Answers to Practice Final Questions on Algebra

Answers to Practice Final Questions on Algebra 1. Margaret writes the following equations in order to solve 87 – 15: 87 – 10 = 77 – 5 = 72. This is incorrect because, for example, 87 – 10 does not equal 77 – 5. Use Margaret’s strategy, but write correct equations showing the steps. 87 – 15 = ( 87 – 10 ) – 5 = 77 – 5 = 72 as in “first subtract 10, then 5" 2. If n is a number, then "the sum of twice that number and 5" can be expressed as © 2n+5. a. 2(n + 5) b. 2n - 5 c. 2n + 5 d. n2 + 5 3. I am thinking of a number. If I add 7, then multiply the sum by 3 and then subtract 12, the result is 36. Use algebra to find the number. ( n+7 )@3 – 12 = 36 3n + 21 – 12 = 36 3n + 9 = 36 3n = 27 So n = 3 4. Write an algebraic expression for the following: a. the number of inches in m yards In 1 yard, there are 36 inches In 2 yds, there are 72 inches In m yds, there are 36@m inches. b. the cost in dollars of a gym membership for t months if it costs $125 just to join (membership fee) and then $75 per month for each month the gym is used one month’s use of the gym costs initial fee plus monthly fee = $125 + $75 two month’s use costs $125 + 2($75) t month’s use costs $125 + t($75) k cm c. the perimeter of a rectangle with length k cm and width s cm Perimeter = distance around = ( k cm + s cm ) @ 2 s cm 5. Illustrate the identity, (a + b)(a + b) = a2 + 2ab + b2, by a rectangular array. a + b (a + b)2 a ax a ax b IS + b bx a bx b a2 + 2ab + b2 6. Let a and b be rational numbers. Fill in the blanks to justify each step. (a + b)2 = (a + b)(a + b) def’n of exponent 2 (Definition of exponents. w2 = w w ) = (a + b)a + (a + b)b Distributive property (of x over +) = (a 2 + ba) + (ab + b2 ) Distributive property (of x over +) 2 2 = a + ba + ab + b Associative property of + (so no parentheses needed) = a 2 + ab + ab + b2 Commutative property of multiplication = a 2 + (ab + ab) + b2 Associative property of + = a 2 + (1ab +1ab) + b2 Identity property of mult. = a 2 + (1+1)ab + b2 Identity property of mult. 2 2 = a + 2ab + b 1+ 1 = 2 7. Factor the algebraic expression, a2 – b2. Use your result to calculate 322 – 282 with only one multiplication. Show your steps. a2 – b 2 = (a+b)(a–b) 322 – 282 = (32 + 28)(32 – 28) = 60 @ 4 = 240 Ans to Practice Final Questions on Algebra— p 2 10. Express )))))) 1015 ·1523 as a product of prime numbers. 307 ·98 15 10 @1523 )))))) 7 30 ·98 15 (2@5) @(3@5)23 )))))))))) 7 (2@3@5) ·(3@3)8 = = 15 15 2)))))))))) @515 @323 @523 = 2)) @ ))) 323 @ )) 538 = 7 7 7 16 7 23 2 @3 @ 5 @3 2 @ 3 @ 57 28 @ 531 11. Give teacher's solutions using algebra. a. The lengths of the sides of a triangle, measured in inches, are consecutive whole numbers. If the perimeter is 45 inches, what is the length of the shortest side? Let the length of the shortest side be called x. Then the other two are x+1 and x+2, because the three lengths are consecutive whole numbers. The perimeter is the sum of the lengths of the sides, x + (x+1) + (x+2), and this is given to be 45 inches. So: x + x+1 + x+2 3x + 3 = 3x = x = = 45 45 42 14 The length of the shortest side is 14 inches. Check: The other two sides are thus 15 in and 16 in. 14 + 15 + 16 = 45. b. Sofia had five times as much money as Diego. After Sofia spent $62 and Diego received his $22 allowance, they had the same amount of money. How much did they each have at the beginning? Let x = the amount of money Diego had at the start of the description. Then 5x = the amount Sofia had at the start. After Sofia spends $62 she will have 5x – 62. These amounts are given to be equal. Diego has x; after he receives $22 allowance he has x + 22. A 5x – 62 4x – 62 4x x = = = = x + 22 22 84 21 Diego had $21.00 to begin. Check: Diego had $21 . Sofia had 5@$21 = $105 After changes, Diego will have 21+22 , Sofia will have $43. 12. Give two teacher's solutions to each of the following problems, one using bar diagrams and the other using algebra. a. There are three times as many boys as girls. If there are 96 children, how many girls are there? Boys Girls 96 ? A 96 altogether 4 x ? = 96 ? = 96 ÷ 4 = 24 Boys Girls Two Alternate bar diagram views.... 96 There are 24 girls. Boys Boys Check: If 24 girls, then 3@24 = 72 boys. 24 + 72 = 96 U With algebra: Let the number of girls = x. Then number of boys = 3x Since the total is 96, 3x + x = 96. Check: 24 girls Y 3@24 = 72 boys 3x + x = 96 4x = 96 x = 24 There are 24 girls. Y 24 + 72 = 96 altogether. U Boys Girls Ans to Practice Final Questions on Algebra— p 3 b. There are three children in a family. Ed is 20 pounds heavier than Fred who weighs twice as much as Ned. If the three children weigh 180 pounds altogether, how much does Ed weigh? Ed ÆÈ Ç Fred Ned 20 ? 180 A 180 — 20 160 ²3 boys’ total weight without Ed’s “extra” 20 pounds 160 ÷ 5 = 32. Ned weighs 32 pounds so Fred weighs 64, and Ed, 84, pounds Check: Ned 32, Fred 64, Ed 84. 32+64+84 = 180 U x2 +20 U With algebra: Let Ned’s weight be x Then Fred’s weight must be 2x and Ed’s weight must be 2x + 20 The total of their weights is 180 pounds, so x + 2x + 2x + 20 = 160 5x + 20 = 180 5x = 160 x = 32 Ned weighs 32 pounds. So Fred weighs 64 pounds, and Ed weighs 84 lbs. Check is same as above. c. John & Wendy have 1012 pennies. Wendy has 134 less than John. How many pennies has John? John has ? Wendy has ÆÈÇ 134 1012 A 1012 — 134 878 total pennies without John’s “extra” 134. 878 ÷ 2 = 439 Wendy has 439 pennies. 439 + 134 573 John has 573 pennies. Check: 573+ 439 = 1012, 5 7 3 —4 3 9 = 1 3 4 U Alternate view: John has ? Wendy has 134 A ? 1012 + 134 1146 If Wendy were to get 134 more pennies, then the total would be 134 more, and she and John would have equal amounts. This would be 1146 ÷ 2 = 573. John must have 573 pennies. With algebra: Let x = number of pennies Wendy has. Then John has x+134 pennies. x + x + 134 = 1012 2x + 134 = 1012 2x = 878 x = 439 The total number of pennies is 1012, so So Wendy has 439 pennies. Therefore John has 439 + 134 = 573 Alternate view: Let x = number of pennies John has = 1012 Then number of pennies Wendy has is x – 134 2x – 134 = 1012 The total number of pennies is 1012, so 2x = 1146 x = 573 John has 573 pennies. Check: 573 – 134 = 439. 573 + 439 = 1012. U x + x – 134 Ans to Practice Final Questions on Algebra— p 4 d. There are 4 times as many CC as OM cookies. 36 more CC than OM cookies, how many cookies total? Choc Chip Oatmeal 3 units = 36 cookies 1 unit = 12 cookies 5 units = 60 cookies ? 36 A There are 60 cookies in total. 36 ÷ 3 = 12 12 x 5 60 (12 are OM, and 48 are CC ) Check: 48 ÷ 12 = 4, 48 — 12 = 36, 48+12 = 60 U With algebra: Let the number of OM (oatmeal) cookies be x. Then the number of CC cookies must be 4x. There are 36 more CC than OM. So 4x – x = 3x = x = 36 36 12 There are 12 OM cookies, and 48 CC cookies, and the total is 12 + 48 = 60 cookies. For those who want a little more challenge, here is one more word problem, another one of those “age” problems: Susan is 10 years older than Jille. In 8 years she will be twice as old as Jille. How old is Susan now? Bar diagram: Susan’s age: Aaargh!!! (What to do with the rest of the info! Starting over....) Jille’s age: 10 yrs ² ² ² ²10 ÷ ÷÷ ÷ Susan’s age 8 yrs. from now: Jille’s age 8 years from now: ? 8 8 Double Aaargh!!! ² ² ² ²¿ ÷ ÷÷ ÷ Conclusion: This problem can be done, but is quite confusing without algebra. Here we go: Once we have the diagram, we can see the difference, indicated by “¿”, must be 18 – 8 = 10. Since Susan’s age at that point will be double Jille’s, from the diagram we see that ? + 8 must also be 10. Therefore, Jille’s age now ( ? ) must be – = . And Susan is 10 years older than Jille, so Susan must be . Check: eight years from now, Jille will be , and Susan will be . Maybe you can find a better way to illustrate this problem! With algebra: Let x = Jille’s age now. Then Susan’s age now is x + 10. In 8 years, Jille’s age will be x + 8, and Susan’s age will be (x + 10) + 8. We know that Susan’s age will be twice Jille’s then: (x + 10) + 8 = 2( x + 8 ) x + 18 = 2x + 16 2 = x So Jille is now 2 years old, and Susan is 2+ 10 = 12 years old. Eight years from now, Jille will be years old, and Susan will be years old. Will Susan be twice as old as Jille then?