Chapter 46: Chapter 46

Problem 1263

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$$\$ 10,000$$ is to be invested for one year. A part of the $$\$ 10,000$$ is invested at $6 \%$ and the remainder at $7 \%$. The income from the $$\$ 10,000$$ is $$\$ 640$$ per year. Find the amount invested at each rate.

Short Answer

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The amount invested at 6% is $$\$ 6,000$$ and the amount invested at 7% is $$\$ 4,000$$.

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TABLE OF CONTENTS

Step 1: Assigning Variables

Let x be the amount invested at 6%, and y be the amount invested at 7%. Our goal is to find the values of x and y.

Step 2: Set up the system of linear equations

First, since the total investment is $10,000, we can write that as the sum of the amount invested at 6% and the amount invested at 7%. That gives us the first equation: $x + y = 10000$ Now, we know that the total income is $640. The income from the amount invested at 6% is 6% of x, or $\frac{6}{100}x$. The income from the amount invested at 7% is 7% of y, or $\frac{7}{100}y$. So the total income can be represented as: $\frac{6}{100}x + \frac{7}{100}y = 640$

Step 3: Solve the system of linear equations

Now we have two equations with two variables. Our system of linear equations looks like this: 1) $x + y = 10000$ 2) $\frac{6}{100}x + \frac{7}{100}y = 640$ To solve the system, we can first multiply equation (2) by 100 to get rid of the fractions: 2) $6x + 7y = 64000$ Now we can solve equation (1) for x: $x = 10000 - y$ And substitute this expression back into equation 2: $6(10000 - y) + 7y = 64000$

Step 4: Solve for y

Now we have an equation with one variable: $60000 - 6y + 7y = 64000$ Solve for y: $y = 4000$

Step 5: Find x

Now we know the value of y, we can plug it back into the expression for x: $x = 10000 - 4000$ $x = 6000$

Step 6: Interpret the result

We have found that $x = 6000$ and $y = 4000$. This means that $$\$ 6,000$$ was invested at 6% and $$\$ 4,000$$ was invested at 7%.

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Chapter 46: Chapter 46

$$\$ 10,000$$ is to be invested for one year. A part of the $$\$ 10,000$$ is invested at \(6 \%\) and the remainder at \(7 \%\). The income from the $$\$ 10,000$$ is $$\$ 640$$ per year. Find the amount invested at each rate.

Short Answer

Step by step solution

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Step 1: Assigning Variables

Step 2: Set up the system of linear equations

Step 3: Solve the system of linear equations

Step 4: Solve for y

Step 5: Find x

Step 6: Interpret the result

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