Problems: Include All Steps And Answers

Problems Need To Include All Required Steps And Answers For Full Cre

Problems need to include all required steps and answer(s) for full credit. All answers need to be reduced to lowest terms where possible. Write the final answer in the terms being asked such as dollars/cents, degrees, tickets, etc.

Paper For Above instruction

1. A firm pays $1.75 for each copy of a magazine and sells each one for $2.50. There is a fixed monthly cost of $40 for printing the publication. If the firm wants to make $550 dollars next month, how many magazines do they have to sell? Write and solve an equation that represents this scenario.

Let x represent the number of magazines sold.

Equation: 2.50x - (1.75x + 40) = 550

Calculations:

  • Revenue from sales: 2.50x
  • Cost of production: 1.75x + 40
  • Profit: 2.50x - (1.75x + 40) = 550
  • Simplify: 2.50x - 1.75x - 40 = 550
  • Combine like terms: 0.75x - 40 = 550
  • Add 40 to both sides: 0.75x = 590
  • Divide both sides by 0.75: x = 590 / 0.75 = 786.67

Answer: The firm must sell approximately 787 magazines.

2. Martin sold his computer and software for $900, receiving three times as much for the computer than the software. What was the selling price of the computer and the software? Write and solve an equation.

Let s = price of software

Then, computer = 3s

Equation: 3s + s = 900

Calculations:

  • Combine like terms: 4s = 900
  • Divide both sides by 4: s = 900 / 4 = 225

Answer: software = $225, computer = 3 × 225 = $675

3. The perimeter of a pool is 64 feet and has a width of x and a length of x - 4. Write an equation and find both the width and length of the pool.

Equation: Perimeter = 2(length + width)

Perimeter: 2((x - 4) + x) = 64

Calculations:

  • Expand: 2(2x - 4) = 64
  • Simplify: 4x - 8 = 64
  • Add 8 to both sides: 4x = 72
  • Divide both sides by 4: x = 18

Width: x = 18 feet

Length: x - 4 = 14 feet

4. The tax on a purchase was $9.33. If the sales tax rate is 6%, how much was the purchase? Write and solve an equation.

Let P = original purchase amount.

Tax = 6% of P: 0.06P = 9.33

Calculations:

  • Divide both sides by 0.06: P = 9.33 / 0.06 ≈ 155.50

Answer: The purchase amount was approximately $155.50.

5. Mike needs at least a 75% average to pass his math course. The class contains 5 exams that are equally weighted. If he scored a 64%, 86%, 71%, and 90% on the first 4 tests, what score does he need on the final test to earn at least a 75% in the class? Write and solve an inequality.

Let x = score needed on the 5th test.

Average: (64 + 86 + 71 + 90 + x) / 5 ≥ 75

Calculations:

  • Sum: (64 + 86 + 71 + 90 + x) ≥ 375
  • Calculate sum of known scores: 311 + x ≥ 375
  • Solve for x: x ≥ 375 - 311 = 64

Answer: Mike needs at least 64% on the final test.

6. The Parker’s are installing a wooden fence in their backyard. They have 330 feet of wood. The length can be no more than 90 feet. Write and solve an inequality to find the maximum width of the fence.

Let w = width and l = length ≤ 90 ft

Perimeter: 2(w + l) ≤ 330

Since length ≤ 90, substitute l = 90 for maximum width:

2(w + 90) ≤ 330

Calculations:

  • Divide both sides by 2: w + 90 ≤ 165
  • Solve for w: w ≤ 165 - 90 = 75

Answer: The maximum width is 75 feet.

7. Paula is an office manager for ABC Advertising. She has been tasked with finding a copy machine that falls within a budget of $750 per month. She finds a company that will lease the machine for $275 a month. Each copy costs 4 cents and a ream of 500 sheets costs $5.00. If she estimates that they will make 10,500 copies per month, is leasing this machine a good choice? Write and solve an inequality and explain your reasoning.

Total costs: lease + cost per copy + cost of paper

Lease: $275

Cost per copy: 4 cents = $0.04 per copy

Total copies: 10,500

Paper cost: (10,500 / 500) reams × $5.00 = 21 × 5 = $105

Total copy cost: 10,500 × 0.04 = $420

Total monthly cost: 275 + 420 + 105 = $800

Since total cost ($800) > budget ($750), leasing is not a good choice.

Alternatively, inequality:

275 + (0.04 × 10,500) + 5 × (10,500 / 500) ≤ 750

Calculations demonstrate total exceeds budget, so leasing is not advisable.

8. Peter is throwing a surprise party for his friend Tammy. He has a budget of $350. If the restaurant charges $20 per person for drinks and food and a cleanup fee of $35, what is the maximum number of people that he can invite to stay within budget? Write and solve an inequality.

Let p be the number of guests (including Peter and Tammy).

Total cost: 20p + 35 ≤ 350

Calculations:

  • Subtract 35 from both sides: 20p ≤ 315
  • Divide both sides by 20: p ≤ 15.75

Since p must be a whole number, maximum number: 15 guests.

9. Sally calculated that she will lose 4.6 calories per minute walking at a rate of 3 miles per hour. How many minutes does she need to walk to burn at least 250 calories? Write and solve an inequality, rounding to the nearest tenth.

Let t be minutes needed.

Calorie burn rate: 4.6 calories/min × t ≥ 250

Calculations:

  • t ≥ 250 / 4.6 ≈ 54.35

Answer: Sally needs to walk approximately 54.4 minutes.

10. When solving an inequality, when is the sign reversed?

The sign is reversed when multiplying or dividing both sides of an inequality by a negative number.

References

  • Clark, K. (2020). Elementary Algebra. Pearson.
  • Gordon, C. (2018). Introductory Algebra. McGraw-Hill Education.
  • Hewitt, P. G. (2019). Conceptual Physics. Pearson.
  • Larson, R., & Hostetler, R. (2016). Precalculus with Limits. Cengage Learning.
  • Princeton Review. (2021). ACT Prep. Princeton Review.
  • Stewart, J. (2018). Calculus: Early Transcendentals. Cengage Learning.
  • Ng, M. (2019). Business Mathematics. Pearson.
  • Anthony, M., & Biggs, N. (2017). Discrete Mathematics. Oxford University Press.
  • Singh, S. (2019). Financial Mathematics. Routledge.
  • Blitzer, R. (2018). Algebra and Trigonometry. Pearson.

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