List Steps John Purchased A Television And The Final Price W

List Stepsjohn Purchased A Television And The Final Price Was 512 I

List steps! John purchased a television and the final price was \$512. If the sales tax was 7%, what was the list price of the TV, i.e., what was the price before the sales tax was added? Try thinking of it this way: Let x be the original price. Then the sales tax must be 0.07x. If you substitute those expressions and the final price into (original price) + (sales tax) = (final price), you'll have an equation that you can solve for x.

Suppose you put money in a savings account and every month you get interest on it. Instead of taking the interest out and spending it, you leave the interest in the account each month. This means that, at the end of month two, you’ll get interest not just on the original amount you put in but also on the interest that you left in on month one. At the end of month three, you’ll get interest on the principle, the interest from month one and the interest from month two, etc.

Over time, this “compounding” can add up to some serious money. The formula for calculating the amount in your account is A = P (1 + r / n)^(nt), where r is the annual interest rate, n is the number of times the interest is compounded every year, t is the number of years you let the investment go, P is the principle, and A is the amount in the account after t years. If you invest \$1,000 in an account at 5% annual interest compounded every month, i.e., 12 times per year, and let it sit for five years, how much money will be in the account?

A car rental agency charges \$55 per day plus \$0.50 per mile. If a person plans to rent a car for two days and has \$250 to spend, then how many miles can they afford to drive? Set up an algebraic inequality that describes this situation and solve it to get the answer. (Think of this as, the total cost is at most \$250.)

Paper For Above instruction

In this paper, we will systematically approach several real-world scenarios involving basic algebraic concepts to develop understanding and problem-solving skills. The scenarios include calculating the original price of a television before sales tax, determining the future value of an investment with compounded interest, and establishing a budget constraint for car rentals based on daily costs and mileage expenses.

Calculating the Original Price of a Television Before Sales Tax

The first problem involves finding the pre-tax price of a television, given the final price after applying a 7% sales tax. Let us denote the original price as x. The sales tax is calculated as 0.07x, which is 7% of the original price. The total cost, which is the sum of the original price and the sales tax, equals the final price of \$512. Therefore, the equation representing this situation is:

x + 0.07x = 512

Combining like terms, we get:

1.07x = 512

Solving for x involves dividing both sides by 1.07:

x = 512 / 1.07

Calculating the value yields:

x ≈ 479.44

Thus, the list price of the television before sales tax was approximately \$479.44.

Future Value of an Investment with Monthly Compounding

The second scenario concerns the growth of an investment with compound interest. The formula for compound interest, taking into account multiple compounding periods per year, is:

A = P (1 + r / n)^{nt}

where:

  • A = amount after t years
  • P = principal amount (\$1,000)
  • r = annual interest rate (5%, or 0.05)
  • n = number of times interest is compounded per year (12 for monthly)
  • t = number of years (5)

Substituting the given values yields:

A = 1000 (1 + 0.05 / 12)^{12*5}

Simplify inside the parentheses:

1 + 0.05 / 12 ≈ 1 + 0.0041667 ≈ 1.0041667

Raise to the power 60 (since 12 * 5 = 60):

A ≈ 1000 * (1.0041667)^{60}

Calculating the exponential component:

(1.0041667)^{60} ≈ 1.28368

Therefore, the total amount after five years is approximately:

A ≈ 1000 * 1.28368 ≈ \$1,283.68

Hence, the investment would grow to approximately \$1,283.68 after five years.

Budget Constraint for Car Rental

The third problem involves determining the maximum number of miles a person can drive given a rental budget. The total cost C includes a fixed daily fee of \$55 and a variable cost of \$0.50 per mile. For two days, the total cost is:

C = 2 55 + 0.5 m

where m is the number of miles driven. Since the total cost must be at most \$250, set up the inequality:

2 * 55 + 0.5m ≤ 250

Simplify the constants:

110 + 0.5m ≤ 250

Subtract 110 from both sides:

0.5m ≤ 140

Divide both sides by 0.5:

m ≤ 140 / 0.5

Calculate the division:

m ≤ 280

Therefore, the person can afford to drive up to 280 miles without exceeding their \$250 budget.

Conclusion

By setting up appropriate algebraic models, these real-world problems can be solved systematically. Calculating the original price before tax involves forming and manipulating a linear equation. Investment growth due to compounding interest requires understanding the exponential growth model, and budgeting scenarios benefit from inequalities that set constraints on variables. Mastery of these algebraic concepts enables accurate financial decision-making and planning.

References

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