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Chapter 4: Problem 8

How many days will it take for a sum of $$\$ 1500$$ to earn $$\$ 25$$ interest if it is deposited in a bank paying \(5 \% /\) year? (Use a 365-day year.)

Short Answer

Expert verified
It will take approximately 122 days for a sum of \( \$ 1500 \) to earn \( \$ 25 \) interest if it is deposited in a bank paying 5% interest per year.

Step by step solution

01

Convert the yearly interest rate to a daily interest rate

First, we'll find the daily interest rate by dividing the annual rate by the number of days in a year (365 days). To do this, we need to convert the annual percentage interest rate to a decimal by dividing by 100: Daily interest rate = \( \frac{Annual\ interest\ rate}{365\ days} \) Daily interest rate = \( \frac{5 \%}{365} \) = \( \frac{5}{100 \cdot 365} \) = \( \frac{1}{7300} \)
02

Calculate the interest earned daily

Next, we will calculate the interest earned each day from the initial sum of $1,500. Daily interest earned = Initial sum 脳 Daily interest rate Daily interest earned = \( 1500 \times \frac{1}{7300} \) Daily interest earned = \( \$\frac{1500}{7300} \)
03

Determine the number of days required

Now, we need to find out how many days it will take for the sum to earn $25 interest. We can set up an equation: Desired interest = Daily interest earned 脳 Number of days We want to solve for the number of days: Number of days = \( \frac{Desired\ interest}{Daily\ interest\ earned} \) Number of days = \( \frac{\$25}{\$ \frac{1500}{7300}} \) Number of days = \( 25 \times \frac{7300}{1500} \) Number of days = \( \frac{25 \times 73}{15} \) Number of days = \( \frac{1825}{15} \) Number of days = 121.67 Since we can't have a fraction of a day, we will round up to the nearest whole number. Number of days = 122
04

Final Answer

It will take approximately 122 days for a sum of \(1,500 to earn \)25 interest if it is deposited in a bank paying 5% interest per year.

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Key Concepts

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

Daily Interest Rate
Understanding the daily interest rate is crucial when calculating how quickly your investment will grow on a day-to-day basis. It is the amount of interest you earn each day on a deposit or the amount charged per day on a loan. To figure out the daily interest rate from an annual rate, divide the annual interest rate by 365, reflecting the number of days in a year.

For example, with an annual interest rate of \( 5\% \), the daily interest rate would be calculated as follows in Latex: \[ Daily\ interest\ rate = \frac{Annual\ interest\ rate}{365\ days} = \frac{5\%}{365} = \frac{5}{36500} \]
The daily interest rate allows us to understand the incremental accrual of interest, providing a granular view of how our investments or loans are performing on a daily basis.
Time Value of Money
The time value of money is a fundamental principle in finance that describes how the value of money changes over time. It is based on the idea that a certain amount of money today is worth more than the same amount in the future due to its potential earning capacity. This principle tells us that receiving money today is preferable because it can be invested to earn interest, generating a higher amount in the future.

To illustrate, if you were to receive \( \(1,000 \) today and save it in an account that pays \( 5\% \) annually, by the end of one year, you would have \( \)1,050 \). It's all about the earning potential of money; the sooner we have it, the more valuable it is, since it can be used to earn more through investments or savings.
Interest Formula
The interest formula is used to calculate the amount of interest earned or paid over a certain period of time. In its simplest form, the formula for simple interest is \( I = PRT \), where \( I \) is the interest, \( P \) is the principal amount, \( R \) is the rate of interest per period, and \( T \) is the time the money is invested or loaned for.

In our exercise, to find the number of days required to earn \( \(25 \) in interest on a \( \)1,500 \) deposit at a \( 5\% \) annual interest rate, we rearrange the formula to solve for \( T \) (the time), which in this case represents the number of days. The steps provided in the solution align perfectly with the application of this simple interest formula.
Mathematics in Finance
Mathematics plays an integral role in finance, helping to quantify economic activities and informing financial decision-making. Calculations involving interest rates, investment growth, loan repayment, and risk assessment all utilize mathematical concepts and formulas. It is important for individuals to understand these mathematical tools in order to make informed personal or business financial decisions.

One practical application of mathematics in finance is in the calculation of interest on savings or loans, as demonstrated by the daily interest example. Knowing how to apply these mathematical principles allows individuals and businesses to accurately forecast future values, understand the cost of borrowing, and make strategic investing decisions based on potential returns.

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Most popular questions from this chapter

The price of a new car is $$\$ 16,000$$. Assume that an individual makes a down payment of \(25 \%\) toward the purchase of the car and secures financing for the balance at the rate of \(10 \% /\) year compounded monthly. a. What monthly payment will she be required to make if the car is financed over a period of 36 mo? Over a period of \(48 \mathrm{mo}\) ? b. What will the interest charges be if she elects the 36 -mo plan? The 48-mo plan?

The Johnsons have accumulated a nest egg of $$\$ 40,000$$ that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively high tax bracket, they have decided to invest a minimum of $$\$ 2400 /$$ month in monthly payments (to take advantage of the tax deduction) toward the purchase of their house. However, because of other financial obligations, their monthly payments should not exceed $$\$ 3000$$. If local mortgage rates are \(7.5 \% /\) year compounded monthly for a conventional 30 -yr mortgage, what is the price range of houses that they should consider?

A state lottery commission pays the winner of the "Million Dollar" lottery 20 installments of $$\$ 50,000$$ /year. The commission makes the first payment of $$\$ 50,000$$ immediately and the other \(n=19\) payments at the end of each of the next 19 yr. Determine how much money the commission should have in the bank initially to guarantee the payments, assuming that the balance on deposit with the bank earns interest at the rate of \(8 \% /\) year compounded yearly. Hint: Find the present value of an annuity.

Martin has deposited $$\$ 375$$ in his IRA at the end of each quarter for the past 20 yr. His investment has earned interest at the rate of \(8 \% /\) year compounded quarterly over this period. Now, at age 60 , he is considering retirement. What quarterly payment will he receive over the next 15 yr? (Assume that the money is earning interest at the same rate and that payments are made at the end of each quarter.) If he continues working and makes quarterly payments of the same amount in his IRA until age 65, what quarterly payment will he receive from his fund upon retirement over the following \(10 \mathrm{yr}\) ?

Mike's Sporting Goods sells elliptical trainers under two payment plans: cash or installment. Under the installment plan, the customer pays $$\$ 22 /$$ month over 3 yr with interest charged on the balance at a rate of \(18 \% /\) year compounded monthly. Find the cash price for an elliptical trainer if it is equivalent to the price paid by a customer using the installment plan.
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