CAN ANYONE HELP ME?
Solve using Polya’s four-step problem solving strategies.
(a) A local furniture store is having a terrific sale. They are marking down every rice 45%. If the couch you have over eye on is $440 after the markdown, what was the original price? How much you save if you bought it at this sale?
(b) The cost C to produce X number of cd’s is C= 50 + 5X. The cd’s are sold wholesale for $15 each, so revenue R is given by R= 15X. Find how many cd’s the manufacturer needs to produce and sell to break even.
(c) One number is 3 less than another number. If the sum of the two numbers is 177, find each number.
Sure, I can help you solve these problems using Polya's four-step problem solving strategies. Here's how we can solve each problem:
(a) Problem analysis:
- The original price of the couch is unknown.
- The couch is marked down by 45% and costs $440 after the markdown.
- We need to find the original price and the amount saved.
Plan:
1. Let's assume the original price of the couch is P.
Solution:
2. Since the couch is marked down by 45%, the sale price is 100% - 45% = 55% of the original price.
So, the sale price is 0.55P = $440.
3. To find the original price, we can set up an equation:
0.55P = $440
Divide both sides of the equation by 0.55 to isolate P:
P = $440 / 0.55
P = $800
4. The original price of the couch was $800.
The amount saved is the original price minus the sale price:
Amount saved = $800 - $440 = $360.
(b) Problem analysis:
- The cost to produce X number of CDs is given by C = 50 + 5X.
- The CDs are sold wholesale for $15 each, so the revenue R is given by R = 15X.
- We need to find the number of CDs needed to break even, where revenue equals cost.
Plan:
1. Set the revenue equal to the cost and solve for X.
Solution:
2. Set up the equation: 15X = 50 + 5X.
3. Move the term with X to one side:
15X - 5X = 50.
4. Combine like terms:
10X = 50.
5. Divide both sides of the equation by 10:
X = 50 / 10
X = 5.
6. The manufacturer needs to produce and sell 5 CDs to break even.
(c) Problem analysis:
- One number is 3 less than another number.
- The sum of both numbers is 177.
- We need to find both numbers.
Plan:
1. Set up equations based on the given information.
Solution:
2. Let's assume one number is N.
3. The other number, which is 3 less than N, can be represented as N - 3.
4. The sum of these two numbers is 177:
N + (N - 3) = 177.
5. Solve the equation:
2N - 3 = 177.
2N = 180.
N = 180 / 2.
N = 90.
6. The first number is 90.
The second number is 90 - 3 = 87.
I hope this helps! Let me know if you have any further questions.
Certainly! Let's solve each of these problems using Polya's four-step problem-solving strategies.
Step 1: Understand the problem
First, let's understand the given information and what is asked in each problem.
(a) We are given that a couch is on sale for $440 after a 45% markdown, and we need to find the original price and the amount saved.
(b) We need to determine the number of CDs the manufacturer needs to produce and sell in order to break even. We are given the cost function and the revenue function.
(c) We need to find two numbers given that one number is 3 less than the other, and their sum is 177.
Step 2: Devise a plan
(a) To find the original price of the couch, we can use the fact that the sale price is 45% off the original price. From there, we can calculate the amount saved.
(b) To find the number of CDs needed to break even, we can set the cost function equal to the revenue function, since revenue needs to be equal to costs.
(c) To find the two numbers, we can set up a system of equations based on the given information and solve for the variables.
Step 3: Execute the plan
(a) Let's calculate the original price:
- Let the original price be x.
- We know that 45% of x is subtracted, leaving us with 55% (100% - 45%) of x.
- So, 55% of x is equal to $440.
- Mathematically: 0.55x = 440.
- Solving for x, we divide both sides of the equation by 0.55: x = 440 / 0.55 = $800.
- Therefore, the original price of the couch was $800.
- To calculate the savings, we subtract the sale price from the original price: Savings = $800 - $440 = $360.
(b) Let's find the number of CDs needed to break even:
- To break even, revenue should equal costs.
- Set the cost function equal to the revenue function:
50 + 5X = 15X.
- Subtract 5X from both sides and simplify:
50 = 10X.
- Divide both sides by 10 to solve for X:
X = 50 / 10 = 5.
- Therefore, the manufacturer needs to produce and sell 5 CDs to break even.
(c) Let's find the two numbers:
- Let x be the first number.
- Since the second number is 3 less than the first number, the second number would be x - 3.
- The sum of the two numbers is 177, so we have the equation: x + (x - 3) = 177.
- Combine like terms: 2x - 3 = 177.
- Add 3 to both sides: 2x = 180.
- Divide both sides by 2 to solve for x: x = 180 / 2 = 90.
- Therefore, the first number is 90, and the second number is 90 - 3 = 87.
Step 4: Review the solution
(a) The original price of the couch was $800, and the amount saved was $360.
(b) The manufacturer needs to produce and sell 5 CDs to break even.
(c) The first number is 90, and the second number is 87, given that their sum is 177.
I hope this helps! Let me know if you have any further questions.
a) .55x = 440
x = 440/.55 = ...
b) solve: 15x = 50 + 5x
c) another number --- x
the one number --- x-3
solve: x + x-3 = 177