^2+1

(x+1)2=m2+1 (x+1)^2=m^2+1

Answer

m2+1=(x+1)2 m^2+1=(x+1)^2

Exercise #3

Given the rectangle ABCD

AB=X the ratio between AB and BC is equal tox2 \sqrt{\frac{x}{2}}

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

Check the correct argument:

XXXmmmAAABBBCCCDDD

Video Solution

Step-by-Step Solution

Let's find side BC

Based on what we're given:

ABBC=xBC=x2 \frac{AB}{BC}=\frac{x}{BC}=\sqrt{\frac{x}{2}}

xBC=x2 \frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}

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

Let's divide by square root x:

2×xx=BC \frac{\sqrt{2}\times x}{\sqrt{x}}=BC

2×x×xx=BC \frac{\sqrt{2}\times\sqrt{x}\times\sqrt{x}}{\sqrt{x}}=BC

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

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

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

AB2+BC2=AC2 AB^2+BC^2=AC^2

Let's substitute what we're given:

x2+(2x)2=m2 x^2+(\sqrt{2}\sqrt{x})^2=m^2

x2+2x=m2 x^2+2x=m^2

Answer

x2+2x=m2 x^2+2x=m^2

Exercise #4

There are 18 balls in a box, 23 \frac{2}{3} of which are white.

How many white balls are there in the box?

Video Solution

Answer

12

Exercise #5

In a box there are 28 balls, 14 \frac{1}{4} of which are orange.

How many orange balls are there in the box?

Video Solution

Answer

7

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