If amount X is invested at ‘r’ percent simple annual interest, then what is the total interest earned by the investment in the 2t years?
An amount of 'X' dollars is invested at 'R%' simple annual interest and yields a total interest of $100 in 2 years.In terms of X, what dollar amount invested at the same rate of interest will yield a total interest of $400 in 4 years?
We can create the equation:
X * R * 2 = 100
X * R = 50
R = 50/X
So to yield 400 dollars in 4 years we have:
N * 50/X * 4 = 400
N * 50/X = 100
N = 100X/50 = 2X
An amount X is invested in a deposit scheme which earns an annual interest rate of 8% compounded quarterly. What is the final amount after 3 years?
Alex invested in a Bumper scheme that pays an interest at an annual rate of 40% compounded quarterly. What is the minimum that Alex needs to invest to earn at least $210 in interest in 6 months?
Alan invested $1,000 at r% simple annual interest and an unknown amount at 2r% simple annual interest. What amount did Alan invest at 2r% simple annual interest?
If i investment were reduced by p percent to the new amount n, which of the following, in terms of n and p, represents the value of i ?
Step 1: Question statement and Inferences
If i investment were reduced by p percent to the new amount n, the question is to represent the original value of i
Step 2: Finding required values
i investment went down p percent to the new amount n.
So, we can write:
Step 3: Calculating the final answer
Solve for i :
Answer: Option (A)
If an investment principal p decreased d percent in each of the first two years and recovered i percent in the third year, which of the following represents the value of the investment at the end of the third year?
Step 1: Question statement and Inferences
The amount p decreased d percent each of the first two years and recovered i percent the third year. The question is to represent the value of the investment at the end of the third year.
Step 2: Finding required values
The tables below show the value of investment at the end of each year and determines the final value at the end of year 3.
Alex bought a $1,000 savings bond that pays interest compounded annually. If at the end of the second year, her account balance was $1210, what is the annual rate of interest?
Step 1: Question statement and Inferences
The bond, which started at $1,000, is worth $1210 after two interest cycles. What is the interest rate?
Step 2: Finding required values
Let x be the percent interest.
Step 3: Calculating the final answer
Solve for x:
The bracket
yields a negative value of x. But that cannot be, because the money is growing over the two years. So, x has to be positive
So, the value of x will be obtained from:
Thus, the annual rate of interest is 10%
Answer: Option (A)
If a total of $10,000 is invested in two separate accounts and it earns a total of $175 simple interest after one year, what are the annual interest rates of these two accounts?
(1) The ratio of the interest rates in two accounts is 3:4
(2) The ratio of the amounts invested two accounts is 1:1
Step 1 & 2 – Understand and Analyze Question Statement
Knowing that $10,000 earns $175 from two different accounts at two different rates, set up the following equation with p as one of the principals, r_{1} and r_{2} as the rates of interest for the two accounts. Since the total money invested is $10,000, second principal amount is (10,000  p)
With one equation and three unknowns (p, r_{1}, and r_{2}), we can’t solve the equation.
Step 3 – Analyze Statement 1
Place these in equation
With one equation and two unknowns (p and r1
), we can’t solve the equation. Statement 1 is not sufficient.
Step 4 – Analyze Statement 2
Statement 2 tells us that $10,000 is evenly split. Place these in equation 1
With one equation and two unknowns, we still can’t solve the equation. Statement 2 is not sufficient.
Step 5 – Analyze Statements 1 and 2 Together
Taken together, we have a single equation with one unknown, and hence we can solve for :
Note that we do not need to do the calculation to answer this question. But if you perform the calculation, the answer is as follows:
r1=1.5%
; r2=2.0%
Answer: Option (C)
$2000 is invested in a new savings account at r percent interest rate compounded annually. If the amount in the account after 3 years is $3456, what is the total interest (in dollars) earned after 2 years of starting the account?
Step 1: Question statement and Inferences
We are given that a sum of $2000 is invested in a new savings account. This savings account offers an interest rate of r percent, which is compounded annually.
The balance in the account at the end of three years is $3456.
We have to find the total interest earned after 2 years.
We know that the formula for compound interest is:
Here,
A = Final amount at the end of time t = $3456
p = Initial Amount = $2000
r = rate of interest
t = number of compounding periods ‘n’ in the total duration = 3 years
Step 2: Finding required values
Upon breaking down 1728 into its factors, the above equation becomes:
Taking cube root on both sides:
Upon simplifying this equation further, we get: r = 20%
Now, we have to find the interest earned at the end of 2 years. To find this, we first need to find the amount that the initial deposit will have grown to after 2 years:
Step 3: Calculating the final answer
Interest earned after 2 years = (Amount after 2 years) – (Initial amount)
= 2880 – 2000
= 880
Thus, the interest earned after 2 years will be $880
Answer: Option (D)
What will be the value of $1000 at the end of 3 years?
(1) The return for the investment is 10% p.a., compounded annually.
(2) The amount at the end of the first year is $1100.
In order to find the value of $1000 at the end of 3 years, we need the rates of return for each of the three years.
Let the rate of return for the first year be r_{1} %
So, the amount at the end of the first year, A_{1} = 1000 + r_{1} % of 1000
Let the rate of return for the second year be r_{2} %
So, the amount at the end of the second year, A_{2} = A_{1} + r_{2} % of A_{1}
Let the rate of return for the third year be r_{3} %
We can similarly prove that the amount at the end of the third year,
Thus, in order to find the amount at the end of the third year, we need to know the values of r_{1}, r_{2 }and r_{3}
Step 3: Analyze statement 1
Statement 1 says that:
The annual rate of return for the investment is 10%
This statement implies that the rate of return for each of the three years is 10%
Putting r_{1} = r_{2} = r_{3} = 10 in Equation 1:
Thus, using Statement 1, we are able to find a unique value for
So, Statement 1 is sufficient
Step 4: Analyze Statement 2
Statement 2 states that
The amount at the end of the first year is $1100
From this statement, we do not know if the interest offered was CI or SI. Also, if it was CI, we do not know the compounding frequency.
Thus, Statement 2 is not sufficient.
Step 5: Analyze Both Statements Together (if needed)
This step is not needed as we do get a unique answer in Step 3
So, the correct answer is Option A.
What is the approximate amount that should be invested today at an annual rate of 10% compounded semiannually so that an amount of $1000 is received at the end of the first year?
Step 1: Question statement and Inferences
This question tests your proficiency in solving problems related to compound interest. Please note the formula used for compound interest problems:
Here A = $1000; r = 10% compounded semiannually; t = compounding time intervals
Step 2: Finding required values
r is given as annual rate of interest. Since we know that the interest is compounded semiannually. So the rate for calculation = r/2 = 10/2 = 5%. Also, the time intervals will be 2 since the compounding will be done 2 times till the end of first year. So t = 2.
Step 3: Calculating the final answer
Substituting these values we get:
Calculating and approximating (1.05)^{2}, we get:
1000 = p × 1.1
Thus, p = 1000/1.1 = 909.09
Thus the closest answer is $910.
Answer: Option (C)
If p were invested at i annual interest rate compounded annually for t years, where t > 3, and at the start of the second last year, the annual interest rate was doubled and then maintained at that level till the end, which of the following represents the amount received at the end of t years?
Step 1: Question statement and Inferences
Based on the compound interest equation:
, you are asked to represent the amount when the rate of interest changes. The interest rate is i for the first (t2) years, then 2i for the last two years.
Step 2: Finding required values
Start with the amount calculation for the first set of years, before the interest changes. If t is the total number of years, this set of years is t – 2.
Next, calculate the amount for the final two years, when the interest becomes 2i. In this case, t equals 2. Note that the principal for this time interval is the Amount calculated above. Thus, the amount at the end of t years is
Answer: Option (E)
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