Problem 45 - Algebra 2

Question

REVIEW Mandy went shopping. She spent two-fifths of her money in the first store. She spent three-fifths of what she had left in the next store. In the last store she visited, she spent three-fourths of the money she had left. When she finished shopping, Mandy had \(\$ 6 .\) How much money in dollars did Mandy have when she started shopping? $$ \begin{array}{lll}{\mathbf{F}} & {\$ 16} & {\mathbf{H}} & {\$ 100} \\\ {\mathbf{G}} & {\$ 56} & {\mathbf{J}} & {\$ 106}\end{array} $$

Step-by-Step Solution

Answer

Mandy initially had $100.

Step 1: Define initial amount of money

Let's denote the initial amount of money Mandy had as \( x \). Our goal is to find the value of \( x \).

Step 2: Calculate money spent and remaining after first store

Mandy spent \( \frac{2}{5} \) of her initial amount in the first store. The amount spent is \( \frac{2}{5}x \). Thus, she is left with \( x - \frac{2}{5}x = \frac{3}{5}x \).

Step 3: Calculate money spent and remaining after second store

In the next store, she spends \( \frac{3}{5} \) of the leftover. Therefore, she spends \( \frac{3}{5} \times \frac{3}{5}x = \frac{9}{25}x \). The amount left is \( \frac{3}{5}x - \frac{9}{25}x = \frac{6}{25}x \).

Step 4: Calculate money spent and remaining after last store

At the last store, Mandy spends \( \frac{3}{4} \) of her remaining money. She spends \( \frac{3}{4} \times \frac{6}{25}x = \frac{18}{100}x = \frac{9}{50}x \). So, she is left with \( \frac{6}{25}x - \frac{9}{50}x = \frac{3}{50}x \).

Step 5: Solve equation for initial amount

We are told that after all her spending, she has $6 left, which equals \( \frac{3}{50}x \). The equation becomes \( \frac{3}{50}x = 6 \). Solve for \( x \):\[ x = 6 \times \frac{50}{3} = 100 \]

Step 6: Verify calculations with options

Verify whether \( x = 100 \) is among the options given. The solution is in option \( H \).

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. In this exercise, Mandy's spending in each store is described as a fraction of the money she has at each stage of her shopping. A fraction consists of two parts:

• The numerator (the top number), which tells us how many parts we have.
• The denominator (the bottom number), which tells us how many parts make up a whole.

For example, when Mandy spends \(\frac{2}{5}\) of her money in the first store, it means out of every five parts, she spends two.
Understanding fractions is crucial for dividing amounts into parts, which is a fundamental skill in many areas, including algebra and finance. The same method is applied throughout the problem to determine how much money remains after she spends in each store.

Linear Equations

Linear equations involve equations of the form \( ax + b = 0 \). These equations are used to find an unknown value. In the context of this problem, the linear equation helps determine Mandy's initial amount of money.
Mandy's remaining balance calculation after all her spending is summarized by the equation \(\frac{3}{50}x = 6\). This equation is linear because it represents a straight-line relationship between \(x\) and the constants.

• To solve, you isolate \(x\) by performing operations such as multiplication or division.
• The goal is to have \(x\) on one side of the equation, resulting in the solution \(x = 100\).

By understanding and applying the principles of linear equations, you can solve problems involving relationships of unknowns and enumerated values effectively.

Word Problems

Word problems are real-life scenarios presented in a textual format. They require translating words into mathematical expressions to find a solution. In Mandy's shopping scenario, we translate the descriptions of her spending into mathematical terms.
When dealing with word problems:

• Identify the unknowns and define them with variables. Here, we define Mandy's initial amount as \(x\).
• Break down the problem step-by-step, reflecting each action in the story with a mathematical operation.
• Utilize the context provided to set up equations that model the situation, like calculating after each shopping step to find the remaining money.

Word problems, although appearing complex, become manageable when each detail is addressed clearly and mathematically.

Monetary Transactions

Monetary transactions refer to the exchange or spending of money, crucial for budgeting and financial planning. This problem illustrates Mandy's transactions over several stores.
To approach transaction problems, consider these steps:

• Calculate how much money is spent and what remains after each transaction.
• Understand the monetary terms: "spend" reduces the total money, while "remaining" refers to what is left.
• Translate descriptions into percentage or fraction terms to facilitate calculations.

This problem shows that even simple monetary spendings can require careful calculation, particularly when multiple transactions affect the remaining balance sequentially, leading to the conclusion.