Interest Problems - GMAT Quantitative

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Question

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?

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Answer

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 , , , so we can plug into the formula and get the value

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

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Question

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?

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Answer

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

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Question

Phillip invests $5,000 in a savings account at 5.64% per year interest, compounded monthly. If he does not withdraw or deposit any money, how much money will he have in the account at the end of five years?

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Answer

Use the compound interest formula

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

where , , , and

$A = 5,000\cdot \left ( 1+ \frac{0.0564}{12} \right )^{12 \cdot 5}$

$A = 5,000\cdot 1.0047 ^{60}$

$A = 5,000\cdot 1.3249 \approx 6624.52$

Phillip will have $6,624.52 in his account.

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Question

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?

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Answer

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$ .

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Question

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?

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Answer

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

Bank A:

Bank B:

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Question

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?

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Answer

Recall the formula for compound interest:

, 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:

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Question

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?

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Answer

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 , 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:

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

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

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

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Question

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?

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Answer

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

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Question

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?

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Answer

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.

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Question

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?

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Answer

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 = $305.54$

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Question

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.)

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Answer

For compound interest, the amount Nick owes is

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

Hence our answer is $1690.

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Question

Casey deposits $\$1000$ in his savings account that pays interest compunded yearly. Two years later, he deposits $\$100$ more into the same saving account. How much money is in Casey's account three years after he started his account?

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Answer

We will make use of the formula where is the accumulated amount, is the starting amount, is the rate of interest, and is the time in year the money is invested.

At the beginning, Casey starts with $1000, at an interest rate of 10% (or .1) and saves his money for 2 years. So after 2 years, he has

dollars in his account.

After he sees the $1210, he deposits $100 more, and then waits one more year.

Now becomes 1310. And after this 3rd year, Casey has

in his account.

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Question

If you invest $\$10,000$ today into a fund which earns a annually compounded interest, what amount of money will you have in the fund years from now?

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Answer

The accumulated amount at the end of 3 years will be .

It is easier to find the correct answer by using the following approach:

Calculate the amount accumulated at the end of each year. (Note that the interest is compounded, so use the amount accumulated at the end of the previous year to calculate the interest for the next year.)

At the end of year 1

At the end of year 2

At the end of year 3

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Question

$\$45,000$ is invested at compounded annual interest rate, how much will the investment yield after one year?

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Answer

This problem simply ask for the amount of interest received one year after having invested this money. We could, for this problem, either use the simple interest formula or the compounded interest formula, which is , where is the principal, the rate, the number of compounding periods and the number of years for which we invest. We substract because we only want the amount of interest, not the total value at the end of periods. Note that is positive for increases and negative for decreases.

Applying this formula we get,

, giving us .

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Question

$\$45,000$ is deposited in an account paying compounded annual interest rate, how much will there be on the account after two years?

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Answer

We apply the compound interest rate formula

where P=principal, r=rate, and t=time.

Pluggin in our values we get

, or .

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Question

We invest $\$100,000$ for two years in an investment paying interest rates. At the end of the two years we end up with $\$132,250$ . What is ?

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Answer

The way we should treat this problem is as an equation and plug in what we know in the compounded interest formula, as follows: . By manipulating the terms we get: . Now, since it would be too complicated to solve this quadratic equation, we should just try with the values in the answers. For example let's try with the square of , which turns out to be , therefore it is too small and we should look for a larger rate. Let's try until we find the right answer. Remember when you test answers to find the right answer, make sure you go slow so you don't have to test twice in case you would make an error.

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Question

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?

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Answer

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}}$

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Question

Five years ago today, Jimmy's grandfather deposited $5,000 into a college fund that yielded interest at an annual rate of 4.8% compounded monthly.

Assuming that no money has been deposited or withdrawn, which of the following expressions would have to be evaluated in order to calculate the amount of money in the account now?

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Answer

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,

Therefore,

$A = 5,000 \left \(1+ \frac{0.048}{12} \right \)^{12 \cdot 5}$

$= 5,000 \left \(1+ 0.004 \right \)^{60}$

$=5,000\cdot 1.004^{60}$

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Question

Veronica's aunt invested $4,000 in some corporate bonds for her niece the day Veronica was born; the bonds paid 4% annual interest compounded continuously. No money was deposited or withdrawn over the next eighteen years.

Which of the following expressions is equal to the amount of money in the account on Veronica's eighteenth birthday?

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Answer

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,

,

so

$A = 4,000 $e^{0.04 \cdot 18}$ = 4,000 $e^{0.72}$$

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Question

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?

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Answer

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 }$$}$

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