Division in a given ratio

🏆Practice reason

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

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Given 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 the circle in drawing 2 than the area of the circle in drawing 1?

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For example

Leo and Romi share a total of 112 112 marbles.

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

How many marbles did Leo and Romi each receive?

In a question of this style, we should divide the defined quantity (112 112 ) 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 5 marbles for Leo, Romi will receive 3 3 .

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

Leo receives 5X 5X marbles

Romi receives 3X 3X marbles

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

5X+3X=112 5X+3X=112

We will solve for X X and obtain:

8X=112 8X=112

x=14 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 5×14=70 5\times14=70

70 70 marbles

Romi will receive 3×14=42 3\times14=42

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 400400 $ to the Animal Protection Association.

For every 33$ that Sharon donated, Ana donated 77$.

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: 400400$.

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

Sharon 3 3 , Ana 7 7 .

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

Good.

Now, let's calculate the total ratio: 3+7 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 10 out of 400 400

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

400:10=40 400:10=40

Now that we know that the total ratio is 40 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 120 120 $ and Ana donated 280 280 $.


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Another simple example

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

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

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

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

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

Let's denote both with a variable X X .

Let's draw a simple equation:

3X+X=100 3X+X=100

4X=100 4X=100

X=25 X=25

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

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


Do you know what the answer is?
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