<Question>
The data in the table below shows the percent of bus riders in a survey of Central H.S. students; 300 males and 300 females in each grade were surveyed. For which grade is the number of its male bus riders closest to $135\%$ of the number of its female bus riders? \begin{tabular}{|l|c|c|}\hline
Grade & Males & Females \\ \hline
$9^{\mathrm{th}}$ grade & $41.1\%$ & $39.4\%$ \\ \hline
$10^{\mathrm{th}}$ grade & $34.4\%$ & $33.1\%$ \\ \hline
$11^{\mathrm{th}}$ grade & $20.6\%$ & $13.8\%$ \\ \hline
$12^{\mathrm{th}}$ grade & $11.6\%$ & $8.6\%$ \\ \hline
\end{tabular}
</Question>

<think>
<Plan_1>
Okay, so I need to figure out for which grade the number of male bus riders is closest to 135% of the number of female bus riders. Let me start by understanding the problem step by step.

First, the table gives the percentage of each grade that are male bus riders and female bus riders. Each grade has 300 males and 300 females surveyed. So, for each grade, I can calculate the actual number of male bus riders and female bus riders by applying the given percentages to 300.

Let me recall that percentage means per hundred, so 41.1% is 41.1 per 100, which translates to 0.411 in decimal. Therefore, for each grade, the number of male bus riders would be 300 multiplied by the male percentage, and similarly, the number of female bus riders would be 300 multiplied by the female percentage.

The question is asking for the grade where the number of male bus riders is closest to 135% of the female bus riders. So, mathematically, for each grade, I need to check if the number of male riders is approximately equal to 135% of the female riders. The grade where this is true (i.e., the difference is the smallest) is the answer.

Let me formalize this. Let’s denote:

- M = number of male bus riders
- F = number of female bus riders

We need to find the grade where M ≈ 135% * F. Since 135% is 1.35 in decimal, the equation becomes M ≈ 1.35 * F. The grade with the smallest difference between M and 1.35*F is the one we need.

Alternatively, we can compute the ratio M/F for each grade and see which one is closest to 1.35.

Given that each grade has 300 males and 300 females surveyed, the actual numbers will be:

For example, in 9th grade:

<prompt_1.1> Calculate the number of male and female bus riders for each grade using the given percentages. </prompt_1.1>
-----

Let me check if I made any errors here. Let's verify with another grade, say 11th grade.

<prompt_1.2> Verify the calculations for 11th grade and ensure the logic is sound. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
Males: 300 * 41.1% = 300 * 0.411 = 123.3
Females: 300 * 39.4% = 300 * 0.394 = 118.2

Similarly for 10th grade:

Males: 300 * 34.4% = 300 * 0.344 = 103.2
Females: 300 * 33.1% = 300 * 0.331 = 99.3

11th grade:

Males: 300 * 20.6% = 300 * 0.206 = 61.8
Females: 300 * 13.8% = 300 * 0.138 = 41.4

12th grade:

Males: 300 * 11.6% = 300 * 0.116 = 34.8
Females: 300 * 8.6% = 300 * 0.086 = 25.8
</execution_1.1>

<execution_1.2>
11th grade: Males 20.6%, Females 13.8%.

Calculating M/F: 20.6 /13.8 ≈ 1.492. Then 1.492 -1.35=0.142. Similarly, absolute difference is 0.142.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, for each grade, compute M and F, then compute |M - 1.35*F| and find which grade has the smallest value.

Alternatively, compute the ratio M/F and see which is closest to 1.35. Let's try both methods.

Starting with 9th grade:

<prompt_2.1> Compute the ratio M/F for 9th grade and compare it to 1.35. </prompt_2.1>
-----

Based on execution_1.2:
12th grade: 34.8/25.8≈1.345, which is very close to 1.35.

Wait, so 12th grade is 1.345, which is 0.005 less than 1.35, and 11th grade is 1.492, which is 0.142 more. So 12th grade is closer.

But let me confirm the calculations again, maybe I made a mistake.

<prompt_2.2> Re-verify the calculations for 11th and 12th grades. </prompt_2.2>

Alternatively, using the difference |M -1.35F|:

For 11th grade:

<prompt_2.3> Recalculate the absolute difference for 11th and 12th grades. </prompt_2.3>

Therefore, 12th grade is the answer.

But wait, let me check once more. Maybe there's a miscalculation.

<prompt_2.4> Perform a final verification of the calculations and confirm the closest grade. </prompt_2.4>
</Plan_2>

<Execution_2>
<execution_2.1>
Males: 123.3
Females: 118.2
Compute 1.35 * 118.2 = 118.2 * 1.35. Let me compute that:

118.2 * 1 = 118.2
118.2 * 0.3 = 35.46
118.2 * 0.05 = 5.91
Adding up: 118.2 + 35.46 = 153.66 + 5.91 = 159.57

So 1.35*F = 159.57. The actual M is 123.3. The difference is |123.3 - 159.57| = 36.27. Alternatively, compute the ratio 123.3 / 118.2 ≈ 1.045. So the ratio is about 1.045, which is quite a bit less than 1.35. So 9th grade is not close.
</execution_2.1>

<execution_2.2>
Let me recompute 12th grade:

Males: 11.6% of 300 = 0.116*300 = 34.8

Females: 8.6% of 300 = 0.086*300 = 25.8

So 34.8 /25.8 = ?

Let me compute that. 25.8 *1.35 = 25.8*1 +25.8*0.35=25.8 +9.03=34.83. So 1.35*25.8=34.83. But the actual male riders are 34.8. So 34.8 is 34.83 -0.03. So that's 34.8 is 0.03 less than 1.35*25.8. Therefore, the ratio is 34.8 /25.8=1.35 - 0.03/25.8≈1.35 -0.00116≈1.3488. So approximately 1.3488, which is 0.0012 less than 1.35.

For 11th grade:

Females:13.8% of 300=41.4

Males:20.6% of 300=61.8

So 61.8 /41.4=1.5. Exactly 1.5. So 1.5 -1.35=0.15 difference.
</execution_2.2>

<execution_2.3>
M=61.8, F=41.4

1.35*F=1.35*41.4=56.49

|M -1.35F|=|61.8 -56.49|=5.31

For 12th grade:

M=34.8, F=25.8

1.35*F=1.35*25.8=34.83

|M -1.35F|=|34.8 -34.83|=0.03

So, 12th grade has an absolute difference of ~0.03, which is much smaller than 12th grade's 0.03, whereas 11th grade's difference is 5.31, which is much larger.

Wait, but 0.03 is in terms of the actual numbers. However, when we computed the ratio, 12th grade was 1.345, which is 0.005 less than 1.35, while 11th grade was 1.492, which is 0.142 more. So, in terms of percentage difference, 12th grade is closer. Therefore, 12th grade is the correct answer.

But let me check the problem statement again. It says "the number of its male bus riders closest to 135% of the number of its female bus riders".

Yes, so it's the absolute difference between M and 1.35*F. So, 12th grade's difference is 0.03, which is much smaller than 12th grade's 5.31. Therefore, 12th grade is the answer.
</execution_2.3>

<execution_2.4>
Wait, 12th grade's male riders are 34.8, female riders 25.8.

Compute 1.35*25.8: 25.8*1.35. Let's compute 25*1.35=33.75, 0.8*1.35=1.08. So total 33.75+1.08=34.83.

So 1.35*F is 34.83, which is very close to M=34.8. The difference is 34.83 -34.8=0.03.

For 11th grade: 1.35*41.4=1.35*40=54, 1.35*1.4=1.89. So total 54+1.89=55.89. The actual M is 61.8. The difference is 61.8 -55.89=5.91.

Whereas for 12th grade, the difference is 0.03, which is much smaller. So yes, 12th grade is the closest.

Therefore, the answer is 12th grade. But wait, let me check if I messed up any grade's numbers.

Wait, 11th grade's female riders are 41.4, which is higher, so 135% of that would be significantly higher. Whereas 12th grade's female riders are 8.6%, which when multiplied by 1.35 gives 11.52%, and the male riders are 11.6%, which is very close. So that makes sense.

Therefore, the grade where male riders are closest to 135% of female riders is 12th grade. So the answer is 12th grade.
</execution_2.4>
</Execution_2>

<Plan_3>
Based on execution_2.1:
For 10th grade:

<prompt_3.1> Compute M and F for 10th grade and compare the ratio to 1.35. </prompt_3.1>
-----

Based on execution_2.4:
**Final Answer**
The number of male bus riders in the 12th grade is closest to 135% of the number of female bus riders, so the answer is \boxed{12}.
</Plan_3>

<Execution_3>
<execution_3.1>
Males: 34.4% of 300 = 103.2
Females: 33.1% of 300 = 99.3
1.35 * 99.3 = ?

Calculating 1.35*99.3:

First, 1*99.3 = 99.3
0.3*99.3 = 29.79
0.05*99.3 = 4.965
Adding up: 99.3 + 29.79 = 129.09 + 4.965 = 134.055

So 1.35*F = 134.055. The actual M is 103.2. The difference is |103.2 - 134.055| = 30.855. The ratio M/F is 103.2 / 99.3 ≈ 1.04, which is still way below 1.35.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1, execution_2.2, execution_2.3, execution_2.4:
**Final Answer**
\boxed{12}
</Plan_4>

<Execution_4>
No prompt to execute.
</Execution_4>

<Final_answer>
Therefore, the grade where the number of male bus riders is closest to $135\%$ of the number of female bus riders is the $\boxed{12^{\mathrm{th}}}$ grade.
</Final_answer>
</think>