Equivalent Ratios - Examples, Exercises and Solutions

To easily solve ratio problems and to gain a better understanding of the topic in general, it is convenient to know about equivalent ratios.

Equivalent ratios are, in fact, ratios that seem different, are not expressed in the same way but, by simplifying or expanding them, you arrive at exactly the same relationship.

Think of it this way,

Detailed explanation

Practice Equivalent Ratios

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

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.

Determine the value of m:

Question 3

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:

Question 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?

Question 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?

Examples with solutions for Equivalent Ratios

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

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.

Determine the value of m:

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 #3

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 #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

Question 1

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?

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 than the radius of the blue circle?

Question 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?

Question 5

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

Exercise #6

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?

Video Solution

Answer

$6\frac{1}{4}$

Exercise #7

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?

Video Solution

Answer

They are equal.

Exercise #8

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

Video Solution

Answer

Exercise #9

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 #10

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

Video Solution

Answer

$2$

Question 1

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?

Question 2

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

Question 3

ABCD is a deltoid with an area of 58 cm².

DB = 4

AE = 3

What is the ratio between the circles that have diameters formed by AE and and EC?

Exercise #11

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?

Video Solution

Answer

Exercise #12

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

Video Solution

Answer

$2\frac{1}{2}$

Exercise #13

ABCD is a deltoid with an area of 58 cm².

DB = 4

AE = 3

What is the ratio between the circles that have diameters formed by AE and and EC?

Video Solution

Answer

3:26

Equivalent Ratios - Examples, Exercises and Solutions

Practice Equivalent Ratios

Examples with solutions for Equivalent Ratios

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Answer

Exercise #5

Video Solution

Answer

Exercise #6

Video Solution

Answer

Exercise #7

Video Solution

Answer

Exercise #8

Video Solution

Answer

Exercise #9

Video Solution

Answer

Exercise #10

Video Solution

Answer

Exercise #11

Video Solution

Answer

Exercise #12

Video Solution

Answer

Exercise #13

Video Solution

Answer

Topics learned in later sections