Country Day's scholarship fund receives a gift of $ 135000. The money is invested in stocks, bonds, and CDs. CDs pay 4.25 % interest, bonds pay 2.4 % interest, and stocks pay 8.7 % interest. Country day invests $ 15000 more in bonds than in CDs. If the annual income from the investments is $ 8112.5 , how much was invested in each vehicle?

Let's define the amounts invested in CDs, bonds, and stocks as follows:

- Let $ x $ represent the amount invested in CDs.
- Since the amount invested in bonds is $15,000 more than the amount invested in CDs, the amount invested in bonds will be $ x + 15000 $.
- The total investment is $135,000, so the amount invested in stocks will be the remaining amount, which is $ 135000 - (x + (x + 15000)) $.

The income from each investment is the principal amount times the respective interest rate:
- The CDs generate an income of $ 0.0425x $.
- The bonds generate an income of $ 0.024(x + 15000) $.
- The stocks generate an income of $ 0.087(135000 - (x + (x + 15000))) $.

The total income from all investments is $8,112.50, so we can set up the following equation:

\[
0.0425x + 0.024(x + 15000) + 0.087(135000 - (x + (x + 15000))) = 8112.5
\]

Now, we will solve this equation step by step.

### Step 1: Expand the equation
First, expand all the terms:

\[
0.0425x + 0.024(x + 15000) + 0.087(135000 - 2x - 15000) = 8112.5
\]

\[
0.0425x + 0.024x + 0.024 \times 15000 + 0.087 \times (135000 - 15000 - 2x) = 8112.5
\]

\[
0.0425x + 0.024x + 360 + 0.087(120000 - 2x) = 8112.5
\]

### Step 2: Simplify the terms
Now combine like terms:

\[
(0.0425 + 0.024)x + 360 + 0.087 \times 120000 - 0.087 \times 2x = 8112.5
\]

\[
0.0665x + 360 + 10440 - 0.174x = 8112.5
\]

\[
0.0665x - 0.174x + 10800 = 8112.5
\]

\[
-0.1075x + 10800 = 8112.5
\]

### Step 3: Solve for $ x $
Now isolate $ x $:

\[
-0.1075x = 8112.5 - 10800
\]

\[
-0.1075x = -1687.5
\]

\[
x = \frac{-1687.5}{-0.1075}
\]

\[
x = 15700
\]

### Step 4: Find the amounts invested in bonds and stocks
- The amount invested in CDs is $ x = 15700 $.
- The amount invested in bonds is $ x + 15000 = 15700 + 15000 = 30700 $.
- The amount invested in stocks is $ 135000 - (15700 + 30700) = 135000 - 46400 = 88600 $.

### Final Answer:
- CDs: $15,700
- Bonds: $30,700
- Stocks: $88,600

Would you like further clarification or details on any step?

---

Here are 5 related questions for further exploration:
1. How would the total income change if the interest rates for bonds were increased by 1%?
2. What would happen if the total investment amount were decreased by 10%?
3. How can you verify that the total interest income matches the given income of $8112.5?
4. If the amount invested in CDs doubled, what would the new income be?
5. How would the solution change if no extra money was invested in bonds?

**Tip:** When solving word problems involving investments, always double-check your interest calculations to ensure they match the problem's conditions.

Country Day's scholarship fund receives a gift of $135,000. The money is invested in stocks, bonds, and CDs. CDs pay 4.25% interest, bonds pay 2.4% interest, and stocks pay 8.7% interest. Country day invests $15,000 more in bonds than in CDs. If the annual income from the investments is $8,112.5, how much was invested in each vehicle?

Learn how to solve an investment problem involving a $135,000 scholarship fund. Calculate the amounts invested in CDs, bonds, and stocks, given different interest rates and conditions. This problem involves algebraic concepts and linear equations, suitable for students in Grades 9-12.

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