Proportionality

Just as we did in the chapter on equivalent ratios, to find out if there is an equivalence relationship (proportionality between the ratios),

we will simplify the ratios.

We will apply the greatest reduction possible (with the highest number by which we can divide without a remainder) and see if we arrive at the same ratio.

We have the following ratios. Check if there is proportionality between them.

$2:3,4:8,6:9$

Solution:

To see if there is proportionality between the ratios, we will reduce each one to its simplest form.

Let's start with $2:3$

This ratio is already simplified and we can't reduce it further without affecting its integrity. Therefore, we will leave it as is.

Moving on to $4:8$

Let's ask ourselves if we can divide both numbers of the ratio by the same divisor to reach a reduced ratio.

The answer is yes. Both $8$ and $4$ can be divided by $4$ without a remainder. So, we will divide both terms by $4$ and get:

Now we know that $4:8$ is equivalent to $1:2$. Therefore, we determine that there is no proportionality between $2:3$ and $4:8$ because after simplifying it we arrive at $1:2$ and not $2:3$.

Let's continue with the third ratio from the question: $6:9$

Let's ask ourselves if we can divide both numbers of the ratio by the same divisor to reach a reduced ratio.

The answer is yes. Both $9$ and $6$ can be divided by $3$ without a remainder.

So, we will divide both terms by $3$ and get:

Pay attention! We got exactly the same ratio that we obtained in the first one!

Therefore, we can say without a doubt that the ratio $2:3$ is proportional to the ratio $6:9$.

It is known that in seventh grade $C$, the ratio of boys to girls is $12:8$

In seventh grade $B$, the ratio of boys to girls is $36:27$.

Is there proportionality in the ratio of boys to girls in both grades?

Solution:

First, we will confirm that the ratios indeed represent the same relationship, boys to girls and not the other way around.

Then, we will simplify the ratios by dividing by the greatest common divisor and see if we arrive at the same simplified ratio.

Let's start with the ratio in seventh grade $C$:

$12:8$.

We will ask ourselves, can we divide both terms by a number without leaving a remainder (we will look for the greatest divisor to reduce the ratio as much as possible)?

The answer is yes, by $4$.

We will divide and find that the ratio is proportional to: $3:2$

Now let's continue with the ratio in seventh grade $B$:

$36:27$.

We will ask ourselves, can we divide both terms by a number without leaving a remainder (we will look for the greatest divisor to reduce the ratio as much as possible)?

The answer is yes, by $9$.

We will divide and find that the ratio is proportional to: $4:3$.

Note:

We always look for the greatest divisor to reach the most reduced fraction possible.

Are the reduced ratios we arrived at equivalent?

The answer is no, $3:2$ is not equal to $4:3$ and therefore, the ratio of boys to girls in the two classes is not proportional.

In a clothing store, there are $100$ blouses and $50$ pants.

Diana has $20$ blouses, and the ratio of blouses to pants in her closet is identical to that of the store.

We must figure out how many pants are in Diana's closet.

In this question, there is a proportionality (equivalence relation) between the store and Diana's closet with respect to the number of items.

Let's write the relationship and we will get:

$100:50=20:X$

$X$ represents the number of pants in Diana's closet.

To maintain proportionality (meaning the ratio remains the same), $X$ must be equal to $10$

In other words, Diana has $10$ pairs of pants in her closet.