Ana invests

​$13 comma 00013,000

received from her grandmother in three ways. With one​ part, she buys savings bonds at an interest rate of

2.52.5​%

per year. She uses the second​ part, which amounts to

66

times the​ first, to buy mutual funds that offer a return of

77​%

per year. She puts the rest of the money into a money market account paying

4.54.5​%

annual interest. The first​ year, her investments bring a return of

$ 715.$715.

How much did she invest each​ way? (Hint: For

2.52.5​%

​interest, the amount of interest paid on x dollars is

0.0250.025​x.) Write system of linear equations

Let's define the following variables to represent the amounts Ana invests in each of the three ways:

- Let $ x $ be the amount Ana invests in the savings bonds (at 2.5% interest).
- Let $ 6x $ be the amount Ana invests in mutual funds (which is 6 times the amount she invests in the savings bonds, at 7% interest).
- Let $ y $ be the amount Ana invests in the money market account (at 4.5% interest).

Ana has a total of $13,000 to invest. So, the first equation is:

\[
x + 6x + y = 13,000
\]
Simplifying this, we get:
\[
7x + y = 13,000 \tag{1}
\]

The interest (or return) Ana earns in the first year is $715. The interest from each investment is calculated as follows:

- The interest from the savings bonds is $ 0.025 \cdot x $.
- The interest from the mutual funds is $ 0.07 \cdot 6x $.
- The interest from the money market account is $ 0.045 \cdot y $.

The total interest earned is $715, so the second equation is:

\[
0.025x + 0.07(6x) + 0.045y = 715
\]
Simplifying the second equation:
\[
0.025x + 0.42x + 0.045y = 715
\]
\[
0.445x + 0.045y = 715 \tag{2}
\]

### System of Equations
We now have the following system of linear equations:
1. $ 7x + y = 13,000 $
2. $ 0.445x + 0.045y = 715 $

These two equations can be solved to find the values of $ x $ (investment in savings bonds), $ 6x $ (investment in mutual funds), and $ y $ (investment in the money market account).

Would you like me to solve these equations for you?

---

### 5 Related Questions:
1. How do you solve a system of linear equations using substitution?
2. How do you solve a system of linear equations using elimination?
3. What is the importance of setting up proper variables when solving real-life problems?
4. How does interest compounding differ from simple interest in investments?
5. How do you calculate the total return on investments over multiple years?

**Tip:** When dealing with investment problems, always ensure the total sum of all parts matches the original investment amount.

Ana invests $13,000 received from her grandmother in three ways. With one part, she buys savings bonds at an interest rate of 2.5% per year. She uses the second part, which amounts to 6 times the first, to buy mutual funds that offer a return of 7% per year. She puts the rest of the money into a money market account paying 4.5% annual interest. The first year, her investments bring a return of $715. How much did she invest each way? (Hint: For 2.5% interest, the amount of interest paid on x dollars is 0.025x.) Write a system of linear equations.

Solve a system of linear equations to find how Ana invested $13,000 in three ways: savings bonds at 2.5%, mutual funds at 7%, and a money market account at 4.5%. Learn the detailed step-by-step solution to calculate how much she invested in each option, suitable for high school algebra students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation