Solving a Ratio Problem: David and Carly’s Shared Amount

Let's delve into a classic ratio problem involving two individuals, David and Carly, who share an amount of money in the proportion of 5:9. Given that Carly has 60 more than David, we will explore the steps to determine how much each person has using algebraic methods and then validate our solution. This problem is not only a fundamental exercise in basic algebra but also a practical example of proportional division.

Detailed Solution and Steps

To begin, we define the amount of money David has as 5x and Carly has as 9x, where x is a common multiplier. The problem states that Carly has 60 more than David. Therefore, we can write the equation:

9x 5x 60

Subtract 5x from both sides to isolate the terms involving x on one side:

9x - 5x 60

Simplify the equation:

4x 60

Divide both sides by 4 to solve for x:

x 15

Now that we have the value of x, we can determine how much money each person has:

David's amount: (5x 5 times 15 75). Carly's amount: (9x 9 times 15 135).

To verify, we check that Carly indeed has 60 more than David:

135 - 75 60

This confirms our solution is correct. Therefore, David has (75) units of money, and Carly has (135) units.

Alternative Methods and Verification

As an additional method, we can use the concept of ratios and the total parts. Since the ratio of David to Carly is 5:9, we have a total of 14 parts, composed of 5 parts for David and 9 parts for Carly. The difference between their shares is 4 parts, which is equal to 60 units. Thus, we can find the value of one part:

4 parts 60 ( Rightarrow ) 1 part 60 / 4 15

Multiplying by the respective parts, we get:

David's share: (5 times 15 75). Carly's share: (9 times 15 135).

This again verifies our solution.

Conclusion

In this problem, we used algebraic methods to resolve the ratio problem, confirming that David has (75) units and Carly has (135) units of money. This approach was both methodical and straightforward, ensuring accuracy and providing a clear understanding of the problem's solution. Such problems are beneficial for honing algebraic skills and understanding proportional reasoning.