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Chapter 6: Problem 12

APPLICATION Lara buys a \(\$ 500\) sofa at a furniture store. She buys the sofa with a new credit card that charges \(1.5 \%\) interest per month, with an offer for "no payments for a year." a. What balance will Lara's credit card bill show after 6 months? Write an exponential expression and evaluate it. (a)

Short Answer

Expert verified
Lara's credit card balance will be $546.72 after 6 months.

Step by step solution

01

Understanding Monthly Interest

Lara's credit card has an interest rate of 1.5% per month. This means that each month, the balance will increase by multiplying it by a factor of \(1 + \frac{1.5}{100} = 1.015\).
02

Setting Up the Exponential Expression

The balance after a certain number of months can be calculated using the compound interest formula: \(\text{Balance} = P \times (1 + r)^n\). Here, \(P = 500\) is the principal, \(r = 0.015\) is the monthly interest rate, and \(n = 6\) is the number of months, so the expression becomes \(500 \times 1.015^6\).
03

Calculating the Balance

Now calculate the balance using the exponential expression: \(500 \times 1.015^6\). First, calculate \(1.015^6\) which is approximately 1.093443, and then multiply by 500 to find the balance: \(500 \times 1.093443 = 546.72\).
04

Final Amount After 6 Months

The calculated balance after 6 months, considering the interest, is approximately \(\\(546.72\). Therefore, Lara's credit card will show a balance of \(\\)546.72\).

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Key Concepts

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

Monthly Interest Rate
A monthly interest rate is a percentage charged each month on borrowed money, calculated using the balance of that month. In Lara's case, the monthly interest rate is 1.5%. This means that each month, the total amount due increases by an amount equal to 1.5% of the outstanding balance.

To find out how much 1.5% increase adds to Lara's sofa purchase, you convert this percentage to a decimal for calculations. This is done by dividing by 100, making it 0.015. The monthly interest, therefore, scales the balance by multiplying it each month by a factor of:
  • Factor: \(1 + 0.015 = 1.015\)
This factor includes the original amount and the added interest. Every month, this new total becomes the starting point for recalculating the next month's interest, making understanding this small percentage crucial.
Exponential Expression
The power of exponential expressions lies in their ability to represent repeated multiplication concisely. When dealing with compound interest such as Lara's credit card, you use the formula:
  • \( \text{Balance} = P \times (1 + r)^n \)
      • Here,
      • \(P\) is the principal, or the initial amount borrowed or owed, \(500\) in this case.
      • \(r\) represents the monthly interest rate expressed as a decimal, which is 0.015 for Lara.
      • \(n\) is the number of compounding periods or months, here 6.
      This expression shows the total effect of applying the interest rate repeatedly, month after month. The exponential part, \((1 + r)^n\), captures how interest compounds over time.

      In practical terms, \(1.015^6\) represents the cumulative effect of the 1.5% interest applied over six months. Once calculated (approximately 1.093443), this factor is multiplied by the principal \(500\), yielding a new balance showing the impact of accumulated interest.
Credit Card Balance
Your credit card balance reflects the amount you owe at any given time, including any new charges and interest accrued over time. For Lara, buying a \(\\(500\) sofa with a credit card at a monthly interest rate of 1.5% impacts her balance significantly over six months.

Initially, Lara's credit card balance is \(\\)500\). With no payments made for a year, the balance increases solely due to monthly compounded interest. By applying the compound interest formula, her balance reaches \(\$546.72\) after 6 months.

Understanding the dynamics of your credit card balance is crucial. It increases not only with additional purchases but also with interest, especially when you "just leave it" to accumulate, as in offers that defer payments. Keeping this picture in mind helps in managing and planning one's finances better, ensuring you're not caught off guard by the effects of compounded interest.

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Most popular questions from this chapter

Look at this "step" pattern. In the first figure, which has one step, each side of the block is \(1 \mathrm{~cm}\) long. a. Make a table showing the number of steps \((x)\) and the perimeter \((y)\) of each figure. (a) b. On a graph, plot the coordinates your table represents. c. Write an equation that relates the perimeter of these figures to the number of steps. d. Use your equation to predict the perimeter of a figure with 47 steps. e. Is there a figure with a perimeter of \(74 \mathrm{~cm}\) ? If so, how many steps does it have? If not, why not?

The population of a town is currently 45,647 . It has been growing at a rate of about \(2.8 \%\) per year. a. Write an expression in the form \(45,647(1+0.028)^{x}\) for the current population. b. What does the expression \(45,647(1+0.028)^{-12}\) represent in this situation? (a) c. Write and evaluate an expression for the population 8 years ago. (A) d. Write expressions without negative exponents that are equivalent to the exponential expressions from \(4 \mathrm{~b}\) and \(\mathrm{c}\). (a)

APPLICATION Tamara works at a bookstore, where she earns \(\$ 7.50\) per hour. a. Her employer is pleased with her work and gives her a \(3.5 \%\) raise. What is her new hourly rate? b. A few weeks later business drops off dramatically. The employer must reduce wages. He decreases Tamara's latest wage by \(3.5 \%\). What is her hourly rate now? c. What is the final result of the two pay changes? Explain. (H)

Use a recursive routine to find the first six terms of a sequence that starts with 100 and has a constant multiplier of \(-1.6\).

APPLICATION Toward the end of the year, to make room for next year's models, a car dealer may decide to drop prices on this year's models. Imagine a car that has a sticker price of \(\$ 20,000\). The dealer lowers the price by \(4 \%\) each week until the car sells. a. Write a recursive routine to generate the sequence of decreasing prices. (a) b. Find the 5 th term and explain what your answer means in this situation. (ii) c. If the dealer paid \(\$ 10,000\) for the car, how many weeks would pass before the car's sale price would produce no profit for the dealer?
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