Lesson 11: Solving Word Problems. Example 1

Transcription

1 Lesson 11: Solving Word Problems Example 1 A computer repair store charges $45 an hour plus parts to repair computers. The parts to repair Mike s computer cost $32. The total bill was $ How many hours did they work on the computer? What I Know Define Your Variable(s). Write a Verbal Model & Substitute Solve Solution

2 Example 2 The perimeter of a flower bed is 56 feet. The length of the flower bed is 4 more than 2 times the width. Find the dimensions of the flower bed. Justify your answer. What I Know Define Your Variable(s). Write a Verbal Model & Substitute Solve Solution

3 Example 3 Kaleigh is selling pizza kits as a fundraiser. She sold 3 less pepperoni kits than supreme kits. She sold 8 less cheese kits than supreme kits. The prices of each are shown below. Cheese Kit - $12.00 Pepperoni Kit - $13.50 Supreme Kit - $15.00 Write an expression to show how many cheese kits were sold. Write an expression to show how many pepperoni kits were sold. The total sales for cheese and pepperoni kits is equal to the total sales for supreme kits. Write an equation that could be used to find how many of each type of pizza kit was sold. How many of each type of kit was sold? What I Know Define Your Variable(s). Write a Verbal Model & Substitute Solve

4 Word Problems Organizer What I Know Define Your Variable(s). Write a Verbal Model & Substitute Solve Solution

5 Lesson 11: Solving Equations Word Problems 1. An automotive repair shop charges $30 an hour for labor plus the price of parts. The parts to repair Jenny s car cost $ The total bill for Jenny s automotive repair was $ How many hours did the automotive technicians work on Jenny s car? 2. Jordan plays on two different basketball teams. He ordered t-shirts for each team. The total cost of each order was the same. The prices are shown below. Team Cost Per Shirt Flat Shipping Rate Bulls $7 $11.00 Terps $6.50 $15.00 Write an expression representing the total cost of shirts for the Bulls. Write an expression representing the total cost of shirts for the Terps. Write an equation that you can use to find how many players are on each basketball team. How many players are on each basketball team? Justify your answer mathematically.

6 3. Terry plans to buy a large screen TV. The perimeter of the screen (which is a rectangle) is 140 inches. The width of the screen is 10 inches longer than the length. Write an equation that could be used to find the dimensions (length and width) of the TV s screen. Find the length and width of the screen. Justify your answers mathematically. 4. John is tiling his kitchen floor. The length of his kitchen is twice as long as the width. The perimeter of the kitchen is 45 feet. Write an equation that could be used to find the dimensions (length and width) of the kitchen. Find the dimensions of the kitchen. Justify your answers mathematically. Find the area of the kitchen. The tile cost $3.50 per square foot. How much money will John have to pay to tile his kitchen? Explain how you determined your answer.

7 5. You need to find a new cell phone plan. Sprint offers $30 flat rate plus 15 cents minute. AT&T offers 25 cents a minute plus $10 flat rate. Write an expression to show how much Sprint charges for a cell phone plan. Write an expression to show how much AT&T charges for a cell phone plan. How many minutes would you have to talk for the price of both cell phone plans to cost the same? Justify your answer mathematically. 6. Jody is selling chocolate candy for a school fundraiser. She sold 40 more Kit Kats than Hershey bars. She sold three times as many M&Ms as Hershey bars. The prices of each Chocolate Candy are shown below. Candy Price Kit Kat Miniature $.50 Extra Large Hershey Bar $1.25 M&Ms $1.00 Let x = the number of Hershey Bars sold. Write an expression to show how many Kit Kat Miniatures were sold. Write an expression to show how many M&Ms were sold.

8 The total amount of money raised was $ Write an equation that could be used to find how many of each type of candy was sold. How many of each type of candy was sold? Justify your answer mathematically. The school keeps 60% of the money raised. How much money does the school keep? Explain how you determined your answer. 7. You are selling tickets to the town carnival. You sold 50 more adult tickets than senior tickets and 5 times as many child tickets as senior tickets. The prices are shown below. Ticket Prices for Carnival Adult - $3.00 Senior - $2.00 Child - $5.00 Let x represent the number of senior tickets sold. Write an expression to represent the number of adult tickets sold and an expression to represent the number of child tickets sold. The total sales from the carnival were $1620. Write an equation that you would use to represent the total amount of sales for the carnival. How many child tickets were sold at the carnival? Justify your answer mathematically.

9 8. Use the following triangle diagram to answer the questions below. x x x+3 Write a formula that could be used to find the perimeter of the triangle. If the perimeter of the triangle is 51 cm, find the length of the longest side of the triangle. Explain how you determined your answer. 9. The side of a square is 3x +2. If the perimeter is 68 cm, what is the length of one side of the square? Justify your answer mathematically.

10 10. The perimeter of a rectangular garden is 54 ft. The length of the garden is 7 more feet than the width. Draw a diagram to illustrate this situation. Find the dimensions of the garden. Justify your answer mathematically. 11. The area of a rectangular piece of carpet of 272 ft 2. The length of the carpet is 16ft. Find the width of the carpet. Justify your answer mathematically. 12. The perimeter of a rectangle is 44 in. The width is 2 more than 3 times the length. Find the dimensions of the rectangle. Justify your answer mathematically.

11 1. Terri plans to build a flower bed. The perimeter of the flower bed (which is a rectangle) is 74 feet. The width of the flower bed is 3 feet shorter than the length. (3 points) Write an equation that could be used to find the dimensions (length and width) of the flower bed. Find the length and width of the flower bed. Justify your answers mathematically. 2. You are selling tickets to the school play. You sold 75 more adult tickets than senior tickets and 3 times as many child tickets as senior tickets. The prices are shown below. (4 points) Ticket Prices for Carnival Adult - $6.00 Senior - $4.00 Child - $2.50 Let x represent the number of senior tickets sold. Write an expression to represent the number of adult tickets sold and an expression to represent the number of child tickets sold. The total sales from the carnival were $ Write an equation that you would use to represent the total amount of sales for the play. How many adult tickets were sold at the play? Justify your answer mathematically.

12 Lesson 11: Solving Equations Word Problems Answer Key 1. An automotive repair shop charges $30 an hour for labor plus the price of parts. The parts to repair Jenny s car cost $ The total bill for Jenny s automotive repair was $ How many hours did the automotive technicians work on Jenny s car? Let h = number of hours Rate Number of hours = Total bill 30h +62 = h = h = 135 $30 an hour +parts = Total Subtract 62 from BOTH sides 30h = 135 Divide both sides by h = 4.5 The technicians worked on Jenny s car for 4.5 hours. Check: 30(4.5) +62 = = Jordan plays on two different basketball teams. He ordered t-shirts for each team. The total cost of each order was the same. The prices are shown below. Team Cost Per Shirt Flat Shipping Rate Bulls $7 $11.00 Terps $6.50 $15.00 Let x= number of shirts cost of shirts number of shirts + shipping = total Write an expression representing the total cost of shirts for the Bulls. 7x +11 Write an expression representing the total cost of shirts for the Terps. 6.50x+15

13 Write an equation that you can use to find how many players are on each basketball team. Since the total cost of the two orders was the same, we can set the two expressions equal to each other and solve for x. 7x+11 = 6.50x +15 Set the two expressions equal to each other How many players are on each basketball team? Justify your answer mathematically. 7x = 6.50x Subtract 11 from both sides. 7x = 6.50x +4 7x 6.50x = 6.50x -6.50x +4 Subtract 6.50x from both sides..5x = 4 Divide both sides by x = 8 There are 8 players on each basketball team. Justify: 7x+11 = 6.50x +15 7(8)+11 = 6.50(8) = Terry plans to buy a large screen TV. The perimeter of the screen (which is a rectangle) is 140 inches. The width of the screen is 10 inches longer than the length. Write an equation that could be used to find the dimensions (length and width) of the TV s screen. P = 2L +2W Let L = Length P = 140 in W = L = 2L +2(L+10)

14 Find the length and width of the screen. Justify your answers mathematically. 140 = 2L +2(L+10) 140 = 2L +2L +20 Distribute. 140 = 4L +20 Combine like terms = 4L Subtract 20 from both sides of the equation. 120 = 4L 120 = 4L Divide BOTH sides by = L W = L +10 W = The length is 30 in. W = 40 in The width is 40in. Justify: P = 2L +2W 140 = 2(30) +2(40) 140 = John is tiling his kitchen floor. The length of his kitchen is twice as long as the width. The perimeter of the kitchen is 45 feet. Write an equation that could be used to find the dimensions (length and width) of the kitchen. Let w = width of kitchen floor P = 45 ft Length = 2w P = 2L +2W 45 = 2(2w) +2w

15 Find the dimensions of the kitchen. Justify your answers mathematically. 45 = 2(2w) +2w Justify: 45 = 4w +2w P = 2L +2W 45 = 6w 45 = 2(15) +2(7.5) 45 = 6w 45 = W = 7.5 ft L = 2w L = 15 ft The width is 7.5 ft and the length is 15 ft. Find the area of the kitchen. A = l w A = 15 ft 7.5 ft A = ft 2 The area of the kitchen is ft 2 The tile cost $3.50 per square foot. How much money will John have to pay to tile his kitchen? Explain how you determined your answer = John will have to pay $ to tile his kitchen. The area was sq. feet. I multiplied the price per sq.ft ($3.50) by to get $

16 5. You need to find a new cell phone plan. Sprint offers $30 flat rate plus 15 cents minute. AT&T offers 25 cents a minute plus $10 flat rate. Let m = the number of minutes price number of minutes + flat rate = Total Write an expression to show how much Sprint charges for a cell phone plan..15 m + 30 Write an expression to show how much AT&T charges for a cell phone plan..25 m + 10 How many minutes would you have to talk for the price of both cell phone plans to cost the same? Justify your answer mathematically..15 m +30 =.25m +10 Justify:.15m =.25m m +30 =.25m m =.25m (200)+30 =.25(200) m -.25m =.25m -.25m = m = m = 200 minutes The two cell phone plans will cost the same after 200 minutes. 6. Jody is selling chocolate candy for a school fundraiser. She sold 40 more Kit Kats than Hershey bars. She sold three times as many M&Ms as Hershey bars. The prices of each Chocolate Candy are shown below. Candy Price Kit Kat Miniature $.50 Extra Large Hershey Bar $1.25 M&Ms $1.00 Let x = the number of Hershey Bars sold.

17 Write an expression to show how many Kit Kat Miniatures were sold. (Since it s just an expression to show how many, we do not need to include the price.) 40 more kit kats than Hershey bars (40 more means add 40) x +40 Write an expression to show how many M&Ms were sold. (Since it s just an expression to show how many, we do not need to include the price.) Three times as many M&M s (3 times as many means multiply by 3) 3x The total amount of money raised was $ Write an equation that could be used to find how many of each type of candy was sold. Now we are talking money, so we need to multiply the number of each candy (expressions written above) by their price. Price KK# + price Hersh # +price M&M# = Total.50(x+40) x + 1(3x) = (x+40) +1.25x +1(3x) = 520 How many of each type of candy was sold? Justify your answer mathematically..50(x+40) +1.25x +1(3x) = Solve for x..50x x + 3x = x x +3x +20 = Rewrite with like terms together. 4.75x +20 = Combine like terms. 4.75x = Subtract 20 from BOTH sides. 4.75x = x = Divide BOTH sides by x = 105 = Hershey Justify: x+40 = 145 = Kit Kat 3x = 315 = M&M/s.50(x+40) +1.25x +1(3x) =

18 105 Hershey bars were sold..50(105+40) +1.25(105)+1(3 105) = Kit Kats were sold = M&M s were sold = The school keeps 60% of the money raised. How much money does the school keep? Explain how you determined your answer. $ = $ % written in decimal form is.60. I multiplied.60 by the total amount of money raised (518.75) to get an answer of is 60% of You are selling tickets to the town carnival. You sold 50 more adult tickets than senior tickets and 5 times as many child tickets as senior tickets. The prices are shown below. Ticket Prices for Carnival Adult - $3.00 Senior - $2.00 Child - $5.00 Let x represent the number of senior tickets sold. Write an expression to represent the number of adult tickets sold and an expression to represent the number of child tickets sold. Since we are just writing an expression to represent the number of tickets, we do not need to include the prices. Senior tickets = x Adult tickets: x+50 (50 more than means add 50) Child tickets: 5x (5 times as many means multiply by 5)

19 The total sales from the carnival were $1620. Write an equation that you would use to represent the total amount of sales for the carnival. Price #of Adult tickets + price # of child tickets + price # of senior tickets = total 3(x+50) + 5(5x) + 2(x) = 1620 How many child tickets were sold at the carnival? Justify your answer mathematically. 3(x+50) + 5(5x) + 2(x) = x x+2x = 1620 Justify: 3x+25x+2x+50 = (x+50) + 5(5x) + 2(x) = x+50 = (49+50) +5(5 49) +2(49) = x = (99) +5(245) +2(49) = x = = x = x = 49 Senior = x There were 49 senior tickets sold. Adult = x +50 (49+50) = 99 Adult tickets sold. Child = 5x (5 49) =245 Child tickets sold. 8. Use the following triangle diagram to answer the questions below. S1 x S2 x x+3 S3 Write a formula that could be used to find the perimeter of the triangle. P = s1 +s2+s3 P = x + x+x+3 P = 3x +3 Perimeter = Side1 + Side 2 +Side3 Combine like terms

20 If the perimeter of the triangle is 51 cm, find the length of the longest side of the triangle. Explain how you determined your answer. P = 51 P = 3x = 3x +3 or 3x +3 = 51 3x +3-3 = 51-3 Subtract 3 from BOTH sides. 3x =48 3x = Divide by 3 on BOTH sides. x = 16 The longest side is x +3, so if I substitute 16 for x, then the longest side is 19 cm. The longest side is 19 and the two other sides are = 51, so I know my answer is correct. 9. The side of a square is 3x +2. If the perimeter is 68 cm, what is the length of one side of the square? Justify your answer mathematically. Perimeter of a square = 4s (4 length of side) - All sides of a square are the same measurement. P = 4s S = 3x +2 P = 68 cm 68 = 4(3x +2) OR 4(3x +2) = 68 4(3x +2) = 68 12x + 8 = 68 Distribute the 2. 12x +8-8 = 68-8 Subtract 4 from BOTH sides. 12x = 60 12x = 60 Divide BOTH sides by x = 5 cm Side = 3x +2 so 3(5)+2 = 17 The length of one side of the square is 17 cm. Justify: 4s = P 4s = 68 cm 4 17in = 68 cm 68 cm = 68 cm

21 10. The perimeter of a rectangular garden is 54 ft. The length of the garden is 7 more feet than the width. Draw a diagram to illustrate this situation. w w +7 Find the dimensions of the garden. Justify your answer mathematically. P = 54 ft width = w length = w +7 P = 2L +2W Formula for perimeter of a rectangle. 54 = 2(w+7) +2w 54 = 2w w Distribute the = 4w = 4w Subtract 14 from BOTH sides 40 = 4w 40 = 4w Divide BOTH sides by W = width width = 10 W +7 = length = 17 The width is 10 ft and the length is 17 ft. Justify: P = 2L +2W 54 ft = 2(17 ft) +2(10 ft) 54 ft = 34 ft + 20 ft 54 ft = 54 ft

22 11. The area of a rectangular piece of carpet of 272 ft 2. The length of the carpet is 16ft. Find the width of the carpet. Justify your answer mathematically. Area of a rectangle = length width A = L W A = 272 ft 2 L = 16 ft W =? Let w = width of carpet. L W = A Justify: 16 w = 272 L W = A 16w = ft 17 ft = 272 ft ft 2 = 272 ft 2 W = 17 ft 12. The perimeter of a rectangle is 44 in. The width is 2 more than 3 times the length. Find the dimensions of the rectangle. Justify your answer mathematically. P = 2L +2W P = 44 in 2 L= length W = 3L +2 (2 more means add 2 and 3 times the length means multiply 3 by the length) 44 = 2L + 2(3L +2) 44 = 2L +6L +4 Distribute the = 8L +4 Combine like terms = 8L +4-4 Subtract 4 from BOTH sides. 40 = 8L 40 = 8L Divide BOTH sides by in= L W = 3L +2 so W = 3(5) +2 W = 17 in The length of the rectangle is 5 in and the width of the rectangle is 17 in. Justify: P = 2L +2W 44 = 2(5) +2(17) 44 = 44

23 1. Terri plans to build a flower bed. The perimeter of the flower bed (which is a rectangle) is 74 feet. The width of the flower bed is 3 feet shorter than the length. (3 points) Write an equation that could be used to find the dimensions (length and width) of the flower bed. What we know: P = 74 ft length = L width = L 3 (3 ft shorter than the length) P = 2L + 2w Perimeter Formula 74 = 2L + 2(L-3) Substitute for P, L and W The equation that can be used to find the dimensions is: 2L + 2(L-3) = 74 Find the length and width of the flower bed. Justify your answers mathematically. 2L + 2(L-3) = 74 Equation from above 2L + 2L 6 = 74 Distribute the 2 throughout the parenthesis 4L 6 = 74 Combine like terms: 2L+ 2L = 4L 4L 6+ 6 = Add 6 to both sides 4L = 80 Simplify: 74+6 = 80 4L/4 = 80/4 Divide by 4 on both sides L = 20 Simplify: 80/4 = 20 The length of the flower bed = 20 ft. The width is 17 ft (20-3 = 17) P = 2L + 2W Justify by substituting 74 = 2(20) + 2(17) 74 = = 74 Yes, my justification is correct, so my answer is correct. 2. You are selling tickets to the school play. You sold 75 more adult tickets than senior tickets and 3 times as many child tickets as senior tickets. The prices are shown below. (4 points) Ticket Prices for Carnival Adult - $6.00 Senior - $4.00 Child - $2.50

24 Let x represent the number of senior tickets sold. Write an expression to represent the number of adult tickets sold and an expression to represent the number of child tickets sold. We know that x represents the number of senior tickets sold. Adult tickets = 75 more adults than senior tickets. This can be translated as x + 75 Child tickets = 3 times as many child as senior tickets. This can be translated as: 3x Adults tickets = x + 75 Child tickets = 3x The total sales from the carnival were $ Write an equation that you would use to represent the total amount of sales for the play. We have to multiply the number of tickets by the price and add then all together. Adults + Senior + Child = Total 6(x+75) + 4x (3x) = How many adult tickets were sold at the play? Justify your answer mathematically. We must solve for x in order to determine how many adult tickets were sold at the play. 6(x+75) + 4x (3x) = Equation from above 6x x x = Distribute the 6 & multiply 2.5(3x) 6x + 4x x = Rewrite like terms together 17.50x = Combine like terms: 6x+4x+7.5x = 17.50x 17.50x = Subtract 450 from both sides 17.50x = Simplify: = x/17.50 = /17.50 Divide by on both sides x = 63 Simplify: /17.5 = 63 We know that x = 63, so that means that the number of senior tickets = 63. Since there were 75 more adult tickets sold than senior tickets, there were 138 adult tickets sold. (63+75 = 138) There were 138 adult tickets sold. Justify: 6(x+75) + 4x (3x) = Equation from above 6(63+75)+4(63)+2.5(3*63) = Substitute 63 for x into the original equation = Multiply = Both sides are equal, so my answer is correct.