GMAT Quantitative - Calculating compound interest

Carl's uncle invested money in some corporate bonds for his nephew the day Carl was born; the bonds paid 4% annual interest compounded continuously. No money was deposited or withdrawn over the next fifteen years. The current value of the bonds is $5,000.

Which of the following expressions is equal to the amount of money Carl's uncle invested initially?

$\frac{5,000 }{e^{0.6 }}$

$\frac{5,000 }{e^{1.6 }}$

$5,000 e^{0.6 }$

$\frac{5,000}{ 1 .04 ^{15 }}$

$5,000 \cdot 1 .04 ^{15 }$

Explanation

The formula for continuously compounded interest is

$A = P e^{rt}$

where is the current, or accrued, value of the investment, is the initial amount invested, or principal, is the annual rate expressed as a decimal, and is the number of years.

In this scenario,

The equation becomes

$5,000 = P e^{0.04 \cdot 15 }$

$5,000 = P e^{0.6 }$

$P = \frac{5,000 }{e^{0.6 }}$

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?

Between $10,000-$10,500

Between $11,000-$11,500

Between $9,000-$9,500

Between $10,500-$11,000

Between $9,500-$10,000

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 . We can begin to see the proper equation is . If we plug in t = 15, we will have our account balance at the end of 15 years. So, our answer is $f(15)= 5000 (1.05)^{15}= 10,394.64$ .

Ten years ago today, Geri's grandmother deposited some money into a college fund that yielded interest at a rate of 3.6% compounded monthly. There is now $6,400 in the account. Assuming that no money has been deposited or withdrawn, which of the following expressions must be evaluated in order to determine the amount of money originally deposited?

$\frac{6,400}{\left $1 .003 \right $^{12 0}}$

$6,400 \cdot 1 .003 ^{12 0}$

$6,400 e ^{0.36}$

$\frac{6,400 }{e ^{0.36}}$

$6,400 - 6,400 \cdot 0.036 \cdot 10$

Explanation

The formula for compound interest is

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

where is the current, or accrued, value of the investment, is the initial amount invested, or principal, is the annual rate expressed as a decimal, is the number of periods per year, and is the number of years.

In this scenario,

so the equation becomes

$6,400 = P \left $1+ \frac{0.036}{12} \right $^{12 \cdot 10}$

$6,400 = P \left $1+0.003 \right $^{12 0}$

$6,400 = P \left $1 .003 \right $^{12 0}$

$P = \frac{6,400}{\left $1 .003 \right $^{12 0}}$

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?

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

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:

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?

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:

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

Second quarter:

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

Third quarter:

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

Fourth quarter:

$$ 886.57 \times 1.03 = $913.16$

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

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?

Explanation

Use the compound interest formula

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

substituting (principal, or amount invested), (decimal equivalent of the 8% interest rate), (four quarters per year), (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

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?

Explanation

Let be the amount Bryan invested in the certificate of deposit. Then he deposited in a savings account. 8% of the amount in the certificate of deposit is , 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:

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

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

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

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?

$Bank\ B;\ $5.12$

$Bank\ A;\ $5.12$

$Bank\ B;\ $1005.12$

$Bank\ A;\ $1005.12$

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:

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

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?