/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 471 Ms. Wong has a total of \(\$ 420... [FREE SOLUTION] | 91影视

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Chapter 19: Problem 471

Ms. Wong has a total of \(\$ 4200\) invested in securities \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). The rates of annual dividends are \(4 \%, 6 \%\) and \(5 \%\) respectively, yielding total annual dividends of \(\$ 214 .\) If the sum of \(\mathrm{A}\) and \(\mathrm{B}\) is twice \(\mathrm{C}\), find the amount invested in each security.

Short Answer

Expert verified
The amounts invested in securities A, B, and C are \(\$500\), \(\$2300\), and \(\$1400\), respectively.

Step by step solution

01

Set up equations based on the given information

Let's use the variables \(x\), \(y\), and \(z\) to represent the amounts invested in securities A, B, and C, respectively. The given information can be represented as equations: 1. Total investment: The sum of the amounts invested in each security should be \(4200: x+y+z=4200\) 2. Ratio of investments: The sum of the amounts invested in A and B is twice the amount invested in C: \(x+y = 2z\) 3. Total annual dividends(\(214\)): The amount earned from each security corresponds to the rate of dividends for the respective security multiplied by the invested amount: \(0.04x + 0.06y + 0.05z = 214\)
02

Solve the system of equations

We have the following system of equations: \( \begin{cases}x+y+z = 4200 \\x+y = 2z \\0.04x + 0.06y + 0.05z = 214\end{cases}\) First, let's eliminate either x or y from the given equations. We can use the second equation to eliminate x by rewriting it as \(x=2z-y\). Now we replace \(x\) in the first and third equations to get: \((2z-y)+y+z = 4200\) \(0.04(2z-y) + 0.06y + 0.05z = 214\) Simplify the above equations: \((2z-y)+y+z = 4200\) can be simplified as: \(3z = 4200\) \(0.04(2z-y) + 0.06y + 0.05z = 214\) can be simplified as: \(0.05z - 0.02y = 214\) Now we can solve for z: \(3z=4200 \Rightarrow z = 4200/3 = 1400\) Next, we'll solve for y using the simplified third equation: \(0.05(1400) - 0.02y = 214\) \(-0.02y = -46\) \(y = 2300\) Lastly, we'll solve for x using the equation \(x = 2z - y\): \(x = 2(1400) - 2300\) \(x = 500\) The amounts invested in securities A, B, and C are \(\$ 500, \)2300\(, and \)\$ 1400$, respectively.

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Key Concepts

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

Systems of Equations
When solving problems involving multiple unknown quantities, systems of equations are invaluable. Here, we had three unknowns: the amounts invested in securities A, B, and C. We used three equations based on the information given:
  • The sum of investments equals \(4200, represented by the equation: \(x + y + z = 4200\).
  • The sum of investments in A and B is twice as much as in C: \(x + y = 2z\).
  • The total dividends earned equate to \)214, described by: \(0.04x + 0.06y + 0.05z = 214\).
These equations facilitate finding the individual investments in each security. To solve the system, we used substitution by expressing \(x\) in terms of \(y\) and \(z\), then substituted into the other equations to focus on two variables initially. Once we solved one variable, we progressed to find the rest. When handling systems of equations, substitution or elimination methods are useful because they simplify the process by reducing the number of unknowns step by step.
Investment Problems
Investment problems often require us to determine variable quantities based on relationships given in the problem. In this scenario, Ms. Wong's overall investment is divided among three securities.
  • Each security has a different rate of annual dividends - a critical factor determining the total return.
  • These problems are composed of determining how much is invested at each rate to achieve a specific dividend return, balancing the logical relationship between different investments.
Investment problems typically require you to set up an equation for the total investment and another for income made from these investments. Solving for unknowns, you uncover how these investments align in real currency to meet the total dividend return specified. Here, understanding these fundamental relationships allowed us to balance Ms. Wong's investments efficiently, calculating precisely how much was allocated into securities A, B, and C.
Dividends Calculation
Dividends calculation plays a vital role in investment problems, as it bridges the investment amount with the eventual returns. For each security, the dividends were a percentage of the total amount invested.
  • Security A provided dividends at a rate of 4%, Security B at 6%, and Security C at 5% annually.
  • To calculate dividends from an investment, we use the formula: \(\text{Dividend} = \text{Investment} \times \text{Dividend Rate}\).
In this exercise, calculating dividends accurately was crucial to find out how Ms. Wong's total invested amount produced $214 annually. By establishing the equation: \(0.04x + 0.06y + 0.05z = 214\), we ensured each investment's contribution to the total dividend. Using these calculations, we verified how much each security contributed to the ongoing income from Ms. Wong's investments, revealing the correct distribution of funds among securities A, B, and C.

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Most popular questions from this chapter

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