Solving a System of Equations to Find the Ratio of Two Numbers

When it comes to solving complex problems in mathematics, one of the most straightforward and effective methods involves the use of a system of equations. In this article, we will explore how to find the ratio of two numbers given their sum and difference. Let's dive into the steps and the algebra involved in solving such a problem.

Problem Statement

The sum of two numbers is 40, and their difference is 4. We need to find the ratio of these two numbers among the provided options: 21:19, 22:9, and 11:18.

Solving the Equations

Step 1: Define the Variables

Let's denote the two numbers by ( x ) and ( y ).

Step 2: Formulate Equations

We have two equations based on the given information:

The sum of the two numbers:

( x y 40 ) ... Equation 1

The difference between the two numbers:

( x - y 4 ) ... Equation 2

Step 3: Solve the Equations Simultaneously

First, let's add Equation 1 and Equation 2 to eliminate ( y ):

( (x y) (x - y) 40 4 )

( 2x 44 )

( x 22 )

Now, substitute ( x 22 ) back into Equation 1 to find ( y ):

( 22 y 40 )

( y 40 - 22 )

( y 18 )

Step 4: Find the Ratio of the Numbers

The two numbers are 22 and 18. The ratio of these numbers can be expressed as:

( frac{22}{18} frac{11}{9} )

Step 5: Compare with Provided Ratios

The ratios provided are:

21:19

( frac{21}{19} approx 1.105 )

22:9

( frac{22}{9} approx 2.444 )

11:18

( frac{11}{18} approx 0.611 )

The ratio (frac{22}{18}) simplifies to (frac{11}{9} approx 1.222), which matches none of the given options directly. However, the same ratio 22:18 simplifies to 11:9.

Conclusion

The resultant ratio of the two numbers is ( 11:9 ). Therefore, the correct ratio among the provided options is:

11:9.

Alternative Methods to Solve the Problem

Here are a couple of alternative methods to solve the problem:

Method 1: Simplified Approach

Simply divide both the sum and the difference by 2:

( frac{40}{2} 20 )

( frac{4}{2} 2 )

The smaller number is ( 20 - 2 18 ), and the larger number is ( 20 2 22 ). The ratio of the smaller to the larger number is ( 18:22 ), which simplifies to ( 9:11 ).

Method 2: Direct Substitution

Assume the numbers are A and B:

( frac{A B}{A - B} frac{40}{4} 10 )

( frac{A}{B} frac{11}{9} )

( A : B 11 : 9 )

Final Thoughts

Both algebraic methods and alternative approaches lead us to the same conclusion: the ratio of the two numbers is ( 11:9 ).