Points to Remember:
- This is a mathematical word problem involving ratios and proportions.
- The solution requires setting up and solving a system of equations.
Introduction:
This question is a classic example of a ratio problem commonly encountered in mathematics. Ratio problems involve comparing the relative sizes of two or more quantities. Understanding ratios is crucial in various fields, from simple everyday calculations to complex scientific analyses. The problem presented involves finding the initial number of students in college N, given a change in the student population of college M and the resulting change in the ratio between the two colleges.
Body:
Setting up the Equations:
Let’s represent the number of students in college M as 4x and the number of students in college N as 3x, reflecting the initial 4:3 ratio. After 100 students join college M, the number of students in M becomes 4x + 100. The ratio then changes to 7:5. This can be expressed as the following equation:
(4x + 100) / 3x = 7/5
Solving for x:
To solve for x, we cross-multiply:
5(4x + 100) = 7(3x)
20x + 500 = 21x
x = 500
Finding the Number of Students in College N:
Since the number of students in college N is represented by 3x, we substitute the value of x we found:
Number of students in college N = 3x = 3 * 500 = 1500
Therefore, there were initially 1500 students in college N.
Verification:
Initially, college M had 4x = 4 * 500 = 2000 students, and college N had 1500 students. The ratio was 2000:1500, which simplifies to 4:3.
After 100 students joined college M, college M had 2100 students, and college N still had 1500 students. The new ratio is 2100:1500, which simplifies to 7:5. This confirms our solution.
Conclusion:
The number of students in college N is 1500. This was determined by setting up and solving a system of equations based on the given ratios before and after the increase in students at college M. The solution demonstrates the practical application of ratio and proportion principles in solving real-world problems. This problem-solving approach can be applied to various scenarios involving comparative analysis and resource allocation. Further, understanding ratios and proportions is fundamental to data analysis and interpretation across numerous disciplines. The accuracy of the solution highlights the importance of precise mathematical reasoning in obtaining reliable results.