Jamaal is planning to invest up to $25,000 in City Bank or State Bank. He wants to invest at least $5000 in City Bank, but not more than $11,000; since State Bank does not insure more than $17,000, he wants to invest no more than this amount in State Bank. The interest at City Bank is 7%, and the interest at State Bank is 12%. How much should he invest in each bank (in dollars) to earn the most interest?

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Since the problem does not state otherwise, I will assume this is simple interest over a period of 1 year.

The simple interest formula is:

I = p * r * t

I is the amount of interest earned

p is the principal (original) amount of money invested

r is the interest rate expressed as a decimal

t is the amount of time the principal remains in the account

Since the assumed time is 1 year, this formula can be written as:

I = p * r

He wants to invest at least $5,000 in City Bank. If he invests $5,000 in City Bank, that leaves $20,000 to invest in State Bank. However, he does not want to invest more than $17,000 in State Bank, so the most he could invest in City Bank is $8,000.

He wants to invest at most $11,000 in City Bank. If he invests $11,000 in City Bank, that leaves $14,000 to invest in State Bank.

This information can be organized into a table:

**City Bank Investment**

**City Bank Interest**

**State Bank Investment**

**State Bank Interest**

**Total Interest**

$8,000

8,000 * 0.07 = $560

$17,000

17,000 * 0.12 = $2,040

560 + 2,040 = $2,600

$9,000

9,000 * 0.07 = $630

$16,000

16,000 * 0.12 = $1,920

630 + 1,920 = $2,550

$10,000

10,000 * 0.07 = $700

$15,000

15,000 * 0.12 = $1,800

700 + 1,800 = $2,500

$11,000

11,000 * 0.07 = $770

$14,000

14,000 * 0.12 = $1,680

770 + 1,680 = $2,450

As you can see from the table, he should invest $8,000 in City Bank and $17,000 in State Bank to earn the most interest of $2,600.

Let x be the amount to invest at City, and y the amount to invest at State.

Then `5000<=x<=11000` and `y<=17000` .

We assume that City and State have the same compounding period, and that there are no other fees.

Then Jamaal will want to invest as much as possible at State (because the interest is so much higher.)

So we have `x+y<=25000` , `5000<=x<=11000` , and `y<=17000` .

If y=17000 and x=8000 all inequalities are met, and the amount of interest earned will be maximized.

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Jamaal should invest $17000 at State bank, and $8000 at City bank.

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