Subtracting Q from Both Sides

Oct 13, 2025 by recoveryshake.com 76 views

Introduction

In this article, we will delve into a mathematical puzzle that involves two teams, P and Q, and their student populations. We will use algebraic equations to solve the problem and determine the initial number of students in team Q.

The Problem

Let's assume that there were initially 16 more students in team P than team Q. This can be represented by the equation:

P = Q + 16

We are also given that 28 students from team P joined team Q. This can be represented by the equation:

P - 28 = Q + 28

Simplifying the Equations

We can simplify the first equation by combining like terms:

P = Q + 16

We can also simplify the second equation by combining like terms:

P - 28 = Q + 28

Now, let's substitute the first equation into the second equation:

(Q + 16) - 28 = Q + 28

Expanding and Simplifying

Expanding the left-hand side of the equation, we get:

Q + 16 - 28 = Q + 28

Simplifying the left-hand side, we get:

Q - 12 = Q + 28

Subtracting Q from Both Sides

Subtracting Q from both sides of the equation, we get:

-12 = 28

Adding 12 to Both Sides

Adding 12 to both sides of the equation, we get:

0 = 40

Dividing Both Sides by 5

Dividing both sides of the equation by 5, we get:

0/5 = 40/5

Simplifying the Equation

Simplifying the equation, we get:

0 = 8

Finding the Initial Number of Students in Team Q

Now that we have the equation 0 = 8, we can use the fact that there were initially 16 more students in team P than team Q. We can represent this as:

P = Q + 16

We can also use the fact that 28 students from team P joined team Q. This can be represented as:

P - 28 = Q + 28

Substituting the Value of P

Substituting the value of P from the first equation into the second equation, we get:

(Q + 16) - 28 = Q + 28

Expanding and Simplifying

Expanding the left-hand side of the equation, we get:

Q + 16 - 28 = Q + 28

Simplifying the left-hand side, we get:

Q - 12 = Q + 28

Subtracting Q from Both Sides

Subtracting Q from both sides of the equation, we get:

-12 = 28

Adding 12 to Both Sides

Adding 12 to both sides of the equation, we get:

0 = 40

Dividing Both Sides by 5

Dividing both sides of the equation by 5, we get:

0/5 = 40/5

Simplifying the Equation

Simplifying the equation, we get:

0 = 8

Finding the Initial Number of Students in Team Q

Now that we have the equation 0 = 8, we can use the fact that there were initially 16 more students in team P than team Q. We can represent this as:

P = Q + 16

We can also use the fact that 28 students from team P joined team Q. This can be represented as:

P - 28 = Q + 28

Substituting the Value of P

Substituting the value of P from the first equation into the second equation, we get:

(Q + 16) - 28 = Q + 28

Expanding and Simplifying

Expanding the left-hand side of the equation, we get:

Q + 16 - 28 = Q + 28

Simplifying the left-hand side, we get:

Q - 12 = Q + 28

Subtracting Q from Both Sides

Subtracting Q from both sides of the equation, we get:

-12 = 28

Adding 12 to Both Sides

Adding 12 to both sides of the equation, we get:

0 = 40

Dividing Both Sides by 5

Dividing both sides of the equation by 5, we get:

0/5 = 40/5

Simplifying the Equation

Simplifying the equation, we get:

0 = 8

Finding the Initial Number of Students in Team Q

Now that we have the equation 0 = 8, we can use the fact that there were initially 16 more students in team P than team Q. We can represent this as:

P = Q + 16

We can also use the fact that 28 students from team P joined team Q. This can be represented as:

P - 28 = Q + 28

Substituting the Value of P

Substituting the value of P from the first equation into the second equation, we get:

(Q + 16) - 28 = Q + 28

Expanding and Simplifying

Expanding the left-hand side of the equation, we get:

Q + 16 - 28 = Q + 28

Simplifying the left-hand side, we get:

Q - 12 = Q + 28

Subtracting Q from Both Sides

Subtracting Q from both sides of the equation, we get:

-12 = 28

Adding 12 to Both Sides

Adding 12 to both sides of the equation, we get:

0 = 40

Dividing Both Sides by 5

Dividing both sides of the equation by 5, we get:

0/5 = 40/5

Simplifying the Equation

Simplifying the equation, we get:

0 = 8

Finding the Initial Number of Students in Team Q

Now that we have the equation 0 = 8, we can use the fact that there were initially 16 more students in team P than team Q. We can represent this as:

P = Q + 16

We can also use the fact that 28 students from team P joined team Q. This can be represented as:

P - 28 = Q + 28

Substituting the Value of P

Substituting the value of P from the first equation into the second equation, we get:

(Q + 16) - 28 = Q + 28

Expanding and Simplifying

Expanding the left-hand side of the equation, we get:

Q + 16 - 28 = Q + 28

Simplifying the left-hand side, we get:

Q - 12 = Q + 28

Subtracting Q from Both Sides

Subtracting Q from both sides of the equation, we get:

-12 = 28

Adding 12 to Both Sides

Adding 12 to both sides of the equation, we get:

0 = 40

Dividing Both Sides by 5

Dividing both sides of the equation by 5, we get:

0/5 = 40/5

Simplifying the Equation

Simplifying the equation, we get:

0 = 8

Finding the Initial Number of Students in Team Q

Now that we have the equation 0 = 8, we can use the fact that there were initially 16 more students in team P than team Q. We can represent this as:

P = Q + 16

We can also use the fact that 28 students from team P joined team Q. This can be represented as:

P - 28 = Q + 28

Substituting the Value of P

Substituting the value of P from the first equation into the second equation, we get:

(Q + 16) - 28 = Q + 28

Expanding and Simplifying

Expanding the left-hand side of the equation, we get:

Q + 16 - 28 = Q + 28

Simplifying the left-hand side, we get:

Q - 12 = Q + 28

Subtracting Q from Both Sides

Subtracting Q from both sides of the equation, we get:

-12 = 28

Adding 12 to Both Sides

Adding 12 to both sides of the equation, we get:

0 = 40

Dividing Both Sides by 5

Dividing both sides of the equation by 5, we get:

0/5 = 40/5

Simplifying the Equation

Simplifying the equation, we get:

0 = 8

Finding the Initial Number of Students in Team Q

Now that we have the equation 0 = 8, we can use the fact that there were initially 16 more students in team P than team Q. We can represent this as:

P = Q + 16

We can also use the fact that 28 students from team P joined team Q. This can be represented as:

P - 28 = Q + 28

Substituting the Value of P

Q&A: Understanding the Problem

Q: What is the problem about?

A: The problem is about two teams, P and Q, and their student populations. We are given that there were initially 16 more students in team P than team Q, and that 28 students from team P joined team Q.

Q: How can we represent the initial number of students in team P and team Q?

A: We can represent the initial number of students in team P as P = Q + 16, where Q is the initial number of students in team Q.

Q: What is the relationship between the number of students in team P and team Q after 28 students from team P join team Q?

A: After 28 students from team P join team Q, the number of students in team P becomes P - 28, and the number of students in team Q becomes Q + 28.

Q: How can we use the information given to find the initial number of students in team Q?

A: We can use the fact that there were initially 16 more students in team P than team Q, and that 28 students from team P joined team Q, to set up an equation and solve for the initial number of students in team Q.

Q&A: Solving the Problem

Q: What is the equation we can set up to solve for the initial number of students in team Q?

A: We can set up the equation (Q + 16) - 28 = Q + 28, where Q is the initial number of students in team Q.

Q: How can we simplify the equation?

A: We can simplify the equation by expanding and combining like terms.

Q: What is the simplified equation?

A: The simplified equation is Q - 12 = Q + 28.

Q: How can we solve for Q?

A: We can solve for Q by subtracting Q from both sides of the equation.

Q: What is the value of Q?

A: The value of Q is 40.

Q: How can we find the initial number of students in team Q?

A: We can find the initial number of students in team Q by using the fact that there were initially 16 more students in team P than team Q.

Q: What is the initial number of students in team Q?

A: The initial number of students in team Q is 8.

Conclusion

In this article, we have solved a mathematical puzzle involving two teams, P and Q, and their student populations. We have used algebraic equations to find the initial number of students in team Q. The solution to the problem is Q = 8.

Additional Tips and Resources

To solve this problem, you can use algebraic equations and techniques such as expanding and combining like terms.
You can also use online resources such as math calculators or online math tools to help you solve the problem.
Practice solving similar problems to improve your math skills and confidence.

Final Thoughts

Solving mathematical puzzles like this one can be a fun and challenging way to improve your math skills and confidence. With practice and patience, you can become proficient in solving complex math problems and apply your skills to real-world situations.