Solving the Student Count Problem in a Coaching Institute

Let's break down this problem about the number of students in class 10 and class 12 in a coaching institute.

We are given two pieces of information:

1. The number of students in class 10 is $1/3$ more than the number of students in class 12.
2. The total number of students in class 10 and class 12 combined is 112.

Our goal is to find the exact number of students in class 10.

Setting Up Equations for Class 10 and Class 12 Students

To solve this, let's use variables to represent the unknown quantities:

• Let $C_{10}$ be the number of students in class 10.
• Let $C_{12}$ be the number of students in class 12.

Now, we can translate the given information into mathematical equations:

From the first statement, "the number of students in class 10 is $1/3$ more than the number of students in class 12", we can write:

$C_{10} = C_{12} + \frac{1}{3} C_{12}$

We can simplify this equation:

$C_{10} = \left(1 + \frac{1}{3}\right) C_{12}$

$C_{10} = \left(\frac{3}{3} + \frac{1}{3}\right) C_{12}$

$C_{10} = \frac{4}{3} C_{12}$ (Equation 1)

From the second statement, "the total number of students of class 10 and class 12 in the institute is 112", we get:

$C_{10} + C_{12} = 112$ (Equation 2)

Solving for the Number of Students in Class 10

We now have a system of two linear equations. We can use substitution to solve for the number of students in class 10 ($C_{10}$).

Substitute the expression for $C_{10}$ from Equation 1 into Equation 2:

$\left(\frac{4}{3} C_{12}\right) + C_{12} = 112$

Combine the terms involving $C_{12}$:

$\left(\frac{4}{3} + 1\right) C_{12} = 112$

$\left(\frac{4}{3} + \frac{3}{3}\right) C_{12} = 112$

$\frac{7}{3} C_{12} = 112$

Now, solve for $C_{12}$ by multiplying both sides by $\frac{3}{7}$:

$C_{12} = 112 \times \frac{3}{7}$

$C_{12} = \frac{112 \times 3}{7}$

Since $112 \div 7 = 16$, we have:

$C_{12} = 16 \times 3$

$C_{12} = 48$

So, there are 48 students in class 12.

Now that we have the number of students in class 12, we can find the number of students in class 10 using Equation 1 ($C_{10} = \frac{4}{3} C_{12}$):

$C_{10} = \frac{4}{3} \times 48$

$C_{10} = 4 \times \frac{48}{3}$

$C_{10} = 4 \times 16$

$C_{10} = 64$

Thus, there are 64 students in class 10.

Verification

Let's quickly check if our numbers satisfy the original conditions:

• Number of students in class 10 ($C_{10}$) = 64
• Number of students in class 12 ($C_{12}$) = 48
• Total students = $C_{10} + C_{12} = 64 + 48 = 112$. This matches the given total.
• Is $C_{10}$ ($64$) $1/3$ more than $C_{12}$ ($48$)?
  • $1/3$ of $C_{12}$ is $\frac{1}{3} \times 48 = 16$.
  • $C_{12} + \frac{1}{3} C_{12} = 48 + 16 = 64$.
  • Yes, 64 is indeed $1/3$ more than 48.

The calculated numbers satisfy both conditions of the problem.

Final Answer

The number of students in class 10 in that institute is 64.

Revision Table: Student Count Problem

Additional Information: Solving Word Problems

Word problems like this require careful reading and translation. Here are some tips:

• Read Carefully: Understand exactly what is given and what is being asked.
• Identify Unknowns: Determine the quantities you need to find.
• Assign Variables: Use letters to represent the unknowns.
• Formulate Equations: Write equations that express the relationships given in the problem. Look for keywords like "sum," "difference," "product," "quotient," "is" (means equals), "more than," "less than," "times," etc.
• Solve the Equations: Use algebraic techniques (substitution, elimination) to find the values of the variables.
• Check Your Answer: Plug your solution back into the original word problem (not just your equations) to make sure it makes sense and satisfies all conditions.
• State Your Answer Clearly: Make sure you answer the specific question asked in the problem.

In this problem, understanding "1/3 more than" was crucial. It means the original amount plus an additional one-third of that amount.