Division in a given ratio

Leo and Romi share a total of $112$ marbles.

The ratio of Leo's marbles to Romi's is $5:3$.

How many marbles did Leo and Romi each receive?

In a question of this style, we should divide the defined quantity ($112$) according to the given ratio between Leo and Romi.

With great ease.

We can choose one of the following ways:

We will simplify the given ratio in the following way:

For every $5$ marbles for Leo, Romi will receive $3$.

Therefore, we can use the variable $X$ and write it as follows:

Leo receives $5X$ marbles

Romi receives $3X$ marbles

Now, we can take the data provided in the question about the total number of marbles being $112$ and write an equation with one variable:

$5X+3X=112$

We will solve for $X$ and obtain:

$8X=112$

$x=14$

Pay attention! We have not yet reached the final answer.

We need to place the new data and it will give us that:

Leo will receive $5\times14=70$

$70$ marbles

Romi will receive $3\times14=42$

$42$ marbles

We will draw a fixed table that will help us organize the data and give us the answer to these types of questions:

Let's learn with this example how to arrange the data in the table and then find the answer.

Question:

Sharon and Ana together donated a total amount of $400$ $ to the Animal Protection Association.

For every $3$$ that Sharon donated, Ana donated $7$$.

How much did each of them donate?

Solution:

We will draw a table:

First, we will write what we have: Sharon and Ana.

Now we will fill in the total amount: $400$$.

Then, we will add the ratio according to the data given in the question:

Sharon $3$, Ana $7$.

Make sure to write it under the ratio column and not the amount column since Sharon and Ana did not donate only $10$$. It's just the ratio.

Good.

Now, let's calculate the total ratio: $3+7$ and it will give us:

We have reached the main phase:

Understanding what is the total ratio within the total amount.

That is:

How much is $10$ out of $400$

Let's divide the $400$ by $10$ and it will give us:

$400:10=40$

Now that we know that the total ratio is $40$, we will apply it to each term separately in the following way:

We will multiply the ratio of each term by the total ratio we found and obtain the amount.

Great! We can take the answers from the table and understand that:

Sharon donated $120$$ and Ana donated $280$$.

In a certain store in the shopping mall, there are $100$ appliances, refrigerators and air conditioners.

The ratio between refrigerators and air conditioners is $3:1$. 

We must find the number of refrigerators and air conditioners in the store.

In this exercise, our task is to divide the $100$ appliances according to the ratio of $3:1$.

We can deduce that $3$ represents the number of refrigerators and, conversely, $1$ represents the number of air conditioners.

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

Let's draw a simple equation:

$3X+X=100$

$4X=100$

$X=25$

From here it follows that the number of refrigerators is $75 (3X)$, and the number of air conditioners is $X=25$.

We can always go back and check our result by verifying that the total number of appliances in the store is $100$, as stated in the first piece of data given.