CP Math Lesson 1-4 “Problem Solving”

Quiz Solve for ‘r’: 1. Solve for ‘w’: 2. Find ‘y’ when x = -3:

Vocabulary Verbal Model: Writing an equation in words before you write it in mathematical symbols.

EXAMPLE 3 You are selling homemade candles at a craft fair for $ 3 each. You spend $ 120 to rent the booth and buy materials for the candles. Write an equation that shows your profit from Write an equation that shows your profit from selling c candles. Verbal Models to solve problems. Evaluate: What does profit mean? Profit is $ from sales subtract costs Verbal model Profit = ($ from sales) – (costs) Costs: rent, insurance, buying the material that will be sold, employee salaries, etc. employee salaries, etc.

EXAMPLE 3 You are selling homemade candles at a craft fair for $ 3 each. You spend $ 120 to rent the booth and to buy materials for the candles. Write an equation that shows your profit from Write an equation that shows your profit from selling c candles. Verbal Models to solve problems. Profit = ($ from sales) – (costs) Verbal model $ from sales = ? $3 per candle * # candles = 3c P = 3c Write the equation: Cost = ? $120

Find your profit if you sell 75 candles. Verbal Models to solve problems. You are selling homemade candles at a craft fair for $ 3 each. You spend $ 120 to rent the booth and buy materials for the candles. P = 3c P = 3 ( ) P = 3 (75) P = 225 – 120 = $105

Your turn: 1. The store you work for buys 20 shirts for $10 each. The store marks the price up to $15 each and sells all 20 shirts. What is the profit from the sale of the shirts? 2. The sales price of a calculator is $100. A total of 20 calculators were sold. If the profit from selling the calculators is $250, what were to total costs associated with the calculators?

Total Income for a salesperson: The total amount of money a person earns as a salesperson. Total income = (wages) + ($$ from commissions) At a company, a salesperson has a base salary of $30,000 per year. She earns an additional 10% of her total sales (commission). year. She earns an additional 10% of her total sales (commission). If her total annual salary was $70,000, what was her total sales? 70,000= 30,000+ (0.10)(total sales) 70,000= 30, x -30,000-30,000 40,000= 0.10x ÷ ,000= 0.10x ÷ ,000 = x

Your turn: 3. An engineering salesman make 5% of his total sale as commission. If he sells $600,000 worth of equipment and his base salary is $50,000 this year, what is total income for the year? 4. A waitress earns $5 per hour plus tips. She averages 15% of her total sales on tips. For an 8 hours shift she earned $100. What was the total cost of the meals her customers ate?

Profit = ($ from sales) – (costs) Your turn: At a hamburger outlet, the manager counted the cash taken in one day and came up with $5000. He had to pay $3000 to the vendor for the supplies used to make the hamburgers, $400 for wages to the workers, $200 for his own salary, and $900 for other costs like electricity, rent, and insurance. How much was the profit for the day? 5.

Vocabulary Total cost: How do all of the products combined cost? combined cost? Total cost of hamburgers = cost of each hamburger * # of hamburgers Total cost of pizzas and drinks = (cost of each pizza) *( # of pizzas) + (cost of each drink)*(# of drinks). Total cost of items A and B = (cost per item A) *( # of items A) + (cost per item B) *( # of items B)

Total cost: How much do all of the products combined cost? combined cost? Total cost = ($ per item A) *( # of items A) + ($ per item B) *( # of items B) + ($ per item B) *( # of items B) Your club wants to sell pizzas to make money. You charge $10 for every large pizza and $6 for every small pizza. A business decides to buy pizza for all of their employees to help your club make money. They want 7 large pizzas and 5 small ones. How much money should they give you? $10 for every large pizza and $6 for every small pizza. A business decides to buy pizza for all of their employees to help your club make money. They want 7 large pizzas and 5 small ones. How much money should they give you? Total cost = ($10 per large pizza) *(7 large pizzas) + ($6 per small pizza) *(5 small pizzas) + ($6 per small pizza) *(5 small pizzas) Total cost = ($10 * 7) + ($6 * 5) Total cost = $70 + $30 $100

Your turn: 6. You and 7 friends went out to have hamburgers and drinks. The bill came to $76. You each had one drink and one hamburger. If drinks cost $2.50 each, what did each of the hamburgers cost?

vocabulary: Unit: how a real world quantity measured. Distance  Time  “seconds”, “hours”, “days” feetmilesmeters Treat units like variables when doing the math.

Your turn: Treat units like variables when doing the math

Speed/Distance Model: (involves time) Distance = (speed) (time) Speed is a “rate” Speed is a “rate” (distance per unit time) This is a ‘gotcha.’ All units (hours, minutes, feet, miles, etc., MUST be CONSISTENT throughout the problem!!!! d = r*t

Speed/Distance Model: (involves time) Distance = (speed) (time) Example: d = r*t It takes you 5 hours to drive to St. George. St. George is 300 miles away. How fast were you going? 1. Write the formula 2.Identify the quantities from the formula that are given in the problem: d = ?, r = ?, t = ? the problem: d = ?, r = ?, t = ? d = 300 miles, r = ?, t = 5 hours d = r*t 3. Replace the values given into the formula. 300 miles = r * 5 hours 4. Solve for the unknown variable. 300 miles = r * 5 hours ÷ 5 hours ÷ 5 hours

Speed/Distance Model: (involves time) Distance = (speed) (time) Example: d = r*t A plane flew at a speed of 300 miles/hr for 7 hours. How far did it fly? 1. Write the formula 2.Identify the quantities from the formula that are given in the problem: d = ?, r = ?, t = ? the problem: d = ?, r = ?, t = ? d = ?, r = 300 miles/hr, t = 7 hours d = r*t 3. Replace the values given into the formula. d= 300 miles/hr * 7 hours 4. Solve for the unknown variable.

Your turn: Distance = (speed) (time) 11. What would the speed have to be to travel 1000 miles in 6 hours? miles in 6 hours? d = r*t 12. How long would it take to travel 1500 miles if your speed was 200 miles per hour? speed was 200 miles per hour?

Problem Solving Strategies Use a formula: D = rt (replace the variables in the formula with numbers from the problem, then solve for the unknown variable. Look for a pattern: make a “table of values” Draw a diagram: make a picture

Problem Solving Strategies Look for a pattern: make a “table of values” Time (min.)01234 Altitude (ft) What is the altitude at the 7 minute point? at the 7 minute point? Time (min.) Altitude (ft) ft

Your turn: Year Lake depth (ft) What will be the depth of the lake in 2009?

Problem Solving Strategies Look for a pattern: Draw a picture, make an equation. You are hanging 5 pictures on the wall. Each picture is 1 ft wide. The wall is 13 feet wide. You want the pictures to be equally spaced. You want the space from the each be equally spaced. You want the space from the each corner of the wall to the nearest picture to be twice the corner of the wall to the nearest picture to be twice the distance between each picture. distance between each picture. What is the space between each picture? 2x 2x xxxx 1 ft 13 ft 13 ft = 2x + 1 ft + x + 1 ft + x + 1 ft + x + 1 ft + x + 1 ft + 2x 13 ft = 8x + 5 ft 8 ft = 8x X = 1 ft

Problem Solving Strategies Replace unknown numbers with variables A board is 26 ft long. You cut it into 3 pieces. The 1 st piece is half as long as the 2 nd piece. The 3 rd piece is 2 ft longer than half as long as the 2 nd piece. The 3 rd piece is 2 ft longer than the first piece. the first piece. a)Draw a picture to represent the problem b)Write an equation for the length of the board with “x” as the length of the first piece. length of the first piece. c) What is the length of each piece? x + 2 x ft 26 ft = x + 2x + (x + 2) 2xx 26 = 4x = 4x 6 = x = 6 = 12 = 8 = 8

Your turn: 14. Wood shop: A piece of wood is 72” long. You cut the wood into 3 pieces. The 2 nd piece is 6” longer than the 1 st piece. The 3 rd piece is 6” longer than the than the 2 nd piece. a. Draw a diagram showing relative lengths of the pieces. b. Write an equation showing the length of each piece (use only one variable: x = length of 1 st piece. c. How long is each piece?

verbal models to solve word problems. Profit = $ received – costs Total cost = sum of the individual costs Total number of items = sum of the individual items Total income = wages + tips Total income = $/hr + (% of the total bill) Profit = ?? Total cost = ?? Total number of items = ?? Total income = salary + commission Total income = $/yr + (% of yearly sales)

Homework 1-5