Question

Divide 315 in the ratio \(6: 7: 8\).

Short answer

Question: Divide the number 315 into three parts in the ratio 6:7:8. Answer: The three parts are 90, 105, and 120.

Step-by-step solution

Find the total of the ratio

Add the three parts of the ratio, i.e., \(6+7+8\). This gives us a total of \(6+7+8=21\).

Divide the given number by the total of the ratio

Divide the given number (315) by the total of the ratio (21) to get the value of one part: \(\frac{315}{21} = 15\).

Multiply the value of one part with each part of the ratio

We need to multiply the value of one part (15) with each part of the ratio to get the required parts of the number 315: First part: \(6 × 15 = 90\) Second part: \(7 × 15 = 105\) Third part: \(8 × 15 = 120\) Thus, the number 315 is divided into three parts in the ratio \(6: 7: 8\) as \(90, 105\), and \(120\).

Key concepts

Dividing Numbers in a Given Ratio

Understanding how to divide numbers in a given ratio is essential in many areas of math and daily life. It helps evenly distribute a quantity into parts that maintain a specific relationship. Thus, if you have been given a task to divide a number, such as 315, into a ratio of 6:7:8, you will first sum up these individual ratio values to find the total ratio count.

In our example, the summed up ratio, 6+7+8, equals 21. This is the combined measure of all parts in the ratio. Now, consider the total amount to be divided (315) as a whole cake that needs to be sliced into 21 even pieces. Each slice represents a 'unit' of the ratio. By dividing 315 by 21, you find that each unit's value is 15. In essence, every single 'slice' or 'unit' of the cake now has an equal value, which is critical for maintaining the fairness of the division according to the given ratio.

Ratio Calculations

After establishing the value of a single unit in the ratio division, the next step involves ratio calculations to distribute the total amount into the respective parts of the given ratio. Apply the calculated unit value to each ratio part to get actual distribution amounts.

For our 6:7:8 ratio, we multiply each ratio part by the unit value, 15. For the first part, multiply 6 (from the ratio) by 15 to get 90; for the second, multiply 7 by 15 to get 105; and for the third, multiply 8 by 15 to get 120. This process grounds you in how each portion of the ratio correlates with the proportional value of the number to be divided. It enables a clear understanding that each part of the ratio holds a value which is multiplied by the same 'unit' value to maintain the proportionality.

Proportionate Division

The concept of proportionate division revolves around ensuring that the ratio is properly maintained when dividing a quantity. It ensures that each share of the given number adheres to the specified ratio so that the proportion is correct.

Importance in Real-life Applications

In practical scenarios, proportionate division can be employed in scenarios such as slicing a cake at a party, dividing profits among business partners, or distributing resources in project management.

The technique of using the unit value for proportional distribution maintains the integrity of the ratio. Each part, 90, 105, and 120 for our given example, represents a proportion of the total based on its importance. Grasping this methodology allows you to evenly distribute any quantity such that the essence and balance of the original ratio are preserved throughout all parts of the division, ensuring fairness and accuracy in numerous real-world situations.