SOLUTION: Giovanni invested N$90 000 in three different account at the beginning of 2018 according to the following table. 2018 yield Saving 6% Unit trust 7% 32 days 5% If he invested

Question 1167408: Giovanni invested N$90 000 in three different account at the beginning of 2018 according to the
following table.
2018 yield
Saving 6%
Unit trust 7%
32 days 5%
If he invested the same amount in the unit trust as well as in the 32 days accounts and his 2018 yield
for the year from the saving and the 32 days amounted to N$400, how much did he invest in each
account? Use Gaussian elimination.
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
This problem can be solved by setting up a system of linear equations and using **Gaussian elimination** to find the investment amounts.
Let:
* $S$ be the amount invested in the **Saving** account.
* $U$ be the amount invested in the **Unit Trust** account.
* $T$ be the amount invested in the **32 Days** account.
## 1. Set Up the System of Equations
We derive three equations from the problem description:
**Equation 1: Total Investment**
The total amount invested is N\$90,000.
$$S + U + T = 90000$$
**Equation 2: Equal Investment**
The amount invested in the Unit Trust ($U$) is the same as the amount invested in the 32 Days account ($T$).
$$U = T \quad \implies \quad 0S + U - T = 0$$
**Equation 3: Specific Yield**
The 2018 yield (interest earned) from the Saving (6%) and the 32 Days (5%) accounts amounted to N\$400.
$$0.06S + 0.05T = 400$$
To work with integers, multiply the equation by 100:
$$6S + 5T = 40000$$
Since $U$ is not in this equation, we include a $0U$ term:
$$6S + 0U + 5T = 40000$$
The system of linear equations is:
1. $S + U + T = 90000$
2. $0S + U - T = 0$
3. $6S + 0U + 5T = 40000$
## 2. Gaussian Elimination
We form the augmented matrix and use row operations to achieve row echelon form.
$$\begin{pmatrix} 1 & 1 & 1 & | & 90000 \\ 0 & 1 & -1 & | & 0 \\ 6 & 0 & 5 & | & 40000 \end{pmatrix}$$
**Step 1: Eliminate $S$ from Row 3**
$$R_3 \leftarrow R_3 - 6R_1$$
$$\begin{aligned} R_3: & \quad 6 & 0 & 5 & | & 40000 \\ -6R_1: & \quad -6 & -6 & -6 & | & -540000 \\ \text{New } R_3: & \quad 0 & -6 & -1 & | & -500000 \end{aligned}$$
The new matrix is:
$$\begin{pmatrix} 1 & 1 & 1 & | & 90000 \\ 0 & 1 & -1 & | & 0 \\ 0 & -6 & -1 & | & -500000 \end{pmatrix}$$
**Step 2: Eliminate $U$ from Row 3**
$$R_3 \leftarrow R_3 + 6R_2$$
$$\begin{aligned} R_3: & \quad 0 & -6 & -1 & | & -500000 \\ 6R_2: & \quad 0 & 6 & -6 & | & 0 \\ \text{New } R_3: & \quad 0 & 0 & -7 & | & -500000 \end{aligned}$$
The matrix is now in row echelon form:
$$\begin{pmatrix} 1 & 1 & 1 & | & 90000 \\ 0 & 1 & -1 & | & 0 \\ 0 & 0 & -7 & | & -500000 \end{pmatrix}$$
## 3. Back-Substitution
The row echelon form corresponds to the following simplified system:
1. $S + U + T = 90000$
2. $U - T = 0$
3. $-7T = -500000$
**From Equation 3 (32 Days Account, $T$):**
$$-7T = -500000$$
$$T = \frac{500000}{7} \approx 71428.5714$$
**From Equation 2 (Unit Trust Account, $U$):**
Since $U - T = 0$, we have $U = T$.
$$U = \frac{500000}{7}$$
**From Equation 1 (Saving Account, $S$):**
$$S + U + T = 90000$$
$$S + \frac{500000}{7} + \frac{500000}{7} = 90000$$
$$S + \frac{1000000}{7} = 90000$$
$$S = 90000 - \frac{1000000}{7}$$
To simplify, use a common denominator:
$$S = \frac{90000 \times 7 - 1000000}{7}$$
$$S = \frac{630000 - 1000000}{7}$$
$$S = \frac{-370000}{7}$$
The result for $S$ is negative, which is impossible for an investment amount. This indicates that **no solution exists** for the given problem parameters.
***
## Conclusion
Based on the required constraints and Gaussian elimination, the resulting investment amount for the Saving account ($S$) is negative: $S = -\frac{370,000}{7}$.
**This means that there is no amount Giovanni could invest in the three accounts that satisfies all the conditions simultaneously.**
### Breakdown of the Contradiction:
The contradiction arises because the required yield is too small for the large investment.
1. **Total Investment:** N\$90,000.
2. **Equal Investments:** $U = T$. This means $S + 2T = 90000$.
3. **Maximum Yield on S & T:** The maximum possible interest from the $S$ and $T$ accounts would occur if $T$ was maximized (close to $45,000$) and $S$ was maximized (close to $90,000$), but constrained by $S + 2T = 90,000$. The yield rate for $S$ (6%) is higher than for $T$ (5%).
4. **Yield Required:** $0.06S + 0.05T = 400$.
The maximum possible value of the yield from $S$ and $T$ given $S+T$ must be less than $90,000$ is significantly higher than N\$400, leading to a mathematically impossible distribution.
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