NUMBER SENSE TRICKS
Some Digits repeating Decimal >>> Fraction (Example: .1242424242424...)
# of repeating digits = # of 9s in denominator. # of non-repeating digits = # of 0s in the denominator, all after 9s. Subtract non-repeating digits from decimal to get numerator. (Example: 124-1 = 123. 123/990)
All Digits Repeating Decimal >>> Fraction (Example: .1313131313...)
# of repeating digits = # of 9s in denominator. One set of repeating digits is numerator. (Example: 13/99)
Multiplying #s just below 100
1) A = 100 minus first number and B = 100 minus second number 2) subtract A from the second number (or vice versa) 3) multiply A and B (Example: 92 x 97 --> A = 8 & B = 3 --> 97 - 8 (or 92 - 3) = 89 --> 8 x 3 = 24 --> 92 x 97 = 8924)
Multiplying #s just above 100
1) A = first number minus 100 and B = second number minus 100 2) add A to the second number (or vice versa) 3) multiply A and B (Example: 112 x 105 --> A = 12 & B = 5 --> 12 + 105 (or 5 + 112) = 117 --> 12 x 5 = 60 --> 11760)
Find the remainder when dividing by 11
1) Beginning with the units digit, add every other digit from left to right. 2) Find the sum of the remaining digits 3) Subtract Step #2 from Step #1 4) If its less than 11, that's the answer. If more, subtract 11 until its less. If its negative, add 11 until its less than 11 and positive. If it is 11, the remainder is 0. (Example: 2537/11 = (7+5) - (3+2) = 7)
# ending in 5 times even # fast method
1) Double the number ending in 5 2) Halve the even number 3) multiply the resulting values (Example: 15*12 = 15*2 + 12/2 = 30*6 = 180. LARGE NUMBERS ONLY)
Multiplying #s ending in 5, first digits differ by an even #
1) Ends in 25, 2) Add first digits and divide by 2, 3) multiply digits and add quotient from step 2. (Example: 65*65 = 6*6 + (6+6/2) = 42, 4225)
Multiplying by 12.5 and 16&2/3 WORKS FOR 125 TOO, JUST SHIFT THE DECIMAL POINT ONE TO THE RIGHT.
1) Mult by 12.5: Divide the other # by 8, add 2 "0"s to the end. (Example: 32*12.5 = 32/8 = 4>>>400) On 125, add 3 "0"s. 2)Mult by 16&2/3: Divide the other # by 6, add 2 "0"s to the end. (Example: 42*16&2/3= 42/6 = 7>>>700)
Least Common Multiple
1) Multiply the #s together 2) Divide by the GCF (Example: LCM(12,20) =
2 digit multiplications
1) Units digit = product of units digit. 2) Tens digit = product of inside and outside digits + carries 3) Hundreds digit = product of tens digits + carries (Example: 23*12 = Units: 2*3 = 6. Tens: 2*2+3*1 = 7. Hundreds: 2*1 = 2. = 276.)
Division by 9 (xyz/9, xyz is one 3-digit # but it works with bigger ones)
1) add x plus y plus z and put sum over 9 (be sure to reduce) 2) add x plus y plus carry 3) bring down x plus carry (Example: 201/9 = (2+C) & (2+0+C) & (2+0+1)/9 = 22 and 3/9 = 22 and 1/3)
Multiplying #s ending in 5, first digits differ by an odd #
1) always ends in 75 2) add first digits and divide by 2 3) multiply first digits and add integer part of quotient (Example: 85(55) = (8+5)/2>>>6.5>>>6+8(5)=4675)
Multiplying by 11
1) bring down units digit 2) add two digits at a time 3) bring down first digit plus any carry (Example: 72*11 = (7+carry) & (7+2) & (2) = 7&9&2 = 792
Finding smaller fraction
1) cross multiply 2) resulting #s tied to numerators. lower result = lower fraction. (Example: 8/11 or 10/13: 8*13 = 104. 10*11 = 110. 8/11 is smaller
Multiplying by 25 or 75
1) divide by 4 2) last two digits 00, 25, 50, or 75 depends on the remainder 3) multiply results by 3 (75 only, NOT ON 25) (Example: 57*25 = 57/4 = 14 R 1 & add remainder digits = 1425. 57*75 = ^^^, so 1425*3 = 4275
Multiplying by 3367 rule
1) divide the other # by 3 2) multiply the quotient by 10101 Example: 18*3367 = 18/3 = 6*10101 = 60606
3 digit # times 101
1) last 2 digits are last 2 digits of the 3 digit # 2) First 3 #s are the 3 digit # plus the hundreds digit. (Example: 934*101: 34 (last 2 digits), 934+9 = 943 (first 3 digits), so 94334)
Dividing by 25
1) multiply by 4 2) place decimal so the answer has 2 decimal places (Example: 64/25 = 64*4 = 256 & place decimal = 2.56)
Multiplying by numbers when units digits total 10
1) multiply first digit times first digit plus 1 2) multiply units digits (Example: 43*47 = 4*(4+1) & 3*7 = 4*5 & 3*7=2021)
Multiplying by numbers when first digits total 10
1) multiply first digits and add the units digit 2) square the units digit (Example: 27*87=2x8+7 & 7*7 = 16+7 & 49 = 2349)
Multiplying by teens
1) multiply units digit of the teen times units digit 2) multiply units digit of the teen times other digits and add back plus carry 3) bring down first digit plus any carry (Example: 72*13 = (7+C) & (3x7+2) & (3x2) = 7&23&6 =(7+2)&3&6 =936)
Prime Numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 87, 89, 97
Divisibility by 2-10 (not 7 though)
2: last digit divisible by 2 3: sum of digits is divisible by 3 4: Last 2 digits divisible by 4 (as one #. So 16 is divisible by 4 and meets, not 1 & 6 individually) 5: Ends in 0 or 5 6: Even and sum of digits is divisible by 3 8: last 3 digits divisible by 8 9: sum of digits divisible by 9 10: ends in 0
1 Bushel = ? pecks
4 pecks per bushel
Set {w,x,y,z...} has ___ subsets
An n-element set has 2^n subsets (Example: Set {M,A,T,H}>>>2^4 = 16
Area of a square with diagonal x
Area of a square = 1/2(diagonal)^2 (Example: Diagonal = 8. 1/2(8^2)= 1/2(64) = 32
Product of GCF and LCM
Equals the product of the 2 #s. (Product of GCF and LCM of 24 and 30 = 24*30 = 720
Multiplying by difference of squares
Example: 53*47 = (50+3)*(50-3)=50^2 + 3^2 = 2500 - 9 = 2491
a% of b is c% of ___
If a% of b is c% of d, then d = ab/c (Example: 72% of 36 is 18% of ___. 72*36/18 = 144)
Roman Numerals (Matt u need to know these)
M = 1000, D = 500, C = 100, L = 50, X = 10, V = 5, I = 1
a*a/b=
Mixed # answer. 1) Fraction = (b-a)^2/b 2) Whole # = a-(b-a)
Miles/hour >>> feet/sec
Multiply by 22/15 (Example: 3omph = 30(22/15) = 44ft/sec
Cartesian Product
The Cartesian product of two sets is a set of ordered pairs where the left term comes from the First set given and the right term comes from the second set given. See #89 here: http://www.rammaterials.com.php53-9.ord1-1.websitetestlink.com/wp-content/uploads/2012/09/psia_number_sense.pdf
Intersection of 2 sets (upside down U)
The intersection of two given sets, is a set whose elements are contained in both sets (Example: A = {2,8,0,5,3} B = {4,3,7,9,5}. Intersection = {5,3}
5-10+15-20...
The sequence consists of multiples with alternating signs. If the last term is negative the answer is equal to the last digit divided by 2. (Example: 5-10+15...-80 = -80/2 = -40
2 digit # times 101
simply write the # twice. (Example: 93*101 = 9393)
1+3+5+...+k
sum = ((k+1)/2)^2 (Example: 1+3+5+...+19: ((19+1)/2)^2 = 100
1+2+3+...+k
sum = n(n+1)/2 (Example: 1+2+...+19: 19(19+1)/2 = 190
a/b + b/a = x
x = 2 + (a-b)^2/ab
LCM(x,y) =, when x < y
x/GCF(x,y) * y
