Chapter 10, Problem 46RE

To determine

To find:the amount invested by the person

Answer to Problem 46RE

The person invested the amount $\$2,500$ in bank $A$ , the amount of $\$40,000$ in bank $B$ and $\$17,500$ in bank $C$ .

Explanation of Solution

Given:

  $\$1.25$ a pound for haddock

  $\$0.75$ a pound for sea bass

  $\$2.00$ a pound for red snapper

  $560$ 1b of fish worth $\$575$

The haddock and red snapper together are worth $\$320$ .

Suppose a person invests $\$60,000$ in three banks $A,B,C$ respectively. Let $x.y$ and $z$ be amounts in banks $A,B,C$ . Then corresponding equation is

  $x+y+z=60,\mathrm{000.......}(1)$

The interest rates per year of three banks $A,B,C$ are $2\%,2.5\%$ and $3\%$ respectively and the person earned $\$1575$ as an interest per year.

  $0.02x+0.025y+0.03z=\mathrm{1575........}(2)$

Also the person invested as much as in bank $B$ as in other two banks. So, the corresponding equation is

  $2x+2z=y\mathrm{........}(3)$

Solve the equations, first multiply equation $(1)$ by $2$

  $\begin{array}{l}\text{2x+2y+2z=1,20,000}\\ \text{y+2y=1,20,000 Substitute 2x+2z=y}\\ \text{3y=1,20,000 Simplify}\\ \text{y=40,000 Divide both sides by 3}\end{array}$

Substitute this value in equation $(2)$ ,

  $\begin{array}{l}\text{0}\text{.02x+0}\text{.025}\left(\text{40,000}\right)\text{+0}\text{.03z=1575}\\ \text{0}\text{.02x+1000+0}\text{.03z=1575}\\ \text{0}\text{.02x+0}\text{.03z=575 Subtract 1000 on both sides}\end{array}$

Equation $(3)$ becomes

  $\begin{array}{l}\text{2x+2z=40,000}\\ \text{x+z=20,000 Divide by 2}\\ \text{z=20,000-x}\end{array}$

Substitute $z=20,000-x$ in the equation $0.02x+0.03z=575$

  $\begin{array}{l}\text{0}\text{.02x+0}\text{.03}\left(\text{20,000-x}\right)\text{=575}\\ \text{0}\text{.02x+600-0}\text{.03x=575}\\ \text{-0}\text{.01x=-25 Combine like terms}\\ \text{x=2,500}\end{array}$

Finally substitute $x=2,500,y=40,000$ in equation $(1)$ ,

  $\begin{array}{l}\text{2500+40000+z=60,000}\\ \text{42500+z=60000 Simplify}\\ \text{z=60000-42500 Subtract 42500non both sides}\\ \text{=17,500}\end{array}$

Therefore, the person invested the amount $\$2,500$ in bank $A$ , the amount of $\$40,000$ in bank $B$ and $\$17,500$ in bank $C$ .

Conclusion:

Therefore, the person invested the amount $\$2,500$ in bank $A$ , the amount of $\$40,000$ in bank $B$ and $\$17,500$ in bank $C$ .