Ratio | Tutorela

What is ratio?

The ratio describes the "relationship" between two or more things.

The ratio connects the given terms and describes how many times greater or smaller a certain magnitude is than another.

Let's see an example from everyday life:

When asked in a class, what is the ratio between boys and girls, it refers to how many girls there are in relation to a certain number of boys.

Or, for example, if in a certain vase there are red and white balls, the ratio between them can describe how many red balls there are in relation to a certain number of white balls or vice versa.

Detailed explanation

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Test yourself on ratio!

If there are 18 balls in a box of which $\frac{2}{3}$ are white:

How many white balls are there in the box in total?

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How is the ratio read?

Just as we read in English, mathematics is also read from left to right. So,

we combine the written words according to their order of appearance and convert them into numbers from left to right.

Let's see an example:

The ratio of purple balls to green balls is: $3:2$

Since the first written word is "purple," it represents the first number on the left

We can really see that, for every $3$ purple balls there are $2$ green balls.

Important

Ratios can also be expressed through fractions: $\frac{3}{2}$

and, in such case, we read it from top to bottom.

Another example:

The ratio of pens to markers in Ariel's school case is $2:1$ .

Which number refers to pens and which number to markers?

Also, which of the two do we have more of in Ariel's school case?

Solution:

Let's observe the phrase, the ratio of pens to markers in Ariel's school case is

The word that appears first is pens.

Therefore, when reading the ratio, we will relate the first number to the term pens.

That is, the $2$ refers to the pens and the $1$ to the markers.

The ratio expresses that, for every $2$ pens found in the case there is one marker.

So, in general, in Ariel's school case there are more pens than markers. (double)

In Ariel's school case there can be:

$4$ pens, $2$ markers

$8$ pens, $4$ markers

and so on.

The ratio always remains as long as the relationship between pens and markers is $2:1$ .

Ratio of a certain part in relation to a whole

We can encounter the ratio of an object to the entire set.

For example, the ratio of apples to all other fruits in the fridge is $3:5$

This means that out of the $5$ fruits in the fridge, $3$ of them are apples.

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Test your knowledge

Question 1

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

How many orange balls are there in the box in total?

Question 2

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

Question 3

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

A simple example

We know that the ratio between apples and oranges in a basket is $2:3$ . The total amount of fruit in the basket is $25$ .
We are asked to calculate the number of apples and oranges in the basket.
We can deduce that the $2$ represents the number of apples and the $3$ represents the number of oranges.
We will denote both fruits with a variable $X$ .

Now let's draw a simple equation:

$2X+3X=25$

$5X=25$

$X=5$

From here we can infer that the number of apples is $10 (2X)$ and the number of oranges is $15 (3X)$ .
We can always go back and check our result by verifying that the total number of apples and oranges is $25$ , as shown in the first piece of data we received.

Example 2

In the dishware cabinet, there is a total of $30$ utensils that include plates and bowls. The ratio between plates and bowls is $7:3$ .

We are asked to determine how many plates and bowls are in the cabinet.

According to what we have learned, we can deduce that the $7$ represents the number of plates and the $3$ the number of bowls.

Let's denote both with a variable $X$ .

Now let's set up a simple equation:

$7X+3X=30$

$10X=30$

$X=3$

From here we can infer that the number of plates is $21 (7X)$ and the number of bowls is $9 (3X)$ .

We can always go back and check our result by verifying that the total number of utensils in the cabinet is $30$ , as seen in the first given data.

Examples and exercises with solutions of Ratio

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

$6\frac{1}{4}$

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

The rectangle ABCD is shown below.

AB = X

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

The length of diagonal AC is labelled m.

Choose the correct answer.

Video Solution

Step-by-Step Solution

We know that:

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

We also know that AB equals X.

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

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

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

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

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

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

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

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

We substitute in our known values:

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

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

Finally, we will 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 #4

If there are 18 balls in a box of which $\frac{2}{3}$ are white:

How many white balls are there in the box in total?

Video Solution

Answer

Exercise #5

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

How many orange balls are there in the box in total?

Video Solution

Answer

Do you know what the answer is?

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

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 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?

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