Solving Money Allocation Problems Through Algebraic Methods

Understanding the principles of money allocation and how to solve related problems using algebraic equations is a crucial skill in financial mathematics. In this article, we will explore different scenarios involving money allocation and solve them using algebraic methods. This will help you gain a deeper understanding of how to approach and solve similar problems efficiently.

Problem 1: If Alan spent 4/9 of his money but has 65 left, how much did he have to begin with?

Let's start by defining the total amount of money Alan had initially as M.

Step-by-step solution:

Define the variables and expressions: M the amount of money Alan had in the beginning Money spent 4/9 M Money left M - 4/9 M 5/9 M Set up the equation based on the given information:

According to the problem, the money left is 65. Therefore,

5/9 M 65

Solve for M:

In order to isolate M, we multiply both sides of the equation by 9/5.

Multiply both sides by 9/5:

M 65 * 9/5

Multiply the numbers:

M 117

Conclusion:

Alan had 117 units of money to begin with.

Problem 2: A man spends 3/8 of his money and saves the rest. He saves €720. How much did he have at the start?

This problem can be solved in a similar manner using algebraic methods.

Step-by-step solution:

Define the variables and expressions: x the amount of money the man had in the beginning Money spent 3/8 x Money saved 5/8 x Set up the equation based on the given information:

According to the problem, the money saved is 720. Therefore,

5/8 x 720

Solve for x:

In order to isolate x, we multiply both sides of the equation by 8/5.

Multiply both sides by 8/5:

x 720 * 8/5

Multiply the numbers:

x 1152

Conclusion:

The man had €1152 at the start.

Interpretation and Different Scenarios

Sometimes, a problem can have different interpretations, leading to multiple solutions. Let's explore one such case.

Interpretation 1:

Let the initial amount of money with Allen be x. According to the problem, x - 0.75x 2.50.

Set up the equation:

x - 0.75x 2.50

Solve for x:

x * (1 - 0.75) 2.50

x * 0.25 2.50

Multiply both sides by 4 to isolate x:

x 2.50 * 4

x 10

Conclusion:

The initial amount of money with Allen is €10.

Interpretation 2:

In this interpretation, 0.75 75/100. Therefore, Allen spent 75% of his money and now has 25 left. Let the initial amount be y.

Set up the equation:

0.25y 2.50

Solve for y:

Multiply both sides by 4 to isolate y:

y 2.50 * 4

y 10

Conclusion:

The initial amount with Allen is also €10.

Conclusion

In this article, we have explored different scenarios involving money allocation and solved them using algebraic methods. Understanding these techniques is essential for anyone dealing with financial mathematics. By applying these methods, you can solve a variety of problems and make informed financial decisions.