HW Letters & Expressions

3. All are algebraic expressions except for b, e and i. Items a, d and h are numeric expressions, and therefore are algebraic expressions. b. 3 +÷ 7 has no meaning, cannot be evaluated, does not represent a number. e. 3x + 2 = 7 is an equation. It does not represent a number. i. y ÷ 0 for any choice of y, this does not represent a number (because ÷ 0 is undefined) 5. Fred is confused about the meaning of the equal sign. He wrote: 3(x+2) = 3x + 6 ! x = 2x + 6 + 8 = 2x + 14 the first two “=” are incorrect... ...because 3(x+2) = 3x + 6 and 3(x+2) 3x + 6 ! x ...because 3x + 6 – x = 2x + 6 and 3x + 6 – x 2x + 6 + 8 . Fred could have written: 3(x+2) – x + 8 Or: 3(x+2) = 3x + 6 = 3x + 6 – x + 8 3x + 6 – x = 2x + 6 = 3x – x + 6 + 8 2x + 6 + 8 = 2x + 14 = 2x + 14 6a. The number of inches in m feet: There are 12m inches in m feet. There are 12 inches in 1 foot, 24 inches in 2 feet, inches in 10 feet.... Inches in m feet. 6b. The perimeter of a square with side s cm is 4s cm. 6c. The value in cents of x nickels and y dimes is 5x + 10y. 6d. The number of pounds in 6z ounces.... is 6z/16 6e. Three consecutive whole numbers, the smallest of which is n... are n, n+1, n+2 (Note, these are THREE expressions, not one.) Start with what you know: The number of pounds in 16 oz is 1 The number of pounds in 32 oz is 2 Work your way up, then ask yourself how you are getting the answer. The number of pounds in 32 oz is 32/16 The number of pounds in w oz is w/16 (w stands for whatever) 6f. The average speed of a plane that travels 600 miles in 2 hours is 300 mph. The average speed of a train that travels w miles in 5 hours is w /5 miles per hour. 6g. Ann is 18 years younger than Bill. Carmen is 1/5 as old as Ann. Dana is 4 yr older than Carmen. 7. A= B – 18 If Bill is B yr old. C = A/5 = (B– 18)/5 D = C + 4 = (B– 18)/5 + 4 The lengths in inches of the sides of a triangle are consecutive integers, P is 27". Let x be the number of inches length of the shortest side. x x+1 The next two are x+1 and x+2. Perimeter = 27 (inches), and perimeter = x + (x+1) + (x+2) [inches], so... x+2 x + (x + 1) + (x + 2) = 27 3x + 3 = 27 3x = 24 x = 8 The shortest side is 8 inches long. 8a. (Re PT5A p 25 #6): P stands for Peter’s stickers, J for Jay’s and E for Emily’s. ¿ U SING ALGEBRA: P: ? ? Let x = # stickers Jay has. Jay has x stickers, J: ? 300+40 so Peter has 2x stickers, E: 40 and Emily has x – 40. If we give E 40 more stickers, she’d have the Altogether they have 300 stickers, so: same as J. And the total would be 40 more. x + 2x + x – 40 = 300 Then 4 units = 340, 4x – 40 = 300 So 1 unit = 340 ÷ 4 = 85 4x = 340 &... Peter has 2 × 85 = 170 stickers x = 85 Peter has 2@85 = 170 A Alternate solution: ¿ P: ? 40 ? 40 J: ? 40 300 ÆÈÇ E: ? 40 Here we think of the basic unit as E’s amount, see that J has 40 more, etc. 3 × 40 = 120 300 – 120 = 180 4 units = 180 1 unit = 180 ÷ 4 = 45 So Peter has 45+40+45+40 = 170 stickers A 9. U SING ALGEBRA: Let x = # stickers Emily has. Then... Jay has x + 40 Peter has 2(Jay’s #) = 2(x+40) They have 300 altogether, so: 2(x+40) + x + 40 + x = 300 2x + 80 + x + 40 + x = 300 4x + 120 = 300 4x = 180 x = 45 So Peter has 2(45+40) = 170 stickers. (Re PT 6A pp 7-11) #1. Alan is 8 yr old. a. How old will he be in 5 yr? b. How old will he be in x years? 13, x+8 (yr); B [Although it may seem in part a that you are evaluating, that is more of a “warm-up” for building the expression requested in part b.] #2 Jim has $2 more than Travis. a. If Jim has $10, how much does Travis have? b. If Jim has $m.... $8, m – 8 ($); B [ Very similar to #1] #3 Tracy bought w kg of flour, used 5 kg. a. Express amount left. b. Tracy bought 8 kg. What’s left? w – 5 ²B, 8– 5 = 3 ²E; B,E [part a is Building, part b is Evaluating.] #4 There are 4 apples in each packet. a. How many in n packets? b. n = 8, # apples =? c. c. n= 11.... 4n (apples) ²B, 32 (apples)²E, 44 (apples)²E; B, E, E #5. 3 boxes, each containing p wings. a. Express total of wings. B. If each box contains 7 wings.... 3p, 7p (wings) ; B, B #6. A rectangular tile is k cm by 8 cm. Express area in terms of k. 8k (sq cm) ; B #7. Ali has 8 boxes, each to have equal # of marbles. a. # = 96 How many in each box? b. # = x.... 96/8 = 12, x/8 ; B [This is another where it may seem we start off E, but the purpose is B. ] 9. Continued #8. Meihua bought 3 books. a. Total cost $12, find average cost. b. If total cost is $m.... $12/3 = $4, m/3 ($); B [see comment above.] #9 Find the value of each of the following when n = 6. 6+4 = 10, 10+6 = 16, 9, 0, 24, 60, 3, 1, ½ ; all parts are E #10. Tyrone puts x marbles in each bag, 5 bags plus 3 marbles. a. total marbles? b. If x = 10.... 5x+3²B 50+3 = 53²E #11. Find the value of 2x – 3 when x = 5. 7; E #12. Jeff had $50, gave $y to his son. 2 daughters split rest equally. a. Girls get ? b. If y=12.... (50 - y)/2 ²B, 19 ²E; B,E [part a is Building, part b is Evaluating.] #13. Find the value of (x–4)/2 when x = 12. 4; E #14. a. Find the value of (4n+3)/5 when n=8 7, 10; E,E b. ...of (45 – 3r)/3 when r = 5. 10c. (Re PT 6A p14 #6–9 ) #6. The admission fee to a bird park is $y, to an amusement park is $1 more. a. Express the admission fee to the amusement park in terms of y. $( y + 1 ) b. If admission fee to bird park is $8, find admission fee to amusement park. $9 #7. Rope is x m long. Iron rod is 3 times as long. a. Express length of iron rod in terms of x. 3x b. If the rope is 9 m long, how long is the iron rod? 27m #8. Henry is x yr old. Betty is 3 times as old as Henry. Peter is 4 yr older than Betty. a. Express Peter’s age in terms of x. 4 + 3x yr b. If Henry is 4 yr old, how old is Peter? 16 yr #9 Huili bought cartons of milk at $2 each. She gave the cashier $50 and received $y change. a. Express the number of cartons of milk Huili bought in terms of y. (50 - y)/2 b. If y = 38, how many cartons of milk did Huili buy? 6 11. Write a short word problem which builds the given expression in the given context. Make clear what each letter represents. 11a. The expression 12c in the context of baking cookies: Margo has 12 cookie sheets. She can put c cookies on each baking sheet. How many cookies will her cookie sheets hold? 11b. The expression 13r+3s in the context of shopping: Incredible Bargains has ruanas on sale for $13 each, and scarves for $3 each. At these prices, what will be the total cost for r ruanas and s scarves? 11c. 2w + 13, allowance: John had $13 when he began saving two dollars each week from allowance. How much will he have after saving $2 per week for w weeks? 11d. (240–x)/50 time It is 240 miles to San Francisco. We have driven x mi, at a steady 50 mph. to complete trip: If we continue at this rate, how long will it take to get to San Francisco? 3. Illustrate the addition of x2+ 3x + 4 and 2x2 + x + 7... 3a. Using apicture involving rectangular arrays of areas x2, x and 1 : x2+ 3x + 4 : x2 + + Sum is 3x2 + 4x + 11 2x2 + x + 7: + x 2 + 2 x 3b. x2+ 3x + 4 + 2x2 + x + 7 = x2 + 2x2 + 3x + x + 4 + 7 = (1+2)x2 + (3 +1) x + (4 + 7) = 3x2 + 4x + 11 4. (a + b)2 = a2 + 2ab + b 2 specifically where b= 1 says: (a + 1)2 = a 2 + 2a + 1 so 21 2 = 20 2 +2@20 + 1 = 400 + 40 + 1 = 441 similarly 31 2 = 30 2 +2@30 + 1 = 900 + 60 + 1 = 961 and 41 2 = (do it mentally) = 1600 + 80 + 1 = and 51 2 = (do it mentally) = __00 + __0 + 1 = 2601 (a – b)2 = a2 – 2ab + b 2 specifically where b= 1 says: (a – 1)2 = a 2 – 2a + 1 so 19 2 = 20 2 +2@20 + 1 = 400 – 40 + 1 = 361 similarly 29 2 = 30 2 +2@30 + 1 = 900 – 60 + 1 = 841 and 39 2 = (do it mentally) = 1600 – 80 + 1 = and 49 2 = (do it mentally) = __00 – __0 + 1 = 2401 152 = (10 + 5)2 = 100 +2@50 + 25 = 225 1502 = (15@10)2 = 152 @ 102 = 225@100 = 22500 151 2 = (150 + 1)2 = 1502 + 2@150 + 1 = 22500 + 300 +1 = 22801 5. By repeated use of commutative and associative properties, ¶ we know these terms can be rearranged in any order. ...by the distributive property applied to x2 + 2x2 and to 3x + x 6a. 14@16 = (15– 1)(15+1) = 152 – 1 = 225 – 1 = 224 6b. 13@17 = (15 – 2)(15 + 2) = 152 – 4 = 225 – 4 = 221 The product of any two numbers that differ by 2 is (n – 1)(n + 1) = n2 – 1 (and n is the average of the two) 8. (a + b)3 = (a + b)(a + b)2 = (a+b)( a2 + 2ab + b2 ) Viewing (a+b) as a number, get 2 2 = (a+b) a + (a+b) 2ab + (a + b) b by the distributive property. 3 2 2 2 2 3 = a + a b + 2a b + 2ab + ab + b also by dist. prop. 3 2 2 3 = a + 3a b + 3ab + b combining like terms, a + b ...which also uses the D.P. 9. b The area of the entire large square at left is (a + b)2 . + The area of the smaller square in the middle is (a – b)2 . The DIFFERENCE between these areas is made up of the a four a-by-b rectangles . So: (a + b)2 – (a – b)2 = 4ab A a–b This can be verified using the distributive A b property of × over + on the integers. Decreasing amounts of work details are shown on higher-numbered problems. If unclear, review lower-numbered problems. 1. a. 32 @ 32 = 25 @ 25 = 25+5 = 210 = 1024 b. 1024 ÷256 = 210 ÷ 28 = 210– 8 = 22 = 4 c. 4096 ÷ 32 = 4@1024 ÷ 32 = 22 210 ÷ 25 = 212 ÷ 25 = 212– 5 = 27 = 28– 1 = 28 ÷ 2 = 256 ÷ 2 = 128 2. a. 28@ 27 ÷211 = 215 ÷ 211 = 24 = 16 b. (23)5 ÷ 29 = 215 ÷ 29 = 26 = 64 c. 256 @128 ÷ 2048 = 28@ 27 ÷211 3. 4. (Re PT 6A p 11 15, 16, 17ghi): #15. 9, 18, 9/5 #16 8, 9, 12 #17 47, 130, 120 a. (52)3 = 52 @ 52 @ 52 = (5@5)(5@5)(5@5) = 5 2 C3 = 5 6 b. (52)m = 52 @ 52 @ 52 @þ@52 = (5 @ 5) (5 @ 5) (5 @ 5) @þ@ (5@5) = 5 2m c. (a2)3 = a2 @ a2 @ a2 = (a@a)(a@a)(a@a) = a 2 C3 = a 6 5. a. 34 54 = (3@3@3@3) @(5@5@5@5) = (3 @ 5) (3 @ 5) (3 @ 5) (3 @ 5) = (3 @ 5)4 = 154 b. 34 b4 = (3@3@3@3) @(b@b@b@b) = (3 @ b) (3 @ b) (3 @ b) (3@b) = (3b)4 6. a. 25 (2@3)2 (2@32 )2 25 22 32 22 34 = ))))))))))) 3 (2 ) 34 2 4 ))))))))))) 4 2 2 = 2 9 36 )) )) 2 4 34 = 25 32 2 9 b6 2 b = 25 b2 25 (2b)2 (2b2)2 b @ (4) = 25 22 b2 22 b4 )))))))))) b4 (2 2 ) 2 c. a5 (ab)2 (ab2 )2 ))))))))))) b4 (a 2 ) 2 = )))))))))) 4 4 d. Letting 3=b and 2=a makes all the above identical. Interesting to note part c is shortest. a. b. c. 53 @242 @100 8 @15 @3 )))))))) 2 a5 a2 b2 a2 b4 = b a 53 (23@3)2 @ 1 2 (3@5) @ 3 )))))))))) 3 2 = a3 @(bc3) 2 @(ac)0 c @(ab) @b ))))))))))) 3 2 = = a3 (c3@b)2 @ 1 c (b@a) @ b ))))))))) 3 2 25 22 b2 22 b4 b 2 = )))))))))) 4 2 b. 7. 25 62 (18 )2 3 @4 )))))))) 4 2 )))))))))) 4 4 a9 b6 a b ))4 )) 4 53 26 32 2 3 5 = Replace a with 5, b with 3, and c with 2. ))4 ))4 = a5 b 2 ))))))) 3 3 2 = = a 3 c6 b 2 c b a ))))))) 3 3 2 2 6 32 53 2 3 5 ))3 )) 3 )) 2 = = c 6 b2 a 3 c b a ))3 ))3 ))2 = 23 5 3 )))) c3 a = ))))) b Steps are combined for brevity. If unclear, review lower-numbered problems. 8. a. 621 @1018 @1522 30 @16 = ))))))))))))))) 11 4 7 (2@3)21 @(2@5)18 @(3@5)22 = (2@3@5) @(2 ) )))) 11+28 ))) 11 63n @10n+11 @1522 = 30 @16 ))))))))))))))) 11 4 n (2@3)3n @(2@5)n+11 @(3@5)22 = (2@3@5) @(2 ) )))) 11+4n ))) 11 (2@3)3n @(2@5)n+m @(3@5)2m = (2@3@5) @(2 ) )))) m+4n ))) m )))))))))) 11 7 221+18 321+22 518+22 2 39 343 540 ))) = )) )) )) 3 5 11 2 39 311 511 2 = 332 529 7 is replaced by n: b. )))))))))) 11 n 24n+11 33n+22 5n+33 2 0 33n+11 5n+22 3n+11 n+22 ))) = =3 5 11 2 3 5 11 is replaced by m: c. 10. 63n @10n+m @152m = 30 @16 )))))))))) m n ))))))))))))))) m 4 n 24n+m 33n+2m 5n+3m 2 0 33n+m 5n+2m 3n+m n+2m ))) = =3 5 m 2 3 5 Write each number in scientific notation: 1.03 × 103 1.56 × 104 3.45 × 108 a. 1030 = 15600 = 345,000,000 = b. 3.4 × 107 + 5.2 × 107 = (3.4 + 5.2) × 107 = 8.6 × 107 6 × 108 = (6 + 9.3) × 108 = 15.3 × 108 = 1.53 × 109 c. d. e. + 9.3 × 108 (2 × 104) × (3.2 × 105) = 2 × 3.2 × 104 × 105 = (8 × 104) × (96 × 1023) = 8 × 96 × 104 × 1023 6 × 109 )))))) 3 × 104 = 6 109 )) × ))) 3 104 = 2 × 105 5.4 × 108 ))))))) 9 × 105 = 5.4 108 )) × ))) 9 105 = .6 × 103 (2 × 107)3 = 23 @ (107)3 = 8 × 1021 (5 × 104)3 = 53 @ (104)3 = 125 × 1012 Then there’s the one they didn’t warn you about: 6.4 × 109 = 768 × 10 27 = 7.68 × 1029 = 6 × 102 = 1.25 × 1014 5.67 x 1014 + 3.33 x 1015 How do you add these? Try it and see if you get the answer below. 3.897 x 1015 Ñ 3.90 x 1015