Let's solve this step-by-step:
1. We know that the number of girls in the school is 180.
2. The ratio of girls to boys is given as \( 3: 4 \).
To find the number of boys, we need to use the given ratio. The ratio \( 3: 4 \) means that for every 3 parts (girls), there are 4 parts (boys).
Let's denote the number of boys as \( B \).
According to the ratio:
[tex]\[
\frac{girls}{boys} = \frac{3}{4}
\][/tex]
We can set up a proportion based on this ratio:
[tex]\[
\frac{180}{B} = \frac{3}{4}
\][/tex]
To solve for \( B \) (the number of boys), we cross-multiply:
[tex]\[
180 \times 4 = 3 \times B
\][/tex]
[tex]\[
720 = 3B
\][/tex]
Now, divide both sides by 3 to isolate \( B \):
[tex]\[
B = \frac{720}{3} = 240
\][/tex]
So, there are 240 boys in the school.
3. Now, we need to find the total number of students in the school. The total number of students is the sum of the number of girls and boys:
[tex]\[
\text{Total students} = \text{number of girls} + \text{number of boys}
\][/tex]
[tex]\[
\text{Total students} = 180 + 240 = 420
\][/tex]
Thus, the total number of students in the school is 420.
Therefore, the correct answer is:
(b) 420
