Calculating Initial Funds A Step-by-Step Mathematical Solution

In this mathematical problem, we delve into the world of fractions and financial calculations. This article aims to solve the question: how much money did a person initially have, given the information about their spending habits and the remaining amount? We'll break down the problem step-by-step, employing a clear and concise approach to ensure understanding for everyone, regardless of their mathematical background. This article provides a detailed solution and explanation, making it a valuable resource for students, educators, and anyone interested in problem-solving techniques in mathematics.

The core of our exploration lies in a scenario where an individual receives a sum of money and proceeds to spend a portion of it on various items. The details of their expenditure are as follows:

• ${ \frac{1}{9} }$ of the money is spent on stationary.
• ${ \frac{4}{9} }$ of the money is used to top up a bus pass.
• ${ \frac{1}{4} }$ of the remaining money is spent on snacks.

After all these expenses, the individual is left with P 27. The central question we aim to answer is: What was the initial sum of money the person had?

To solve this problem effectively, we'll employ a step-by-step approach, breaking it down into manageable parts. This will allow us to track the money spent and the money remaining after each transaction. Understanding the order of operations and how each expenditure affects the remaining balance is crucial. We'll begin by focusing on the fractions and their relationship to the initial sum.

Step 1: Calculate the Total Fraction Spent on Stationary and Bus Pass

The first step in solving this problem is to determine the total fraction of money spent on stationary and the bus pass. The individual spent ${ \frac{1}{9} }$ of their money on stationary and ${ \frac{4}{9} }$ on the bus pass. To find the total fraction spent on these two items, we need to add these fractions together. This involves adding fractions with the same denominator, which is a straightforward process. This calculation will give us a clear picture of how much of the initial sum was used for these essential expenses. Understanding this combined fraction is crucial for determining the remaining amount before the snack purchase.

Calculation:

To calculate the total fraction spent on stationary and the bus pass, we simply add the individual fractions:

$\frac{1}{9} + \frac{4}{9} = \frac{1 + 4}{9} = \frac{5}{9}$

This means that the individual spent a total of ${ \frac{5}{9} }$ of their money on stationary and the bus pass.

Step 2: Determine the Fraction of Money Remaining After Stationary and Bus Pass Purchases

Following the expenditure on stationary and the bus pass, it's crucial to determine the fraction of money that remains. Since the individual initially had the whole amount, which can be represented as ${ \frac{9}{9} }$ (since ${ \frac{9}{9} }$ is equal to 1), we need to subtract the fraction spent (${ \frac{5}{9} }$) from this whole. This calculation will reveal the fraction of the initial sum that was available before the individual considered buying snacks. Understanding this remaining fraction is key to calculating the amount spent on snacks and, ultimately, the initial sum of money.

Calculation:

To find the fraction of money remaining, we subtract the fraction spent from the whole:

$\frac{9}{9} - \frac{5}{9} = \frac{9 - 5}{9} = \frac{4}{9}$

Therefore, ${ \frac{4}{9} }$ of the initial sum remained after the stationary and bus pass purchases.

Step 3: Calculate the Amount Spent on Snacks

The next part of the problem states that the individual spent ${ \frac{1}{4} }$ of the remaining money on snacks. From the previous step, we know that ${ \frac{4}{9} }$ of the initial sum was remaining. To find the amount spent on snacks, we need to calculate ${ \frac{1}{4} }$ of ${ \frac{4}{9} }$. This involves multiplying these two fractions together. The result will represent the fraction of the initial sum that was used for snacks. This step is vital in determining how much money was left after all the expenses.

Calculation:

To calculate the fraction of money spent on snacks, we multiply the fractions:

$\frac{1}{4} \times \frac{4}{9} = \frac{1 \times 4}{4 \times 9} = \frac{4}{36}$

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

$\frac{4}{36} = \frac{4 \div 4}{36 \div 4} = \frac{1}{9}$

So, the individual spent ${ \frac{1}{9} }$ of the initial sum on snacks.

Step 4: Determine the Total Fraction of Money Spent

Now that we know the fractions spent on stationary, the bus pass, and snacks, we can calculate the total fraction of the initial sum that was spent. This involves adding the fractions representing each expenditure: ${ \frac{1}{9} }$ for stationary, ${ \frac{4}{9} }$ for the bus pass, and ${ \frac{1}{9} }$ for snacks. Adding these fractions will give us a comprehensive view of the proportion of the initial sum that was used, which is crucial for determining the remaining fraction and, ultimately, the initial amount.

Calculation:

To find the total fraction of money spent, we add the individual fractions:

$\frac{1}{9} + \frac{4}{9} + \frac{1}{9} = \frac{1 + 4 + 1}{9} = \frac{6}{9}$

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

$\frac{6}{9} = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}$

Therefore, the individual spent a total of ${ \frac{2}{3} }$ of the initial sum.

Step 5: Calculate the Fraction of Money Remaining

After determining the total fraction of money spent, we need to calculate the fraction of the initial sum that remains. This is a crucial step because we know the actual amount of money remaining (P 27), and we can use this information to find the value of the initial sum. To find the remaining fraction, we subtract the total fraction spent (${ \frac{2}{3} }$) from the whole, which is represented as ${ \frac{3}{3} }$. This calculation will give us the fraction that corresponds to the P 27 remaining.

Calculation:

To find the fraction of money remaining, we subtract the fraction spent from the whole:

$\frac{3}{3} - \frac{2}{3} = \frac{3 - 2}{3} = \frac{1}{3}$

So, ${ \frac{1}{3} }$ of the initial sum remained after all the expenses.

Step 6: Determine the Initial Sum of Money

We've reached the final step in solving the problem: determining the initial sum of money. We know that ${ \frac{1}{3} }$ of the initial sum is equal to P 27. To find the whole amount (the initial sum), we need to determine what one whole unit (${ \frac{3}{3} }$) represents. This can be achieved by multiplying the remaining amount (P 27) by the inverse of the remaining fraction (which is 3). This calculation will provide us with the total amount of money the individual had at first.

Calculation:

To find the initial sum, we multiply the remaining amount by the inverse of the remaining fraction:

P 27 ${ \times 3 = P 81 }$

Therefore, the individual initially had P 81.

After meticulously working through the problem step by step, we have arrived at the solution. The individual initially had P 81. This answer was derived by carefully tracking the expenditures, calculating the fractions spent and remaining, and using the final amount left to determine the initial sum. This problem highlights the importance of understanding fractions and their applications in real-world financial scenarios.

In conclusion, this problem provides a valuable exercise in applying mathematical principles to everyday situations. By breaking down the problem into smaller, manageable steps, we were able to systematically calculate the initial sum of money. This approach not only solves the specific problem at hand but also reinforces the importance of logical thinking and problem-solving skills. Understanding fractions and their operations is crucial in various aspects of life, and this example serves as a practical illustration of their significance. This problem-solving approach can be applied to a wide range of similar scenarios, making it a valuable tool for anyone seeking to improve their mathematical abilities.