Solving Mathematical Problems: A Beginner’s Guide with Examples and Solutions

Problem Statement

Imagine the scenario where Peter has five times as much money as David. If Peter gives David $28, both of them end up with the same amount. This involves setting up and solving a linear equation to find the initial amount of money that Peter had. Let's break this down step-by-step.

Let's denote the amount of money David has as D. Since Peter has five times as much money as David, we can express Peter's amount as 5D.

Mathematical Setup

According to the problem, if Peter gives $28 to David, both of them will have equal amounts of money. After the transaction, Peter's amount will be: 5D - 28

David's amount will be: D 28

Setting these two expressions equal, we have the equation:

5D - 28 D 28

Solving the Equation

Subtract D from both sides:

5D - D - 28 D 28 - D 4D - 28 28

Add 28 to both sides:

4D - 28 28 28 28 4D 56

Divide both sides by 4:

D 14

Now that we have D, we can find Peter's initial amount:

5D 5 × 14 70

Thus, Peter had $70 at the beginning.

Alternative Problem Statements and Solutions

Let's solve a similar but different problem using the same problem-solving technique.

Problem 2:

Consider the situation where the total amount of money is $240, and Peter has twice as much money as Andy subtracted from $240. Peter gives 2/5 of his money to Andy. Then, Andy gives 3/7 of his new total amount to Peter. After these transactions, both have $120 each.

Solution 2:

Let Peter's money be P and Andy's money be 240 - P.

Peter has left with 3/5P. Andy received 3/7(240 - P) from Peter.

After these transactions, Andy has (240 - P 3/7(240 - P)).

Peter gives 2/5 of his money, so he has 3/5P

Then, Andy gives 3/7 of his new total to Peter, which means Andy has 4/7(240 - P 3/7(240 - P)).

Setting these to be equal to 120:

3/5P 3/7(240 - P) 120

Also, 4/7(240 - P 3/7(240 - P)) 120.

Solving, we find:

P 100 (Peter's initial amount)

240 - P 140 (Andy's initial amount)

After calculations, both end up with $120 each.

Problem 3:

Let's analyze another problem where initially, Peter has 50 dollars, and Andy has 190 dollars. Peter gives Andy $20, leaving Peter with $30, and Andy with $210. Then, Andy gives Peter $90, making both have $120 each.

Solution 3:

Calculating, we deduce that initially, Andy had 210 dollars and Peter had 30 dollars.

Problem 4:

Finally, let's consider the problem where Peter has 50 dollars, and Andy has 190 dollars. Peter gives Andy $20, leaving Peter with $30, and Andy with $210. Then, Andy in return gives 90 dollars to Peter, making both have $120 each.

By solving the equation, we find that initially Andy had 210 dollars, and Peter had 30 dollars.

Conclusion

Solving such mathematical problems requires setting up the correct equations and solving them step-by-step. By breaking down complex problems into simple parts, we can find the right solution. These problems also help in strengthening our understanding of algebra and number theory concepts.