Shopping List Using Multi-Step Equations
Shopping List Using Multi-Step Equations Wr
Write a multi-step equation for each of the three scenarios below to find the answer to the question. Show all the steps needed. Then write two more shopping scenarios (in the Solve Your Own Problem section) where multi-step equations can be written and solved. Explain all your work.
Write a shopping list to show what you will purchase, including the amounts and costs for each item and the total cost.
Scenario 1: The technical crew of the play needs shirts. Each shirt color matches the job the crew will be working on. Different color shirts have different designs and therefore cost different amounts. You will need: 6 red shirts at $10.90 each for the lighting crew; 4 yellow shirts at $10.50 each for the props crew; 5 purple shirts at $12.50 each for the make-up crew. You have $200.00 to spend on all the shirts. How many green shirts can you purchase for the ushers if those shirts cost $11.50 each? If you spend less than your budget, how much money do you have left over?
Scenario 2: Refreshments will be sold during the play’s intermission. The servers estimate they need 74 bottles of water at $0.75 each and 150 bags of chips at $0.40 each. You have $150.00 total to spend. Find out how much you will spend on water and chips. In addition, how many boxes of gummy bears, costing $0.50 each, can you buy with the remaining budget? If you spend less than your budget, how much money do you have left over?
Scenario 3: Actors need make-up for the play. You estimate they will need 10 bottles of foundation at $5.25 each, 7 containers of eyeshadow at $3.25 each, and 5 tubes of lipstick at $4.75 each. The total budget for make-up is $124.00. Find out how much you will spend on the foundation, eyeshadow, and lipstick. In addition, how many containers of face powder at $2.50 each can you buy? If you spend less than your budget, how much money do you have left over?
Solve Your Own Problem: Write two more shopping scenarios where multi-step equations can be written and solved. Explain all your work. Write a shopping list to show what you will purchase, including amounts and costs for each item and the total cost. Text1: Text2: Text3: Text4: Text5:
Paper For Above instruction
In this assignment, the focus is on translating real-world shopping and scenario problems into multi-step algebraic equations, solving them systematically, and interpreting the results. The three initial scenarios involve calculating costs and quantities given budgets, prices, and requirements, which demonstrates the practical application of algebra to everyday situations. Additionally, creating new scenarios encourages deeper understanding of how to model real-world problems with algebraic expressions, fostering critical thinking and problem-solving skills.
To effectively solve each scenario, set up equations that account for the total costs, quantities, and budgets. For example, in Scenario 1 about purchasing shirts, multiply the number of shirts needed by their prices and sum these to compare with the total budget. Then, find the maximum number of green shirts affordable within the remaining budget. Similarly, in Scenario 2, compute the total costs for water and chips, and then determine how many gummy bear boxes can be bought with remaining funds. In Scenario 3, sum the costs of different make-up items and compute how many face powders can be purchased within the budget.
Furthermore, the process of creating two additional shopping problems will involve defining variables for quantities and costs, formulating algebraic expressions representing the total expenditure, and solving for unknowns like the maximum number of items affordable or the total expense. This exercise reinforces skills in setting up and manipulating multi-step equations, as well as applying arithmetic operations accurately.
Overall, this assignment not only develops proficiency in solving multi-step equations but also enhances the ability to model real-life situations algebraically, interpret solutions in context, and communicate mathematical reasoning clearly and effectively.
Answer to the assigned problems
Scenario 1: Shirts Purchase
Let us denote the number of green shirts as \( g \). The costs for the existing shirts are:
- Red shirts: 6 shirts at \( \$10.90 \) each, total \( 6 \times 10.90 = \$65.40 \).
- Yellow shirts: 4 shirts at \( \$10.50 \) each, total \( 4 \times 10.50 = \$42.00 \).
- Purple shirts: 5 shirts at \( \$12.50 \) each, total \( 5 \times 12.50 = \$62.50 \).
Total spent on these shirts:
\[
65.40 + 42.00 + 62.50 = \$169.90
\]
Remaining budget:
\[
\$200.00 - \$169.90 = \$30.10
\]
Price per green shirt is \( \$11.50 \). To find how many green shirts \( g \) can be bought:
\[
11.50 \times g \leq 30.10
\]
\[
g \leq \frac{30.10}{11.50} \approx 2.62
\]
Since the number of shirts must be a whole number, the maximum number of green shirts possible is 2.
Remaining money after purchasing 2 green shirts:
\[
\$30.10 - 2 \times 11.50 = \$30.10 - \$23.00 = \$7.10
\]
Answer: You can purchase 2 green shirts, leaving you with \$7.10.
---
Scenario 2: Refreshments Cost Calculation
Total cost of water:
\[
74 \times \$0.75 = \$55.50
\]
Total cost of chips:
\[
150 \times \$0.40 = \$60.00
\]
Total spent on water and chips:
\[
55.50 + 60.00 = \$115.50
\]
Remaining budget for gummy bears:
\[
\$150.00 - \$115.50 = \$34.50
\]
Number of gummy bear boxes:
\[
\frac{\$34.50}{\$0.50} = 69
\]
Since we cannot buy a fraction of a box, buy 69 boxes.
Remaining money:
\[
\$34.50 - 69 \times 0.50 = \$34.50 - \$34.50 = \$0
\]
Answer: Spend \$115.50 on water and chips; purchase 69 boxes of gummy bears, with no money left over.
---
Scenario 3: Make-up Purchase
Costs:
- Foundation:
\[
10 \times \$5.25 = \$52.50
\]
- Eyeshadow:
\[
7 \times \$3.25 = \$22.75
\]
- Lipstick:
\[
5 \times \$4.75 = \$23.75
\]
Total cost:
\[
52.50 + 22.75 + 23.75 = \$99.00
\]
Remaining budget:
\[
\$124.00 - \$99.00 = \$25.00
\]
Number of face powder containers:
\[
\frac{\$25.00}{\$2.50} = 10
\]
Remaining money after purchasing 10 face powders:
\[
\$25.00 - 10 \times 2.50 = \$0
\]
Answer: Spend \$99.00 on specified make-up items; buy 10 containers of face powder, with no money left over.
---
Creating Your Own Scenarios
Scenario 1: Renting Movie Equipment
Suppose a rental service charges a flat fee of \$15 plus \$3.50 per hour to rent a projector. If a group has \$50 to spend and rents the projector for \( h \) hours, then the total cost \( C \) in dollars is:
\[
C = 15 + 3.50h
\]
Solve for \( h \) if the total cost cannot exceed \$50:
\[
15 + 3.50h \leq 50
\]
\[
3.50h \leq 35
\]
\[
h \leq 10
\]
Hence, they can rent it for up to 10 hours.
---
Scenario 2: Buying Fruits at a Market
A fruit stand sells apples for \$2.00 per pound and bananas for \$1.50 per pound. If a customer has \$10 to spend and wants to buy \( a \) pounds of apples and \( b \) pounds of bananas, then total cost \( T \):
\[
T = 2.00a + 1.50b
\]
Suppose the customer wants to buy exactly 4 pounds of fruit:
\[
a + b = 4
\]
Express \( T \) in terms of one variable \( a \):
\[
b = 4 - a
\]
\[
T = 2.00a + 1.50(4 - a) = 2a + 6 - 1.5a = (2 - 1.5)a + 6 = 0.5a + 6
\]
To stay within \$10:
\[
0.5a + 6 \leq 10
\]
\[
0.5a \leq 4
\]
\[
a \leq 8
\]
Since buying more than 4 pounds in total isn’t possible with this constraint, choose any \( a \leq 8 \).
---
Conclusion
This exercise demonstrates how real-world problems involving costs, quantities, and budgets can be modeled using multi-step algebraic equations. By translating scenarios into equations, systematically solving for unknowns, and interpreting the results, students develop critical skills in algebraic reasoning. The additional scenarios encourage creative thinking while reinforcing mathematical concepts, ensuring a comprehensive understanding of solving multi-step equations in practical contexts.
References
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