Directions: Study the information below and answer the question.

A Grade 11 class is divided into four sections: P, Q, R, and S. The overall ratio of boys to girls is 8:9. In section Q, the ratio of boys to girls is 2:5. Sections P and Q have an equal number of girls. In section R, the number of boys is 15 fewer than the number of girls. The number of girls in section S is 2 more than the number of boys in section Q. There are 67 girls in section R, out of a total of 261 girls. The ratio of boys in sections P and S is 3:1.

By what percentage is the number of girls in section S greater than the number of boys in section Q?

The correct answer is 6.25%.

Step 1: Determine the total number of boys and girls in the class.
We are given that the total number of girls in the Grade 11 class is 261.
The overall ratio of boys to girls is given as 8:9.

Using this ratio, we can calculate the total number of boys in the class:

$\text{Total Boys}=\frac{8}{9}\times 261=8\times 29=232$

Step 2: Express the number of girls in each section in terms of the number of boys in section Q.
Let $B_Q$ represent the number of boys in section Q.
Let $G_P$, $G_Q$, $G_R$, and $G_S$ represent the number of girls in sections P, Q, R, and S, respectively.

1. In section Q, the ratio of boys to girls is 2:5:

$\frac{B_Q}{G_Q}=\frac{2}{5}\Rightarrow G_Q=\frac{5}{2}{B}_Q=2.5B_Q$

2. Sections P and Q have an equal number of girls:

$G_P=G_Q=2.5B_Q$

3. The number of girls in section S is 2 more than the number of boys in section Q:

$G_S=B_Q+2$

4. There are 67 girls in section R:

$G_R=67$

Step 3: Solve for the number of boys in section Q ($B_Q$) and girls in section S ($G_S$).
The sum of the girls across all four sections equals the total of 261 girls:

$G_P+G_Q+G_R+G_S=261$

Substitute the algebraic expressions for each section:

$2.5B_Q+2.5B_Q+67+(B_Q+2)=261$

Combine like terms:

$6B_Q+69=261$

Subtract 69 from both sides:

$6B_Q=261-69=192$

Divide by 6:

$B_Q=\frac{192}{6}=32$

Now substitute $B_Q=32$ to find the number of girls in section S:

$G_S=B_Q+2=32+2=34$

Step 4: Calculate the required percentage increase.
We need to calculate by what percentage the number of girls in section S ($G_S=34$) is greater than the number of boys in section Q ($B_Q=32$):

$\text{Percentage Greater}=\frac{G_S-B_Q}{B_Q}\times 100\%$

Substitute the values into the formula:

$\text{Percentage Greater}=\frac{34-32}{32}\times 100\%=\frac{2}{32}\times 100\%$

$\text{Percentage Greater}=\frac{1}{16}\times 100\%=6.25\%$

Thus, the number of girls in section S is 6.25% greater than the number of boys in section Q.