QuestionNicole and Chris each deposit \$20,000 at 2\% interest. Find their earnings for 3 years and compare.

Studdy Solution

STEP 1

Assumptions1. Nicole and Chris both deposit $20,000 into their respective accounts. . Both accounts pay an interest rate of% per year.
3. Nicole's account compounds interest annually.
4. Chris's account accrues simple interest.
5. There are no withdrawals and no additional deposits.

STEP 2

First, we need to understand the difference between simple interest and compound interest. Simple interest is calculated only on the initial amount (principal) that was deposited into the account. Compound interest, on the other hand, is calculated on the initial amount and also on the accumulated interest of previous periods.

STEP 3

Let's calculate the interest Nicole earns in the first year. Since it's the first year, the compound interest is the same as the simple interest. We can calculate this by multiplying the initial deposit by the interest rate.
$Interest_{Nicole, Year1} = Initial\, deposit \\times Interest\, rate$

STEP 4

Now, plug in the given values for the initial deposit and interest rate to calculate the interest.
$Interest_{Nicole, Year1} = \$20,000 \\times2\%$

STEP 5

Convert the percentage to a decimal value.
$2\% =0.02$ $Interest_{Nicole, Year1} = \$20,000 \\times0.02$

STEP 6

Calculate the interest amount for Nicole in the first year.
$Interest_{Nicole, Year1} = \$20,000 \\times0.02 = \$400$

STEP 7

Now, let's calculate the interest Chris earns in the first year. Since Chris's account accrues simple interest, the calculation is the same as for Nicole's first year.
$Interest_{Chris, Year1} = Initial\, deposit \\times Interest\, rate = \$20,000 \\times2\% = \$400$

STEP 8

For the second year, Nicole's interest is compounded, so we calculate the interest based on the initial deposit plus the interest from the first year.
$Interest_{Nicole, Year2} = (Initial\, deposit + Interest_{Year1}) \\times Interest\, rate$

STEP 9

Plug in the values for the initial deposit, the interest from the first year, and the interest rate to calculate the interest for Nicole in the second year.
$Interest_{Nicole, Year2} = (\$20,000 + \$400) \\times2\%$

STEP 10

Calculate the interest amount for Nicole in the second year.
$Interest_{Nicole, Year2} = (\$20,000 + \$400) \\times0.02 = \$408$

STEP 11

The interest Chris earns in the second year remains the same as the first year, as his account accrues simple interest.
$Interest_{Chris, Year} = \$400$

STEP 12

For the third year, Nicole's interest is again compounded, so we calculate the interest based on the initial deposit plus the interest from the first and second years.
$Interest_{Nicole, Year} = (Initial\, deposit + Interest_{Year} + Interest_{Year2}) \\times Interest\, rate$

STEP 13

Plug in the values for the initial deposit, the interest from the first and second years, and the interest rate to calculate the interest for Nicole in the third year.
$Interest_{Nicole, Year3} = (\$20,000 + \$400 + \$408) \\times2\%$

STEP 14

Calculate the interest amount for Nicole in the third year.
$Interest_{Nicole, Year3} = (\$20,000 + \$400 + \$408) \\times0.02 = \$416.16$

STEP 15

The interest Chris earns in the third year remains the same as the first and second years, as his account accrues simple interest.
$Interest_{Chris, Year3} = \$400$

STEP 16

Now, let's compare the interest earned by Nicole and Chris each year. In the first year, they both earn the same amount of interest.
$Interest_{Nicole, Year} = Interest_{Chris, Year} = \$400$

STEP 17

In the second year, Nicole earns more interest than Chris.
$Interest_{Nicole, Year2} = \$408 > Interest_{Chris, Year2} = \$400$

STEP 18

In the third year, Nicole again earns more interest than Chris.
$Interest_{Nicole, Year3} = \$416.16 > Interest_{Chris, Year3} = \$400$ So, Nicole earns more interest in the second and third years, while they earn the same amount in the first year.

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