# Equivalent Ratios

🏆Practice ratio

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,

## Test yourself on ratio!

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

How many white balls are there in the box?

If you have two fractions in front of you:
$\frac{2000}{4000}$

and

$\frac{2}{4}$

we can simplify the larger fraction and arrive at

and even more, we can simplify the smaller fraction and arrive at:

In fact, we can say that:

$\frac{4000}{2000}=\frac{2}{4}=\frac{1}{2}$

All the expressions are equivalent ratios.

Do you remember we said that a ratio can be shown in the form of a fraction?

Therefore, the same rule also applies to the ratios we have learned.

We can reduce both terms of the ratio or amplify them and arrive at equivalent ratios.

To solve this type of problem easily we will always try to arrive at the smallest ratio.

We will ask ourselves by what number we can divide both terms of the ratio, in this way we will arrive at the most reduced equivalent ratio possible.

## How can you tell if they are equivalent ratios?

We will ask ourselves: Will we arrive at the same ratio by reduction or by amplification?

Let's see some examples:

$2:5$

$6:16$

We have managed to demonstrate that by multiplying both terms by $3$ we arrive at the same ratio. Therefore, they are equivalent ratios!

Are these ratios equivalent?

$1:3$

$2:6$

$6:18$

Yes! The first ratio is equivalent to the second: we multiply both terms by $2$.

The first ratio is also equivalent to the third, we multiply by $6$.

The second ratio is equivalent to the third: multiplication of both terms by $3$.

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### Example

José and Dani have notebooks and pencils. José has $4$ notebooks and $8$ pencils.

The ratio between the notebooks and pencils that Dani has is the same as José's. Dani has $6$ notebooks. We are asked to calculate how many pencils Dani has.

We see that the number of pencils José has is double the number of notebooks he has. Since we already know that the ratio between notebooks and pencils that José and Dani have is identical, we can deduce that Dani has $12$ pencils ($6$ times $2$, so that the number of pencils is double the number of notebooks).

## Examples and exercises with solutions of Equivalent Ratios

### Exercise #1

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

How many white balls are there in the box?

12

### Exercise #2

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

How many orange balls are there in the box?

7

### Exercise #3

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

$6\frac{1}{4}$

### Exercise #4

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?

They are equal.

### Exercise #5

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

$6\frac{1}{4}$