When 5 children from class A join class B, the number of children in both classes is the same. If 25 children from B, join A, then the number of children in A becomes double the number of children in B. The ratio of the number of children in A to those in B is:
19 : 17
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
| Concept | Description | Application Here |
|---|---|---|
| Variables | Symbols representing unknown quantities (e.g., A, B). | Representing the initial number of children. |
| Linear Equation | An equation where variables have a power of 1 (e.g., \(Ax + By = C\)). | Derived from the problem scenarios. |
| System of Equations | A set of two or more equations with the same variables. | Used to find the values of A and B simultaneously. |
| Elimination Method | A method to solve systems of equations by adding or subtracting equations to eliminate a variable. | Used to solve for B and then A. |
| Ratio | A comparison of two quantities using division (e.g., a : b or a/b). | Expressing the relationship between the number of children in Class A and Class B. |
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.
Similar Questions
What is the solution of the following equations ?
2x + 3y = 12 and 3x − 2y = 5
Simplify \(\frac{1}{2+2p} + \frac{1}{2+2q}+\frac{1}{2+2r}\), where p = \(\frac{x}{y+z}\), if q = \(\frac{y}{z+x}\) and r = \(\frac{z}{x+y}\).
Simplify \(\rm {256x^4 - 16y^4} \over {(80x^2 - 20y^2)}{(16x^2 + 4y^2)}\).
For what value of m will the system of equations 17x + my + 102 = 0 and 23x + 299y + 138 = 0 have infinite number of solutions ?
If two LED TVs and one mobile phone cost ₹31,000, while two mobile phones and one LED TV cost ₹35,000, then the value of one mobile phone is:
Cost of 4 pens, 6 notebooks and 9 files is Rs. 305. Cost of 3 pens, 4 notebooks and 2 files is Rs. 145. What is the cost (in Rs) of 5 pens, 8 notebooks and 16 files?
Two positive numbers differ by 1280. When the greater number is divided by the smaller number, the quotient is 7 and the remainder is 50. The greater number is:
If x + y + 3 = 0, then find the value of x 3+ y 3- 9xy + 9.
If x = 4 + \( {\sqrt{15} }\) , what is the value of \(\rm \left( {{x^2}\, + \,\frac{1}{{{x^2}}}} \right)\) ?
The greater of the two numbers whose product is 900 and the sum exceeds their difference by 30 is:
Important Questions from Linear Equation in 2 Variable
If 2 x + 3 y = 17;
2 x+2 - 3 y+1 = 5
then the values of x and y are:
The solution of pair of linear equations \(\dfrac{1}{2}x+\dfrac{2}{3}y=-1,x-\dfrac{1}{3}y=3\) by the elimination method, is:
Kumar tried his skill at shooting at a fun fair. He has to hit the target and if he hits the target he gets 1 Rs. and if he misses he has to pay 50 paise. He attempted 25 shots and won 10 Rs. In how many did he hit the target?
The sum of two numbers is 66 and their difference is 22. What is the ratio of the two numbers?
Shyam spent half of his money and was left with as many as he had rupees before, but with half as many rupees as he had paise before. Which of the following is a possible amount of money he is left with?
