Translations: Comparison Sentences A comparison sentence is a special form of translation, a single sentence within a word problem that provides information about two different things: two unknowns. In prealgebra, one of these unknowns will be identified by a letter (a variable, like x) and the other will be identified by an expression (like x + 5). Later in algebra, each unknown will be identified by a different letter. Jim s yard has a perimeter of 124 feet. Its length is 2 more than twice its width. What are the dimensions of his yard? In this problem, the second sentence is the comparison sentence, describing the length as compared to the width. In many problems, a specific strategy can help you identify your two unknowns. 1. Copy the comparison sentence (not the whole word problem). 2. Go to the last word/phrase in the sentence. This item gets the letter (variable). 3. Off to the side, write (word/phrase) = x (or whatever letter you choose) 4. Cross out that last word/phrase, and write the letter you chose in its place. 5. Translate the rest of the sentence. 6. Off to the side, write (the other unknown) = (expression)
Now you are ready to do the problem. This one involves perimeter, for which you can add up all the sides (use a picture if you do this!) or use the formula P = 2l + 2w. Are we done yet? No The problem asked what are the dimensions of his yard? What did w mean? Width. So the width of the yard is 20. What about the length? But there are a few translation sentences that are different. Usually these involve two items of specific rates that add up to a total. Mixture problems are like that. I ll talk about those in a different handout.
Practice Page (1) The perimeter of a rectangle is 30 inches. The width is five inches less than the length. What are the dimensions of the rectangle? The length of a rectangle is twice its width. Its perimeter is 42 cm. What are the length and width? (width is 7 cm, length is 14 cm) The width is five inches less than the length. Length is l length = l Width is five less than l. Width is 5 (-) l Width is (l 5) width = (l 5) Perimeter is 30 P = l + w + l + w w l w l 30 = l + (l 5) + l + (l 5) (parentheses needed?) 30 = l + l 5 + l + l 5 (combine like terms) 30 = 4l 10 (add 10 to both sides) 40 = 4l (divide both sides by 4) 10 = l What did l menn? length is 10 inches width = (l 5) width = 10 5 width is 5 inches
Practice Page (2) The sum of two numbers is 22. One number is six more than three times the other number. What are the numbers? Paul is thinking of two numbers. One is 8 less than twice the other. They add up to 25. What are the numbers? (11 and 14) One number is 6 more than 3 times the other. Other number is x other = x (2 nd number) One number is 6 more than 3 times x. One number is 6 + 3 * x One number is (6 + 3x) nmbr = (6 + 3x) (1 st number) Sum is 22 Sum means add them One number plus other number = sum (6 + 3x) + x = 22 (parentheses needed?) 6 + 3x + x = 22 (combine like terms) 6 + 4 x = 22 (subtract 6 from both sides) 4 x = 16 (divide both sides by 4) x = 4 What did x mean? 2 nd number = 4 1 st number = (6 + 3 x) = 6 + 3(4 ) 1 st number = 18
Practice Page (3) 1. Arnold is going to build a yard for his horses. He has 160 feet of fencing. He wants the length of the yard to be 8 feet more than the width. What will the dimensions of the yard be? (did you get a length of 44 feet and a width of 36 feet?) 2. A rectangle has a perimeter of 52 cm. The width is 4 cm less than the length. What are the length and width of the rectangle? (did you get a length of 15 cm and a width of 11 cm?) 3. The sum of two numbers is 30. The first number is 4 times as big as the second. What are the numbers? (did you get 24 for the first number and 6 for the second?) 4. Sarah is thinking of two numbers. One is 7 less than twice the other. The sum of these numbers is 17. What are the numbers? (did you get 9 and 8?) 5. Jim is buying crown molding for the ceiling of his living room. His living room s width is 3 feet less than its length. He needs 70 feet of crown molding. What are the dimensions of his living room? (did you get 19 feet for the length and 16 feet for the width?) 6. Karen hopes that putting barbed wire around her garden will keep out rabbits. A roll of 100 feet of barbed wire will fit her garden perfectly. If the garden is 10 feet longer than triple its width, what are the width and length of her garden? (did you get a width of 10 feet and a length of 40 feet?) 7. An isosceles triangle has two equal sides. Each side of the triangle is 4 inches more than double the base. If the perimeter of the triangle is 68 inches, what are the measures of each side and the base? (did you get 28 inches for each side and 12 inches for the base?) 8. Susan and Paula s birthdays are both in April. Susan tells Paula that if they added their birth date numbers together, it would total the 30 days in April. If Susan s birth date is 2 more than 3 times Paula s, what would each one s birth date in April be? (did you get Paula s birth date is the 7 th and Susan s is the 23 rd?) 9. Karen is buying tickets to take her daughter to go to a Wiggles concert. Adults seats cost $6.00 less than twice the cost of seats for children. Karen pays $30.00 for tickets for herself and her daughter. How much did each ticket cost? (did you get children s seats are $12 and adult s seats are $18?) 10. Your teacher asks you to draw a specific size trapezoid. A trapezoid has a short top base, wide bottom base, and equal sides. For this trapezoid, the teacher wants the bottom base to measure 4 cm less than twice the top base. The sides will measure 8 cm and the perimeter should be 39 cm. What will the top and bottom bases measure? (did you get 9 cm and 14 cm?)