YYYY
FILE INFORMATION
FILENAME(S): YYYY
FILE TYPE(S): PRG
FILE SIZE: 35.7K
FIRST SEEN: 2025-11-30 16:59:42
APPEARS ON: 1 disk(s)
FILE HASH
762d2e697c26ab9858ed1f8f2ff62fcb623523bb7c3c5580105f7c97ca3fc726
FOUND ON DISKS (1 DISKS)
| DISK TITLE | FILENAME | FILE TYPE | COLLECTION | TRACK | SECTOR | ACTIONS |
|---|---|---|---|---|---|---|
| MATHS TUTOR 1 | YYYY | PRG | High Voltage Complete C64 Collection | 19 | 3 | DOWNLOAD FILE |
FILE CONTENT & ANALYSIS
> DETECTED: COMMODORE BASIC PROGRAM (LOAD ADDRESS $0801)
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00005460: 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
00005470: 31 30 20 20 20 20 20 20 20 31 30 22 3A 99 00 AA |10 10":...|
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00005490: 20 20 20 20 20 20 20 32 33 37 34 36 20 20 20 20 | 23746 |
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000054C0: 20 3D 20 C0 C0 C0 C0 C0 20 D6 20 C0 C0 20 3D 20 | = ..... . .. = |
000054D0: C0 C0 C0 C0 C0 20 3D 22 00 05 5D 71 31 99 22 20 |..... ="..]q1." |
000054E0: 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
000054F0: 20 20 20 31 30 20 20 20 20 31 30 20 20 20 20 31 | 10 10 1|
00005500: 30 30 22 3A 99 00 33 5D 7E 31 99 22 20 20 32 33 |00":..3]~1." 23|
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00006A90: 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | |
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000071F0: 22 52 41 54 49 4F 4E 41 4C 20 4E 55 4D 42 45 52 |"RATIONAL NUMBER|
00007200: 53 22 B0 42 31 24 B2 22 52 41 54 49 4F 4E 41 4C |S".B1$."RATIONAL|
00007210: 20 4E 55 4D 42 45 52 22 89 31 34 36 36 30 00 39 | NUMBER".14660.9|
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00007270: 20 20 12 4E 55 4D 42 45 52 53 92 2C 20 9B 49 46 | .NUMBERS., .IF|
00007280: 20 59 4F 55 20 41 52 45 20 43 4F 4E 46 55 53 45 | YOU ARE CONFUSE|
00007290: 44 20 41 53 20 54 4F 22 3A 99 00 C9 7A 3A 39 99 |D AS TO":...z:9.|
000072A0: 22 20 20 52 41 54 49 4F 4E 41 4C 20 4E 55 4D 42 |" RATIONAL NUMB|
000072B0: 45 52 53 20 59 4F 55 20 53 48 4F 55 4C 44 20 52 |ERS YOU SHOULD R|
000072C0: 45 56 49 45 57 20 22 3A 99 00 FC 7A 3F 39 99 22 |EVIEW ":...z?9."|
000072D0: 20 20 50 41 52 54 20 4F 4E 45 21 9A 20 4F 4B 2C | PART ONE!. OK,|
000072E0: 20 4C 45 54 53 20 53 54 41 52 54 20 41 47 41 49 | LETS START AGAI|
000072F0: 4E 2E 22 3A 99 3A 89 31 34 35 37 30 00 07 7B 44 |N.":.:.14570..{D|
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00007320: 54 9A 2C 20 46 52 41 43 54 49 4F 4E 53 20 4F 52 |T., FRACTIONS OR|
00007330: 20 12 52 41 54 49 4F 4E 41 4C 20 4E 55 4D 42 45 | .RATIONAL NUMBE|
00007340: 52 53 92 2E 22 3A 99 22 A0 22 00 5A 7B 4C 39 81 |RS..":.".".Z{L9.|
00007350: 4C B2 31 A4 32 30 30 30 3A 82 00 88 7B 4E 39 99 |L.1.2000:...{N9.|
00007360: 22 20 20 57 45 20 48 41 56 45 20 41 4C 52 45 41 |" WE HAVE ALREA|
00007370: 44 59 20 43 4F 56 45 52 45 44 20 43 4F 4E 56 45 |DY COVERED CONVE|
00007380: 52 54 49 4E 47 22 3A 99 00 B8 7B 58 39 99 22 20 |RTING":...{X9." |
00007390: 20 54 45 52 4D 49 4E 41 54 49 4E 47 20 44 45 43 | TERMINATING DEC|
000073A0: 49 4D 41 4C 53 20 54 4F 20 46 52 41 43 54 49 4F |IMALS TO FRACTIO|
000073B0: 4E 53 2C 20 46 4F 52 22 00 E8 7B 62 39 99 22 20 |NS, FOR"..{b9." |
000073C0: 20 45 58 41 4D 50 4C 45 20 30 2E 32 20 3D 20 32 | EXAMPLE 0.2 = 2|
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000073E0: 57 49 4C 4C 20 22 3A 99 00 1B 7C 6C 39 99 22 20 |WILL ":...|l9." |
000073F0: 20 4E 4F 57 20 4C 4F 4F 4B 20 41 54 20 43 4F 4E | NOW LOOK AT CON|
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00007480: 3A 99 00 B3 7C 8F 39 99 22 20 20 30 2E 33 33 33 |:...|.9." 0.333|
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000074C0: 2E 33 91 9D 2E 11 2E 20 54 4F 20 43 4F 4E 56 45 |.3..... TO CONVE|
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000074E0: 43 54 49 4F 4E 22 3A 99 00 1D 7D 99 39 99 22 20 |CTION":...}.9." |
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00007500: 54 48 45 4E 20 31 30 58 20 3D 20 33 2E 33 91 9D |THEN 10X = 3.3..|
00007510: 2E 11 20 3D 20 33 20 2B 20 58 22 3A 99 00 2D 7D |.. = 3 + X":..-}|
00007520: 9B 39 81 4C B2 31 A4 36 30 30 30 3A 82 00 55 7D |.9.L.1.6000:..U}|
00007530: 9E 39 99 22 20 20 53 4F 20 31 30 58 3D 33 2B 58 |.9." SO 10X=3+X|
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00007550: 3D 33 22 3A 99 00 80 7D A0 39 99 22 20 20 28 53 |=3":...}.9." (S|
00007560: 55 42 54 52 41 43 54 49 4E 47 20 58 20 46 52 4F |UBTRACTING X FRO|
00007570: 4D 20 42 4F 54 48 20 53 49 44 45 53 29 22 3A 99 |M BOTH SIDES)":.|
00007580: 00 90 7D A1 39 81 4C B2 31 A4 33 30 30 30 3A 82 |..}.9.L.1.3000:.|
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000075A0: 20 3D 20 33 20 53 4F 20 58 20 3D 20 33 2F 39 20 | = 3 SO X = 3/9 |
000075B0: 3D 20 31 2F 33 22 3A 99 3A 99 00 CA 7D A4 39 81 |= 1/3":.:...}.9.|
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000075D0: A6 32 29 22 53 4F 20 30 2E 33 91 9D 2E 11 20 3D |.2)"SO 0.3.... =|
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00007600: 45 20 47 45 4E 45 52 41 4C 20 50 52 49 4E 43 49 |E GENERAL PRINCI|
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00007670: 30 91 50 11 58 3D 52 2B 52 91 9D 2E 11 20 57 48 |0.P.X=R+R.... WH|
00007680: 45 52 45 20 50 20 49 53 22 3A 99 00 BB 7E BC 39 |ERE P IS":...~.9|
00007690: 99 22 20 20 54 48 45 20 4E 55 4D 42 45 52 20 4F |." THE NUMBER O|
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000076B0: 20 28 50 45 52 49 4F 44 22 3A 99 00 F6 7E C1 39 | (PERIOD":...~.9|
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000076E0: 20 58 20 3D 20 52 2E 22 3A 99 3A 81 4C B2 31 A4 | X = R.":.:.L.1.|
000076F0: 37 30 30 30 3A 82 00 2E 7F C6 39 99 22 20 20 9F |7000:.....9." .|
00007700: 45 58 41 4D 50 4C 45 3A 9A 20 45 58 50 52 45 53 |EXAMPLE:. EXPRES|
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00007720: 41 20 46 52 41 43 54 49 4F 4E 2E 22 3A 99 00 6A |A FRACTION.":..j|
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00007740: 3A 9A 20 4C 45 54 20 58 20 3D 20 30 2E 32 31 91 |:. LET X = 0.21.|
00007750: 9D 9D 2E 2E 11 20 54 48 45 4E 20 22 3A 99 3A 81 |..... THEN ":.:.|
00007760: 4C B2 31 A4 34 30 30 30 3A 82 00 9E 7F D0 39 99 |L.1.4000:.....9.|
00007770: 22 20 20 31 30 30 58 3D 32 31 2E 32 31 91 9D 9D |" 100X=21.21...|
00007780: 2E 2E 11 20 28 98 50 3D 32 20 26 20 31 30 91 32 |... (.P=2 & 10.2|
00007790: 11 3D 31 30 30 9A 29 20 53 4F 20 22 3A 99 00 DB |.=100.) SO ":...|
000077A0: 7F DA 39 99 22 20 20 31 30 30 58 3D 32 31 2B 58 |..9." 100X=21+X|
000077B0: 2C 20 54 48 55 53 20 39 39 58 3D 32 31 20 26 20 |, THUS 99X=21 & |
000077C0: 58 3D 32 31 2F 39 39 3D 37 2F 33 33 22 3A 99 3A |X=21/99=7/33":.:|
000077D0: 81 4C B2 31 A4 34 30 30 30 3A 82 00 0D 80 DF 39 |.L.1.4000:.....9|
000077E0: 99 22 20 20 4E 4F 57 2C 20 96 55 53 49 4E 47 20 |." NOW, .USING |
000077F0: 50 45 4E 43 49 4C 20 41 4E 44 20 50 41 50 45 52 |PENCIL AND PAPER|
00007800: 9A 2C 20 59 4F 55 20 54 52 59 22 3A 99 00 23 80 |., YOU TRY":..#.|
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00007850: 54 49 4F 4E 20 12 D2 2F D2 22 3B 42 31 24 3A 99 |TION ../.";B1$:.|
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000078D0: 20 49 53 20 43 4F 52 52 45 43 54 22 3A 99 00 0F | IS CORRECT":...|
000078E0: 81 FD 39 99 22 20 20 4C 45 54 53 20 47 4F 20 4F |..9." LETS GO O|
000078F0: 56 45 52 20 54 48 49 53 20 41 47 41 49 4E 20 22 |VER THIS AGAIN "|
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00007940: A3 A3 A3 A3 A3 A3 A3 A3 A3 A3 A3 9A 22 00 7D 81 |............".}.|
00007950: 20 3A 99 22 20 20 49 4E 20 52 45 41 4C 20 4C 49 | :." IN REAL LI|
00007960: 46 45 20 57 45 20 44 4F 20 4E 4F 54 20 41 4C 57 |FE WE DO NOT ALW|
00007970: 41 59 53 20 45 4E 44 20 55 50 22 3A 99 00 A3 81 |AYS END UP":....|
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000079C0: 20 4F 46 46 9A 20 4E 55 4D 42 45 52 53 20 54 4F | OFF. NUMBERS TO|
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000079E0: 41 50 50 52 4F 50 52 49 41 54 45 20 4E 55 4D 42 |APPROPRIATE NUMB|
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00007B40: 20 42 31 24 B2 22 32 2E 38 30 22 89 31 35 30 32 | B1$."2.80".1502|
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00007C20: 20 4F 4E 45 21 20 54 52 59 20 41 47 41 49 4E 3A | ONE! TRY AGAIN:|
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00007DE0: 45 43 2D 22 3A 99 00 1C 86 1A 3B 99 22 20 20 49 |EC-":.....;." I|
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00007E30: 20 54 4F 20 32 20 44 45 43 49 4D 41 4C 20 50 4C | TO 2 DECIMAL PL|
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00008DF0: B2 31 A4 31 30 30 30 3A 82 00 02 96 E1 F6 99 22 |.1.1000:......."|
00008E00: 93 22 00 0E 96 E5 F6 81 20 4C B2 31 A4 37 00 26 |."...... L.1.7.&|
00008E10: 96 EA F6 99 3A 99 3A 99 3A 99 3A 99 3A 99 3A 99 |....:.:.:.:.:.:.|
00008E20: 3A 99 3A 99 3A 99 00 43 96 F4 F6 99 A6 39 29 22 |:.:.:..C.....9)"|
00008E30: 1E 50 45 52 46 45 43 54 20 53 43 4F 52 45 20 22 |.PERFECT SCORE "|
00008E40: 41 31 24 00 52 96 FE F6 81 54 B2 31 A4 35 30 30 |A1$.R....T.1.500|
00008E50: 3A 82 00 6F 96 03 F7 99 22 93 22 3A 99 3A 99 3A |:..o....".":.:.:|
00008E60: 99 3A 99 3A 99 3A 99 3A 99 3A 99 3A 99 3A 99 00 |.:.:.:.:.:.:.:..|
00008E70: 8C 96 08 F7 99 A6 39 29 22 9F 50 45 52 46 45 43 |......9)".PERFEC|
00008E80: 54 20 53 43 4F 52 45 20 22 41 31 24 00 A5 96 12 |T SCORE "A1$....|
00008E90: F7 81 54 B2 31 A4 35 30 30 3A 82 3A 99 22 9A 93 |..T.1.500:.:."..|
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".".14)"..COMPUTER TUTOR...."......"
.BASIC ARITHMETIC AND ALGEBRA..".....
." PART TWO..."...#...11
)".COPYRIGHT (C) 1983."...(...20)".BY.."
...-.A1$."LEARNER"...2.." JOHN C
PAYNTER B.SC.,MAPSS.".Q.<.."
&":." "...F.." MARY ANN
PAYNTER B.A.(HONS) SYD...."...P.." PLEA
SE TYPE IN YOUR FIRST NAME FOR ME":..V.U
..198,0:. " .RRRRRR";A1$:."."...V.A1$.H
(A1$,8)...Z.." NICE TO SEE YOU AGAIN "A
1$"..".?._.." .I AM ASSUMING THAT YOU H
AVE COMPLETED.".T.D.." PART ONE!.."._.G
..40000...I.." THE FIRST TOPIC FOR PART
TWO IS:."...N.." .MULTIPLICATION & DIVI
SION OF FRACTIONS."...S.." .............
..........................."...X.. 10000
...}.X.0./...." .QUIZ ONE - OPERATIONS
ON FRACTIONS.".^...." ...............
....................."......" .BE VERY
CAREFUL TYPING IN YOUR ANSWERS"......"
SINCE ONE ERROR STARTS THE QUIZ OVER."..
. .." AND TWO ERRORS MEANS YOU HAVE TO
REV-."......" IEW THE SECTION. .YOU MAY
USE PENCIL.".J...." PAPER AND CALCULAT
OR IF YOU WISH...".{....".1 TYPE THE REC
IPROCALS IN LOWEST TERMS.."...H.." .OF:
.."...R.Z.0:.198,0:. " 3/8 .R/R";B1
$:.".".U...C1$."8/3":.B1$."8/3"..8090...
...Z.0.240......8020.....Z.0:.198,0:. "
13/5 .R/R";B1$:.".".9...C1$."5/13":.
B1$."5/13". .8090.F....Z.0.270.P....8020
.|...Z.0:.198,0:. " -3/2 .R/R";B1$:.
"..".....C1$."-2/3 OR 2/-3"..."..B1$."-2
/3".B1$."2/-3"..8090.A.,..Z.0.310.K.1..8
020...6.."..2 SIMPLIFY THE FOLLOWING:...
"...@.Z.0:.198,0:. " 2/3 V 2/7 .R/R";B
1$:.".".A.E.C1$."4/21":.B1$."4/21". .809
0.N.J..Z.0.340.X.O..8020...T.Z.0:.198,0:
. " 2/3 - 1/2 V 4/7 .R/R";B1$:."."...^
.C1$."8/21":.B1$."8/21"..8090...H..Z.0.3
70.D.M..8020...R.Z.0:.198,0:. " (2/3 +
1/4) V 1/6 .R/R";B1$:."."...|.C1$."11/72
":.B1$."11/72"..8090.)....Z.0.400.3....8
020.N...Z.0:.198,0:. " 2/3 DIVIDED BY 1
/2 - 2 .R/R";B1$:.".".....C1$."-2/3 OR
2/-3"......B1$."-2/3".B1$."2/-3". .8090.
..... Z.0 . 440.A....8020......:.:."
.END OF QUIZ ONE."...... 6320
0......".".)...." .TOPIC TWO - D
ECIMALS.".P...." ...............
.......".\.B.. 11500.D.G.X.0...L.."
.QUIZ TWO - DECIMALS."...Q.."
....................."...V.." .BE V
ERY CAREFUL TYPING IN YOUR ANSWERS".....
." SINCE ONE ERROR STARTS THE QUIZ OVER
.".B...." AND TWO ERRORS MEANS THAT YOU
MUST RE-".S...." VIEW THE SECTION. .YO
U MAY USE PENCIL,."......" PAPER AND CA
LCULATOR IF YOU WISH...."......A2.10. ..
.....".1 EXPRESS AS FRACTIONS IN THEIR L
OWEST."......" TERMS:..".....Z.0:.198,0
:. " 0.4 .R/R";B1$:.".".;...C1$."2/
5":.B1$."2/5"..8190.H....Z.0.540.R....81
20.}...Z.0:.198,0:. " 0.05 .R/R";B1$
:."."...!.C1$."1/20":.B1$."1/20"..8190..
.&..Z.0.560...+..8120...0.Z.0:.198,0:. "
0.72 .R/R";B1$:.".."...5.C1$."18/25
":.B1$."18/25"..8190...:..Z.0.580...?..8
120.F.D..:.:.".2 SIMPLIFY THE FOLLOWING:
.":.:..W.N.Z.0:.198,0:. " 2.674 V 100 =
.RRR";B1$:."."...S.C1$."267.4":.B1$."26
7.4".B1$."267.40"..8190...T..B1$."267.40
0"..8190.M.X..Z.0.610.W.]..8120...B.Z.0:
.198,0:. " 532.7 DIVIDED BY 10 = .RR";B
1$:.".".2.G.CI$."53.27":.B1$."53.27"..81
90.?.L..Z.0.630.I.Q..8120.|.V.Z.0:.198,0
:. " 12.38 + 11.63 = .RRR";B1$:."."...{
.C1$."24.01":.B1$."24.01"..8190......Z.0
.650......8120.....D.0.....Z.0:.198,0:.
" 0.3 + 12.6 V 2 = .RR";B1$:.".".....C1
$."25.5":.B1$."25.5"..8190......Z.0.670.
3....B1$."25.8".660.=....8120.K....D.0.8
120....." REMEMBER .V. BEFORE .+. UNLE
SS THERE ARE":......." .( ).. I'LL GIVE
YOU ONE MORE CHANCE":.:D.D.1:.650.....D
.0:Z.0:.198,0:. " (0.3 + 12.6) V 2 = .R
R";B1$:.".".....C1$."25.8":.B1$."25.8"..
8190.!....Z.0.700.+....8120.\...Z.0:.198
,0:. " 11.37 - 10.2 = .RR";B1$:.".".}.A
.C1$."1.17":.B1$."1.17"..8190...F..Z.0.7
20...K..8120.L.P.Z.0:.198,0:. " .049 DI
VIDED BY 1.4 = .RR";B1$:."."...U.C1$."0.
035":.B1$."0.035".B1$.".035"..8190...Z..
Z.0.740......8120.?...Z.0:.198,0:. " 2.
34 V .2 = .RR";B1$:.".".M...C1$."0.468":
.B1$."0.468".B1$.".468"..8190.Z....Z.0.7
57......8120......63200......".. THE NEX
T TOPIC, "A1$", IS:.".Z...."
.RELATED TOPICS."......" .............
........................."... ..14000...
%.X.0.F.*.." .QUIZ THREE - RELATED TO
PICS.".O.4.." .......................
......".~.>.A2.10:.470...C.A2.0...H..".1
EXPRESS AS DECIMALS:.":.:..S.R..198,0:Z
.0:. " 12% .RRR";B1$:."."...\.C1$."0.1
2":.B1$."0.12".B1$.".12"..8290...F..Z.0.
890...P..8220.>.Z..198,0:Z.0:. " 1.7% .
RRR";B1$:."..".L...C1$."0.017":.B1$."0.0
17".B1$.".017"..8290.Y....Z.0.930......8
220......".2 EXPRESS AS FRACTIONS IN LOW
EST TERMS:.":.:......Z.0:.198,0:. " 75%
.RRR";B1$:."..".....C1$."3/4":.B1$."3/
4"..8290......Z.0.980...J..8220.3.T..".3
EXPRESS AS %.":.:..[...Z.0:.198,0:. "
1/5 .RRR";B1$:.".".....C1$."20%":.B1$."
20%".B1$."20"..8290......Z.0.1030......8
220.E...Z.0:.198,0:. " 0.163 .RRR";B1$:
."..".....C1$."16.3%":.B1$."16.3%".B1$."
16.3"..8290......Z.0.1070...$..8220.;...
.".4 EXPRESS AS DECIMALS OR INTEGERS:.":
.:..J.8.Z.0:.198,0:. " 12% OF 200 .RRR
";B1$:."."...B.C1$."24":.B1$."24".B1$."2
4.0"..8290. .L..Z.0.1120...V..8220...`.Z
.0:.198,0:. " 15 INCREASED BY 10% .RRR
";B1$:."."...J.C1$."16.5":.B1$."16.5"..8
290...T..Z.0.1160...~..8220.T...Z.0:.198
,0:. " 12 DECREASED BY 20% .RRR";B1$:.
"..".S...C1$."9.6":.B1$."9.6"..8290.....
.Z.0.1200......8220......".5 ARE REPEATI
NG DECIMALS RATIONAL NUM-.".....Z.0:.198
,0:. " BERS. (YES OR NO) .RRR";B1$:.".
."...D.C1$."YES":.B1$."YES"..8290...N..Z
.0.1250.).X..8220.^....".6 EXPRESS AS FR
ACTIONS IN LOWEST TERMS:.":.:......Z.0:.
198,0:. " 0.6.... .R/R";B1$:.".".....C
1$."2/3":.B1$."2/3". .8290......Z.0.1300
.B....8220.....Z.0:.198,0:. " -0.3....
.R/R";B1$:."..".....C1$."-1/3":.B1$."-1
/3".B1$."1/-3"..8290.*.(..Z.0.1340.4.2..
8220.F.<..".7 EXPRESS THE FOLLOWING CORR
ECT TO 2 ":.:....F.." DECIMAL PLACES:."
:.:....P.Z.0:.198,0:. " 5.417 .RRRRR";
B1$:.".".Q.Z.C1$."5.42":.B1$."5.42"..829
0...D..Z.0.1400...N..8220...X.Z.0:.198,0
:. " -1.008 .RRRR";B1$:.".".8...C1$."-
1.01":.B1$."-1.01"..8290.F....Z.0.1440.P
....8220... .Z.0:.198,0:. " 1.26734 V 1
0.2. .RRRR";B1$:.".".....C1$."126.73":.
B1$."126.73"..8290......Z.0.1480.B....82
20...H.Z.0:.198,0:. " 1.276 V 10.-2. .
RRRR";B1$:.".."." R.C1$."0.01":.B1$."0.0
1".B1$.".01"..8290.0 ...Z.0.1520.: ...82
20.I ...".8 EXPRESS THE FOLLOWING CORREC
T TO 2":... ..." SIGNIFICANT FIQURES:."
:.:... ..Z.0:.198,0:. " 5.012 .RRR";B1
$:.".".U ..C1$."5.0":.B1$."5.0"..8290..
...Z.0.1580.. "..8220..!,.Z.0:.198,0:. "
-0.0135 .RRRRR";B1$:.".".L!6.C1$."-0.
014":.B1$."-0.014".B1$."-.014"..8290.Z!@
..Z.0.1620.D!J..8220..!T.Z.0:.198,0:. "
1.276 V 10.-2. .RRR";B1$:."..".F!^.C1$
."0.013":.B1$."0.013".B1$.".013"..8290.T
!H..Z.0.1660..!R..8220..!|..63200.."P...
6)".THIS COMPLETES PART TWO...".>"U.."IT
'S BEEN A REAL PLEASURE WORKING WITH.".B
"Z.."YOU. SEE YOU LATER "A1$"....".."...
.25)".TO EXIT.":.9300:.64738.."....."D..
"..LOADING TEST - CHECK BYTES:...".."X..
" ".(0).(.(0).0).65536" BYTES FREE...":.
2020..#..."......NOTE: THE 'RUN/STOP' KE
Y WILL NOT WORK.".H#B.."WHILE THIS PROGR
AM IS RUNNING. TO EXIT.".V#L.."BEFORE TH
E END OF THE PROGRAM, SIMPLY."..#Q.."REM
OVE THE DISK FROM THE DISK DRIVE AND.".U
#V.."TURN OFF YOUR COMPUTER. ALSO, YOU W
ILL."..$..."NOT BE ABLE TO CHANGE TO LOW
ER CASE..".2$..."DON'T WORRY! ALL WILL B
E BACK TO NORMAL.".M$..."WHEN WE FINISH.
....".W$...9300.]$....P$T..".":.X.0.8040
..$^.." NO THATS WRONG. ."C1$". IS RIGH
T.":..Z$C.." WE'LL START AGAIN. .YOU MA
Y USE PENCIL,":X.X.1:. 180..%H.." I'M S
ORRY, BUT THATS TWO ERRORS. YOU":..<%R..
" WILL HAVE TO REVIEW THE SECTION:":.:.
120.F%...9000.V%..Z.Z.1:.:.:..\%....O%.
..".":.X.0.8140..%B.." NO THATS WRONG.
."C1$". IS RIGHT.":...%D..:.:...%G.." W
E'LL START AGAIN. .YOU MAY USE PENCIL,":
X.X.1:.490..&L.." I'M SORRY, BUT THATS
TWO ERRORS. YOU":..B&V.." WILL HAVE TO
REVIEW THE SECTION":.:.445.L&...9000.\&.
Z.Z.1:.:.:..B&. ..U&. .".":.X.0.8240..&
& ." NO THATS WRONG. ."C1$". IS RIGHT."
:.:.:.:...&+ ." WE'LL START AGAIN. .YOU
MAY USE PENCIL,":." "..&, X.X.1..'- ."
PAPER & CALCULATOR!.":." ":." ":.840.H'
0 ." I'M SORRY, BUT THATS TWO ERRORS. Y
OU":." ".}': ." WILL HAVE TO REVIEW THI
S SECTION.":." ":.800..'B .9000..'G Z.Z.
1:.:.:...'L ...'(#A..(.(0).4)..')#.A.0.9
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.9030..(-#." .CORRECT "A1$"...":.63010.
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:.63010.A(7#A.0:..E(<#." .VERY GOOD "A1
$"!..":.63010.O(>#A.0:...(F#." .WELL DO
NE "A1$", GOOD ON YOU..":.63010..(H#A.0:
...(P#." .TREMENDOUS! VERY WELL DONE "A
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)".PRESS ANY KEY."..)^$.A$:.A$."".9310./
)H$.".":A$."":..X).'."
A C"..).'." TO MULTIPLY TWO
FRACTIONS . & . WE"..).'."
B D":..Z)"'."
A C A V C"..*$'." PROCE
ED AS FOLLOWS: . V . = ...... SO".6*&'."
B D B V D":..V
*,'." 3 4 3 V 4 12 3"..*.'." .
V . = ..... = .. = .. NOW YOU TRY:"..*0
'." 4 5 4 V 5 20 5":.:..T*8'.19
8,0:. " 1/3 V 2/7 = .R/R";B1$:.".."..*B
'.B1$."2/21".10080..+L'." NO, LETS GO O
VER THIS "A1$":":..*+T'." 1 2 1 V 2
".Z+V'." . V . = ..... = .2/21. NOW YO
U TELL ME".Z+X'." 3 7 3 V 7":.:.100
40..+`'. 9000..+J'." REMEMBERING THAT M
ULTIPLICATION IS":...+T'." .COMMUTATIVE
. (A V B = B V A) LETS."..,~'." LOOK AT
3/4 V 4/5 AGAIN:":.:..5,.'." 3 4 3
V 4 4 V 3 4 3 3".D,.'." .
V . = ..... = ..... = . V . = 1 V ."..,.
'." 4 5 4 V 5 4 V 5 4 5
5":..E,.'." =3/5. .THERE IS A COMMON F
ACTOR OF 4.."..,.'." AND 4/4 = 1. (THIS
PROCEDURE IS REF-.".$-.'." ERED TO AS
'CANCELLING' IN SOME TEXTS)".L-.'." KEE
PING THIS IN MIND, "A1$":.."..-.'.198,0:
. " WHAT IS 1/237 V 237/5 .R/R";B1$:.".
"..-D'.B1$."1/5".10210..-I'." NO, "B1$"
IS NOT CORRECT:.."..-L'."
1 237 237 1"...N'." .1/5. I
S CORRECT. ... V ... = ... V .".F.P'."
237 5 237 5":..{
.S'." =1 V 1/5 = 1/5. LETS GO BACK A BI
T..":.10000....'.9000....'."
A C".Y..'." TO .DIVIDE. .
BY A FRACTION . PROCEED"../.'."
B D":.../.'." AS FOLL
OWS:":." ".B/.'." A C A
D A V D".O/.(." . DIVIDED BY . = . V
. = ...... LET"../.(." B D
B C B V C":..E/.(." ME SHOW YOU WH
Y THIS IS SO "A1$":."../.(."
A"..0.(."
.".30.(." A C B
A 1".`0.(." . DIVIDED BY . MEANS
... OR . V ..."..0 (." B D
C B C"..0#(."
. ."..0((."
D D":...12(." (
A/B TIMES THE .RECIPROCAL. OF C/D)":...1
<(.9300.*1A(."...".W1F(." BUT:
.D D D."..1P(." 1
1 . 1 V . ."..1Z(."
.C C C .D."..1_
(." ... = ... V ... = ..... = .. =
."..2D(." .C C .D .C
.D .C.".C2I(." . .
. . V . 1".N2N(." .D
D .C .D .C.".V2S(.:...2X(."
A 1"..2}(." SO: . V ... = A/B V .
D./.C.. 6/12 DIVIDED".V2.(." B .C
./.D."..2.(.:...2.(." 6 1"
.&3.(." BY .1./.3. = .. V ... = 6/12 V
.R/R.".G3.(." 12 .1./.3.":..
D3.(.198,0:. " .";B1$:.". ".Y3 (.B1$."3
/1".10440..3.(.B1$."3".." YES 3/1 WHICH
= 3":.:.10445.J3.(."
1"..3.(." THATS NOT RIGHT. 6/12
V ... = 6/12 V"..4.(."
1/3.".P4.(." .3/1.. LETS TAKE
ANOTHER LOOK AT THIS..".Z4C(.9300.L4D(.
".":.:.10218.{4H(.".":.9000..4J(.10450..
4M(.9020..4R(." SO TO DIVIDE ANY NUMBER
BY A FRACTION."..4W(." (A/B) WE .MULTI
PLY. BY ITS .RECIPROCAL.":...4Z(." 1".
,5.(." ... (WHICH = B/A). NOW YOU TELL
ME:".:5.(." A/B.".J5.(." TO DIVIDE A N
UMBER BY A FRACTION (A/B)"..5.(." WE MU
LTIPLY BY B/A WHICH IS THE ......".D5.(.
198,0:. " OF A/B. .RRRRRR";B1$:."."..5.
(.B1$."RECIPROCAL".10520..6.(." NO, THE
CORRECT ANSWER IS .RECIPROCAL...".86.).
" CHECK YOUR SPELLING, "A1$"..".N6.)."
HAVE ANOTHER GO, AND BE CAREFUL! OK,.":
.10450..6.)." .THATS RIGHT., "A1$" TO D
IVIDE":...6.).63010.U6.)." A NUMBER BY
A FRACTION (.A/B.) WE MULT-."..7")." IP
LY BY ITS RECIPROCAL WHICH IS .R/R..".37
').198,0:. " .";B1$:." .":.B1$."B/A".10
560.<7*).".".O7,)." NO! THE .REIPROCAL.
OF A/B IS .B/A.. YOU."..76)." WILL HAV
E TO GO BACK TO THE START."..7;).10000..
7@).".":.9000..7J)." .EXAMPLE QUESTION:
. TYPE IN THE RECIP-."..8O)." ROCALS OF
THE FOLLOWING:."..8T).198,0:. " 3/7 .
R/R";B1$:.". ".L8Y). " 2/5 .R/R";B2$:.
". ".J8^). " -3/4 .R/R";B3$:.". ".S8C).
"."..8H).B1$."7/3".B2$."5/2"..B3$."-4/3"
.B3$."4/-3".10650.M8M)." NO, I'M SORRY
"A1$" BUT:":...8R).B1$."7/3".10620..9W).
" THE RECIPROCAL OF 3/7 IS .7/3..":.."9
|).B2$."5/2".10630.M9.)." THE RECIPROCA
L OF 2/5 IS .5/2..":..N9.).B3$."-4/3".B3
$."4/-3".10640..9.)." THE RECIPROCAL OF
-3/4 IS .-4/3. OR ":." ":." .4/-3..":.
" "..9.)." YOU HAD BETTER GO OVER THIS
AGAIN."..9.).9300:.10218..9.).9000.,:.).
" WHEN INTEGER PARTS APPEAR IN FRACTION
S":..C:.)." LIKE 1...1...2., THESE ARE
CALLED .MIXED.":.:...:.)." .NUMBERS.. I
N SIMPLIFYING IT IS BEST TO":..G:.)." C
ONVERT .MIXED NUMBERS. TO IMPROPER":.:..
.;B)." FRACTIONS. SO 1...1...2. = ...2.
..2. + ...1...2. =3/2. YOU":." "..;D)..F
;L)." TELL ME: IN SIMPLIFYING IT IS BES
T TO.".V;V)." CONVERT RRRRR RRRRRR TO I
MPROPER FRAC-"..;.).198,0:. " TIONS. .R
RRR RRRRRR";B1$:."..".D;.).B1$."MIXED NU
MBERS".10760..;.)." WRONG, THE CORRECT
ANSWER IS .MIXED..".&<.)." .NUMBERS.. L
ETS GO OVER IT AGAIN:.":.10670.0<.*.9000
.P<.*." 1 2".<.*."
SO TO MULTIPLY 1. V 3. YOU WOULD CON-".
.<.*." 2 3":..T<.*.1
98,0:. " VERT 1...1...2. TO .R/R";B1$:.
". "..<.*.:...=!*.198,0:. " AND 3...2..
.3. TO .R/R";B2$:.".."..=&*.B1$."3/2".B2
$."11/3".10820.X=(*." NO, THATS NOT COR
RECT "A1$":":.:..T=+*.B1$."3/2"..:.:.:.1
0805..=0*." 1...1...2. = .3/2..":.:...
=5*.B2$."11/3".10815.S=:*..14)"...3...2.
..3. = .11/3..":.:...=?*.L.1.3000:.:.106
70..=D*.9000. >N*." .COMBINED OPERATION
S ON FRACTIONS.".L>S*." ...............
...................".V>X*." AS YOU MAY
KNOW, "A1$" WE COMP-."..>]*." UTERS PER
FORM CALCULATIONS IN A CER-.".U>B*." TA
IN ORDER. THIS ORDER IS THE SAME IN."..?
G*." MATHEMATICS. THE ORDER OF PRECEDEN
CE."..?L*." IS AS FOLLOWS:.".I?Q*." .F
IRST.: (),[],OR POWERS & ROOTS;.".V?V*."
.NEXT. : MULTIPLICATION & DIVISION."..
?{*." WORKING FROM LEFT TO RIGHT
;.".S?.*." .THEN .: ADDITION & SUBTRACT
ION WORK-."..?.*." ING FROM LEFT
TO RIGHT.."..@.*.9300..@.*.".".<@.*."
LETS LOOK AT AN EXAMPLE, CONSIDER:.".M@.
*." 6 - 1 V 5 = 6 - 5 = 1 .NOTE. THE OP
ER-."..@.*." ATION .V. WAS PERFORMED FI
RST. NOW CON-.".O@.*." SIDER (6 - 1) V
5 = 5 V 5 = 25 .NOTE.."..@.*." THE OPER
ATION IN THE GROUPING SYMBOLS."..A.*."
.(). WAS PERFORMED FIRST. AS YOU CAN.".^
A.*." SEE, THE ANSWERS ARE QUITE DIFFER
ENT!."..A.*." HERE ARE SOME EXAMPLES. Y
OU CAN USE."..A.*." PENCIL & PAPER OR E
VEN YOUR CALCULATOR".HA.*. L.1.3..A.*X..
(.(1).9):Y..(.(1).9)..AA*Z..(.(1).9)..BF
*." "X" + "Y" V "Z" = ":." ".4BK*.198,
0:. " .RR";B1:.". ".HBP*.B1.X.Y.Z.10975
..BU*.".":." NO, ."X.Y.Z". IS CORRECT.
WE'LL GO BACK":...BZ*.10830..B.*." .COR
RECT. AND ("X"+"Y") V"Z"=":." ".UB.*.198
,0:. " .RR";B1:.". "..B.*.B1.(X.Y).Z.11
000..C.*.".":." NO, ."(X.Y).Z". IS CORR
ECT. REVIEW:":..+C.*. 10830.6C.*. 9000.>
C.*. L.SC.+.".":." LETS LOOK AT SOME EX
AMPLES COMBINING":...C.+." EVERYTHING S
O FAR:."..C.+." 1 2 1"
.RC.+." .SIMPLIFY:. 3. + 5. V ."..C.+.
" 4 3 2"..D.+.L.1.4000
:...D +." .SOLUTION:..".GD2+." 1 2
1 13 17 1 13 17V1".ZD4+."
3. + 5. V . .=. .. + .. V . .=. .. + ...
."..D6+." 4 3 2 4 3 2
4 3V2":...D9+.L.1.4000:...D<+." 13
17 13 3 17 2 39 34 73".!E>+.
" .=... + .. .=. ..V. + ..V. .=. .. + .
. .=. ..".RE@+." 4 6 4 3 6 2
12 12 12":..CEC+. L.1.4000:...EH+
." YOUR TURN "A1$", USE PEN & PAPER.":.
..EK+." 1 1 3".PEM+."
.SIMPLIFY:. 2. + 1. V ."..ER+."
3 6 4".)FW+.198,0:. "
.RR/RR";B1$:."..".@FF+.B
1$."77/24".11140.RFP+." NO, .77/24. IS
CORRECT - YOU WILL HAVE":...FZ+." TO RE
VIEW THIS SECTION.":.:.10000..F.+.9040.P
F.+." OK "A1$", .I. THINK YOU KNOW IT."
..F.+." BUT WE HAD BETTER MAKE SURE..".
.F.+..(G.,.". FIRST LETS REVIEW WHAT IS
MEANT.".]G.,." BY A .POWER.. A..X.. ME
ANS A RAISED TO THE."..G.-." POWER .X.
WHICH MEANS A V A V A V ....".AG.-." X
TIMES. FOR EXAMPLE A..3.. MEANS A RAISED
"..G.-." TO THE POWER .3. WHICH MEANS A
V A V A."."H.-." SO 4.2. MEANS 4 RAISE
D TO THE POWER 2.".OH(-." WHICH MEANS 4
V 4 WHICH EQUALS 16.."..H2-.198,0:. "
3.3. MEANS 3 RAISED TO THE POWER .R";B1$
:.".."..H<-.B1$."3".B1$."THREE".11600..H
F-." NO, 3.3. MEANS 3 RAISED TO THE POW
ER .3..":...IK-." "A1$", A..X.. MEANS A
RAISED TO THE.":.11520.!IP-.9000.MIZ-."
AND 3 RAISED TO THE POWER 3 MEANS.".|I
D-." 3 V 3 V 3 = 27. SO 10.2. = 10 TO T
HE."..IN-.198,0:. " POWER 2 = 10 V 10 =
.RRR";B1$:.".".DIX-.B1$."100".11660.MI}
-."."..J.-." NO, 10.2.=.100.. A..X.. ME
ANS A RAISED TO THE.":.11520..J.-.9010.G
J.-." SO 2.3. EQUALS 8; 5.2. = 25; AND
10.3..".HJ -.198,0:. " = .RRR";B1$:."."
.~J.-.B1$."1000".11700..J.-." NO, 10.3.
= .1000.. PLEASE REMEMBER":.:.11595..J.
-." RIGHT! NOW ONE MORE THING, RECALL T
HAT"..K.-." .ANY NUMBER RAISED TO THE P
OWER 0 IS 1..".WKH-.198,0:. " SO 2.0. =
1; 5.0. = 1; AND 10.0. = .R";B1$:."..".
TKR-.B1$."1".B1$."ONE".11750..K.-." NO,
10.0. = .1.. REMEMBER "A1$",.":.11710..
K.-.9000..K..." .A .DECIMAL. IS A FRACT
ION WHOSE DEN-."..L..." OMINATOR IS SOM
E POWER OF 10.. SO A.".;L..." FRACTION
WHOSE DENOMINATOR IS SOME.".QL...198,0:.
" POWER OF 10 IS A ..RRRRRR";B1$:."...
"..L./.B1$."DECIMAL".12058..L./." NO .D
ECIMAL. IS CORRECT. "A1$",.":.12000..L./
." 1 1 1 "..M./." .RI
GHT., SO .., ..., ...., ETC. ARE".,M./."
10 100 1000":..7M!/.63010.
MM&/." ALL DECIMALS SINCE .10., .100. &
.1000. ARE."..M0/." ALL POWERS OF 10.
IN OUR BASE 10 NUM-.".HM:/." BER SYSTEM
, THE NUMBER 100 MEANS."..MD/." 1 HUNDR
ED, 0 TENS, AND 0 UNITS. IN."."NN/." TH
E SAME WAY THE NUMBER 0.01 MEANS.".NNX/.
" 0 UNITS, 0 TENTHS AND 1 HUNDREDTH.".Z
NB/." OK "A1$", YOU TELL ME. THE NUMBER
."..NG/.198,0:. " 0.01 MEANS RR
UNITS .RR";B1$:."."..NL/. "
AND RR TENTHS .RR";B2$:."."..OQ/. "
AND RR HUNDREDTHS .RR";B3$:."
..".?OV/.B1$."0".B2$."0"..B3$."1".12170.
QO{/." NO, 0.01 MEANS .0. UNITS, .0. TE
NTHS &":...O./." .1. HUNDREDTH. "A1$",
IN OUR NUM-.":.12090..O./.9030..O./." T
HE .POWER. TO WHICH 10 IS RAISED IN."..P
./." THE DENOMINATOR TELLS YOU THE NUMB
ER.".8P./." OF .DECIMAL PLACES.. FOR EX
AMPLE:.".HP./." 1/10.0. = 1 SINCE 10.0.
= 1 & 1/1 = 1."..P./." SO THERE ARE 0
DECIMAL PLACES. 1/10.1..".IP./." = 0.1
SINCE 10..1.. = 10 & 1/10 = 0.1."..P./."
SO THERE IS .1. DECIMAL PLACE. GENERAL
LY".1Q./." 1/10..X.. MEANS THERE ARE .X
. DECIMAL PLACES".AQF/." SO 1/10.2. WIL
L HAVE HOW MANY DECIMAL."..QP/.198,0:. "
PLACES? .RR";B1$:.".."..QZ/.B1$."2".B1
$."TWO".12270..Q./." NOT "B1$". 1/10.2.
HAS .2. DECIMAL PLACES.":.:...R./." 1/
10.2. = 0.01 WE HAD BETTER GO BACK.".6R.
/." OVER DECIMALS "A1$":":." ":.12000.@
R./.9000.OR./." AND 1/10.3. HAS 3 DECIM
AL PLACES AND."..R.0.198,0:. " EQUALS .
RRRRR";B1$:.".."..R.0.B1$."0.001" . 1232
0.GR.0.B1$.".001".12320..R.0." NO 1/10.
3. HAS .3. DECIMAL PLACES AND.".-S.0."
EQUALS .0.001. WE HAD BETTER REVIEW:.":.
12000.8S 0. 9000.JS*0." .ADDITION & SUB
TRACTION. OF DECIMALS IS."..S40." VERY
SIMPLE. IT IS JUST LIKE ORDINARY.".JS>0.
" NUMBERS. FOR EXAMPLE: 2.343 + 1.756 =
."..SC0." 2.343"..SH0."
+ 1.756"..TJ0." .....".(TM0."
4.099":..UTR0." 6+3=9;5+4=9;7+3
=10 (0 CARRY 1) & ":...TW0." 2+1+1(CARR
IED)=4. OK, NOW YOU TRY ONE":...T\0.198,
0:. " 1.53+2.46= .RRRR";B1$:.".".GTF0.B
1$."3.99".12420..UK0.B1$."3.990".." RIG
HT EXCEPT FOR THE 0 TRY AGAIN.":.12380.=
UZ0." NO, .3.99. IS CORRECT. WE'LL REVI
EW:":.:.12000.GU.0.9000.SU.0.198,0:. "
3.75-2.40= .RRRR";B1$:.".."..U.0.B1$."1.
35".12450..U.0." NO, .1.35. IS CORRECT.
LETS REVIEW:":.:.12000.JU.0. 9020..U.0.
" THATS ALL THERE IS TO IT "A1$" NOW.".
)V.0." LETS LOOK AT .MULTIPLICATION & D
IVISION.".SV.0." WE WILL COVER THREE SI
TUATIONS:."..VE0.".1 MULTIPLYING OR DIVI
DING DECIMALS BY."..VJ0." A POWER OF TE
N.".KVO0."2 MULTIPLYING DECIMALS BY OTHE
R DECIMALS"..VT0."3 DIVISION INVOLVING D
ECIMALS.."."W.0.".1. SINCE DECIMALS ARE
FRACTIONS WITH A.".QW.0." DENOMINATOR A
POWER OF 10, MULTIPLY-."..W.0." ING OR
DIVIDING BY A POWER OF 10 SIM-."..W.0."
PLY .MOVES THE DECIMAL POINT TO THE.".
OW.0. " RIGHT OR LEFT..":.9300..W.0."
.LETS LOOK AT SOME EXAMPLES:..".$X.0."
0.02 V 10 = 0.2 .THIS IS BECAUSE:..".4X.
1.L.1.4000:.._X.1." 2 2
20 2"..X.1." 0.02 = ... SO ..
. V 10 = ... = ..=0.2"..X.1." 10
0 100 100 10":..KX.1.L.1.400
0:...X.1." SO IN .MULTIPLYING BY 10 WE
MOVED THE.".,Y.1." DECIMAL POINT ONE PL
ACE TO THE RIGHT...".\Y.1." IF WE MULTI
PLIED BY 100 WE WOULD MOVE."..Y.1." THE
DECIMAL POINT .TWO. PLACES TO THE"..Y#1
." 2".YY$1." RIGH
T. 0.02 V 100 = ... V 100 = 2.00"..Y&1."
100":...Z)1.L.1.80
00:..=Z.1." NOTICE 100 IS 10..2.. AND W
E MOVED THE.".NZ31." DECIMAL POINT .2.
PLACES TO THE RIGHT..".Z51. L.1.5000:..
.Z81." OK "A1$", WITHOUT PENCIL & PAPER
,."..ZB1.198,0:. " WHAT IS 1.0175 V 100
0? .RRR";B1$:.".."..ZG1.B1$."1017.5".126
30.+[L1." NO 1.0175 V 1000 = 1.0175 V 1
0..3.. SO.".[[Q1." YOU MOVE THE DECIMAL
POINT .3. PLACES."..[S1." TO THE RIGHT
GIVING .1017.5., "A1$:.:.12510..[V1.903
0.I[[1.:." NOW CONSIDER 2374.6 .DIVIDED
BY. 10."..[`1." 2374.6 DIVIDED BY 10 =
237.46 .THIS IS.".&\D1."
.23746 23746.".R\E1." BECAUSE.:
2374.6 = ..... SO ..... ".~\G1."
10 10":...\M1."
23746 1 23746".X\O1.
" DIVIDED BY 10 = ..... V .. = ..... ="
..]Q1." 10 10 1
00":..3]~1." 237.46 .DIVIDING BY 10 WE
MOVED THE.".C].1." DECIMAL POINT ONE PL
ACE TO THE LEFT...".M].1.9300..].1." IF
WE WERE TO DIVIDE BY 100 WE WOULD.".M].
1." MOVE THE DECIMAL POINT .TWO. PLACES
TO."..].1." THE LEFT. 2374.6 DIVIDED B
Y 100 =."..^.1." 23746 1 23746".C
^.1." ..... V ... = ..... = 23.746 NOTI
CE".E^.1." 10 100 1000":." "..^
.1." 100 IS 10..2.. & WE MOVED THE DECI
MAL":..H^.1." POINT .2. PLACES TO THE L
EFT. ALRIGHT,":...^.1." WITHOUT PENCIL
& PAPER, WHAT IS 2374.6".+_.1.198,0:. "
DIVIDED BY 1000 .RRRRRR";B1$:."..".C_.1
.B1$."2.3746".12760.T_.1." NO, 2374.6 D
IVIDED BY 1000 = 2374.6 V":..._B1." 1"
.._D1." ...3. SO YOU MOVE THE DECIMAL P
LACE .3.".._F1." 10":..._N1." PLACES T
O THE LEFT, GIVING .2.3746.":.:.12510.._
X1.9010.1`.1.".2. TO MULTIPLY DECIMALS B
Y OTHER DECIMALS".``.1." .1) MULTIPLY T
HE NUMBERS AS IF THEY":...`.1." WERE
INTEGERS.":.:.L.1.4000:...`.1." 2) POS
ITION THE DECIMAL POINT BY FIND-"..`.1."
ING THE TOTAL NUMBER OF DECIMAL ":.
..A.1." PLACES..":.:.L.1.3000:..2A.2
." .EXAMPLE:. 0.3 V 0.2 = 0.06":..<A.2.
9300.GA.2." 3
2 6"..A.2." .THAT IS:. 0.3 V 0.2 = .
. V .. = ...".CA.2."
10 10 100":...A.2." AND 6/100 =
6/10.2. = 0.06 IN THIS EX-":..'B.2." AM
PLE 3 V 2 = 6 AND .THERE ARE A TOTAL":..
XB.2." OF 2 DECIMAL PLACES. (1 FOR 0.3
& 1 FOR"..B#2." 0.6) SO IF YOU .MULTIPL
Y AS IF THE":...B(2." DECIMALS WERE INT
EGERS. YOU GET .6. & IF":...B-2." .YOU
FIND THE TOTAL NUMBER OF DECIMAL":...C22
." PLACES. YOU ARE LEFT WITH .0.06. NOW
YOU":..SC72.198,0:. " TRY ONE. 0.4 V 0
.2 = .RRR";B1$:.".".SC<2.B1$."0.08".B1$.
".08".12880.|C?2."."..CA2." NO NOT "B1$
" THE CORRECT ANSWER IS":..ZCF2." .0.08
. 4 V 2 = 8 & THERE ARE A TOTAL":...DK2.
" OF .2. DECIMAL PLACES. LETS LOOK AGAI
N":.:.12770..DP2.9030.MDU2." .GET YOUR
PENCIL & PAPER "A1$".. OK,":...DZ2.198,0
:. " HOW ABOUT 1.03 V 2.4 = .RRRR";B1$:
.".."..D_2.B1$."2.472".12920.IDD2." NO,
THE CORRECT ANSWER IS .2.472.":...DI2."
.1. MULTIPLY LIKE YOU WOULD WITH INTEGER
S:".+EN2." 103 V 24 = 2472: .2.) THEN C
OUNT THE":..\ER2." DECIMAL PLACES (2 IN
1.03 & 1 IN 2.4)":...EU2." S0 1.03 V 2
.4 = .2...472. REMEMBER:":.:.12770..EX2.
9040.LE.2.".3. IN DIVISIONS INVOLVING DE
CIMALS WE":...E.2." MAKE THE DIVISOR (D
ENOMINATOR) AN IN-":...F.2." TEGER BY .
MULTIPLYING THE NUMERATOR &":..^F.2." D
ENOMINATOR BY AN APPROPRIATE POWER ":...
F.2." OF TEN.. .FOR EXAMPLE:.":...F.2."
15 100 1500"
..F 2." 15 DIVIDED BY 0.03 = .... V ...
=...."..G.2." 0.0
3 100 3":..>G.2." AND 1500/3 = 500
NOW ITS YOUR TURN:":..QG.2.198,0:. " 2
DIVIDED BY 0.04 = .RR";B1$:.".."..G.2.B
1$."50".B1$."50.0".B1$."50.00".13000.OG.
2." NO, 2 DIVIDED BY 0.04 = 200/4 = .50
.00.":...H.2." LETS JUST GO BACK OVER T
HIS "A1$".":.:. 12930..HH2.9000.>H.2."
WHEN YOU ARE LEFT WITH A DECIMAL IN":..M
H.2." THE NUMERATOR, .BE CAREFUL TO KEE
P ":...H.2." TRACK OF THE DECIMAL POINT
.. FOR EX-":..MH.2." AMPLE: 0.063 DIVID
ED BY O.7 = 0.63/7 ="..H.3." 0.09"..H
.3." CCCCC"..I.3." 7)0.63 NOW YOU TR
Y THIS ONE: 0.1164":..PI.3.198,0:. " DI
VIDED BY 0.12 = .RRRR";B1$:."..".PI.3.B1
$."0.97".B1$.".97".13100..I.3." NO .0.9
7. IS CORRECT. 0.1164/0.12 =":..SI"3."
11.64/12 = .0.97. REMEMBER "A1$",":.:.12
930..I,3.9000..J@3." WELL "A1$", YOU SE
EM TO HAVE A":..:JJ3." GOOD GRASP OF TH
IS. LETS CHECK IT OUT..":..VJ.6." ":.".1
PERCENTAGES.".MJ.6.". ............"..J
.6." .A FRACTION WITH A DENOMINATOR OF
100":..PJ.6." IS A PERCENTAGE.. SO A PE
RCENTAGE IS A":...KD6.198,0:. " FRACTIO
N WITH DENOMINATOR = ..RRR";B1$:.".."."K
I6.B1$."100".14040.TKN6.".":." NO "A1$"
LETS START AGAIN:":.:.14000.^KX6.9000..
K.6.198,0:. " SO 50%=50/100 & 75%= .RR/
RR";B1$:.".."..K.6.B1$."75/100".14080..K
.6.B1$."15/20".." .VERY GOOD., 75% = 75
/100 = 15/20":." ":.14090./L.6.B1$."3/4"
.." .VERY GOOD., 75% = 75/100 = 3/4.":.
:.14090.GL.6." NO, 75%= .75/100.. WE'LL
START AGAIN.":.:.14000..L.7." .THATS R
IGHT., & 75/100 = 15/20 OR 3/4.":...L.7.
63010.UL.7." PERCENTAGES CAN ALSO BE EX
PRESSED AS":...M.7." DECIMALS VERY EASI
LY SINCE PERCENTAGES".4M.7." ARE FRACTI
ONS WITH A DENOMINATOR OF":..DM#7." 100
& .100 IS A POWER OF 10!. SO 24%":...M(
7.198,0:. " =24/100=0.24 & 12%=12/100 =
.RR";B1$:.".."..M27.B1$."0.12".B1$.".12
".14150..M<7." NO, "A1$". YOU HAD BETTE
R GO BACK":...NA7." OVER THE SECTION ON
DECIMALS:":.:.445.$NF7.9030.UNK7." SIM
ILARLY, WE CAN CONVERT FRACTIONS &":...N
P7." DECIMALS TO PERCENTAGES SIMPLY BY
EX-":...NU7." PRESSING THEM AS FRACTION
S WITH DEN-":...NZ7." OMINATOR 100. .FO
R EXAMPLE.: 1/2 = ":...O_7." 50/100 = 5
0% THERE IS NO NEED TO HAVE":..FOD7." A
N INTEGER IN THE DENOMINATOR, FOR EX-".X
OI7." AMPLE: 0.013 = 1.3/100 = 1.3%, 1/
7 =":.:...OQ7." 1 100 100 1
2 1 2".UOS7." . V ... = ... V .
.. = 14. V ...= 14.%"..PU7." 7 100
7 100 7 100 7":.:...PV7.930
0.APX7." NOW ITS YOUR TURN. EXPRESS THE
FOLLOW-".PP.7." ING AS % USING THE FOR
MAT SHOWN FOR":...P.7." YOUR ANSWER. 1/
8 = .RR%?. .12.5.":..NP.7.198,0:. "
2/5 = .RR%";B1$:."."..P.7.B1$.
"40".B1$."40%".14250.$Q 7.".":." NO, .4
0. IS CORRECT. 2/5 = 40/100 = 40%":..YQ.
7." YOU HAD BETTER GO OVER THIS "A1$"."
:.:.14000.CQ.7.9000..Q.7.198,0:. "
0.932 = .RR%";B1$:.".."..Q.7.B1$.
"93.2".B1$."93.2%".14280..Q.7." NO, .93
.2. IS CORRECT. 0.932 = 93.2/100":.:.142
45..QH7.9000..RR7." OFTEN WE ARE ASKED
TO FIND SOME % .OF.":..^R.7." SOMETHING
. THIS IS FOUND AS FOLLOWS:":...R.7." .
EXAMPLE:. FIND 30% OF 600":...R.7."
30".VR.7." .SOLUTI
ON:. 30% OF 600 = ... V 600 ="..R.7."
100":...S.7." 3/1
0 V 600 = 3 V 60 = 180. SO .TO FIND":..`
S.7." X% OF Y WE MULTIPLY Y BY .X/100..
. NOW":...S.8." YOU TELL ME, TO FIND X%
OF Y WE MULT-":...S.8.198,0:. " IPLY Y
BY ..R/RR";B1$:."...".US.8.B1$."X/100".
14370..T.8." N0, X% OF Y = .X/100. V Y.
LETS GO BACK:":.14290.>T"8." .THATS RI
GHT.. NOW THERE IS ONE MORE":..IT'8.6301
0.ZT,8." TYPE OF QUESTION WE SHOULD DEA
L WITH,":...T18." THAT IS: .FINDING THE
AMOUNT OBTAINED":...T68." BY INCREASIN
G OR DECREASING SOMETHING":...U;8." BY
X%. .TO INCREASE P BY X% WE TAKE":..<U@8
." P + X% OF P = P + X/100 V P.. FOR EX
-":..LUJ8." AMPLE: .DAD, INCREASE MY $1
0.00 ALL-":...UT8." OWANCE BY 20%. THE
NEW ALLOWANCE WOULD":...UX8."
20"..UY8." BE $10.00 + ... V $10.00
= $10.00 +"..UZ8." 100":..
+VC8." $2.00 = $12.00. NOW YOU TRY ONE:
":..AVH8.198,0:. " 16 INCREASED BY 10%
= .RRR";B1$:."..".WVR8.B1$."17.6".14470.
.V|8." NO,.17.6. IS CORRECT. LETS REVIE
W:":.:.14000..V.8.9030..V.8." .TO DECRE
ASE P BY X% WE TAKE P - X% OF":...W.8."
P = P - X/100 V P.. FOR EXAMPLE: .SON,"
:..JW.8." I'LL DECREASE YOUR $10.00 ALL
OWANCE BY".WW.8." 20%.. THE NEW ALLOWAN
CE WOULD BE:":...W.8." 20"..W
.8." $10.00 - ... V $10.00 = $10.00 -$2
.00":.UW.8." 100":...X.8." =
$8.00. NOW ITS YOUR TURN "A1$".":..8X.8.
198,0:. " 16 DECREASED BY 10% = .RRR";B
1$:."..".ZXB8.B1$."14.4".B1$."14.40".145
50..XL8." NO, 16 - 10/100 V 16 = 16 - 1
.6 = .14.4."..XQ8." WE'D BETTER REVIEW
"A1$".":.:.14000.CXV8. 9000..X.8.".2 REP
EATING DECIMALS."..X.8.". .............
......".2Y.8." FRACTIONS ( .RATIONAL NU
MBERS. ) CAN BE":..`Y.8." EXPRESSED AS
.TERMINATING. OR .RE-":...Y.9." PEATING
. DECIMALS. SO TERMINATING OR":..BY.9."
REPEATING DECIMALS REPRESENT .RRRRRRRR"
:...Y.9.198,0:. " .RRRRRRR";B1$:.".."..
Z&9.B1$."RATIONAL NUMBERS".B1$."RATIONAL
NUMBER".14660.9Z(9.B1$."FRACTIONS".1466
7.IZ+9." NO, THE CORRECT ANSWER IS .RAT
IONAL.":...Z09." .NUMBERS., .IF YOU ARE
CONFUSED AS TO":..IZ:9." RATIONAL NUMB
ERS YOU SHOULD REVIEW ":...Z?9." PART O
NE!. OK, LETS START AGAIN.":.:.14570..{D
9. 9000..{F9.14670.J{K9." .RIGHT., FRAC
TIONS OR .RATIONAL NUMBERS..":." ".Z{L9.
L.1.2000:...{N9." WE HAVE ALREADY COVER
ED CONVERTING":...{X9." TERMINATING DEC
IMALS TO FRACTIONS, FOR"..{B9." EXAMPLE
0.2 = 2/10 =1/5, SO WE WILL ":...|L9."
NOW LOOK AT CONVERTING .REPEATING. DEC-
":..K|V9." IMALS. LETS LOOK AT A FAMILI
AR EXAMPLE".U|.9.9300..|.9." YOU PROBAB
LY REMEMBER THAT 1/3 = ":...|.9." 0.333
33333(REPEATED) WHICH IS WRITTEN":...|.9
." AS 0.3..... TO CONVERT THIS TO A FRA
CTION":...}.9." LET X=0.3...., THEN 10X
= 3.3.... = 3 + X":..-}.9.L.1.6000:..U}
.9." SO 10X=3+X THEREFORE 10X-X=3":...}
9." (SUBTRACTING X FROM BOTH SIDES)":.
..}.9.L.1.3000:...}.9." THUS 9X = 3 SO
X = 3/9 = 1/3":.:..J}.9.L.1.3000:...}.9.
.2)"SO 0.3.... = 1/3":.:.:...}.9.9300.$~
.9." THE GENERAL PRINCIPLE HERE WILL BE
":..U~.9." COVERED LATER IN THE COURSE
. FOR NOW:":...~.9." USE THE FORMULA .1
0.P.X=R+R.... WHERE P IS":...~.9." THE
NUMBER OF DECIMAL PLACES (PERIOD":...~A9
." LENGTH) IN R.. THEN 10.P.X - X = R."
:.:.L.1.7000:...F9." .EXAMPLE:. EXPRES
S 0.21...... AS A FRACTION.":..JK9." .
SOLUTION:. LET X = 0.21...... THEN ":.:.
L.1.4000:...P9." 100X=21.21...... (.P=
2 & 10.2.=100.) SO ":...Z9." 100X=21+X
, THUS 99X=21 & X=21/99=7/33":.:.L.1.400
0:.....9." NOW, .USING PENCIL AND PAPER
., YOU TRY":..#..9." ONE "A1$":":..D..9
.198,0:. " EXPRESS 0.12...... AS A FRAC
TION .R/R";B1$:.".."....9.B1$."12/99".."
CORRECT BUT REDUCE FURTHER":.:.14830..
..9.B1$."4/33".14860....9." NO NOT "B1$
" .4/33. IS CORRECT":.....9." LETS GO O
VER THIS AGAIN "A1$":":.:.14570....:.900
0.4..:.".3 APPROXIMATIONS:.".M..:." ...
............".}. :." IN REAL LIFE WE DO
NOT ALWAYS END UP":....*:." WITH EXACT
ANSWERS, "A1$".":..S.4:." OFTEN WE MUS
T .ROUND OFF. NUMBERS TO":....>:." THE
APPROPRIATE NUMBER OF DECIMAL ":..1.H:."
PLACES OR SIGNIFICANT FIQURES. TO DO":
..].R:." THIS, "A1$" IF THE NEXT DIGIT
IS":....\:." >..D. (GREATER THAN OR EQU
AL TO) 5 WE":....F:." 'ROUND UP'. THAT
IS, 2.35 TO ONE DEC-":....P:." IMAL PLA
CE BECOMES 2.4 AND 2.75 TO ONE".$.Z:.198
,0:. " DECIMAL PLACE BECOMES .RR";B1$:.
"..".:..:. B1$."2.8".15020.Q..:. B1$."2.
80".15025....:." NO, NOT "B1$". .2.8. I
S CORRECT. READ":.....:." THIS AGAIN CA
REFULLY "A1$".":.....:.L.1.2500:.:.14880
.J..:.9000.U..:.15040....:." NO .2.8. I
S CORRECT. 2.80 IS TO 2 DEC-":..:..:."
IMAL PLACES, NOT ONE! TRY AGAIN:":.:.150
15.J..:." IF THE NEXT DIGIT IS < (LESS
THAN) 5":....J:." WE 'ROUND DOWN'. THAT
IS 2.74 TO ONE":..K.T:." DECIMAL PLACE
BECOMES 2.7 AND 2.53 TO":.....:.198,0:.
" ONE DECIMAL PLACE BECOMES .RR";B1$:.
".."....:.B1$."2.5".15120./..:.B1$."2.50
".15125.`..:." NO, NOT "B1$". .2.5. IS
CORRECT. READ":.....;." THIS AGAIN CARE
FULLY "A1$".":.....;.L.1.2500:.:.14880..
..;.9000....;.15140....;." NO, .2.5. IS
CORRECT. 2.50 IS TO 2 DEC-":.....;." I
MAL PLACES, NOT ONE! LETS GO BACK.":.:.1
4880.O.$;." NOW ..00627 TO 2 DECIMAL PL
ACES IS .01.":.....;." .BUT .00627 TO 2
SIGNIFICANT FIQURES IS"...3;." .0063.
NOW ITS YOUR TURN, .01467 TO 2":....8;.1
98,0:. " .DECIMAL PLACES. IS .RRR";B1$:
."."...=;." WHILE .01467 TO 2 .SIGNIFIC
ANT FIQURES.":..I.B;.198,0:. "
IS .RRR";B2$:.".."...L;.B1$.".01"
.B1$."0.01".B2$.".015".B2$."0.015"..1522
0...V;." NO, .01467 TO 2 DECIMAL PLACES
IS ..01.":....[;." .01467 TO 2 SIGNIFI
CANT FIQURES IS ":....`;." ..015. WE'LL
GO BACK A BIT "A1$:.:.14870...T;.9030.N
.~;." IN SCIENCE & TECHNOLOGY VERY LARG
E &":..}..;." VERY SMALL NUMBERS OFTEN
OCCUR. TO ":.....;." DEAL WITH THESE WE
USE .SCIENTIFIC":.....;." .NOTATION. W
HICH PUTS NUMBERS IN THE":.....;." FORM
M V 10.N. WHERE M IS ANY REAL NUM-":..B
..;." BER BETWEEN 1 & 9.9.... INCLUSIVE
& N IS":..S..;." AN INTEGER. FOR EXAMP
LE 20356 IN SCI-":....D;." ENTIFIC NOTA
TION BECOMES 2.0356 V 10.4.":..T.N;." S
O 2.0356 V 10.4. IS EXPRESSED IN:":....X
;.198,0:. " .RRRRRRRR RRRRRRR";B1$:."..
".&..;.B1$."SCIENTIFIC NOTATION".15360.?
..;." NO NOT "B1$".":..S..;." .SCIENTI
FIC NOTATION. IS CORRECT.":.:.15230.~..<
. 9000....<." OK "A1$" SO TO PUT 475 IN
TO":..S..<." SCIENTIFIC NOTATION WE WOU
LD HAVE:":.....<." 4.75 V 10.2. NOW ITS
YOUR TURN. 3129 IN":..6.(<." SCIENTIFI
C NOTATION IS RRRRR V 10.3.":..X.2<.198,
0:. " .RRRRR";B1$:."..".O.<<.B1$."3.129
".15450...A<." NO "A1$" 3129 EXPRESSED
IN SCIEN-":..W.F<." TIFIC NOTATION IS .
3.129. V 10.3. REMEMBER":.15230...Z<.904
0...D<." THE SAME PRINCIPLE HOLDS FOR V
ERY ":..=.N<." SMALL NUMBERS, "A1$". FO
R INSTANCE":..N.X<." 0.001 IN SCIENTIFI
C NOTATION IS 1.0 V":.....<."
1"....<." 10.-3. THAT IS: 1 V
..3 SO 0.0146 BE-"....<."
10":.....<." COMES 1.46 V 10.-2.
AND 0.25 BECOMES":.:..=..<.198,0:. " RR
RRR V 10.-1. .RRRR";B1$:."..".]. <.B1$."
2.5".B1$."2.50".15550....<." NO, 0.25 I
N SCIENTIFIC NOTATION IS:":.....<." .2.
5. V 10.-1., REMEMBER "A1$:.:.15230.E..<
.9000...H<." AND 0.0025 BECOMES 2.5 V 1
0.?. (TYPE":..'.R<.198,0:. " IN THE POW
ER OF 10) .RR";B1$:.".".;..<.B1$."-3".15
610.D..<.".".Y..<." NOT ."B1$". 0.0025
= 2.5 V 10..-3.. LETS":.....<." GO OVE
R THIS AGAIN "A1$".":.....<.L.1.2500:.:.
15230....<.9000....=." WHILE 2673 BECOM
ES 2.673 V 10.?. (TYPE":..$..=.198,0:. "
IN THE POWER OF 10) .RR";B1$:."..".7..
=.B1$."3".15670.G."=." NO, 2673 = 2.673
V 10..3.. READ THIS":....,=." AGAIN CA
REFULLY, "A1$".":.:.15605...6=." .SPOT
ON "A1$".. NOW 1.46 V 10.-2.":..J.;=.630
10...@=." .CORRECT TO 2 DECIMAL PLACES.
= 0.01":..,.J=." WHILE 1.46 V 10.-2. .
CORRECT TO 2 SIG-":..`.T=." NIFICANT FI
QURES. = 0.015, NOW YOU TRY":.:....^=."
2.57 V 10.-2. CORRECT TO .2 DECIMAL ":.
...H=.198,0:. " PLACES. IS .RRR";B1$:."
.."...R=.B1$.".03".B1$."0.03".15760...|=
." NO, 2.57 V 10.-2. = 0.0257 WHICH EQU
ALS":..E..=." .0.03. CORRECT TO .2 DECI
MAL PLACES.":.:.15605.O..=.9000....=."
AND 2.57 V 10.-2. CORRECT TO .2 SIGNIF-"
:.....=.198,0:. " ICANT FIQURES. IS .RR
R";B1$:."..".V..=.B1$.".026".B1$."0.026"
.15820....=." NO, 2.57 V 10.-2. = 0.025
7 WHICH EQUALS":..D.B=." .0.026. CORREC
T TO .2 SIGNIFICANT FIQURES.":.15605.N.L
=.9040...V=." .AND THAT COMPLETES THIS
SECTION!. LETS":.....=." JUST MAKE SURE
"A1$".."....=..T.@.." .THE TOPICS FO
R PART TWO ARE:":.:....J.."1 MULTIPLICAT
ION & DIVISION OF FRACTIONS"...T.."2 DEC
IMALS":..2.^.."3 RELATED TOPICS":..C.H..
" .HAVE YOU ALREADY COMPLETED ANY OF THE
":....R..198,0:." ABOVE? YES OR NO .RRR"
;B1$:."."...|..B1$."NO"..".":.105......B
1$."YES".40100......".":." ANSWER .YES.
OR .NO..":.:.40000......" DO YOU WISH TO
REVIEW TOPIC 1 .RR";B1$:.".".0....B1$."
YES".110.D....B1$."NO".40140.O.B..40080.
|.L.." DO YOU WISH TO START WITH TOPIC 2
?":....V.." .RRR";B1$:.".."......B1$."YE
S".40200......B1$."NO".40220.G....40080.
....." OK "A1$", JUST TO MAKE SURE THAT"
:..(...." YOU KNOW THE MATERIAL WE'LL ST
ART WITH":.:.125.U...." THAT LEAVES RELA
TED TOPICS, BUT TO":....&.." MAKE SURE Y
OU'RE READY WE'LL START WITH":.455..."..
M.1.7...,.. 53280,13...6..T.1.100:..B.@
.. 53280,5.S.J..T.1.100:.:....T.. 53280,
14...^........ T.1.1000:......."."......
L.1.7.&....:.:.:.:.:.:.:.:.:..C.....9)"
.PERFECT SCORE "A1$.R....T.1.500:..O....
".":.:.:.:.:.:.:.:.:.:........9)".PERFEC
T SCORE "A1$......T.1.500:.:."..":.:....
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