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.

What is the ratio of the total boys in sections Q and R to the total girls in sections P and S?

The correct option is 14:19.

Step 1: Find the total number of boys in Grade 11
We are given that the total number of girls across all four sections (P, Q, R, and S) is 261.
The overall ratio of boys to girls in the class is 8:9.
Using this ratio, we can calculate the total number of boys:

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

Step 2: Determine the number of girls in each section
Let the number of girls in sections P, Q, R, and S be represented as $G_P$, $G_Q$, $G_R$, and $G_S$ respectively.
From the given information:

1. The number of girls in section R is $G_R=67$.
2. Sections P and Q have an equal number of girls: $G_P=G_Q$.
3. In section Q, the ratio of boys ($B_Q$) to girls ($G_Q$) is 2:5:
$B_Q=\frac{2}{5}{G}_Q$
4. The number of girls in section S is 2 more than the number of boys in section Q:
$G_S=B_Q+2=\frac{2}{5}{G}_Q+2$

Since the total number of girls is 261, we can write:

$G_P+G_Q+G_R+G_S=261$

Substituting all values in terms of $G_Q$:

$G_Q+G_Q+67+(\frac{2}{5}{G}_Q+2)=261$

$2G_Q+0.4G_Q+69=261$

$2.4G_Q=261-69$

$2.4G_Q=192$

$G_Q=\frac{192}{2.4}=80$

Now, calculating the girls in each section:
• Girls in Section P: $G_P=80$
• Girls in Section Q: $G_Q=80$
• Girls in Section R: $G_R=67$
• Girls in Section S: $G_S=\frac{2}{5}\left(80\right)+2=32+2=34$

Step 3: Determine the number of boys in each section
Let the number of boys in sections P, Q, R, and S be represented as $B_P$, $B_Q$, $B_R$, and $B_S$ respectively.

1. Boys in section Q: $B_Q=32$
2. Boys in section R is 15 fewer than the girls in section R:
$B_R=G_R-15=67-15=52$
3. Total boys across all four sections is 232:
$B_P+B_Q+B_R+B_S=232$

We are given that the ratio of boys in section P to section S is 3:1, which means $B_P=3B_S$.
Substituting these values into the total boys equation:

$3B_S+32+52+B_S=232$

$4B_S+84=232$

$4B_S=232-84=148$

$B_S=\frac{148}{4}=37$

Therefore, boys in section P: $B_P=3\times 37=111$.

Step 4: Calculate the required ratio
We need to find the ratio of total boys in sections Q and R to total girls in sections P and S.

• Total boys in sections Q and R:
$B_Q+B_R=32+52=84$

• Total girls in sections P and S:
$G_P+G_S=80+34=114$

• Required Ratio:
$\text{Ratio}=84:114$

Simplifying the ratio by dividing both terms by 6:

$\frac{84}{6}:\frac{114}{6}=14:19$

Thus, the final ratio is 14:19.