Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
What are the challenges? • To know how to translate word problems into a workable mathematical equation. • To know where to start and how to go about figuring out the answer. • To know the strategies needed to break down a word problem so it is clear and easy to solve. 6 Steps for Solving Math Word Problems Step 1 Convert the words into an equation. •Read through the entire problem first. •Don’t try to solve it when you’ve only read one sentence. •Read the problem completely in order to get the whole picture. •Effectively translate and solve the problem. Step 2 Go back to the beginning. •Reread the first sentence. •Write down what you know and what you don’t know. •Use variables to stand for the unknowns and clearly label what they stand for. •Do the same for the second sentence and each following sentence. Step 3 Look for Signal Words that mean certain mathematical operations. Signal Words Addition Subtraction Multiplication Division Equals Step 4 Perform the required operations. Step 5 Make sure you are answering the question that is being asked of you. •One step may seem like it’s the answer, but it may take several steps to arrive at the final answer. •Don’t lose sight of what the word problem is asking for or asking you to do. •Label the answer with the unit of measurement or with the object the question is requesting. Step 6 Mentally or on paper check your answer to make sure it is correct and makes sense. •Replace the variable in each step with your answer. •Calculate the equation with the now known variable to see if it satisfies the equation. If it does, then you’re done! Practice, Practice, Practice! Mastering word problems requires practice! Understanding Algebra Word Problems *Represent a given situation using algebraic expressions or equations in one variable. *Simplify and evaluate algebraic expressions. The biggest challenge to solving word problems is figuring out whether to add, subtract, multiply or divide. Words Meaning Addition Words Meaning Subtraction Words Meaning Words Meaning Multiplication Division and increased more more than plus sum total decreased difference less less than left lower than minus double half (× ½) product times triple twice divide into half (÷ 2) divide by divide among quotient To solve a word problem, you must first interpret the action you need to perform and then choose the correct operation for that action. Action Operation Possible Question Clues Combine Add How many all together? Combine equal groups Multiply How many all together? Separate Subtract How many more? How many less? Separate into equal groups Divide How many equal groups? Word Phrases + – • a number plus 5 • add 5 to a number • sum of a number and 5 • 5 more than a number • a number increased by 5 • a number minus 11 • subtract 11 from a number • difference of a number and 11 • 11 less than a number • a number decreased by 11 Expression n+5 x 11 • 3 times a number • 3 multiplied by a number • product of 3 and a number 3m • a number divided by 7 • 7 divided into a number • quotient of a number and 7 a or a 7 7 Word Problems Solve the word problem: A number increased by 14 is 49. Answer: n + 14 = 49 n = 35 Solve the word problem: Find the number Twenty-nine more than a number is 52. Answer: n + 29 = 52 n = 23 Solve the word problem: Find the number One-half of a number is 9. Answer: ½(n)= 9 n = 18 Solve the word problem: Find the number The sum of a number and 79 is 143. Answer: n + 79 = 143 n = 64 Solve the word problem: Find the number A number decreased by 28 is 76. Answer: n - 28 = 76 n = 104 Solve the word problem: Lisa is cooking muffins. The recipe calls for 7 cups of sugar. She has already put in 2 cups. How many more cups does she need to put in? Answer: 2+c=7 c=5 Solve the word problem: How many packages of diapers can you buy with $40 if one package cost $8? Answer: p(8) = 40 p=5 Word Problems Multiple step Solve the word problem: 331 students went on a field trip. Six buses were filled and 7 students traveled in cars. How many students were in each bus? (assume each bus was equally filled) Answer: 6b + 7 = 331 b = 54 Solve the word problem: Carter had $24 to spend on seven pencils. After buying them he had $10. How much did each pencil cost? Answer: 7p + 10 = 24 or 24 – 7p = 10 p = $2 Solve the word problem: The sum of three consecutive numbers is 72. What is the smallest of these numbers? Answer: x + (x+1) + (x+2) = 72 x = 23 Solve the word problem: The sum of three consecutive even numbers is 48. What is the smallest of these numbers? Answer: x + (x+2) + (x+4) = 48 x = 14 Solve the word problem: You bought a magazine for $5 and four erasers. You spent a total of $25. How much did each eraser cost? Answer: 5 + 4e = 25 e = $5 Solve the word problem: The Math Team made some pies to sell at a basketball game to raise money for the new math books. The cafeteria contributed four pies to the sale. Each pie was then cut into five pieces and sold. There were a total of 60 pieces to sell. How many pies did the Math Team make? Answer: 5(p+4)= 60 p=8 Solve the word problem: When the magic number is increased by 32 and this result is tripled, the final result is 246. What is the magic number? Answer: 3(n + 32) = 246 n = 50 Solve the word problem: On a 30-question test, Makenzie earns 5 points for each correct answer and loses 3 points for every incorrect answer. If she answers all the questions, and her score was 6, how many did she answer correctly? Answer: 5c – 3(30 – c) = 6 c = 12 The ratio of kids to adults is 5 to 2. If there is 14 adults, how many kids are there? Kids:adults=5:2 adults=14 Or lets try… Find the factor: 5x kids 2x adults 2x = 14 x = 7 (factor) Therefore: 5x (kids) 5·7 =35 Answer: 35 kids The ratio of kids to adults is 3 to 5. If there are 26 more adults than kids, how many kids are there? Kids:adults=3:5 Find the factor: 3x kids & 5x adults adults – kids=26 5x-3x=26 2x=26 x = 13 (factor) Therefore: 3x (kids) 3·13 =39 Answer: 39 kids 30 boys and girls are planning for a picnic. There are a ratio of 3 girls to 7 boys. How many boys are there? Answer: 3x + 7x = 30 10x = 30 x=3 3·3 3·7 9/21 (girls/boys) I have 70 friends. Among them are 11 girls for every 3 boys. How many friends are boys? How many are girls? Answer: find the factor 11x + 3x = 70 14x=70 x=5 (factor) 11·5=55 girls 3·5=15 boys The ratio of dogs to cats is 9 to 2. If there are 72 dogs, how many total dogs and cats are there? Answer: 88 To get a shade of orange paint, we mix white, red, & yellow in the ratio 2:5:4. If we need 6 gallons of orange paint, then how many gallons of each color do we need? Answer: White=2x, red=5x, yellow=4x 2x+5x+4x=6 X=6/11 White=12/11, red=30/11, yellow=24/11 The top 3 cash prizes in the math tournament are in the ratio 5:2:1. The 1st place prize rd is $20,000 more than the 3 place prize. How much is the second place prize? Answer: 1st=5x, 2nd=2x, 3rd=1x 5x-1x=20000 X=5000 nd 2 place = 2x=2∙5000=$10,000 Lauren divided her money in the ratio 4:2 between Haley and Makenzie. Haley got the smaller amount of $1256. How much did Makenzie receive? 4/2 = M/1256 2M = 4·1256 M =2512 12 miles is approximately equal to 6 km. How many miles are equal to 42 km? Answer: x⁄ 42 ¹²⁄₆ = ¹²⁄₆ · 42 = 84 12 miles is approximately equal to 6 km. How many km are equal to 18 miles? Answer: 9 Direct Variation if y varies directly as x and y = 10 as x = 2.4, find x when y =15 10 15 2.4 x Direct Variation If y varies directly with x and y = 12 when x = 2, find y when x = 8. 12 y 2 8 Inverse Variation If y varies inversely with x and y = 12 when x = 2, find y when x = 8. x1y1 = x2y2 2(12) = 8y 24 = 8y y=3 Inverse Variation If y varies inversely as x and x = 18 when y = 6, find y when x = 8. 18(6) = 8y 108 = 8y y = 13.5 How much tax will Carter pay on a surround-sound system that cost $1885.00 if the tax is 5% Answer: Equation: n = 0.05 ∙ 1885 n = $94.25 Bob now earns $9.50 per hour. This is 125% of what she earned last year. What did she earn per hour last year? Answer: Equation: 9.5 = n ∙ 1.25 n = $7.60 I paid $8400 in federal and state income taxes last year, which amounted to 28% of my annual income. What was my income last year? Answer: Equation: 8400 = 0.28 ∙ n n = $30,000 Competition Scoring: 5 points – 1st Interval (0-30 seconds) 3 points – 2nd Interval (30-60 seconds) 1 points – 3rd Interval (60-90 seconds) Ready…? What is the quotient of 3.5316 divided by 3.27? Answer: 1.08 Every large pack of gum has 8 pieces and every small pack of gum has 3 pieces. Mary has 7 small packs and 4 large packs. How many pieces of gum does Mary have altogether? Answer: 53 Jason has 17 pencils and Shannon has 43 pencils. How many pencils must Shannon give Jason in order for them to have the same number of pencils? Answer: 13 Ravi's age is double Ranu's age. The product of their ages in years is 72. How old is Ranu? Answer: 6 Billy brought a large box of chocolates to school. In his first class, he gave half of the chocolates in the box to his teacher. In his second class, he gave half of his remaining chocolates to his best friend. At lunch, he ate half of his remaining chocolates. If he had 3 pieces of chocolate left at the end of lunch, then how many pieces were in the box when Billy arrived at school? Answer: 24 Your brother traveled 117 miles in 2.25 hours to come home for school break. What’s the average speed that he was traveling? Answer: 52 mph What is thirty-four hundredths minus fifty-two thousandths? Answer: 0.288 You have 15 yards of ribbon for your gift boxes. Each box gets the same amount of ribbon. How much ribbon will each of your 20 gift boxes get? Answer: 0.75 yards What is two and three thousandths times twenty-one hundredths? Answer: 0.4206 Sam gave Jen 1/2 of his jujubes. Jen ate 1/2 of the jujubes and gave the rest to Kyle. Kyle kept 8 of the jujubes and gave the last 10 to Kim. How many jujubes did Jen eat? Answer: 18 You finally get an allowance! You put $2 away in January, $4 away in February, $8 away in March, $16 away in April and followed this savings pattern through to December. How much money do you have in 12 months? Answer: $4096.00 Jeff has 1/2 pizza left in the fridge. At breakfast he at 1/3 of it. What fraction of the original pizza does he have left for lunch? Answer: 1/3 What is one-sixth of 678? Answer: 113 I’m thinking of two number, 12 and another number. 12 and my other number have a greatest common factor of 6 and their least common multiple is 36. What’s the other number I’m thinking of? Answer: 18 Tom had a platter of chocolate wafers. He ate 5 of them and then gave his brother 3, he then handed them to his ball team of 8 members. The first player to arrive took 1, the second player took 3, the third player took 5 and so on. When the last player took his, the platter was empty. How many chocolate wafers did Tom start with? Answer: 72 Jasmine has 50 marbles in a bag. 20% of the marbles are blue. How many marbles are blue? Answer: 10 An Olympic runner set a record for the 100m dash. The time was ten and sixty-two hundredths seconds. How would you write this as a number? Answer: 10.62 The tables at the party are shaped like the hexagon. If you put the tables together, how many would you need for 50 people? Answer: 12 At your birthday party, you had 7 8slice pizzas. 41 slices were eaten. What fraction of pizza is left? Answer: 1 7/ 8 Peter uses quick fix glue to hold a bamboo rod. One piece of the rod measures 1 ⅕ meters and another piece measures 2 ⅔ meters. What is the total length of bamboo rod after it has been glued together? Answer: 3 13/ meters 15 What is the reciprocal of eight and one-third? Answer: 3/25 Which property states: (5)(17)(23) = (5)(23)(17) Answer: Commutative Property of multiplication Find the product of three and fortyeight thousandths with four and six hundredths. (round your answer to the nearest hundredth) Answer: 12.37 A triangle has one angle that is 62 degrees and one angle that has a measure of 41 degrees. What is the measure of the third angle? Answer: 77 What is the sum of the GCF of 17 and 51 and the first seven prime numbers? Answer: 75 Write the reciprocal of 2⅛ Answer: 8/17 What is 75.5% of 60? Answer: 45.3 What is the base of a triangle with a height of 4 cm and an area of 28 cm²? Answer: 14 A student had the following scores in math: 89, 97, 100, 85, 93, 73, and 100. What is the positive difference in the median and mode? Answer: 7 What is the 7th number in the following sequence? 1, 3, 9, 27, 81, __, __ Answer: 729 What is the measure of the complement of a 60° angle? Answer: 30° If a + 6 = 12 and 7b = 49, what is the value of 3ab? Answer: 126 What is the product of all the positive odd number factors of 50? Answer: 125 What is the area of a square with perimeter of 48 cm? Answer: 144 2 cm What is 25% of 60% of 180? Answer: 27 Find the sum of the first 10 positive integers that are not prime. Answer: 95 a b = 5b – a, then what does -7 -4 equal? Answer: -13 What is the smallest number divisible by 8, 12, and 20? Answer: 120 Of the numbers 2, 15, 19, 41, 55, 87 and 91, what is the sum of the composite numbers? Answer: 248 Find the total cost of a camera marked 30% off the original price of $59.99, plus a 7% sales tax. Answer: $44.93 What is the mean for the given data: 36, 25, 28, 39, and 30. Answer: 31.6 The scale of a map is 1 in. = 20 mi. Find the actual distance, in miles, represented by 2 ¾ inches on the map? Answer: 55 A right triangle has side measurements of 8 in., 15 in., and 17 in. What is the area of this triangle? Answer: 60 In the equation 2x + y = 8, find the value of x when y = -16. Answer: 12 Using the diagram below, how many students studied exactly two days for a test? Mon. 3 Tues. 6 5 4 1 2 3 Wed. Answer: 6 A rectangular room measures 19 ft. by 18 ft. How many square yards of tile are required to cover the floor? Answer: 38 Mary spent $21 on a taxi ride, $43.30 on a concert ticket, and $32.50 on food. If she started with $130.00 , how much is left? Answer: $33.20 Tom spent $4.25 to wash his car. If one quarter operates the car wash for 60 seconds, how many minutes did he was his car? Answer: 17 Two angles of a triangle measure 50° and 33°. Find the measure of the third angle. Answer: 97° Ford makes 7 red cars for every 4 white cars made. If it makes 16 white cars a day, how many red cars will it make a day? Answer: 28 John has twice as many dimes as nickels. If the value of the dimes is $5.00, find the value of the nickels. Answer: $1.25 Two dozen pencils are in a bag. 5 are blue, 3 are red, and the rest are yellow. What is the probability of drawing a yellow pencil? Answer: 2/3 What is 60% of 75? Answer: 45 Find the sum of the next two numbers in the following sequence: 1, 8, 27, 64, … Answer: 341 Six times a number is eight more then twice the number. What is the number? Answer: 2 How many of the following numbers are perfect squares? 1, 2, 4, 6, 9, 15, 16, 27, 32, 36, 48, 50, 64, 89 Answer: 6