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3.1 Changing App Probs into Equations How do different words and phrases relate to mathematical operations? What words do you know that indicate a certain operation? Refer to the chart on p 180 1 3.1 Changing App Probs into Equations One of the things you will be asked to do in this section is translate from words to symbols and from symbols to words: Given a mathematical expression, provide an expression in words that would be equivalent: 2x + 3 2 Given a mathematical expression, provide an expression in words that would be equivalent: 2x + 3 • Three more than twice a number • The sum of twice a number and three • Twice a number, increased by three • Three added to twice a number 3 Given an expression in words, provide a mathematical expression that would be equivalent: • Four less than three • 3x – 4 times a number • Five more than twice • 5 + 2x a number • Three times the sum • 3(x + 8) of a number and 8 • Twice the difference of a number and 4 • 2(x – 4) 4 Consider this: • 2(x – 4) Two times a number minus 4 Which expression does that represent? • 2x – 4 What if I read it with inflection and pauses? Be careful!!! 5 Given a sentence in words, provide a mathematical equation that would be equivalent: Three times the difference between a number and two is four less than six times the number 3(x – 2) = 6x – 4 When four is added to a number, the sum will be twenty x +4 = 20 6 Steps for solving a word problem 1. Read and reread the problem until you understand it (not until you know how to solve it, but until you understand what the story is.) 2. Take notes. (they only have to make sense to you, but they should include all the important information in the problem) 3. Define the variable (and any other unknown quanities. 4. Write an equation. 5. Solve the Equation. 6. Label your answer. 7. Ask yourself, Does my answer make sense? 7 3.1 Changing App Probs into Equations In this section, you are learning how to pick out the single variable and how to define your unknown quantities. 8 Steps for solving a word problem 1. Read and reread the problem until you understand it (not until you know how to solve it, but until you understand what the story is.) 2. Take notes. (they only have to make sense to you, but they should include all the important information in the problem) 3. Define the variable (and any other unknown quanities. 4. Write an equation. 5. Solve the Equation. 6. Label your answer. 7. Ask yourself, Does my answer make sense? 9 3.2 Solving Application Problems Two subtracted from 4 times a number is 10. Find the number. n = number 4n – 2 = 10 +2 +2 4n = 12 4 4 n=3 10 3.2 Solving Application Problems The sum of two numbers is 26. Find the two numbers if the larger number is 2 less than three times the smaller number. 11 NOTES: sum is 26 Larger is 3(sm) -2 ?find both numbers? 3n-2 =Larger number n =Smaller number smaller + larger = 26 n + (3n – 2)= 26 4n – 2 = 26 4n = 28 n=7 smaller is 7; larger is 19 12 3.2 Solving Application Problems A company currently produces 1200 widgets per year. They plan to increase production by 550 widgets each year until they reach their goal of 4500 widgets produced in a year. How many years will it take them to reach their goal? 13 1200/yr starting point 550/yr increase 4500/year goal ?how long will it take? y = number of years starting + inc = goal 1200 + 550y = 4500 550y = 3300 y=6 it will take 6 years. 14 3.2 Solving Application Problems Joe is moving and needs to rent a truck. It will cost him a flat fee of $60/day plus $0.40/mile. He has $92 budgeted for the truck rental. How many miles can he drive without going over the amount he has budgeted? 15 $60/day $0.40/mile $92 total ?how many miles? m = miles 60 + 0.40 m = 92 0.40 m = 32 m = 80 he can drive 80 miles. 16 3.2 Solving Application Problems If I went out to eat and spent exactly $20 including tax (7.5%), how much was the pretax price of my meal? 17 spent 20 including 0.075 tax ?pretax price? p = pretax price price + tax = total spent p + 0.075p = 20 1.075p = 20 p = 18.60 my meal was $18.60 before tax. 18 3.2 Solving Application Problems I received a raise in 2007 of 18% over what I made in 2006. If I made $43,000 in 2007, what did I make in 2006? 19 18% raise (2007) 43,000 in 2007 ?make in 2006? m = 2006 salary 06salary + raise = 07salary m + 0.18m = 43,000 1.18m = 43,000 m = 36,440.68 I made $36,441 in 2006 20 3.2 Solving Application Problems I am considering two sales jobs: one where I would earn $450 base and 3% commission on my sales. The second I would earn 0 base but 10% commission on sales. What would my weekly sales need to be in order for my earnings to be equal? 21 450 base + 3% com 0 base + 10% com ?sales to be equal? w=weekly sales 450 + 0.03w = 0+0.10w 450=0.07w 6428=w if I had weekly sales of $6428, my earnings would be equal. 22 3.3 Geometric Problems I plan to build a rectangular patio where the length is 8 feet more than the width. The perimeter of the patio will be 56 feet. Find the length and width. P=2L + 2W w w+8 23 Length is 8 more than width Perimeter is 56 feet ?find L and W? W + 8 = length W = width 2L + 2W = P 2(w + 8) + 2w = 56 2w + 16 + 2w = 56 4w + 16 = 56 4w = 40 w = 10 Width =10 ft; length = 18 ft 24 3.3 Geometric Problems A triangle has two congruent angles. The third angle is 30°greater than the two equal angles. Find all three angles. (angles of a triangle add up to 180°) 2 1 3 25 There are two = angles 3rd angle is 30 more than other two ?find all three angles? Angle1=x Angle2=x Angle3=30 + x Angles add up to 180 x + x + (30 + x) = 180 3x + 30 = 180 3x = 150 x = 50° The angles measure 50°, 50° and 80° 26 3.3 Geometric Problems In a trapezoid, the bottom angles are 15 degrees less than twice the top angles. Find all the angles. (in a quadrilateral, the angles add to 360°) 27 Bottom angles are 15 less than twice the top Quad adds up to 360° Bottom angle = 2x - 15 Bottom angle = 2x - 15 Top angle = x Top angle = x Sum of all angles = 360 x+x+(2x-15)+(2x-15)=360 6x – 30 =360 6x = 390 X=65° The angles are 65°, 65°, 115° and 115° 28 3.4 Motion, Money, Mixture Problems A family went canoeing. The parents are in one canoe and the children in another. Both canoes start at the same time from the same point and travel in the same direction. The parents paddle at 2 miles per hour and the children at 4 miles per hour. How long until the canoes will be 5 miles apart? parents 2 mph 5 miles kids 4 mph 29 3.4 Motion, Money, Mixture Problems ?how long until they are 5 miles apart R x T = D (P) 2 mph X T = 2T (K) 4 mph X T = 4T 5 miles 4T – 2T = 5 2T = 5 T = 2.5 hours It will take 2 ½ hours for them to be 5 miles apart. 30 3.4 Motion, Money, Mixture Problems My husband and I go running at the same time but starting from different locations and running toward each other. We are 6 miles apart when we start . My husband runs 0.5 mph faster than I run. After twenty minutes, we meet. How fast were we each running? Me Hubby 20 min R R + 0.5 6 miles 31 ?how fast is each running? R x T = D R mph X 1/3 = (1/3)R R+0.5mph X 1/3 = (1/3)(R+0.5) 6 miles My distance + his distance = 6 miles (1/3) R + (1/3) (R + 0.5) = 6 R + R + 0.5 = 18 (after clearing fractions) 2R + 0.5 = 18 2R = 17.5 R = 8.75 mph (my rate) r+ 0.5 = 9.25 (hubby’s rate) 32 3.4 Motion, Money, Mixture Problems I have $15,000 to invest. The two investments I am considering are: -a mortgage investment that has an 11% return -a cd investment that has a 5% return How much should I put into each investment if I want to earn $1500 interest in one year? 33 ?how much should I put into each investment? P X R x T = I (M) X 0.11 1 0.11x (CD) 15000-x 0.05 1 0.05(15000-x) $1500 Int(m) + int(cd) = 1500 0.11x + 0.05(15000-x) = 1500 11x + 5(15000-x) = 150000 (after clearing dec) 11x + 75000 – 5x = 150000 6x + 75000 = 150000 6x = 75000 x = 12,500 $12,500 goes into the mortgage and $2,500 into the CD 34 3.4 Motion, Money, Mixture Problems The community playhouse is putting on a play. Tickets are $15 for adults and $8 for students. They have sold 361 tickets for a total of $4841. How many adult tickets and how many student tickets have they sold? 35 ?how many adult tickets and student tickets were sold? price x number sold = income from tickets (A) 15 361-x 15(361-x) (S) 8 x 8x $4841 15(361-x) + 8x = 4841 5415 – 15x + 8x = 4841 5415 – 7x = 4841 -7x = -574 X= 82 student tickets sold 361-x = 279 adult tickets sold 36 3.4 Motion, Money, Mixture Problems A candy store sells licorice sticks for $4.50/lb and licorice balls for $2.75/lb. If they want to sell a new mix of the two for $3.75/lb, how many pounds of licorice balls will they need to put with 5 pounds of licorice sticks? 37 ?how many pounds of licorice balls will they put with 5 pounds of licorice sticks? price x quantity = value (st) 4.50 5 4.5(5) (bl) 2.75 x 2.75x (mx)3.75 5+x 3.75(5+x) 4.5(5) + 2.75x = 3.75(5+x) 450(5) + 275x = 375(5+x) 2250+275x = 1875 + 375x 2250 = 1875 + 100x 375 = 100x 3.75 = x 3.75 pounds of licorice balls would be needed. 38 3.4 Motion, Money, Mixture Problems A Chemistry teacher is preparing for tomorrow’s lab when he realizes that he needs a 15% base solution but only has on hand the same type base solution in 10% and 25%. How much of the 10% solution will he need to add to 6 liters of the 25% solution in order to make a 15% base solution? 39 ?how much 10% solution shall be added to the 6 liters of 25% solution to create 15%? strength x quantity (L) = total 0.10 x 0.10x 0.25 6 0.25(6) 0.15 x+6 0.15(x+6) 0.10x + 0.25(6) = 0.15(x+6) 10x + 25(6) = 15(x+6) 10x + 150 = 15x + 90 150 = 5x + 90 60 = 5x 12=x so 12 liters of 10% solution 40

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