Sure, let's solve this step-by-step.
### Given Data:
- The total number of teachers is 50.
- The average weight of all the teachers is 46 kg.
- The average weight of gents teachers is 50 kg.
- The average weight of ladies teachers is 40 kg.
### Step-by-Step Solution:
1. Identify variables:
- Let [tex]\( g \)[/tex] represent the number of gents teachers.
- Let [tex]\( l \)[/tex] represent the number of ladies teachers.
2. Set up the relationships:
- The total number of teachers is the sum of gents and ladies teachers:
[tex]\[
g + l = 50
\][/tex]
3. Calculate total weights:
- The total weight of all teachers combined can be found using the average weight:
[tex]\[
\text{Total weight of all teachers} = \text{Average weight} \times \text{Total number of teachers} = 46 \times 50 = 2300 \text{ kg}
\][/tex]
- The total weight of gents teachers:
[tex]\[
\text{Total weight of gents teachers} = g \times 50
\][/tex]
- The total weight of ladies teachers:
[tex]\[
\text{Total weight of ladies teachers} = l \times 40
\][/tex]
4. Set up the equation based on total weights:
- The sum of the weights of gents and ladies teachers must equal the total weight of all teachers:
[tex]\[
g \times 50 + l \times 40 = 2300
\][/tex]
5. Substitute [tex]\( l \)[/tex] using the total number of teachers:
- From the total number of teachers equation [tex]\( g + l = 50 \)[/tex], we can express [tex]\( l \)[/tex] as:
[tex]\[
l = 50 - g
\][/tex]
- Substitute [tex]\( l \)[/tex] in the total weight equation:
[tex]\[
g \times 50 + (50 - g) \times 40 = 2300
\][/tex]
6. Simplify and solve for [tex]\( g \)[/tex]:
- Expand the equation:
[tex]\[
g \times 50 + 50 \times 40 - g \times 40 = 2300
\][/tex]
[tex]\[
50g + 2000 - 40g = 2300
\][/tex]
[tex]\[
10g + 2000 = 2300
\][/tex]
[tex]\[
10g = 300
\][/tex]
[tex]\[
g = 30
\][/tex]
7. Find the number of lady teachers [tex]\( l \)[/tex]:
- Substitute [tex]\( g = 30 \)[/tex] in [tex]\( l = 50 - g \)[/tex]:
[tex]\[
l = 50 - 30 = 20
\][/tex]
### Conclusion:
- The number of gents teachers is [tex]\( \boxed{30} \)[/tex].
- The number of ladies teachers is [tex]\( \boxed{20} \)[/tex].
So, there are 30 gents teachers and 20 ladies teachers.
