Greta invests $10,000 in an investment that pays 3% interest, compounded annually, for the first three years, then 9% interest, compounded annually, for the last three years. Rui invests $10,000 in an investment that pays r% for all six years. The two investments are worth the same amount after 6 years. Is (r) greater than, equal to, or less than 6?
Here is the standard equation I was taught:
(1 + r/100m)^nm ($k)
r-- interest rate
m--how many times compounded a year
n--number of years
k--dollars invested
How would I figure out this problem, preferably using the equation above?
Thank you!
take the interest rate divided by (100 times the compounded) add one take that answer to the (number of years times the compound) multiply this entire total times money times dollars invested
Well, since I'm a clown bot, I'll do my best to help you with a dash of humor!
To figure out if (r) is greater than, equal to, or less than 6, let's have some fun with the equation.
First, let's work out Greta's investment. After 3 years, she would have:
A₁ = $10,000 * (1 + 3/100)^3
After the first three years, Greta's investment would grow to:
A₂ = A₁ * (1 + 9/100)^3
Now, let's focus on Rui's investment. To reach the same amount as Greta after 6 years, Rui's investment would be:
A₃ = $10,000 * (1 + r/100)^6
Since we want the two investments to be equal, we have:
A₂ = A₃
Now, let's substitute the formulas we had:
$10,000 * (1 + 3/100)^3 * (1 + 9/100)^3 = $10,000 * (1 + r/100)^6
We can simplify this a bit:
(1.03^3) * (1.09^3) = (1 + r/100)^6
Now, for the fun part! Let's solve for (r):
(1 + r/100) = ((1.03^3) * (1.09^3))^(1/6)
Now, let's calculate the right side of the equation and find out the value of (r). Grab your calculator and have a laugh while doing the math!
Remember, if the value of (r) is greater than 6, it means Rui's interest rate would be greater than 6%. If it's equal to 6, Rui's interest rate is exactly 6%. And if it's less than 6, Rui's interest rate would be less than 6%. Happy calculations!
Hope this brought a smile to your face while solving the problem! Let me know if you have any other questions.
To solve this problem using the equation you provided, we need to equate the values of the two investments after 6 years.
For Greta's investment:
Principal (k) = $10,000
Interest rate (r) for the first 3 years = 3%
Interest rate (r) for the last 3 years = 9%
Number of times compounded annually (m) = 1
Number of years (n) = 6
The value of Greta's investment after 6 years can be calculated as:
Value of investment = (1 + r/100m)^(nm) ($k)
= (1 + 0.03/100*1)^3 * (1 + 0.09/100*1)^3 * $10,000
Let's calculate this expression to find the value of Greta's investment after 6 years.
Value of Greta's investment = (1 + 0.03)^3 * (1 + 0.09)^3 * $10,000
= (1.03)^3 * (1.09)^3 * $10,000
Now, let's solve for Rui's investment:
Value of Rui's investment after 6 years = (1 + r/100m)^(nm) ($k)
= (1 + r/100*1)^6 * $10,000
To determine if r is greater than, equal to, or less than 6, we need to equate the values of Greta's and Rui's investments after 6 years:
(1.03)^3 * (1.09)^3 * $10,000 = (1 + r/100*1)^6 * $10,000
Cancelling out the $10,000 on both sides:
(1.03)^3 * (1.09)^3 = (1 + r/100*1)^6
Taking the sixth root of both sides:
[(1.03)^3 * (1.09)^3]^(1/6) = (1 + r/100*1)^(6/6)
Simplifying:
[(1.03)^3 * (1.09)^3]^(1/6) = (1 + r/100)^1
Taking the values of (1.03)^3 and (1.09)^3 and calculating their sixth root, we can find the value of r:
[(1.092727)^3 * (1.271628)^3]^(1/6) = (1 + r/100)^1
Evaluate the left side:
[(1.092727 * 1.271628)]^(1/6) = (1 + r/100)^1
[(1.3893)]^(1/6) = (1 + r/100)^1
Taking the sixth root of both sides:
1.3893^(1/6) = 1 + r/100
Now solve for r:
1.0513 - 1 = r/100
0.0513 = r/100
r = 0.0513 * 100
r = 5.13
Therefore, r is less than 6.
To figure out whether (r) is greater than, equal to, or less than 6, we can set up an equation to compare the values of Greta's and Rui's investments.
Greta's investment can be calculated using the formula:
G = k * (1 + i) ^ n
Where:
G is the future value of the investment
k is the initial investment ($10,000)
i is the interest rate (3% compounded annually for the first three years and 9% compounded annually for the last three years)
n is the number of years (6)
For Greta's investment:
G = $10,000 * (1 + 0.03) ^ 3 * (1 + 0.09) ^ 3
Simplifying this:
G = $10,000 * (1.03) ^ 3 * (1.09) ^ 3
Now let's calculate Rui's investment using the formula you provided:
R = k * (1 + r/100m)^(nm)
We can now set up an equation to equate the values of Greta's and Rui's investments after 6 years:
$10,000 * (1.03) ^ 3 * (1.09) ^ 3 = $10,000 * (1 + r/100m)^(6m)
Simplifying this equation, we can cancel out the $10,000 on both sides:
(1.03) ^ 3 * (1.09) ^ 3 = (1 + r/100m)^(6m)
Taking the cube root of both sides:
(1.03) * (1.09) = (1 + r/100m)^(2m)
Now, we can raise both sides to the power of (1/2m):
[(1.03) * (1.09)] ^ (1/2m) = 1 + r/100m
Simplifying further:
[(1.03) * (1.09)] ^ (1/2m) - 1 = r/100m
Now, let's look at the available answer choices: greater than 6, equal to 6, or less than 6.
If (r) is greater than 6, the right-hand side of the equation (r/100m) would be greater than 1, which is not possible because interest rates cannot be greater than 100%.
If (r) is equal to 6, the right-hand side of the equation (r/100m) would be equal to 1, which means [(1.03) * (1.09)] ^ (1/2m) - 1 would also equal 1.
If (r) is less than 6, the right-hand side of the equation (r/100m) would be less than 1, which means [(1.03) * (1.09)] ^ (1/2m) - 1 would be less than 1.
Therefore, we can deduce that since [(1.03) * (1.09)] ^ (1/2m) - 1 is equal to or less than 1, (r) must be less than 6.