Question

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An algebra student has won $390,000 in a lottery and wishes to deposit it in savings accounts in two financial institutions. One account pays 12% simple interest, but deposits are insured to only $250,000. The second account pays 6.6% simple interest, and deposits are insured to $500,000. Determine whether the money can be deposited so that it is fully and earns annual interest of $39,740.

a. The money can be fully insured and earn annual interest of $39,720
b. The money cannot be fully insured and earn annual interest of $39,740.

Answer 1

b. The money cannot be fully insured and earn annual interest of $39,740.

Explanation:

Simple interest is calculated by multiplying the principal deposited by the interest rate (in decimal form) and the time (in years).

Let x be the amount invested in the savings account that pays 12% simple interest and where deposits are insured to only $250,000.

Therefore:

• 
  `\textsf{Interest} = 0.12x`
• 
  `x \leq 250000`

As the total amount the student is investing is $390,000, the amount invested in the second account that pays 6.6% simple interest is (390000 - x). Therefore:

• 
  `\textsf{Interest} = 0.066(390000 - x)`

To determine if the money can be deposited so that it earns annual interest of $39,740, equate the sum of the two expressions for interest to $39.740 and solve for x:

`\begin{aligned}0.12x+0.066(390000 - x)&=39740\\0.12x+25740-0.066x&=39740\\0.054+25740&=39740\\0.054&=14000\\x&=259259.26\;\sf(2\;d.p.)\end{aligned}`

As x is not less than or equal to $250,000, it is not possible to deposit the money so that it is fully insured and earns the desired interest.