Name: Date: More Word Problems 1) Tickets to a concert were $19 for the seats near the front and $14 for the rear seats. There were 525 more rear seats sold than front seats, and sales for all tickets totaled $31,770. How many of each kind of ticket were sold? What do we know? Total sales is the sum of the front plus rear tickets. Make a table to organize your information (Price)(Number Sold) = Sales Front 19 x 19x Rear 14 x + 525 14(x + 525) Solve: 19x + 14(x + 525) = 31,770 770 front and 1265 rear 2) My daughter had two pieces of wire of equal length. She shaped one piece into a square and the other in the shape of an isosceles triangle (this is what we do for fun). The base of the isosceles triangle is 4 cm shorter than a side of the square and the legs are 9 cm longer than the side of the square. How long was each piece of wire? Sketch each figure. We should see that the perimeter of the square is 4s and the perimeter of the triangle is (s - 4) + 2(s + 9). Because the wire lengths are equal, we can set the two perimeters equal to one another. Perimeter of square = 56 cm 3) If the base of the isosceles triangle in example 2 (above) were 6 cm shorter than a side of the square and each leg was 3 cm longer, find the length of the wire. 4s = (s 6) + 2(s + 3) 4s = 3s.what does this answer mean 4) At noon a cargo plane leaves the airport at Celladelphia, (the homeliest town in America) and heads east at 180 mph. Its destination is Beckerville (where all the men are handsome and have a full head of hair). At 1 pm a jet takes off from the same airport and flies east (same direction as the cargo plane) at 450 mph. At what time will the jet overtake the cargo plane? Let t = hours after noon that the jet overtakes the cargo plane (Rate)(Time) = Distance Cargo 180 t 180t Jet 450 (t 1) 450(t 1) Because they have traveled the same distance we can set them equal to one another to see when they will meet (1 and 2/3 hrs)
5) Amy has $8 less than Maria, Together they have $30. How much money does each girl have? Set Amy s money in terms of the Maria s. Amy = M - 8 and Maria = M, so M 8 + M = 30 2M 8 = 30 2M = 38 M = 19, so A = 11 6) Jim s weekly pay is two thirds of Alicia s. Together they earn $600 per week. How much is each person s weekly pay? J = 2/3(A), so 2/3(A) + A = 600 5/3(A) = 600 A = 1800/5 A = 360 and 2/3 A = 240 7) A music store that sells used records and tapes (ask your parents what they are). The records sell for $7 each and the tapes sell for $7.50. The dealer sold 60 more records than tapes for a total sale of $2160. How many records were sold? Because we know the total sales are 2160, set the rest of the problem equal to 2160. Total sales are ($$)(# sold). Since the total sales are the result of two different products (tapes + records) we should do the sum of the ($$)(# sold) for tapes and records. ($$) (# sold). Records = 7 t + 60 Tapes = 7.5 t 7.5(t) + 7(t+60) = 2160 Solving gives us t = 120 and Records = t + 60 = 180. If we check we get 7(180) + 7.5(60) = 2160. We could have also done ($$) (# sold). Records = 7 r Tapes = 7.5 r 60 (60 less than the number or records) 7.5(r 60) + 7(r) = 2160 Solve to get r = 180 and t =180 60 = 120 8) The perimeter of a basketball court is 266 feet and its length is 35 more feet than its width. What are the dimensions of the court? 2(w) + 2(w + 35) = 266 4w = 196, w = 49 and length = 49 + 35 = 84: 49x84 feet 9) The degree measures of the angles of a pentagon are consecutive even integers. Find the measure of the largest angle? (There are 540 degrees in a pentagon). Even numbers are two integers apart so x + (x+2) + (x+4) + (x+6) + (x+8) = 540
10) In a walkathon to raise money for a charity, Molly walked a certain distance at 5 mi/h and then jogged twice that distance at 8 mi/h. Her total time walking and jogging was 2h and 15 minutes. How many miles long was the walkathon? Total time is what we know; therefore time is what we set the equation equal to. Also, we know that Time = Distance/Rate and that the total time is made of the walking and jogging distances so we set up: 2.25 = D/5 + 2D/8: In words, 2.25 hours is the walking distance (D) at a rate of 5 mph and the jogging distance -- which is twice the walking distance -- (2D) at a rate of 8 mph Solving this gives a total Distance of 5 miles. What happens if we put time in minutes? 11) A road cyclist travels 120 miles with the wind in 3 hours and 120 miles against the wind in 5 hours. Find the speed of the wind. Against the wind DATA: distance = 120 miles; time = 5 hrs ; rate = 120/5 = 24 mph With the wind DATA: distance = 120 miles; time = 3 hrs ; rate = 120/3 = 40 mph Equations: Let "c" be speed of the cyclist in still air Let "w" be speed of the wind c + w = 40 c - w = 24 Add the two equations to solve for "c": 2c = 64 c = 32 mph (speed of the plane in still air) ---Substitute into c + w = 40 to solve for "w" 32 + w = 40 w = 8 mph (speed of the wind) 12) An airplane travels 400 miles against the wind in 5 hours, and makes the return trip with the same wind in 2 hours. Find the speed of the wind. ------------------------- Against the wind DATA: distance = 400 miles; time = 5 hrs ; rate = 400/5 = 80 mph With the wind DATA: distance = 400 miles; time = 2 hrs ; rate = 400/2 = 200 mph
Equations: Let "p" be speed of the plane in still air Let "w" be speed of the wind p + w = 200 P - w = 80 Add the two equations to solve for "p": 2p = 280 p = 140 mph (speed of the plane in still air) ---Substitute into P+w = 200 to solve for "w" 140 + w = 200 w = 60 mph (speed of the wind) 13) Jon averages 30mph when he drives on the highway to his house and 50mph on the interstate. If both routes are the same length, and he saves 2 hours by traveling on the interstates, how far away is his house? Assume the time if he drives on the interstate is x hrs, then it takes x+2 for him to drive on highway. Since both routes are the same length we can set them equal to one another, we have 50x = 30(x+2) or 50x = 30x + 60, or 20 x = 60, so x = 3 (hrs) Hence,the required distance is 50*3= 150 miles. 14) On Monday, Voltaire drove to town at 60 miles per hour. On Tuesday, he drove to town at 40 miles per hour. If the total traveling time for both trips was 15 hours, how far was it to town? The daily driving distance to town site is x miles. On Monday, he spent x/60 hrs and on Monday,he spent x/40 hrs in driving to town. Since the total travelling time was 15 hrs, we get x/60 + x/40 = 15, Multiplying by 120 on both side: 2x + 3x = 1800, Or 5x = 1800, x = 360 miles
15) If two cars leave the same location at the same time, AND car one is traveling 20 MPH faster than car two in opposite directions, and they are 168 miles apart 1 1/2 hours later...how fast is each one going? Let x be the slower car x = slower car x+20 = faster car 168/1.5 = 112 = how much they are apart in one hour (We make distance in terms of 1 hour because time is in miles per hour. Units need to match) x+x+20 = 112 2x = 92 x = 46 So the speed of the slower car is 46mph, the speed of the faster car is 66mph 18) Antoine rode his bike the 4 miles to the fair at a relaxed pace. He stayed to long and had to double his speed on the way back in order to get home in time for dinner. If his total time traveling was 3 hours, how fast did he travel in each direction? Let S be the speed of his going to the fair, then his speed of the return trip was 2S. Now the total distance of the round trip is 2*4 = 8 miles. By Speed = Distance / Time, and so Time = Distance /Speed, we have 3 = 8/S + 8/(2S), Cancel the denominator by multiple by 2, 6S = 16 + 8 = 24. so, S =4 and 2S = 8. Hence, his speed going to the fair was 4 miles/hr, and his speed backing home was 8 miles/hr.