Interest Problems - GMAT Quantitative
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Grandpa Jack wants to help pay for college for his grandson, Little Jack. Little Jack is currently 8 years old. Grandpa Jack makes a one-time deposit into an account that earns simple interest every year. Grandpa Jack invests $10,000 now and in ten years, that will grow to $15,000. What rate of simple interest did Grandpa Jack receive?
Grandpa Jack wants to help pay for college for his grandson, Little Jack. Little Jack is currently 8 years old. Grandpa Jack makes a one-time deposit into an account that earns simple interest every year. Grandpa Jack invests $10,000 now and in ten years, that will grow to $15,000. What rate of simple interest did Grandpa Jack receive?
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To calculate simple interest, the formula is
 "FV= PV (1+it)")
where 
stands for Future Value, 
stands for Present Value, 
stands for the interest rate, and 
stands for the number of periods (in this case years). So plugging in,
Solving this we get 
or 5%
ALTERNATE SOLUTION:
Another way of finding this is to calculate the amount of interest per year. Since this is simple interest, Grandpa Jack earns the same amount of interest per year. The total interest earned is 15,000-10,000= 5,000. $5,000 over 10 years, equates to $500 per year. $500 divided by the original $10,000 is .05, or 5%.
To calculate simple interest, the formula is
where
Solving this we get
or 5%
ALTERNATE SOLUTION:
Another way of finding this is to calculate the amount of interest per year. Since this is simple interest, Grandpa Jack earns the same amount of interest per year. The total interest earned is 15,000-10,000= 5,000. $5,000 over 10 years, equates to $500 per year. $500 divided by the original $10,000 is .05, or 5%.
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Jessica deposits $5,000 in a savings account that collects 6% simple interest. How much money will she have accumulated after 5 years?
Jessica deposits $5,000 in a savings account that collects 6% simple interest. How much money will she have accumulated after 5 years?
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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?
Cherry invested
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The monthly rate is 
3 years = 36 months
According to the compound interest formula
and here 
, 
, 
, so we can plug into the formula and get the value
The monthly rate is
3 years = 36 months
According to the compound interest formula
and here
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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?
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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where 
is the principal, 
is the number of times per year interest is compounded, 
is the time in years, and 
is the interest rate.
where
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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?
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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Use the compound interest formula
where 
, 
, 
, and 
Phillip will have $6,624.52 in his account.
Use the compound interest formula
where
Phillip will have $6,624.52 in his account.
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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?
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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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 
.
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
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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?
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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Calculate the total amount from each bank using the following formula:
Bank A:
Bank B:
Calculate the total amount from each bank using the following formula:
Bank A:
Bank B:
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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?
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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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:
Recall the formula for compound interest:
Plug in the values given in the question:
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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?
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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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 
; add these interest amounts to get $365.00. Therefore, we can set up and solve the equation:
Let
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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?
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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Use the compound interest formula
substituting 
(principal, or amount invested), 
(decimal equivalent of the 8% interest rate), 
(four quarters per year), 
(one year).
Subtract 9,000 from this figure - the interest earned is $741.89
Use the compound interest formula
substituting
Subtract 9,000 from this figure - the interest earned is $741.89
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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?
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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12% annual interest compounded quarterly is, effectively, 3% interest per quarter.
Over the course of one quarter, Gary pays off 
, 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:
Second quarter:
Third quarter:
Fourth quarter:
The thriteenth payment, with which Gary will pay off the loan, will be $913.16.
12% annual interest compounded quarterly is, effectively, 3% interest per quarter.
Over the course of one quarter, Gary pays off
First quarter:
Second quarter:
Third quarter:
Fourth quarter:
The thriteenth payment, with which Gary will pay off the loan, will be $913.16.
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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?
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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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:
$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:
Subtract $5,000 to get the interest:
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:
$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:
Subtract $5,000 to get the interest:
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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.)
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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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.
For compound interest, the amount Nick owes is
where
We have
Hence our answer is $1690.
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Casey deposits 
in his savings account that pays 
interest compunded yearly. Two years later, he deposits 
more into the same saving account. How much money is in Casey's account three years after he started his account?
Casey deposits
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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.
We will make use of the formula
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
After he sees the $1210, he deposits $100 more, and then waits one more year.
Now
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If you invest 
today into a fund which earns a 
annually compounded interest, what amount of money will you have in the fund 
years from now?
If you invest
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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
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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is invested at 
compounded annual interest rate, how much will the investment yield after one year?
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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 
.
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
Applying this formula we get,
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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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We apply the compound interest rate formula
where P=principal, r=rate, and t=time.
Pluggin in our values we get
, or 
.
We apply the compound interest rate formula
where P=principal, r=rate, and t=time.
Pluggin in our values we get
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We invest 
for two years in an investment paying 
interest rates. At the end of the two years we end up with 
. What is 
?
We invest
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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.
The way we should treat this problem is as an equation and plug in what we know in the compounded interest formula, as follows:
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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?
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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The formula for compound interest is
,
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
The formula for compound interest is
where
In this scenario,
so the equation becomes
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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?
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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The formula for compound interest is
,
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,
The formula for compound interest is
where
In this scenario,
Therefore,
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