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?

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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.