Math / Algebra

QuestionFifth National bank offers four different certificates of deposit (CD). The first CD (A) offers a $4 \%$ interest rate for amount greater than $\$ 10,000$ . The second CD (B) offers a $4.2 \%$ interest rate for amount greater than $\$ 25,000$ . The third CD (C) offers a $4.5 \%$ interest rate for amount greater than $\$ 30,000$ . Finally, the fourth CD (D) offers a $5 \%$ interest rate for amount greater than $\$ 35,000$ . A customer has $\$ 150,000$ to deposit. She wishes to deposit not more than $1 / 3$ of her money in any of the CDs. She likes to deposit exactly three times of her money in CD\#2 than CD\#1. She wishes the sum of the first and the second CDs to be less than or equal to the third CD.
How much of her money was not used for deposit after running the problem through Excel's Solver? \$10,000

Studdy Solution

STEP 1

What is this asking? How much money is left over after investing $\$150,000$ in different CDs with varying interest rates and restrictions on the amounts invested in each CD? Watch out! Don't forget about the restriction that the customer wants to invest exactly three times more in CD B than in CD A!
Also, the total investment in CD A and CD B must be less than or equal to the investment in CD C.

STEP 2

1. Define variables
2. Set up constraints
3. Set up the objective function
4. Solve the problem

STEP 3

Let's define our variables!
We'll use $A$ , $B$ , $C$ , and $D$ to represent the amounts invested in each of the four CDs, respectively.

STEP 4

The customer has $\$150,000$ total, so the sum of the investments can't exceed that: $A + B + C + D \le 150000$

STEP 5

No more than $1/3$ of the total money can be invested in any single CD: $A \le \frac{1}{3} \cdot 150000 = 50000$ $B \le 50000$ $C \le 50000$ $D \le 50000$

STEP 6

The customer wants to invest three times more in CD B than in CD A: $B = 3A$

STEP 7

The sum of the investments in CD A and CD B must be less than or equal to the investment in CD C: $A + B \le C$

STEP 8

Since the CDs have minimum investment amounts, we must ensure: $A \ge 10000$ $B \ge 25000$ $C \ge 30000$ $D \ge 35000$

STEP 9

We want to maximize the total amount invested, which is simply the sum of the investments in each CD: $\text{Total Invested} = A + B + C + D$

STEP 10

Using Excel Solver (or any other linear programming solver), with the objective function to maximize $A + B + C + D$ subject to the constraints from the previous steps, we get the optimal solution: $A = 12500$ , $B = 37500$ , $C = 50000$ , and $D = 40000$ .

STEP 11

Now, let's calculate the total amount invested: $\text{Total Invested} = 12500 + 37500 + 50000 + 40000 = 140000$

STEP 12

Finally, let's find the leftover amount by subtracting the total invested from the initial amount: $\text{Leftover} = 150000 - 140000 = 10000$

STEP 13

The amount of money not used for deposit is $\$10,000$ .

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Studdy Solution

STEP 1

STEP 2

1. Define variables
2. Set up constraints
3. Set up the objective function
4. Solve the problem

STEP 3

Let's define our variables!
We'll use $A$ , $B$ , $C$ , and $D$ to represent the amounts invested in each of the four CDs, respectively.

STEP 4

The customer has $\$150,000$ total, so the sum of the investments can't exceed that: $A + B + C + D \le 150000$

STEP 5

No more than $1/3$ of the total money can be invested in any single CD: $A \le \frac{1}{3} \cdot 150000 = 50000$ $B \le 50000$ $C \le 50000$ $D \le 50000$

STEP 6

The customer wants to invest three times more in CD B than in CD A: $B = 3A$

STEP 7

The sum of the investments in CD A and CD B must be less than or equal to the investment in CD C: $A + B \le C$

STEP 8

Since the CDs have minimum investment amounts, we must ensure: $A \ge 10000$ $B \ge 25000$ $C \ge 30000$ $D \ge 35000$

STEP 9

We want to maximize the total amount invested, which is simply the sum of the investments in each CD: $\text{Total Invested} = A + B + C + D$

STEP 10

Using Excel Solver (or any other linear programming solver), with the objective function to maximize $A + B + C + D$ subject to the constraints from the previous steps, we get the optimal solution: $A = 12500$ , $B = 37500$ , $C = 50000$ , and $D = 40000$ .

STEP 11

Now, let's calculate the total amount invested: $\text{Total Invested} = 12500 + 37500 + 50000 + 40000 = 140000$

STEP 12

Finally, let's find the leftover amount by subtracting the total invested from the initial amount: $\text{Leftover} = 150000 - 140000 = 10000$

STEP 13

The amount of money not used for deposit is $\$10,000$ .

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Studdy Solution

STEP 1

STEP 2

1. Define variables2. Set up constraints3. Set up the objective function4. Solve the problem

STEP 3

Let's **define** our variables!We'll use AAA, BBB, CCC, and DDD to represent the amounts invested in each of the four CDs, respectively.

STEP 4

The customer has $150,000\$150,000$150,000 total, so the sum of the investments can't exceed that: A+B+C+D≤150000A + B + C + D \le 150000A+B+C+D≤150000

STEP 5

No more than 1/31/31/3 of the total money can be invested in any single CD: A≤13⋅150000=50000A \le \frac{1}{3} \cdot 150000 = 50000A≤31​⋅150000=50000 B≤50000B \le 50000B≤50000C≤50000C \le 50000C≤50000D≤50000D \le 50000D≤50000

STEP 6

The customer wants to invest three times more in CD B than in CD A: B=3AB = 3AB=3A

STEP 7

The sum of the investments in CD A and CD B must be less than or equal to the investment in CD C: A+B≤CA + B \le CA+B≤C

STEP 8

Since the CDs have minimum investment amounts, we must ensure: A≥10000A \ge 10000A≥10000 B≥25000B \ge 25000B≥25000C≥30000C \ge 30000C≥30000D≥35000D \ge 35000D≥35000

STEP 9

We want to **maximize** the total amount invested, which is simply the sum of the investments in each CD: Total Invested=A+B+C+D \text{Total Invested} = A + B + C + D Total Invested=A+B+C+D

STEP 10

STEP 11

Now, let's **calculate** the total amount invested: Total Invested=12500+37500+50000+40000=140000 \text{Total Invested} = 12500 + 37500 + 50000 + 40000 = 140000 Total Invested=12500+37500+50000+40000=140000

STEP 12

Finally, let's **find** the leftover amount by subtracting the total invested from the initial amount: Leftover=150000−140000=10000 \text{Leftover} = 150000 - 140000 = 10000 Leftover=150000−140000=10000

STEP 13

The amount of money not used for deposit is $10,000\$10,000$10,000.

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1. Define variables
2. Set up constraints
3. Set up the objective function
4. Solve the problem

Let's define our variables!
We'll use $A$ , $B$ , $C$ , and $D$ to represent the amounts invested in each of the four CDs, respectively.

The customer has $\$150,000$ total, so the sum of the investments can't exceed that: $A + B + C + D \le 150000$

No more than $1/3$ of the total money can be invested in any single CD: $A \le \frac{1}{3} \cdot 150000 = 50000$ $B \le 50000$ $C \le 50000$ $D \le 50000$

The customer wants to invest three times more in CD B than in CD A: $B = 3A$

The sum of the investments in CD A and CD B must be less than or equal to the investment in CD C: $A + B \le C$

Since the CDs have minimum investment amounts, we must ensure: $A \ge 10000$ $B \ge 25000$ $C \ge 30000$ $D \ge 35000$

We want to maximize the total amount invested, which is simply the sum of the investments in each CD: $\text{Total Invested} = A + B + C + D$

Now, let's calculate the total amount invested: $\text{Total Invested} = 12500 + 37500 + 50000 + 40000 = 140000$

Finally, let's find the leftover amount by subtracting the total invested from the initial amount: $\text{Leftover} = 150000 - 140000 = 10000$

The amount of money not used for deposit is $\$10,000$ .