Points to Remember:
- The problem involves solving a system of two linear equations.
- We need to find the two numbers and then calculate their ratio.
Introduction:
This question is a simple algebra problem involving the solution of simultaneous linear equations. We are given that the sum of two unknown numbers is 40 and their difference is 4. Our goal is to find the values of these two numbers and then determine their ratio. This type of problem is frequently encountered in basic mathematics and has applications in various fields, from simple resource allocation to more complex mathematical modeling.
Body:
1. Setting up the Equations:
Let’s represent the two numbers as ‘x’ and ‘y’. We can translate the given information into two equations:
- Equation 1 (Sum): x + y = 40
- Equation 2 (Difference): x – y = 4
2. Solving the Equations:
We can solve this system of equations using several methods. One simple approach is the elimination method. Adding Equation 1 and Equation 2 eliminates ‘y’:
(x + y) + (x – y) = 40 + 4
2x = 44
x = 22
Now, substitute the value of x (22) into either Equation 1 or Equation 2 to solve for y. Using Equation 1:
22 + y = 40
y = 40 – 22
y = 18
Therefore, the two numbers are 22 and 18.
3. Calculating the Ratio:
The ratio of the two numbers is x:y, which is 22:18. This ratio can be simplified by dividing both numbers by their greatest common divisor (GCD), which is 2:
22/2 : 18/2 = 11:9
Conclusion:
In summary, by setting up and solving a system of two linear equations representing the sum and difference of two numbers, we determined that the numbers are 22 and 18. The ratio of these two numbers is 11:9. This simple problem demonstrates the practical application of basic algebraic principles in solving real-world problems. Further applications of this type of problem-solving could involve more complex scenarios with multiple variables and equations, requiring more advanced algebraic techniques. The ability to solve such problems is crucial for developing a strong foundation in mathematics and its applications in various fields. The solution highlights the importance of clear problem definition and systematic application of mathematical methods to arrive at a precise and accurate answer.
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