## Distribution of Students in Colleges in the Year 2000
In the year 2000, the total student strength of three colleges, denoted as [tex]\(X\)[/tex], [tex]\(Y\)[/tex], and [tex]\(Z\)[/tex], in a city were in the ratio [tex]\(2:5:1\)[/tex]. To determine the number of students in each college, we follow the given ratio and assume a common factor [tex]\(k\)[/tex] which represents the multiple of the ratio parts.
Let’s break down the problem step-by-step.
### Given Ratios
1. College [tex]\(X\)[/tex]: [tex]\(2\)[/tex] parts
2. College [tex]\(Y\)[/tex]: [tex]\(5\)[/tex] parts
3. College [tex]\(Z\)[/tex]: [tex]\(1\)[/tex] part
### Total Parts
The total parts representing all students across the three colleges is:
[tex]\[ \text{Total Parts} = 2 + 5 + 1 = 8 \][/tex]
### Determining Student Strengths
Using the ratio parts, we can determine the number of students in each college if we assume the simplest multiple [tex]\(k = 1\)[/tex].
1. Students in College [tex]\(X\)[/tex] (Strength [tex]\(X\)[/tex]):
[tex]\[ \text{Strength}_X = 2 \times k = 2 \times 1 = 2 \][/tex]
2. Students in College [tex]\(Y\)[/tex] (Strength [tex]\(Y\)[/tex]):
[tex]\[ \text{Strength}_Y = 5 \times k = 5 \times 1 = 5 \][/tex]
3. Students in College [tex]\(Z\)[/tex] (Strength [tex]\(Z\)[/tex]):
[tex]\[ \text{Strength}_Z = 1 \times k = 1 \times 1 = 1 \][/tex]
### Total Strength
The total strength of students in all the colleges can be calculated by summing the individual strengths:
[tex]\[ \text{Total Strength} = \text{Strength}_X + \text{Strength}_Y + \text{Strength}_Z \][/tex]
[tex]\[ \text{Total Strength} = 2 + 5 + 1 = 8 \][/tex]
### Table of Distribution
Below is the complete table summarizing the number of students in each college based on the provided ratio:
| College | Strength (Number of Students) |
|-----------------|-------------------------------|
| College [tex]\(X\)[/tex] | 2 |
| College [tex]\(Y\)[/tex] | 5 |
| College [tex]\(Z\)[/tex] | 1 |
| Total | 8 |
This table shows the distribution of students in the three colleges in the year 2000, maintaining the given ratio of [tex]\(2:5:1\)[/tex].
