Equivalent Ratios

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.

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 $3$.

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).