Understanding the Problem of Children in Classes

The problem describes the number of children in two different classes, let's call them Class A and Class B. We are given two scenarios where children move between the classes, and based on these movements, we need to find the initial ratio of the number of children in Class A to the number of children in Class B.

Let's denote the initial number of children in Class A as \(A\) and the initial number of children in Class B as \(B\).

Setting Up Equations from Class Scenarios

We can translate the given information into algebraic equations based on the two scenarios described:

Scenario 1: 5 children from Class A join Class B.

• Initial number of children: Class A has \(A\), Class B has \(B\).
• After movement: Class A has \(A - 5\), Class B has \(B + 5\).
• Condition: The number of children in both classes is the same.
• Equation from Scenario 1: \(A - 5 = B + 5\)
• We can simplify this equation: \(A - B = 5 + 5\), which gives us \(A - B = 10\). Let's call this Equation (1).

Scenario 2: 25 children from Class B join Class A.

• Initial number of children: Class A has \(A\), Class B has \(B\).
• After movement: Class A has \(A + 25\), Class B has \(B - 25\).
• Condition: The number of children in Class A becomes double the number of children in Class B.
• Equation from Scenario 2: \(A + 25 = 2 \times (B - 25)\)
• We can simplify this equation: \(A + 25 = 2B - 50\). Rearranging the terms to group A and B: \(A - 2B = -50 - 25\), which gives us \(A - 2B = -75\). Let's call this Equation (2).

Solving the System of Equations

Now we have a system of two linear equations with two variables, \(A\) and \(B\):

Equation (1): \(A - B = 10\)

Equation (2): \(A - 2B = -75\)

We can solve this system using methods like substitution or elimination. Let's use the elimination method by subtracting Equation (2) from Equation (1).

• Subtract Equation (2) from Equation (1):
• \((A - B) - (A - 2B) = 10 - (-75)\)
• \(A - B - A + 2B = 10 + 75\)
• \(B = 85\)

So, the initial number of children in Class B is 85.

Now substitute the value of \(B\) (which is 85) into Equation (1) to find the value of \(A\):

• Substitute \(B = 85\) into \(A - B = 10\):
• \(A - 85 = 10\)
• \(A = 10 + 85\)
• \(A = 95\)

So, the initial number of children in Class A is 95.

Calculating the Ratio of Children

The question asks for the ratio of the number of children in Class A to those in Class B. This is the ratio \(A : B\).

Ratio \(A : B = 95 : 85\)

To simplify this ratio, we find the greatest common divisor (GCD) of 95 and 85. Both numbers are divisible by 5.

• Divide 95 by 5: \(95 \div 5 = 19\)
• Divide 85 by 5: \(85 \div 5 = 17\)

The simplified ratio is \(19 : 17\).

Conclusion

Based on the given conditions and our calculations, the initial ratio of the number of children in Class A to those in Class B is 19 : 17.

Revision Table: Key Concepts

Additional Information: Systems of Linear Equations

A system of linear equations involves two or more linear equations that are considered together. The solution to a system is a set of values for the variables that satisfies all equations in the system simultaneously.

Common methods for solving a system of two linear equations with two variables include:

• Substitution Method: Solve one equation for one variable in terms of the other, then substitute that expression into the other equation.
• Elimination Method: Multiply one or both equations by constants so that the coefficients of one variable are opposites, then add the equations to eliminate that variable.
• Graphical Method: Graph both equations on the same coordinate plane. The point of intersection of the lines is the solution.

In this problem, we used the elimination method, which was straightforward because subtracting the two equations directly eliminated the variable A.