The average weight of 50 teachers in a school is 46 kg. If the average weight of male teachers is 50 kg and the average weight of female teachers is 40 kg, what are the numbers of male and female teachers?



Answer :

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.