Question

Walter earns a year-end bonus of $\$5000$ and puts it in $3$ one-year investments that pay $\$243$ in simple interest.

Part is invested at $3\%$, part at $4\%$ and part at $6\%$.

There is $\$1500$ more invested at $6\%$ than at $3\%$.

Find the amount invested at each rate.

Answer 1

Explanation:

Step-1: Framing two equations:

Walter earns a year-end bonus of $\$5000$ .

Let $\mathrm{x},\mathrm{y},\mathrm{z}$ be the amount invested at rate $3\%$, $4\%$, $6\%$ respectively.

$\Rightarrow x+y+z=5000.....\left(1\right)$

The total interest he gets is $\$243$.

Formula for calculating simple interest is,

$\mathrm{SI}=\frac{\mathrm{PRT}}{100}$

Therefore,

$$\begin{gathered} \Rightarrow \frac{3x}{100}+\frac{4y}{100}+\frac{6z}{100}=243 \\ \Rightarrow 3x+4y+6z=24300.....\left(2\right) \end{gathered}$$

The amount invested at $6\%$ is $\$1500$ more than the amount invested at rate $3\%$.

$\Rightarrow z=x+1500$

Substituting this in equation $\left(1\right)$and $\left(2\right)$.

$$\begin{gathered} \Rightarrow x+y+x+1500=5000 \\ \Rightarrow 2x+y=3500.......\left(3\right) \end{gathered}$$

and

$$\begin{gathered} \Rightarrow 3x+4y+6(x+1500)=24300 \\ \Rightarrow 3x+4y+6x+9000=24300 \\ \Rightarrow 9x+4y=15300....\left(4\right) \end{gathered}$$

Step-2: Finding the value of $x$

Multiply equation (3) with $4$.

$$\begin{gathered} \Rightarrow 4\left(2x+y=3500\right) \\ \Rightarrow 8x+4y=4\left(3500\right) \\ \Rightarrow 8x+4y=14000.....\left(5\right) \end{gathered}$$

Subtracting equation $\left(5\right)$ from equation $\left(4\right)$

$$\begin{gathered} \Rightarrow x=15300-14000 \\ \Rightarrow x=1300 \end{gathered}$$

Step-3: Calculating $y,\mathrm{z}$

Substitute the value of $x$ in equation (3)

$$\begin{gathered} \Rightarrow 2\left(1300\right)+y=3500 \\ \Rightarrow 2600+y=3500 \\ \Rightarrow y=3500-2600 \\ \Rightarrow y=900 \end{gathered}$$

Substituting the value of $x$ in equation $z=x+1500$.

$$\begin{gathered} \Rightarrow z=\left(1300\right)+1500 \\ \Rightarrow z=2800 \end{gathered}$$

Therefore, Walter invests $\$1300$, $\$900$, $\$2800$ in one-year investment with rates $3\%$, $4\%$, $6\%$ respectively.