Solving Number Problems Using Ratios and Differences

When dealing with number problems that mention both a difference and a ratio, a systematic approach using algebraic equations can help us find the exact values. In this article, we will explore a specific problem where the difference between two numbers is given along with their ratio, and we will solve it using both the algebraic method and the unitary method.

Problem Statement

The difference between two numbers is 33, and their ratio is 5:2. We need to find the actual numbers.

Step-by-Step Solution

Method 1: Algebraic Equations

Let the two numbers be denoted as x and y.

According to the problem, we have two main equations:

The difference of the two numbers:

x - y 33

The ratio of the two numbers:

x/y 5/2

From the ratio equation, we can express x in terms of y:

x (5/2)y

Now substitute this expression for x into the difference equation:

(5/2)y - y 33

To simplify, write y as (2/2)y:

(5/2)y - (2/2)y 33

(3/2)y 33

Solve for y:

y 33 * (2/3) 22

Now, find x:

x (5/2) * 22 55

Therefore, the two numbers are 55 and 22.

Method 2: Unitary Method

Let the two numbers be 5x and 2x. According to the problem, we have:

5x - 2x 33

Hence, x 11. The numbers are then:

5x 5 * 11 55

2x 2 * 11 22

Conclusion

The two numbers are 55 and 22.

Further Explanations and Examples

In solving these problems, it's important to understand that the algebraic method involves setting up equations based on the provided conditions and solving them systematically. The unitary method involves breaking down the problem into simpler parts using the given ratios and differences.

For additional practice, consider similar problems such as:

1. Two numbers are in the ratio 4:3, and they have a difference of 14. Find the numbers.

2. The difference between two numbers is 27, and their ratio is 7:5. What are the numbers?

By practicing such problems, you can develop a better understanding of how to handle problems involving ratios and differences.

Key Takeaways

Always identify and set up the given conditions as equations. Use the unitary method to simplify the solution process. Practice solving similar problems for better grasp and application.

References

Example Algebraic Equations Problems