Chegg Which Option is Better if the Interest Rate is 6 Compounded Continuously

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Grandpa Jack wants to help his grandson, Little Jack, with college expenses. Little Jack is currently 3 years old. If Grandpa Jack invests $5,000 in a college savings account earning 5% compounded yearly, how much money will he have in 15 years when Little Jack is 18?

Possible Answers:

Between $11,000-$11,500

Between $9,000-$9,500

Between $10,000-$10,500

Between $9,500-$10,000

Between $10,500-$11,000

Correct answer:

Between $10,000-$10,500

Explanation:

To solve this, we can create an equation for the value based on time. So if we let t be the nmbers of years that have passed, we can create a function f(t) for the value in the savings account.

We note that f(0) =5000. (We invest 5000 at time 0.) Next year, he will have 5% more than that. To find our total value at the end of the year, we multiply 5,000 * 1.05 = 5,250. f(1) = 5000(1.05)=5,250. At the end of year 2, we will have a 5% growth rate. In other words, f(2) = (1.05)* f(1). We can rewrite this as f(2) = 5000(1.05)(1.05) = 5000 (1.05)^2 . We can begin to see the proper equation is f(t)= 5000 (1.05)^t . If we plug in t = 15, we will have our account balance at the end of 15 years. So, our answer is $f(15)= 5000 (1.05)^{15}= 10,394.64$ .

Cherry invested dollars in a fund that paid 6% annual interest, compounded monthly. Which of the following represents the value, in dollars, of Cherry's investment plus interest at the end of 3 years?

Correct answer:

$(1.005)^{36} * m$

Explanation:

The monthly rate is $\frac{6\ percent}{12}=0.5\ percent=0.005$

3 years = 36 months

According to the compound interest formula

$P=C\left ( 1+r \right )t$

and here C=m , r=0.005 , t=36 , so we can plug into the formula and get the value

$m( 1+0.005)^{36} = (1.005)^{36} * m$

Scott wants to invest $1000 for 1 year. At Bank A, his investment will collect 3% interest compounded daily while at Bank B, his investment will collect 3.50% interest compounded monthly. Which bank offers a better return? How much more will he receive by choosing that bank over the other?

Correct answer:

$Bank\ B;\ \$5.12$

Explanation:

Calculate the total amount from each bank using the following formula:

$A=P\left(1+\frac{r}{n}\right)^{nt}$

Bank A:

$A=1000\left(1+\frac{0.03}{365}\right)^{(1)(365)}=1030.45$

Bank B:

$A=1000\left(1+\frac{.035}{12}\right)^{(1)(12)}=1035.57$

1035.57-1030.45=5.12

Bryan invests $8,000 in both a savings account that pays 3% simple interest annually and a certificate of deposit that pays 8% simple interest anually. After the first year, Bryan has earned a total of $365.00 from these investments. How much did Bryan invest in the certificate of deposit?

Correct answer:

$\$2,500$

Explanation:

Let be the amount Bryan invested in the certificate of deposit. Then he deposited $\$ 8,000-C$ in a savings account. 8% of the amount in the certificate of deposit is 0.08C , and 3% of the amount in the savings account is $0.03 \left ( 8,000-C \right )$ ; add these interest amounts to get $365.00. Therefore, we can set up and solve the equation:

$0.08C + 0.03 \left ( 8,000-C \right ) = 365$

$0.08C + 0.03 \cdot 8,000 -0.03 \cdot C = 365$

0.05C + 240 = 365

0.05C + 240-240 = 365 -240

0.05C = 125

$0.05C \div 0.05 = 125 \div 0.05$

C = 2,500

Barry invests $9000 in corporate bonds at 8% annual interest, compounded quarterly. At the end of the year, how much interest has his investment earned?

Correct answer:

$\$741.89$

Explanation:

Use the compound interest formula

$A =P \left ( 1+ \frac{r}{n}\right )^{nt}$

substituting P = 9,000 (principal, or amount invested), r = 0.08 (decimal equivalent of the 8% interest rate), n = 4 (four quarters per year), t = 1 (one year).

$A =9,000 \left ( 1+ \frac{0.08}{4}\right )^{4 \cdot 1}$

$=9,000 \left ( 1.02 \right )^{4 }$

$\approx 9,741.89$

Subtract 9,000 from this figure - the interest earned is $741.89

Tom deposits his $10,000 inheritance in a savings account with a 4% annual interest rate, compounded quarterly. He leaves it there untouched for six months, after which he withdraws $5,000. He leaves the remainder untouched for another six months.

How much interest has Tom earned on the inheritance after one year?

Correct answer:

$\$305.54$

Explanation:

Since in each case the interest is compounded quarterly, the annual interest rate of 4% is divided by 4 to get 1%, the effective quarterly interest rate.

The $10,000 remains in the savings account six months, or two quarters, so 1% is added twice - equivalently, the $10,000 is multiplied by 1.01 twice:

$\$ 10,000.00 \times 1.01 = \$ 10,100.00$

$\$ 10,100.00 \times 1.01 = \$ 10,201.00$

$5,000 is withdrawn from the savings account, leaving $\$ 10,201.00 - \$ 5,000.00 = \$ 5,201.00$

This money is untouched for six months, or two quarters, so again, we multiply by 1.01 twice:

$\$ 5,201.00 \times 1.01 = \$ 5,253.01$

$\$ 5,253.01 \times 1.01 =\$ 5,305.54$

Subtract $5,000 to get the interest:

$\$ 5,305.54 - \$5,000.00 = \$305.54$

On January 1, Gary borrows $10,000 to purchase an automobile at 12% annual interest, compounded quarterly beginning on April 1. He agrees to pay $800 per month on the last day of the month, beginning on January 31, over twelve months; his thirteenth payment, on the following January 31, will be the unpaid balance. How much will that thirteenth payment be?

Correct answer:

$\$913.16$

Explanation:

12% annual interest compounded quarterly is, effectively, 3% interest per quarter.

Over the course of one quarter, Gary pays off $\$800 \times 3 = \$2,400$ , and the remainder of the loan accruses 3% interest. This happens four times, so we will subtract $2,400 and subsequently multiply by 1.03 (adding 3% interest) four times.

First quarter:

$\$10,000 - \$2,400 = \$7,600$

$\$7,600 \times 1.03 = \$7,828$

Second quarter:

$\$7,828 - \$2,400 = \$5,428$

$\$ 5,428 \times 1.03 = \$5,590.84$

Third quarter:

$\$5,590.84 - \$2,400 =\$ 3,190.84$

$\$ 3,190.84 \times 1.03 = \$3,286.57$

Fourth quarter:

$\$3,286.57 - \$2,400 =\$ 886.57$

$\$ 886.57 \times 1.03 = \$913.16$

The thriteenth payment, with which Gary will pay off the loan, will be $913.16.

Jessica deposits $5,000 in a savings account at 6% interest. The interest is compounded monthly. How much will she have in her savings account after 5 years?

Possible Answers:

$\$6744$

$\$2744$

$\$6500$

None of the other answers are correct.

$\$1500$

Correct answer:

$\$6744$

Explanation:

$A=P(1+\frac{r}{n})^{nt}$

where is the principal, is the number of times per year interest is compounded, is the time in years, and is the interest rate.

$A=5000(1+\frac{0.06}{12})^{(12)(5)}=6744$

A real estate company is considering whether to accept a loan offer in order to develop property. The principal amount of the loan is $400,000, and the annual interest rate is 7% compounded semi-anually. If the company accepts the loan, what will be the balance after 4 years?

Correct answer:

526,720

Explanation:

Recall the formula for compound interest:

$Amount=P(1+r/n)^{nt}$ , where n is the number of periods per year, r is the annual interest rate, and t is the number of years.

Plug in the values given in the question:

$400,000(1+0.07/2)^{2\cdot 4}=400,000(1.035)^{8}=$

400,000(1.3168)=526,720

Nick found a once-in-a-lifetime opportunity to buy a rare arcade game being sold at a garage sale for $5730. However, Nick can't afford that right now, and decides to take out a loan for $1000. Nick didn't really read the fine print on the loan, and later figures out that the loan has a 30% annualy compounded interest rate! (A very dangerous rate). How much does Nick owe on the loan 2 years from the time he takes out the loan? (Assume he's lazy and doesn't pay anything back over those 2 years.)

Correct answer:

$\$1690$

Explanation:

For compound interest, the amount Nick owes is

P(1+i)^t

where is the principal, or starting amount of the loan ($1000), is the interest rate per year (30% = .3). and is the time that has passed since Nick took out the loan. (2)

We have

1000(1+.3)^2 =1000(1.3)^2 = 1000(1.3)(1.3)=1000(1.69)=1690

Hence our answer is $1690.

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Chegg Which Option is Better if the Interest Rate is 6 Compounded Continuously

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All GMAT Math Resources

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