Question

Divide 250 in the ratio \(1: 9\).

Short answer

Answer: The two parts of the given quantity are 25 and 225.

Step-by-step solution

Understand the given ratio

We are given the ratio as \(1:9\) which means that the first part should have 1 unit and the second part should have 9 units.

Calculate the sum of the ratio's parts

Add the two parts of the ratio together to find the sum: \(1 + 9 = 10\).

Find the value of each part

Divide the given quantity (250) by the sum of the ratio's parts (10) to find the value of each part: \(\frac{250}{10} = 25\).

Calculate the individual parts of the given ratio

Now, multiply the value of each part by the ratio's parts to find the individual parts: Part 1: \(1 \times 25 = 25\) Part 2: \(9 \times 25 = 225\)

Verify the results

Check if the sum of the individual parts is equal to the given quantity: \(25 + 225 = 250\) The given quantity, 250, has been successfully divided into the ratio \(1:9\) as the two parts, 25 and 225.

Key concepts

Ratio Calculation

Understanding ratio calculation is fundamental to divide quantities proportionally. A ratio represents how many times one amount contains another, and it is expressed using the colon symbol (:).

For example, the ratio of 1 to 9 (noted as 1:9) implies that for every 1 unit of the first quantity, there are 9 units of the second quantity. When dividing a number like 250 in a 1:9 ratio, you are allocating the total quantity into parts that respect this relative proportion.

To effectively calculate the parts in a given ratio, follow these steps:

• Determine the total number of parts by adding the ratio numbers (1+9=10 in our example).
• Divide the total quantity by the total number of parts (250/10=25 in our example).
• Multiply each ratio number by the value of each part to find the amounts for each (1×25=25 and 9×25=225 in our example).

By systematically applying these steps, the concept of ratio calculation becomes clearer and easier to execute.

Proportional Division

Proportional division is a technique used to break down a quantity into parts that are in a specified ratio to one another. This method ensures that each part of the division relates to the others in a way that is fair and consistent with the given ratio.

To apply proportional division, start by understanding that each part of the division represents a piece of the whole. Therefore, the key is to determine the size of a single proportionate 'part' before multiplying it by the right quantities to achieve the ratio.

In our example, dividing 250 in a 1:9 ratio, the 'whole' is made up of 10 parts (the sum of 1 and 9). Each 'part' is equivalent to 250/10 or 25. So, one unit of the ratio is worth 25, and nine units are worth 9×25. This method of division maintains the original ratio, ensuring each share is proportionate to the others.

Arithmetic Operations

Arithmetic operations, including addition, subtraction, multiplication, and division, are the building blocks of mathematics that allow us to solve a wide range of problems, including those involving ratios.

In the context of dividing in a ratio, arithmetic operations are used sequentially to find the proportionate parts. For instance, addition helps to amalgamate the parts of the ratio (1+9=10). Division is then used to find out what each unit is worth (250/10=25). Finally, multiplication gives us the size of each part according to the ratio (1×25 for the first part and 9×25 for the second part).

By mastering these operations and understanding their role within the process, arithmetic becomes a valuable tool for solving ratio problems efficiently. It's important to carefully execute each step to ensure accuracy, always verifying the results by checking that the total sums back to the original quantity.