The total number of students in three classes of a school is 333. The numbers of students in class 1 and class 2 are in the ratio 3 : 5, and the numbers of students in class 2 and class 3 are in the ratio 7 : 11. What is the strength of the class that has the highest number of students?

Introduction / Context:
This question is about combining two ratios involving three related quantities and then using the total sum to find individual values. Here, the three quantities are the numbers of students in three different classes. We know the total number of students and two pairwise ratios. The goal is to determine which class has the highest strength and what that number is.

Given Data / Assumptions:

• Total students in three classes = 333.
• Let the numbers in classes 1, 2 and 3 be C1, C2 and C3.
• C1 : C2 = 3 : 5.
• C2 : C3 = 7 : 11.
• We must find the largest of C1, C2 and C3.

Concept / Approach:
The idea is to express all three class strengths using a single variable. From the first ratio, express C1 and C2 in terms of one parameter. From the second ratio, express C2 and C3 in terms of another parameter. Equating the two expressions for C2 links the parameters and allows us to write C1, C2 and C3 in terms of a single variable. With these expressions, we use the total of 333 to find the variable, then compute the exact strengths and identify the maximum.

Step-by-Step Solution:
From C1 : C2 = 3 : 5, let C1 = 3k and C2 = 5k. From C2 : C3 = 7 : 11, let C2 = 7m and C3 = 11m. Equate C2 values: 5k = 7m. So m = (5k) / 7. Then C3 = 11m = 11 * (5k / 7) = 55k / 7. Total students C1 + C2 + C3 = 333. So 3k + 5k + 55k / 7 = 333. Combine terms: (3k + 5k) = 8k, so we have 8k + 55k / 7. Express with denominator 7: (56k + 55k) / 7 = 333. So 111k / 7 = 333. Multiply both sides by 7: 111k = 2331. Thus k = 2331 / 111 = 21. Now C1 = 3k = 63, C2 = 5k = 105, C3 = 55k / 7 = 55 * 3 = 165. The highest strength is C3 = 165.

Verification / Alternative check:
Add the class sizes 63, 105 and 165 to confirm the total: 63 + 105 = 168, and 168 + 165 = 333, which matches the given total. The ratios C1 : C2 = 63 : 105 simplify to 3 : 5, and C2 : C3 = 105 : 165 simplify to 7 : 11. Since both pairwise ratios and the total are satisfied, the values are correct.

Why Other Options Are Wrong:
Numbers such as 125, 135 or 155 do not match the computed strengths of any class. They arise if the algebra is done incorrectly or if a wrong class is chosen as the largest. The value 105 is the size of class 2, not the maximum strength. Only 165 correctly corresponds to the largest class size.

Common Pitfalls:
Students may attempt to combine ratios by guessing or by simply adding them without aligning the common term C2. Others may forget to express all classes in terms of a single variable before applying the total. Carefully equating C2 and then using the total prevents such errors.

Final Answer:
The strength of the class with the highest number of students is 165.