Question

A man withdrew \(25.0 \%\) of his bank deposits and spent \(45.0 \%\) of the amount withdrawn on a car costing \(\$ 31,100 .\) How much money was originally in the bank?

Short answer

Originally, there was $138,000 in the bank.

Step-by-step solution

Determine the Fraction of the Withdrawn Amount Spent on the Car

The man spent 45% of the amount he withdrew on the car. First convert the percentage to a decimal by dividing by 100: 45% = 0.45. To find out how much of the original deposit this represents, we will denote this fraction of the withdrawn amount as X, where X is the amount spent on the car.

Calculate the Amount Withdrawn Using the Cost of the Car

Since the car cost \(31,100 and this represents 45% of the withdrawn amount, we can calculate the amount withdrawn (W) using the equation: W = \)31,100 / 0.45.

Calculate the Initial Deposit

The withdrawn amount represents 25% of the initial deposit. Convert this percentage to a decimal to get 0.25. Let's denote the initial deposit as D. From step 2 we have W, which is 25% of D, therefore we can write the equation D = W / 0.25 to solve for the initial deposit.

Key concepts

Percentage to Decimal Conversion

Understanding how to convert percentages to decimals is fundamental in mathematics as it simplifies various calculations, specifically when dealing with proportions and ratios. To convert a percentage to a decimal, you simply divide the percentage by 100. This process takes the percentage, which denotes a part out of 100, and turns it into a decimal, which can be used for further calculations. For example, converting 25.0% into a decimal involves dividing 25.0 by 100, resulting in 0.25.

This conversion is essential, especially when trying to determine parts of a whole in financial applications, as seen in the exercise related to bank deposits. Remember, moving the decimal two places to the left effectively divides a number by 100, making it a quick mental calculation that doesn't always require a calculator.

Calculating Initial Value Problems

When facing a problem that requires you to find an original amount or 'initial value', a structured approach can be invaluable. Calculating initial value problems generally involves reversing a process. First, identify what percentage of the initial value you are given or can calculate. Then, you convert this percentage to a decimal form to use it in equations. This allows you to work backwards from the known value to determine the unknown original amount.

For instance, if you know that 25.0% of the initial bank deposit was withdrawn, you convert that percentage to a decimal (0.25) to compute the total initial deposit. By dividing the withdrawn amount by this decimal, the initial value is retrieved. It's like retracing your steps to find a point you passed earlier when the percentage is indicative of that pass point.

Mathematical Problem-Solving

Tackling mathematical problems involves a series of logical steps and a clear understanding of the concepts at hand. Each problem is a puzzle, and using a step-by-step method can greatly enhance your ability to find the correct solution. Start by defining what you need to find and then identify the data provided. Next, convert any percentages into decimals as they are easier to work with in equations. Then, determine the relationships between the known values and what you need to find, often requiring you to set up an equation or set of equations.

Problems such as the bank deposit scenario offer practice in problem-solving by drawing on knowledge of percentages, decimals, and basic algebra. By breaking down the problem into smaller parts—converting the car's price into a percentage of the withdrawn amount, then converting the withdrawn amount into a percentage of the initial deposit—you create a clear path to the solution. Each step builds upon the previous one, ensuring a structured solution process.