Investment Problem: CD & Bond Interest
What This Problem Teaches
- Setting up a system of linear equations from word problems involving money allocation
- Converting percentage rates to decimal form for mathematical calculations
- Using substitution method to solve two-equation systems efficiently
- Distinguishing between principal (investment amount) and interest (earnings) in financial contexts
- Verification techniques for investment problems by checking both total amount and total return constraints
Visualizing the Problem
Let's organize the given information in a table to see the relationship between investments and returns:
| Investment Type | Amount | Interest Rate | Interest Earned |
|---|---|---|---|
| CD Account | x | 4% | 0.04x |
| Savings Bond | 7000 - x | 7% | 0.07(7000 - x) |
| Total | $7000 | — | $420 |
Solution: The System of Equations Approach
Step 1 — Define Variables
Let x = amount invested in the CD account (what we're solving for)
Then 7000 - x = amount invested in the savings bond (the remainder)
Step 2 — Set Up the Interest Equation
The total interest earned equals the sum of interest from both investments:
0.04x + 0.07(7000 - x) = 420
Step 3 — Solve for x
Expand the equation and combine like terms:
0.04x + 490 - 0.07x = 420
-0.03x + 490 = 420
Step 4 — Isolate x
Subtract 490 from both sides and divide by the coefficient:
-0.03x = -70
x = -70 ÷ (-0.03) = 2333.33
Step 5 — Find Both Investment Amounts
CD amount: x = $2333.33
Bond amount: 7000 - 2333.33 = $4666.67
Solution: Method 2 — The Interest Rate Difference Method
Step 1 — Consider What Happens If All Money Earned 7%
If the entire $7000 were invested at 7%, the total interest would be:
Step 2 — Calculate the Interest "Shortfall"
Since the actual interest is only $420, there's a shortfall of:
Step 3 — Find the Rate Difference Per Dollar
Each dollar in the CD earns 3% less than if it were in the bond:
Step 4 — Calculate CD Amount
The amount in the CD is the shortfall divided by the rate difference:
This method gives us the same answer by thinking about the "opportunity cost" of choosing the lower-rate investment.
Verification
Let's check our answer by verifying both the total investment and total interest:
Check Total Investment
$2333.33 + $4666.67 = $7000 ✓
Check Total Interest
Bond interest: $4666.67 × 0.07 = $326.67
Total interest: $93.33 + $326.67 = $420.00 ✓
Both constraints are satisfied, confirming our solution is correct.
Common Pitfalls
Writing the equation as
4x + 7(7000 - x) = 420 instead of 0.04x + 0.07(7000 - x) = 420. This leads to an impossibly large solution because you're treating 4% as 4 instead of 0.04.
Some students write
x + (7000 - x) = 420, confusing the total investment with the total interest. The $420 is interest earned, not principal invested.
Defining
x as the bond amount but then calculating CD interest as 0.04x. Stay consistent with your variable definitions throughout the problem.
When combining
0.04x - 0.07x, getting +0.03x instead of -0.03x. Double-check the sign when the higher rate coefficient is being subtracted.
The Pattern Behind This
All two-investment problems with different interest rates follow the same general structure:
Let (Total - x) = amount in second investment
Interest equation: r₁ × x + r₂ × (Total - x) = Total Interest
Solving: x = (r₂ × Total - Total Interest) ÷ (r₂ - r₁)
In our problem: x = (0.07 × 7000 - 420) ÷ (0.07 - 0.04) = 70 ÷ 0.03 = $2333.33
Real Applications
- Personal Finance: Diversifying retirement savings between conservative CDs and higher-yield mutual funds while targeting a specific income goal
- Corporate Treasury: Companies split cash reserves between safe money market accounts and higher-risk investments to balance liquidity needs with return targets
- Portfolio Management: Financial advisors use similar calculations to allocate client funds between bonds and stocks based on risk tolerance and income requirements
Four "What-If?" Problems
The mother still invests $7000 split between a 4% CD and 7% bond, but the total interest earned is only $385. How much was invested in the CD account?
Let x = CD amount. Interest equation: 0.04x + 0.07(7000 - x) = 385
0.04x + 490 - 0.07x = 385-0.03x = 385 - 490 = -105
x = -105 ÷ (-0.03) = 3500
CD: $3500 at 4% = $140 interest
Bond: $3500 at 7% = $245 interest
Total: $140 + $245 = $385 ✓
Answer: $3500 in the CD account
The mother has $9000 to split among three accounts: CD at 4%, bond at 7%, and stock fund at 10%. She invests twice as much in the bond as in the CD. Total interest is $600. Find all three amounts.
Let x = CD amount, 2x = bond amount, 9000 - 3x = stock amount
0.04x + 0.07(2x) + 0.10(9000 - 3x) = 600
0.04x + 0.14x + 900 - 0.30x = 600-0.12x = -300
x = 2500
CD: $2500, Bond: $5000, Stock: $1500
Interest: $100 + $350 + $150 = $600 ✓
Total: $2500 + $5000 + $1500 = $9000 ✓
Answer: CD $2500, Bond $5000, Stock $1500
The mother invests $3000 in a 4% CD and $4000 in a bond with unknown interest rate r%. If the total interest earned is $450, what is the bond's interest rate?
CD interest + Bond interest = Total interest3000(0.04) + 4000(r) = 450
120 + 4000r = 450
4000r = 450 - 120 = 330r = 330 ÷ 4000 = 0.0825
r = 0.0825 = 8.25%
CD: $3000 × 4% = $120
Bond: $4000 × 8.25% = $330
Total: $120 + $330 = $450 ✓
Answer: The bond's interest rate is 8.25%
The mother splits $8000 between a 4% CD and 6% bond, both compounding annually. After 2 years, the total value of both investments is $8960. How much was initially in the CD?
Let x = CD amount. After 2 years:x(1.04)² + (8000-x)(1.06)² = 8960
x(1.0816) + (8000-x)(1.1236) = 8960
1.0816x + 8988.80 - 1.1236x = 8960-0.042x = 8960 - 8988.80 = -28.80
x = -28.80 ÷ (-0.042) ≈ 686
CD: $686 grows to $686(1.04)² = $742.06
Bond: $7314 grows to $7314(1.06)² = $8217.94
Total: $742.06 + $8217.94 = $8960 ✓
Answer: $686 was initially invested in the CD
Frequently Asked Questions
2026-06-10