Points to Remember:
- This is a mathematical word problem requiring an algebraic approach to solve.
- The core concept involves setting up equations based on ratios and solving for the unknowns.
- We need to find the total number of members in the assembly.
Introduction:
This question involves a classic ratio problem often encountered in algebra. It tests the ability to translate a word problem into a mathematical equation and solve for the unknown variable. Ratio problems are frequently used to represent proportions and relationships between different quantities. In this case, the ratio represents the proportion of ruling party members to opposition party members in a legislative assembly. The problem introduces a change in the ratio due to members switching parties, requiring us to solve for the original number of members in each party.
Body:
Setting up the Equations:
Let’s denote:
- ‘x’ as the common factor in the initial ratio of ruling party members to opposition party members (7:3).
- Therefore, initially, there were 7x ruling party members and 3x opposition party members.
After 18 members of the ruling party joined the opposition, the new numbers are:
- Ruling party: 7x – 18
- Opposition party: 3x + 18
The new ratio is given as 3:2. This allows us to set up the following equation:
(7x – 18) / (3x + 18) = 3/2
Solving the Equation:
Cross-multiplying the equation, we get:
2(7x – 18) = 3(3x + 18)
14x – 36 = 9x + 54
14x – 9x = 54 + 36
5x = 90
x = 18
Finding the Total Number of Members:
Initially, there were 7x + 3x = 10x members in the assembly. Substituting x = 18, we get:
Total members = 10 * 18 = 180
Therefore, there were 180 members in the assembly.
Verification:
Initially: Ruling party = 7 * 18 = 126; Opposition party = 3 * 18 = 54. Total = 180. Ratio = 126:54 = 7:3 (Correct)
After the shift: Ruling party = 126 – 18 = 108; Opposition party = 54 + 18 = 72. Total = 180. Ratio = 108:72 = 3:2 (Correct)
Conclusion:
By setting up and solving a system of equations based on the given ratios and the change in party membership, we determined that the total number of members in the legislative assembly was 180. This problem highlights the practical application of algebraic techniques in solving real-world problems involving proportions and changes in quantities. The ability to translate word problems into mathematical models is a crucial skill in various fields, including political science, economics, and statistics. This approach ensures accuracy and provides a clear, systematic method for solving similar ratio problems. The solution emphasizes the importance of precise mathematical reasoning and problem-solving skills in analyzing and interpreting data related to political representation.