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Chapter 2: Problem 44

Sarah's stock investment grew \(28 \%\) to \(\$ 448 .\) How much did she invest?

Short Answer

Expert verified
Sarah originally invested \( \$350 \).

Step by step solution

01

Understand the Problem

You are given the final value of Sarah's stock investment after it has grown by 28%. You need to determine the original amount she invested.
02

Set Up the Equation

Let the initial investment be represented by \( x \). The problem states that her investment grew by 28% to become \( \$448 \). This can be written as \( x + 0.28x = 448 \).
03

Simplify the Equation

Combine like terms in the equation: \( 1.28x = 448 \).
04

Solve for \( x \)

To find \( x \), divide both sides of the equation by 1.28: \[ x = \frac{448}{1.28} \]
05

Calculate the Result

Perform the division: \[ x = 350 \]
06

Confirm the Result

Sarah's original investment is \( \$350 \). This can be confirmed by calculating 28% of 350 (which is 98) and adding it to 350, resulting in 448.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

percentage increase
When dealing with investments and savings, understanding percentage increase is crucial. In Sarah's case, her investment grew by 28%. This means that the value of her investment became 28% more than it originally was.
To understand the concept of percentage increase:
  • Calculate 28% of the original amount.
  • Add this amount to the original investment.

In mathematical terms, if the original amount is represented by 鈥榵鈥�, then the total amount after a 28% increase would be:
\text {Total Amount} = \text {Original Amount} + \text {Percentage Increase} \times \text {Original Amount}.
In this case, this can be written as:
\(x + 0.28x = 448\).
solving equations
Solving equations is an essential skill in mathematics, often used to find unknown values. For instance, in Sarah's investment problem, we need to solve the equation: \(x + 0.28x = 448\) to find \(x\), the initial investment amount.
  • First, combine like terms to simplify the equation: \(1.28x = 448\).
  • Next, isolate the variable \(x\) by dividing both sides of the equation by 1.28: \(x = \frac{448}{1.28}\).

Now, we just need to perform the division, which gives us \(x = 350\). This means Sarah's initial investment was \(\$350\).
investment growth
Investment growth refers to the increase in value of the investment over time. For Sarah, the growth amounted to a 28% increase from her original investment.
To determine the initial investment from the final amount, we work backwards from the known percentage increase.
  • First, denote the original investment by \(x\).
  • Add the growth percentage to the initial amount: \(x + 0.28x = 448\).
  • Combine like terms to get \(1.28x = 448\).
  • Isolate \(x\) by dividing by 1.28.

This straightforward process helps us find that Sarah's initial investment was \(\$350\). Thus, by understanding investment growth, we can make more informed financial decisions.

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