# Division in a given ratio

🏆Practice ratio

In a division according to a given ratio, we will have a defined quantity that we must divide according to said ratio.

## 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?

## For example

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.

### How is it solved?

With great ease.

We can choose one of the following ways:

#### First way: With one unknown

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

#### Second way: With a table

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 $280$\$.

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## Another simple example

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.

## Examples and exercises with solutions for division according to a given ratio

### 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

$6\frac{1}{4}$