Solving and Explaining a Ratio Puzzle in a Party Setting

This article provides a detailed solution and explanation for the algebraic puzzle involving the ratio of ladies to gentlemen in a party. We'll walk through the steps to determine the number of gentlemen in the party and explore the mathematical reasoning behind the solution. This content is designed to be SEO-optimized for Google search visibility, making it conform to Google's high standards for search engine optimization.

Understanding the Problem

The initial scenario states that the ratio of the number of ladies to the number of gentlemen in a party is 3:4. This means that for every 3 ladies, there are 4 gentlemen. Then, it is observed that if 8 more ladies join the party, the new ratio becomes 5:4. The challenge is to determine the number of gentlemen in the party.

Setting up the Equation

Let's denote the number of ladies in the party as ( L ) and the number of gentlemen as ( G ). According to the initial ratio requirement, we can write:

( L frac{3}{4} G )

When 8 more ladies join, the new number of ladies becomes ( L 8 ), and the ratio of ladies to gentlemen becomes 5:4. Therefore, we can write the equation as:

( frac{L 8}{G} frac{5}{4} )

Substituting ( L frac{3}{4} G ) into the equation, we get:

( frac{frac{3}{4} G 8}{G} frac{5}{4} )

Multiplying both sides by ( G ) to clear the denominator:

( frac{3}{4} G 8 frac{5}{4} G )

Subtracting ( frac{3}{4} G ) from both sides:

( 8 frac{5}{4} G - frac{3}{4} G )

( 8 frac{2}{4} G )

( 8 frac{1}{2} G )

Multiplying both sides by 2:

( G 16 )

Hence, the number of gentlemen in the party is 16.

Verification and Explanation

For verification, we can check the initial condition. Initially, the number of ladies is ( frac{3}{4} times 16 12 ). Therefore, the ratio is ( 12 : 16 3 : 4 ), which is correct. After adding 8 ladies, the number of ladies becomes ( 12 8 20 ). The new ratio is then ( 20 : 16 5 : 4 ), which also matches the described condition.

Conclusion

In conclusion, the solution to the puzzle involves setting up and solving algebraic equations based on the given ratio and changes in the number of attendees. This method not only provides a clear answer to the question but also demonstrates a systematic approach to problem-solving in mathematics.