Mathematics, 08.03.2020 09:59 paige1616

Perry has an opportunity to invest with an APR of 5.25%.

Using the rule of 70, how long will it take his investment to double? (Round to the hundredths.)

If you had a bank that offered 6.1% interest and compounded money 4 times a year, how many years would go by until you could turn $6000 into enough to buy a brand new car ($12,000)?

For this problem, you will want to guess and check until you find the right solution.

How many years does it take for $100 to reach $1 million dollars? The rate is at 5% with it being compounded twice a year. Round to a whole year. Hint: You may want to start with numbers above 100 years!

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Answer from: dchannakhone84

1. 13.33 years

2. 11 years

3. 186 years

Explanation:

Problem 1. Perry has an opportunity to invest with an APR of 5.25%.

Using the rule of 70, how long will it take his investment to double? (Round to the hundredths.)

Solution

The rule of 70 permits to make a quick estimation of the number of years an investment would double its value, depending on the annual percentage rate (APR).

The formula used for the rule of 70 is:

$\text{Number of Years to Double}=\dfrac{70}{\text{APR}}$

Substitute with APR = 5.25 and compute:

$\text{Number of Years to Double}=\dfrac{70}{\text{0.0525}}=13.33years$

Problem 2. If you had a bank that offered 6.1% interest and compounded money 4 times a year, how many years would go by until you could turn $6000 into enough to buy a brand new car ($12,000)?

For this problem, you will want to guess and check until you find the right solution.

Solution

You have to use the formula for monthly compound interest:

$A=P\times \bigg(1+\dfrac{r}{n}\bigg)^{(n\times t)}$

Where:

A is the the value after adding the interests: $12,000P is the value invested: $6,000r is the APR: 6.1% = 0.061n is the number of times the interest is compounded per year: 4t is the number of years: your unknown

Substitute:

$\$12,000=\$6,000\times \bigg(1+\dfrac{0.061}{4}\bigg)^{(4\times t)}\\\\\\2=1.01525^{4t}$

Guess and check until you find the right solution:

Your first educated guess may be using the rule of 70.

Number of years = 70/6.1 = 11.48

Use t = 12 ⇒ 4t = 4(12) = 48

1.01525⁴⁸ = 2.067 . . . pretty close to 2

Use t = 11 to verify which is closer to 2:

4t = 4(11) = 441.01525⁴⁴ = 1.946

t = 11 is closer because 2 - 1.946 = 0.054 and 2.067 - 2 = 0.067

Hence, the best calculation is 11 years.

Problem 3. How many years does it take for $100 to reach $1 million dollars? The rate is at 5% with it being compounded twice a year. Round to a whole year. Hint: You may want to start with numbers above 100 years!

Solution

Use the same formula for monthly compound interest:

$A=P\times \bigg(1+\dfrac{r}{n}\bigg)^{(n\times t)}$

With:

A: $100P: $1,000,000r: 5% = 0.05n: 2t: your unknown

Substitute:

$\$1,000,000=\$100\times \bigg(1+\dfrac{0.05}{2}\bigg)^{(2\times t)}\\\\\\10,000=1.025^{2t}$

Guess and check until you find the right solution.

Start with t = 100 years:

1.025²⁰⁰ ≈ 140 . . . very far from 10,000

Try t = 200 years:

1.025⁴⁰⁰ = 19,478

t = 150

1.025³⁰⁰ = 1,648

t = 186

1.025³⁷² = 9,756

t = 187

1.025³⁷⁴ = 10,250

9,756 is closer to 10,000 than 10,250, thus the answer is 186 years.

Answer from: Quest

step-by-step explanation:

Answer from: Quest

It will take him 5 days

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Perry has an opportunity to invest with an APR of 5.25%.

Using the rule of 70, how long...

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