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Chapter 2: Problem 11

Imagine a certain savings account started out with a balance of \(\$ 5250.00\) on day-one, and today has a current balance of \(\$ 5780.23\) a. Exactly how much more money does the account have today, compared with day- one? b. Rounding to the nearest tenth of a percent: By what percentage amount has the account balance grown? c. If instead, the bank balance today was exactly double the starting balance, then by what exact percentage amount would the bank balance have grown? d. If the bank balance today had instead grown by \(15.5 \%\) since day-one, then what would be the exact amount of today's balance?

Short Answer

Expert verified
a. 530.23; b. 10.1%; c. 100%; d. 6063.75

Step by step solution

01

Calculating the Increase in Balance

To find out how much more money the account has today compared to day-one, subtract the initial balance from the current balance: \[5780.23 - 5250.00 = 530.23\]
02

Calculating the Percentage Increase

To find the percentage increase, use the formula: \(\text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{Original Amount}} \right) \times 100\). Here, \[\frac{530.23}{5250.00} \times 100 = 10.099\%\]Rounding to the nearest tenth of a percent gives us 10.1\%.
03

Calculating Double Original Balance

If the current balance is double the starting balance, it would be \[2 \times 5250.00 = 10500.00\]The amount of growth is \[10500.00 - 5250.00 = 5250.00\]
04

Percentage Increase When Doubling

For a balance that has doubled, the percentage increase is \[\left( \frac{5250.00}{5250.00} \right) \times 100 = 100\%\] So, doubling the amount reflects a 100\% increase.
05

Calculating New Balance with 15.5\% Increase

If the balance grew by 15.5\%, we first find the increase amount by calculating 15.5\% of the original balance:\[0.155 \times 5250.00 = 813.75\]Thus, the new balance would be: \[5250.00 + 813.75 = 6063.75\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculating Balance Increase
To figure out the increase in balance, we subtract the starting amount from the current amount. It鈥檚 like figuring out how much extra money a friend has given you! In this case:
  • Starting Balance: $5250.00
  • Current Balance: $5780.23
  • Balance Increase: $5780.23 - $5250.00 = $530.23
The balance increase is simply the difference between the current and initial amounts. It tells us how much more we have today compared to the starting point.
Percentage Growth on Investments
When it comes to investments, knowing how much your investment has grown in percentage terms can be enlightening. To find this percentage growth, you can use this handy formula:

\(\text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{Original Amount}} \right) \times 100\)

Using our example:
  • Increase: \(530.23
  • Original Amount: \)5250.00
  • Calculation: \(\frac{530.23}{5250.00} \times 100 = 10.099\%\)
Then, rounding to the nearest tenth of a percent gives us 10.1%. This shows the growth in the account expressed as a relative comparison to the original amount.
Doubling Investment Calculation
Doubling an investment is an exciting milestone, symbolizing a major growth achievement. To determine what it means for an investment to double, consider this calculation:
  • Original Balance: \(5250.00
  • Double Balance: \(2 \times 5250.00 = 10500.00\)
So, the account balance would have increased by \)5250.00 when it doubled.Now, converting this increase into a percentage involves this calculation:

\(\left( \frac{5250.00}{5250.00} \right) \times 100 = 100\%\)

A 100% increase confirms a doubling of the original investment, marking an impressive growth.
Interest Rate Adjustment
Adjusting for an interest rate can mean recalibrating how much growth you expect or receive over time. To illustrate, if a savings account grows by 15.5% over a period, this rise can be calculated by:
  • Original Balance: \(5250.00
  • Interest Rate: 15.5%
  • Interest Increase: \(0.155 \times 5250.00 = 813.75\)
Adding this growth to the original balance gives us the final total:
  • New Balance: \)5250.00 + \(813.75 = \)6063.75
This demonstrates the effect of an interest rate on the overall savings, showing how much more we have thanks to the increased percentage.

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Most popular questions from this chapter

Suppose another mortgage lender offers you a fixed-rate 15 -year mortgage at \(2.95 \%\) APR. You have \(\$ 20,000\) saved as a down payment, and you can afford a maximum mortgage payment of \(\$ 950\) per month. You are interested in a certain house for sale, with firm selling price of \(\$ 200,000\). a. Find the monthly payment for this house. Can you afford it, under the terms of this lender? b. (Challenge): Suppose this same lender offers to increase the APR by only \(0.05 \%,\) for each additional year added to the loan period beyond 15 years (so that a 16-year loan would have \(3.00 \%\) APR, and a 17-year loan would have \(3.05 \%\) APR, and so on ), up to a maximum loan period of 25 years. Given these terms, does any combination of APR and loan period exist that would let you afford the house? If so, state the minimum number of additional years needed, the total resulting loan period, the resulting APR, and the resulting monthly payment.

How much will \(\$ 1,000\) deposited in an account earning \(7 \%\) APR compounded weekly be worth in 20 years?

You are in the \(12 \%\) tax bracket and get a credit of \(\$ 500 .\) How has the amount of taxes owed changed?

Imagine that at the start of a certain month, you will make an opening deposit of \(\$ 500\) into a savings account, and you will then leave the account alone (meaning you will make no further deposits or withdrawals). Also, for this account: Every month after the opening deposit, the amount in the account will grow to be \(101 \%\) of its previous month's balance. a. Use a spreadsheet to enter 500 in cell A1. Using a formula and a cell- reference: Compute in cell A2, the amount in the account after one month has passed. Then using the fill down feature, continue the pattern for another eleven full months (you should end at cell A13). Format all the cells to show dollar signs. What is the amount in the account after one year? b. Now continue the pattern in column A of your spreadsheet to extend for a second full year (you should end at cell A25). What is the amount in the account after two years? c. What overall percentage growth occurred in the account between the opening deposit and one year later? (Compute using a formula and cell references) d. What overall percentage growth occurred in the account between the end of year one, and the end of year two? (Compute using a formula and cell references) e. (Challenge) The annual percentage growth that you found in part (d) for the second year, should be identical to the annual percentage growth that you found in part (c) for the first year. Can you mathematically explain why this is true? Do you think this pattern of identical overall annual percentage growth would continue, if you extend the pattern for even more years?

You wish to have \(\$ 3,000\) in 2 years to buy a fancy new stereo system. How much should you deposit each quarter into an account paying \(6.5 \%\) APR compounded quarterly?
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