To find the height of the 5th girl, we need to carefully analyze the given information and apply basic arithmetic principles.
Let's break down the given information step-by-step:
1. Total number of girls: 9
2. Average height of the first 5 girls: 120 cm
- If the average height of the first 5 girls is 120 cm, then the total height of these 5 girls can be calculated as:
[tex]\[
\text{Total height of first 5 girls} = 120 \times 5 = 600 \text{ cm}
\][/tex]
3. Average height of the last 5 girls: 140 cm
- Similarly, if the average height of the last 5 girls is 140 cm, then the total height of these 5 girls can be calculated as:
[tex]\[
\text{Total height of last 5 girls} = 140 \times 5 = 700 \text{ cm}
\][/tex]
4. Average height of all 9 girls: 125 cm
- If the average height of all 9 girls is 125 cm, then the total height of all 9 girls can be calculated as:
[tex]\[
\text{Total height of all 9 girls} = 125 \times 9 = 1125 \text{ cm}
\][/tex]
Now, we know the 5th girl is included in both the first group of 5 girls and the last group of 5 girls. Therefore, we must adjust for this overlap.
We need to find the height of the 5th girl (let's denote her height as [tex]\( h_5 \)[/tex]). The total height of the first 5 girls plus the total height of the last 5 girls includes the height of the 5th girl twice. Thus, we need to subtract the height of the 5th girl once to find the actual total height of all 9 girls:
[tex]\[
\text{Total height of all 9 girls} = \text{Total height of first 5 girls} + \text{Total height of last 5 girls} - h_5
\][/tex]
Substituting the known values:
[tex]\[
1125 = 600 + 700 - h_5
\][/tex]
Now, solve for [tex]\( h_5 \)[/tex]:
[tex]\[
1125 = 1300 - h_5
\][/tex]
Isolating [tex]\( h_5 \)[/tex]:
[tex]\[
h_5 = 1300 - 1125 = 175 \text{ cm}
\][/tex]
Therefore, the height of the 5th girl is:
[tex]\[
\boxed{175 \text{ cm}}
\][/tex]
