Division in a given ratio

What is division in a given ratio?

Division in a given ratio means splitting a total quantity into parts that maintain a specific proportional relationship, based on the ratio provided.
In a division according to a given ratio, we will have a defined quantity that we must divide according to said ratio. The process ensures that the ratio between the parts stays consistent, regardless of the total amount being divided. This concept is frequently used in various scenarios, such as dividing an inheritance, sharing resources, or solving problems in geometry.

Let's use an Example:

We want to divide $100$ Dollars in a $2:3$ ratio.
So, the quantity is $100$ , and the ratio provided is $2:3$ .

In order to do so, let's follow there simple steps:

Add the parts of the ratio. In our case: $2 + 3 = 5$ .
Now we know that we need to divide the quantity to $5$ .
Divide the total amount by $5$ . In our case: $100:5=20$
So we get $20$ Dollars per part.
Multiply each of the ratio side by the part.
So: $20\cdot2=40$ , $3\cdot20=60$ .

And so the $100$ Dollars is divided into $40$ Dollars and $60$ Dollars , maintaining the $2:3$ ratio.

Diagram demonstrating division in a given ratio: total quantity of $100 divided in the ratio 2:3, resulting in $40 and $60. A simple visualization for understanding ratio-based division, featured in a math tutorial on dividing quantities in specific ratios.

Detailed explanation

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Test yourself on ratio!

If there are 18 balls in a box of which $\frac{2}{3}$ are white:

How many white balls are there in the box in total?

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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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Test your knowledge

Question 1

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

How many orange balls are there in the box in total?

Question 2

How many times longer is the radius of the red circle than the radius of the blue circle?

Question 3

How many times longer is the radius of the red circle than the radius of the blue circle?

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 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

Step-by-Step Solution

The area of a circle is calculated using the following formula:

where r represents the radius.

Using the formula, we calculate the areas of the circles:

Circle 1:

π*4² =

π16

Circle 2:

π*10² =

π100

To calculate how much larger one circle is than the other (in other words - what is the ratio between them)

All we need to do is divide one area by the other.

100/16 =

6.25

Therefore the answer is 6 and a quarter!

Answer

$6\frac{1}{4}$

Exercise #2

Given the rectangle ABCD

AB=X the ratio between AB and BC is equal to $\sqrt{\frac{x}{2}}$

We mark the length of the diagonal $A$ with $m$

Check the correct argument:

Video Solution

Step-by-Step Solution

Let's find side BC

Based on what we're given:

$\frac{AB}{BC}=\frac{x}{BC}=\sqrt{\frac{x}{2}}$

$\frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}$

$\sqrt{2}x=\sqrt{x}BC$

Let's divide by square root x:

$\frac{\sqrt{2}\times x}{\sqrt{x}}=BC$

$\frac{\sqrt{2}\times\sqrt{x}\times\sqrt{x}}{\sqrt{x}}=BC$

Let's reduce the numerator and denominator by square root x:

$\sqrt{2}\sqrt{x}=BC$

We'll use the Pythagorean theorem to calculate the area of triangle ABC:

$AB^2+BC^2=AC^2$

Let's substitute what we're given:

$x^2+(\sqrt{2}\sqrt{x})^2=m^2$

$x^2+2x=m^2$

Answer

$x^2+2x=m^2$

Exercise #3

The rectangle ABCD is shown below.

AB = X

The ratio between AB and BC is $\sqrt{\frac{x}{2}}$ .

The length of diagonal AC is labelled m.

Choose the correct answer.

Video Solution

Step-by-Step Solution

We know that:

$\frac{AB}{BC}=\sqrt{\frac{x}{2}}$

We also know that AB equals X.

First, we will substitute the given data into the formula accordingly:

$\frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}$

$x\sqrt{2}=BC\sqrt{x}$

$\frac{x\sqrt{2}}{\sqrt{x}}=BC$

$\frac{\sqrt{x}\times\sqrt{x}\times\sqrt{2}}{\sqrt{x}}=BC$

$\sqrt{x}\times\sqrt{2}=BC$

Now let's look at triangle ABC and use the Pythagorean theorem:

$AB^2+BC^2=AC^2$

We substitute in our known values:

$x^2+(\sqrt{x}\times\sqrt{2})^2=m^2$

$x^2+x\times2=m^2$

Finally, we will add 1 to both sides:

$x^2+2x+1=m^2+1$

$(x+1)^2=m^2+1$

Answer

$m^2+1=(x+1)^2$

Exercise #4

If there are 18 balls in a box of which $\frac{2}{3}$ are white:

How many white balls are there in the box in total?

Video Solution

Answer

Exercise #5

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

How many orange balls are there in the box in total?

Video Solution

Answer

Do you know what the answer is?

Question 1

How many times longer is the radius of the red circle, which has a diameter of 24, than the radius of the blue circle, which has a diameter of 12?

Question 2

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?

Question 3

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?

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