Number Sense Tricks:
1. Subtracting Reverses
a. 3 digit or 4 digit (work in pairs): subtract first and last digit/pair, then multiply by 100 and subtract difference
i. 634-436 = 198 (6-4 = 2, 100x2 – 2 = 198)
ii. 1495-9514 = -8019 (95-14=81, 81x100-81 = 8019, make negative)
b. 4 digit number: subtract first and last digit, then multiply by 1000 and subtract difference
i. 1668-8661 = -6993 (8-1=7, 7x1000-7 = 6993, make neg)
2. Adding fractions of form a/b + b/a
a. Find the difference of a and b and square it for the numerator, the product of a and b is denominator; write 2
for the whole number
b. 3/5 + 5/3 = 2 4/15 (5-3 = 2, 22 = 4; 5x3 = 15)
c. 2/7 + 7/2 = 3 11/14 (7-2 = 5, 52 = 25; 7x2 = 14; since 25/14 is improper, make it mixed and add the 2)
3. Multiply by 11
a. Write last digit, add backwards and write digit one at a time until one digit left then write it; if >10, carry 1 and
add, don’t write commas
b. 53 x 11 = 583 (write 3, add 5+3, write 8, write 5)
c. 57 x 11 = 627 (write 7, add 5+7, write 2, carry 1 and add 5, write 6)
d. 143x 11 = 1573 (write 3, add 4+3, write 7, add 1 +4, write 5, write 1)
4. Multiply by 12
a. Writing backwards, multiply 2 x last digit and write down, multiply 2 x first digit and add to original #
b. 12x31 = 372 ( 2x1 = 2; 2x3 = 6; add 6 to 31 = 37)
c. 12x47 = 564 (2x7 = 14, 2x4 = 8, so add carried 1 +8 + 47 = 56)
5. Multiply by 25
a. Divide number by 4 and * 100
b. 25 x 12 = 300 (12/4 = 3, 3*100)
c. 25 x 28 = 700
d. 25 x 37 = 925 (37/4 = 9 ¼ * 100)
6. Multiply by 75
a. Divide number by 4, multiply by 3, then *100
b. 75 x 36 = 2700 (36/4 = 9, 9 * 3 = 27)
7. Multiply by 50
a. Divide number by 2 and x100, when the remainder is 1, write down 50 instead of x100
b. 50x26 = 1300 ( 26 /2 = 13)
c. 50x93 = 4650 (93/2 = 46 R 1)
8. Multiply by 101
a. Repeat the number
b. 27 x 101 = 2727
9. Multiply by 143
a. Divide number by 7 then use that digit for 1st and last, fill in 2 zeros
b. 143x35 = 5005
(35/7 = 5)
c. 143x63 = 9009 ( 63/7=9)
10. Multiply numbers close to 100
a. Take product of differences from 100 as last 2 digits (if 3 digits, carry extra to next step); take difference of
original # and other number’s difference from 100 as 1st digits if #s are less than 100, add if #s are more than
100; doesn’t work if 1 above 100 and 1 below 100
b. 97x95 = 9215 (97 is 3 away from 100 & 95 is 5 away; 3x5 = 15, 97-5 (or 95 – 3)= 92)
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c. 89 x 98 = 8722 (11 and 2 units away from 100; 11x2 = 22, 89 – 2 = 87)
d. 85x84 = 7140 (15 x16 = 240, write 40, carry 2; 85 -16 = 69 + 2 = 71)
e. 108x103 = 11124 (8x3 = 24, 108+3 = 111)
Multiply using double & half
a. Works for any, but good for number ending in 5 and an even number
b. Double 5 number and half the even number and multiply
c. 15 x 12 = 180 (15 x 2 = 30, 12/2 = 6, 30 x 6 =180)
d. 35x18 = 630 (70 x 9)
e. 4 ½ x 18 = 9 x 9 = 81
Multiplying numbers ending in 5
a. Squaring: 25 is last two digits, add 1 to first digit(s) and multiply to first digit(s)
i. 652 = 4225 (write 25, 7 x 6 = 42)
ii. 1452 = 21025 (write 25, 15 x 14 = 30 x 7 (double & half rule) = 210)
b. 10s digit both even: write 25, add 10s digits and divide by 2, add to product
i. 45 x 85 = 3825 (4 + 8 = 12, 12/2 = 6, 6 + 32 = 38)
c. 10s digits even and odd: write 75, same process as above
i. 35 x 85 = 2975 (3 + 8 = 11; 11/2 = 5.5 – just take whole number, 5 + 24 = 29)
Squaring 2 digit number
a. Writing backwards, write the square of the last digit, the write double the product of 2 digits, then carry the
tens digit from that product and add to the square of the first digit
b. 312 = 961 (12 = 1, 3x1x2 = 6, 32 = 9)
c. 432 = 1849 (32 = 9, 4x3x2 = 24, write 4, carry 2, 2+42 = 18)
Squaring a 3 digit number
a. Same rule as above but treat 1st 2 digits as the tens digit
b. 1232 = 15129 (32 = 9, 12x3x2 = 72, write 2, carry 7, 7 + 122 = 151)
Multiply two numbers using FOIL
a. Writing backwards, multiply last digits L, multiply “rainbow OI” and add, carry tens digit and add to product of
first F
b. 31x45 = 1395 (1x5 = 5, 3x5 = 15 and 1 x 4 = 4, 15 + 4 =19, write 9, carry 1, 1 + 3x4 = 13)
c. 42x23 = 966 (2x3 = 6, 2x2 = 4 and 4x3 = 12, 12 + 4 = 16, write 6, 1 + 4x2 = 9)
d. 102x91 = 9282 (2x1 = 2, 10x1 + 2x 9 = 28, write 8, 2 + 10x 9 = 92)
Multiplying numbers equidistant from some number
a. Use difference of squares – square the common number and subtract the square of the difference; (x + y)(x – y)
= x2 – y 2
b. 51 x 49 = (50 + 1)(50 – 1) = 502 – 12 = 2499
c. 17x13 = (15+2)(15-2) = 152 – 22 = 225 – 4=221
d. Use this in reverse as well
e. 732 – 272 = (73 + 27) x (73 – 27) = 100 x 46 = 4600
Reverse distribution
a. 5 x 17 + 17 x 12 = 17 x 17 = 289 (know your squares)
Multiply mixed numbers
a. Equal fractions
i. add whole numbers and multiply by fraction; multiply whole numbers and add result, tack on product of
fractions
ii. 2 1/3 x 4 1/3 = 10 1/9 (2 + 4 = 6, 6/3 = 2; 4 x 2 = 8, 8 + 2 = 10)
iii. 12 3/8 x 4 3/8 = 54 9/64 (12 + 4 = 16, 16x 3/8 = 6; 12x4 = 48, 48 + 6 = 54)
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b. Equal whole numbers and fractions add to 1
i. Multiply whole number x whole number + 1; tack on product of fractions
ii. 5 1/3 x 5 2/3 = 30 2/9 (5 x 6 = 30)
Double double to divide
a. Double both numbers then divide
b. 8 / 2.5 = 16 /5
Divisibility Rules
a. Divisible by 2 if even
b. Divisible by 3 if sum of digits is div by 3
c. Divisible by 4 if last 2 digits is div by 4
d. Divisible by 5 if ends in 0 or 5
e. Divisible by 6 if even and sum of digits div by 3
f. Divisible by 8 if last 3 digits div by 8
g. Divisible by 9 if sum of digits div by 9
h. Divisible by 12 if divisible by 3 and 4
i. These are good to use for reducing fractions
i. 51/87 = 17/29 (divide by 3 since both are div by 3: 5+1 =6 and 8+7 = 15)
Remainders dividing by 9 (or 3)
a. Add up digits; when reach 9 start over, remainder is what’s left
b. 412/9 has remainder of 7 (4+1+2)
c. 4753/9 has remainder of 1 (4 + 7 = 11, cast out 9, 2 + 5 = 7, 7 + 3 = 10, 10-9= 1)
d. 643152/9 has remainder of 3
Remainders dividing by 4
a. Use the last 2 digits
b. 8743 /4 has reminder of 3since 43 has a remainder of 3
Remainders dividing by 11
a. Add every other digit starting with ones, then subtract every other digit starting with tens
b. 564722 has remainder of 4 (2+7+6 – 2 – 4 – 5 = 4)
Remainders of x2/n
a. Find remainder of x/n, then square it and find remainder of that
b. 732/7 has remainder of 2 (73/7 is R3, 32 = 9, 9/7 has R2)
Dividing by 9 to Mixed Number
a. Writing backwards, add up digits to write over 9 for fraction, reduce if necessary; add 1st 2 digits and write
down (carrying any whole number from fraction first), write 1st digit digit
b. 215/9 = 23 and 8/9 (2+1+5 = 8 for numerator, 2+1 = 3, write 3, first digit is 2)
c. 426/9 = 47 and 1/3 (4+2+6 = 12, 12/9 = 1 and 1/3, carry 1; 4+2 = 6 + 1 = 7)
Dividing by decimals
a. Memorize 1/8 ths as decimals or percents
i. 1/8 = 12.5%; 3/8 = 37.5%; 5/8 = 62.5%; 7/8 = 87.5%
b. 12/.375 = 32 (12 divided by 3/8 or 12 x 8/3 = 4 x 8 )
c. 875 x 888 = 7/8 x 1000 x 888 = 1000 x 777 = 777,000
Approximations *
a. Acceptable range is 5% on either side of correct answer
b. 124 x 162 = think of 125 which is 1/8 x 160 = 20, tack on zeros to get 20000
28. Averages
a. Pick a middle # and add or subtract difference from there; find mean of differences and add to middle #
b. 84,92,89,100,90,79,97,97: Pick 90, so 6 +2-1+10+0-11+7=8 8/8#s = 1; 90 + 1 = 91
29. Comparing Fractions
a. Use cross products
b. Which is larger: 5/13 or 7/18? 5 x 18 = 90; 7 x 13 = 91; 91 is larger so 7/18 is larger
30. Roman Numerals
a. I = 1; V = 5; X = 10; L = 50; C = 100; D = 500; M = 1000
31. GCF or GCD (Greatest Common Factor/Divisor)
a. List factors of smaller number largest to smallest, stop when you get a factor of larger #
b. Double smallest number so bigger than other #, then subtract, if not GCF divide by prime factor or double and
subtract again
c. 12 & 20 = 4 (12x2 -20 = 4)
d. 20 & 72 = 4 (80 -72 = 8/2 = 4)
e. 126 & 198 = 18 (198-126=72, double 72,144-126 = 18)
32. LCM (Least Common Multiple)
a. List multiples of larger number and stop when you get a multiple of smaller #
b. LCM = axb/GCF
c. 21 & 35 = 105 (21x35/7 = 3x35 = 165)
33. LCM & GCD
a. The product of GCD and LCM of 2 numbers a and b is axb
b. 12 & 20 = 240 (12 x 20; GCF = 4, LCM = 60, 4x60 =240)
34. Percents
a. Divide 1st percent into product of other 2 numbers and cancel out factors and multiply
b. 21% of what is 35% of 18 = 30 (35x18/21 = 5x6 = 30)
c. 24% of what is 30% of 28 = 35 ( 30x28/24 = 5x7 = 35)
35. Percents with less than and 3 numbers or more than
a. Add %s, subtact A% of B (add if “more than”)
b. If A is 20% less than B and B is 40% less than C, then A is what % less than C? 52 (20+40 =60, 20% of 40 is 1/5 of
40 or 8; 60 – 8 = 52)
c. If A is 10% more than B and B is 40% more than C, then A is what % more than C? 54 (10+40 = 50; 10% of 40 is
4; 50 + 4 = 54)
36. Prime numbers less than 100
a. 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97
37. Conversions
a. 1 inch = 2.54 cm; 231 in3 = 1 gallon
b. 5280 ft = 1 mile; 1 square mile = 640 acres
c. 16 oz = 1 pound; 2000 pounds = 1 ton
d. 1 Tbl = 3 tsp; 1 cup = 16 Tbl
e. 8 oz = 1 cup; 2 cups = 1 pint; 2 pints = 1 qt; 4 quarts = 1 gallon = 128 oz
f. Kilo = 1000; hector = 100; deka = 10; deci = .1 ;centi =.01; milli = .001
38. Changing Bases
a. Each unit is power of base
b. 1278 = ?10: 1x82 + 2x81 + 7x80 = 64 + 16 + 7 = 87
c. 3110 = ?5: think in powers of 5: 31 = 25 + 5 + 1 =1115
d. Base 2 to base 8: take 3 digits at a time and use base 10
i. 101110012 = ?8: ( 01)(111)(001) = 2718 ( 012 = 210, 1112 = 710, 0012 = 110)
39. Arithmetic Series – finding missing number
a. Average two numbers surrounding
b. 3,7,11,?,19,23,27 = 15 (11 + 19 and divide by 2)
c. -2.25,-1,x,1.5,2.75,… = .25 ( -1 + 1.5 and divide by 2)
40. Series
a. Add first and last and divide by 2; subtract first and last and add amount of increase, then divide by amount of
increase; multiply results
i. 3 + 5 + 7 + … + 15 = 63 (3 + 15 = 18, 18/2 = 9; 15 – 3 = 12, 12 + 2 = 14, 14/2 = 7; 9 x 7 = 63)
b. Sum of numbers starting with 1 is n(n+1)/2
i. 1+2+3+…+10 = 10(11)/2 = 55
c. Sum of odd numbers starting with 1 is [(n+1)/2]2
i. 1+3+5+…+11 = (12/2)2 = 36
41. Squares & alternating patterns
a. Difference by 1: add numbers; Difference by 2,3, or 4: then multiply by sum 2,3 ,or 4
b. 232 – 222 + 212 -202 = 86 (23+22+21+20)
c. 362 -342 +322 -302= 264 ( 36+34+32+30= 132, 132x2)
d. If numbers increase before alternating then take opposite sign of sum
e. 512-522+532-542 = -210 (51+52+53+54 = 210, then make neg)
42. Fibonacci (sum for 8 terms is 21a +33b or 3(7a+11b))
a. The sum of the first 8 terms in 1,1,2,3,5,8,..=54 ( 21x1 + 33x1)
b. Sum of first 8 terms in 2,5,7,12,19,… = 207 ( 21x2 + 33x5)
43. Repeating Decimals as Fractions
a. Any repeating # as a fraction goes over 9; for every digit that doesn’t repeat, think of as messed up mixed
number with decimal # in front, convert mixed number to fraction disregarding decimal, multiply denom by 10
b. .666.. = 6/9 = 2/3
c. .272727… = 27/99 = 3/11
d. .1333… = 2/15 (.1 and 3/9 = .1 and 1/3; 3 *1 = 4; 4/30, reduce to 2/15)
e. .3444… = (.3 and 4/9; 9x3 + 4 = 31; 31/90)
f. .185185185.. = 185/999 = 5/27 (need to memorize that 999=27x37)
44. Sets & Elements
a. Multiply elements in each set for total number of elements
b. How many elements does the product of {1,2,3} and {a,b} have? 2x3 = 6
45. Power Set
a. A Power set’s elements are all possible subsets including null, number of elements or subsets = 2n
b. The power set of {a,b,c} contains x elements. = 8 (23 = 8; { } {a} {b} {c} {ab} {ac} {bc} {abc})
c. Proper subsets have 1 less
46. Polynomials
a. Sum of coef = Add coef and raise to power
b. The sum of the coefficients of (6x + 4y)2 is: 100 ( (6+4)2 = 100, 36 + 16 + 48 = 100)
c. The sum of the coefficients of (10a + 5b)3 is: 3375 (153 = 3375)
47. Exponents – powers of 2 and 5
a. Use exponent rules
b. 27 x 54 = 23 x 24 x 54 = 23 x 104 = 80000
48. Geometry
a. Number of diagonals in a polygon is n(n-3)/2
i. Pentagon = 5 ( 5x2/2)
ii. Undecagaon = 44 (11x8/2)
b. Triangle inequality: subtract and add to find possible range of third side exclusive
i. An obtuse triangle has integer sides 5, x, and 9. The smallest value of x is? 5 (4 < x < 14)
ii. A triangle has integer sides 8, x, and 17. The largest value of x is? 24 (9 < x < 25)
c. Pythagorean triples
i. 3,4,5 (and multiples); 5,12,13; 7,24,25