Question

A 20-year old hopes to retire by age 65. To help with future expenses, they invest $6 500 today at an interest rate of 6.4% compounded annually. At age 65, what is the difference between the exact accumulated value and the approximate accumulated value (using the Rule of 72)?

232

likes
1159 views

Answer to a math question A 20-year old hopes to retire by age 65. To help with future expenses, they invest $6 500 today at an interest rate of 6.4% compounded annually. At age 65, what is the difference between the exact accumulated value and the approximate accumulated value (using the Rule of 72)?

Expert avatar
Jon
4.6
55 Answers
To find the exact accumulated value, we can use the formula for compound interest:

A = P(1 + r)^n

Where:
- A is the accumulated value
- P is the principal amount (initial investment)
- r is the interest rate per compounding period
- n is the number of compounding periods

In this case, the principal amount is $6,500, the interest rate is 6.4% or 0.064 as a decimal, and the number of compounding periods is 65 - 20 = 45 (since the investment is made for 45 years).

Let's calculate the exact accumulated value:

A = 6500(1 + 0.064)^{45}

Now, to find the approximate accumulated value using the Rule of 72, we can use the following formula:

A \approx P \times \frac{72}{r}

Where:
- A is the approximate accumulated value
- P is the principal amount
- r is the interest rate per compounding period

In this case, the principal amount is still $6,500 and the interest rate is 6.4%.

Let's calculate the approximate accumulated value:

A \approx 6500 \times \frac{72}{6.4}

Now, let's calculate both the exact accumulated value and the approximate accumulated value.

Answer:
Exact accumulated value: \ 105995.3
Approximate accumulated value: \ 73125

The difference between the exact accumulated value and the approximate accumulated value is:

\text{Difference} = \text{Exact accumulated value} - \text{Approximate accumulated value}

\text{Difference}=105995.3-73125

\text{Difference}=32870.3

Therefore, the difference between the exact accumulated value and the approximate accumulated value is approximately $32870.3.

Frequently asked questions (FAQs)
What is the limit of (sin(x) + cos(x)) / x as x approaches 0?
+
What is the total surface area of a rectangular solid with length L, width W, and height H?
+
What is the value of 'a' in the equation y = ax^2 if the vertex of the parabola is (3, -1)?
+
New questions in Mathematics
What is the amount of interest of 75,000 at 3.45% per year, at the end of 12 years and 6 months?
Find the equation of the normal to the curve y=x²+4x-3 at point(1,2)
Exercise 4 - the line (AC) is perpendicular to the line (AB) - the line (EB) is perpendicular to the line (AB) - the lines (AE) and (BC) intersect at D - AC = 2.4 cm; BD = 2.5 cm: DC = 1.5 cm Determine the area of triangle ABE.
58+861-87
4.2x10^_6 convert to standard notation
What is the r.p.m. required to drill a 13/16" hole in mild steel if the cutting speed is 100 feet per minute?
You are planning to buy a car worth $20,000. Which of the two deals described below would you choose, both with a 48-month term? (NB: estimate the monthly payment of each offer). i) the dealer offers to take 10% off the price, then lend you the balance at an annual percentage rate (APR) of 9%, monthly compounding. ii) the dealer offers to lend you $20,000 (i.e., no discount) at an APR of 3%, monthly compounding.
If f(x,y)=6xy^2+3y^3 find (∫3,-2) f(x,y)dx.
prove that if n odd integer then n^2+5 is even
20% of 3500
What’s the slope of a tangent line at x=1 for f(x)=x2. We can find the slopes of a sequence of secant lines that get closer and closer to the tangent line. What we are working towards is the process of finding a “limit” which is a foundational topic of calculus.
Use a pattern to prove that (-2)-(-3)=1
TEST 123123+1236ttttt
In a company dedicated to packaging beer in 750 mL containers, a normal distribution is handled in its packaging process, which registers an average of 745 mL and a standard deviation of 8 mL. Determine: a) The probability that a randomly selected container exceeds 765 mL of beer b) The probability that the beer content of a randomly selected container is between 735 and 755 mL.
The mass of 120 molecules of X2C4 is 9127.2 amu. Identify the unknown atom, X, by finding the atomic mass. The atomic mass of C is 12.01 amu/atom
Translate to an equation and solve. Let x be the unknown number: What number is 52% of 81.
Emile organizes a community dance to raise funds. In addition to paying $300 to rent the room, she must rent chairs at $2 each. The quantity of chairs rented will be equal to the number of tickets sold. She sells tickets for $7 each. How much should she sell to raise money?
Find the symmetric point to a point P = (2,-7,10) with respect to a plane containing a point Po = (3, 2, 2) and perpendicular to a vector u = [1, -3, 2].
g(x)=3(x+8). What is the value of g(12)
15=5(x+3)