Understanding the Class Student Problem

The problem asks for the total number of students in a class based on two conditions related to grouping the remaining students when some are absent. Let's denote the total number of students in the class as \(T\).

Analyzing the Given Conditions

We are given two conditions:

1. If 3 students are absent, the rest of the students can be divided equally into 7 groups. This means that the number of students remaining (\(T - 3\)) is a multiple of 7. Mathematically, this can be written as \(T - 3 = 7k\), where \(k\) is an integer (the number of students in each of the 7 groups). This also implies \(T \equiv 3 \pmod{7}\).
2. If 7 students are absent, the rest of the students can be divided equally into 9 groups. This means that the number of students remaining (\(T - 7\)) is a multiple of 9. Mathematically, this can be written as \(T - 7 = 9m\), where \(m\) is an integer (the number of students in each of the 9 groups). This also implies \(T \equiv 7 \pmod{9}\).

We need to find a number \(T\) from the given options that satisfies both of these conditions.

Testing the Options

Let's check each option against the two conditions:

• Option 1: 115 students
• Condition 1: If 3 are absent, students remaining = \(115 - 3 = 112\). Is 112 divisible by 7? \(112 \div 7 = 16\). Yes, it is divisible by 7.
• Condition 2: If 7 are absent, students remaining = \(115 - 7 = 108\). Is 108 divisible by 9? \(108 \div 9 = 12\). Yes, it is divisible by 9.
• Since 115 satisfies both conditions, it is a possible answer.

• Option 2: 140 students
• Condition 1: If 3 are absent, students remaining = \(140 - 3 = 137\). Is 137 divisible by 7? \(137 \div 7 \approx 19.57\). No, it is not divisible by 7.
• Since the first condition is not met, 140 is not the correct answer. We don't need to check the second condition.

• Option 3: 87 students
• Condition 1: If 3 are absent, students remaining = \(87 - 3 = 84\). Is 84 divisible by 7? \(84 \div 7 = 12\). Yes, it is divisible by 7.
• Condition 2: If 7 are absent, students remaining = \(87 - 7 = 80\). Is 80 divisible by 9? \(80 \div 9 \approx 8.89\). No, it is not divisible by 9.
• Since the second condition is not met, 87 is not the correct answer.

• Option 4: 101 students
• Condition 1: If 3 are absent, students remaining = \(101 - 3 = 98\). Is 98 divisible by 7? \(98 \div 7 = 14\). Yes, it is divisible by 7.
• Condition 2: If 7 are absent, students remaining = \(101 - 7 = 94\). Is 94 divisible by 9? \(94 \div 9 \approx 10.44\). No, it is not divisible by 9.
• Since the second condition is not met, 101 is not the correct answer.

Based on checking the options, only 115 satisfies both given conditions.

Detailed Calculation Steps (Using Modular Arithmetic)

Let the total number of students be \(T\).

From the first condition, \(T - 3\) is a multiple of 7. So, \(T - 3 = 7k\) for some integer \(k\). This means \(T = 7k + 3\).

From the second condition, \(T - 7\) is a multiple of 9. So, \(T - 7 = 9m\) for some integer \(m\). This means \(T = 9m + 7\).

We are looking for a number \(T\) such that:

\(T \equiv 3 \pmod{7}\)

\(T \equiv 7 \pmod{9}\)

We can substitute the first equation into the second congruence:

\(7k + 3 \equiv 7 \pmod{9}\)

Subtract 3 from both sides:

\(7k \equiv 7 - 3 \pmod{9}\)

\(7k \equiv 4 \pmod{9}\)

To solve for \(k\), we need to find the multiplicative inverse of 7 modulo 9. We look for a number \(x\) such that \(7x \equiv 1 \pmod{9}\). By checking values: \(7 \times 1 = 7\), \(7 \times 2 = 14 \equiv 5\), \(7 \times 3 = 21 \equiv 3\), \(7 \times 4 = 28 \equiv 1\). So, the inverse is 4.

Multiply both sides of \(7k \equiv 4 \pmod{9}\) by 4:

\(4 \times 7k \equiv 4 \times 4 \pmod{9}\)

\(28k \equiv 16 \pmod{9}\)

Since \(28 \equiv 1 \pmod{9}\) and \(16 \equiv 7 \pmod{9}\):

\(1k \equiv 7 \pmod{9}\)

\(k \equiv 7 \pmod{9}\)

This means \(k\) can be written as \(k = 9m + 7\) for some integer \(m\).

Now substitute this expression for \(k\) back into the equation for \(T\):

\(T = 7k + 3\)

\(T = 7(9m + 7) + 3\)

\(T = 63m + 49 + 3\)

\(T = 63m + 52\)

The total number of students \(T\) must be of the form \(63m + 52\). We are looking for a positive number of students, so we test values for \(m \geq 0\).

• If \(m = 0\), \(T = 63(0) + 52 = 52\). 52 is not among the options.
• If \(m = 1\), \(T = 63(1) + 52 = 63 + 52 = 115\). 115 is among the options.
• If \(m = 2\), \(T = 63(2) + 52 = 126 + 52 = 178\). 178 is not among the options.

The smallest positive solution is 115, which is one of the options provided.

Summary of Results

Only the option with 115 students satisfies both conditions.

The total number of students in the class is 115.

Revision Table: Key Concepts

Additional Information: Solving Word Problems

Solving word problems often involves translating the given information into mathematical equations or inequalities. Here are some general steps:

1. Read Carefully: Understand what the problem is asking and identify all the given information and conditions.
2. Define Variables: Assign variables to the unknown quantities you need to find.
3. Translate to Math: Write equations or inequalities based on the relationships described in the problem. Look for keywords like "is," "sum," "difference," "product," "quotient," "more than," "less than," "can be divided equally into" (indicating divisibility).
4. Solve the Equations: Use algebraic techniques, checking options, or number theory methods (like modular arithmetic) to find the value(s) of the variable(s).
5. Check the Solution: Make sure your answer makes sense in the context of the original word problem and satisfies all the given conditions.

In this specific problem, the condition "can be divided equally into N groups" directly implies that the number being divided is a multiple of N. The conditions "if 3 students are absent" means we consider the total number minus 3, and so on.