UIL Number Sense: Shortcuts
SHORTCUT #32: adding fractions
(N+1)/N + N/(N+1) = 2+(1/N(N+1)) (N+2)/N + N/(N+2) = 2+(4/N(N+2)) (N+3)/N + N/(N+3) = 2+(9/N(N+3)) ex: 13/10 + 10/13 = 2 9/130
SHORTCUT #33: subtracting fractions
(N+1)/N - N/(N+1) = (1*(N+(N+1)))/(N*(N+1)) (N+2)/N - N/(N+2) = (2*(N+(N+2)))/(N*(N+2)) (N+3)/N - N/(N+3) = (3*(N+(N+3)))/(N*(N+3)) ex: 13/10 - 10/13 = 69/130
SHORTCUT #49: x+y=# xy=# Find x³+y³.
(x+y)*((x+y)²-3xy) ~signs: +, +, -~ ex: x+y=1/3; xy=3; find x³+y³ (1/3)*((1/9)-9)=(1/3)(-80/9)=-80/27
SHORTCUT #48: x-y=# xy=# Find x³-y³.
(x-y)*((x-y)²+3xy) ~signs: -, -, +~ ex: x-y=½; xy=2; find x³-y³ ½(¼+6)=½(25/4)=25/8
SHORTCUT #22: remainder test for 11
1. beginning with the ones place, add every other digit 2. add the remaining digits and subtract from step 1 (or starting with the ones digit and moving left: subtract, add, subtract, add, etc.) ex: 94635/11 5+6+9=20 3+4=7 20-7=13-11=2 ~or~ 5-3=2+6=8-4=4+9=13 13-11=2
SHORTCUT #12: finding the number of integers between two numbers that are divisible by an integer
1. divide 4 into each number (omit the remainder) 2. subtract ex: How many numbers between 6 and 70 are divisible by 4? 70/4=17; 6/4=1 17-1=16 ~or~ subtract the two numbers and then divide by 4?
SHORTCUT #11: multiplying a number by 37
1. divide by 3 2. multiply by 111
SHORTCUT #15: multiplying a number by 75
1. divide by 4 2. multiply by 3 3. multiply by 100 (move decimal 2 places) (¾ of the number * 100)
SHORTCUT #9: multiplying by 125
1. divide by 8 2. multiply by 1000 (move decimal 3 places)
SHORTCUT #18: multiplying by 12
1. double the ones digit (carry if needed) 2. double the tens digit and add to the ones digit (carry if needed) 3. double the hundreds digit and add to the tens digit ex: 57*12 7*2=14 (write down 4, carry 1) 4 5*2+7=17+1=18 (write down 8, carry 1) 84 0*2+5=5+1=6 684
SHORTCUT # 26: multiplying a 100s number (near 100) by a 90s number (near 100)
1. find the difference between each number and 100 and multiply the results 2. subtract this from 100 (write down using two places) 3. add one to the difference between the 90s number and 100 4. subtract this from the 100s number (write down) ex: 103*98 103-100=3; 100-98=2; 2*3=6 100-6=94 (write down) 100-98=2+1=3 103-3=100 (write down) 10094
SHORTCUT #34: finding the remainder of two multiplied fractions with the same denominator
1. find the remainder of each fractions (usually given) 2. multiply the remainders 3. divide this product by the denominator ex: If A÷7 has a remainder of 6 and B÷7 has a remainder of 4, then AB÷7 has a remainder of what? 6*4=24 24/7=3 with a remainder of 3
SHORTCUT #31: multiplying a power of 2 by a power of 5
1. make the larger exponent into the same as the smaller by breaking it up 2. multiply those bases, then evaluate with the exponent 3. multiply by the other broken up term ex: 2⁶*5³=2³*2³*5³=2³*10³=8*1000=8000
SHORTCUT #10: dividing a number by 125
1. multiply by 8 2. divide by 1000 (move decimal 3 places)
SHORTCUT #35: multiplying mixed numbers (same whole number, fractions add up to one)
1. multiply the fractions 2. add 1 to the whole number and multiply by the original whole number ex: 7 1/3 * 7 2/3 = 56 2/9
SHORTCUT #8: squaring 50-59
1. multiply the ones (write down using 2 places) 2. add the ones digit to 25 (write down) ex: 53²=2809
SHORTCUT #6: multiplying numbers close to but greater than 100
1. multiply the ones digits (write down using 2 places) 2. add the ones digits (write down using 2 places) 3. put a one in front ex: 103*104=10712
SHORTCUT #24: multiplying two numbers that are equal distance from a number ending in 5
1. multiply the ones digits (write down using two places) 2. add one to the tens digit and multiply by the original tens digit) ex: 62*68=4216
SHORTCUT #54: finding GCD (one way)
1. multiply the smaller number by any constant to get it close to the larger number 2. take the difference and see what factors of this will divide into both original numbers
SHORTCUT #1: multiplying two numbers that differ by the same amount from a common number
1. square the average 2. square the difference 3. subtract ex: 54*46 50²-4²=2500-16=2484
SHORTCUT #27: multiplying two two digit numbers with the same units digit and the sum of the tens digit is 10
1. square the units digit (write down using two places) 2. multiply the tens digits and add the units digit (write down) ex: 47*67 7*7=49 (write down) 4*6+7=31 (write down) 3149
multiplying numbers close to but less than 100
1. subtract each number from 100 2. multiply those numbers together (write down) 3. subtract one of the differences from the other original number (write down) ex: 93*95 differences: 7 and 5 7*5=35 (write down) 93-5 or 95-7 = 88 (write down) 8835
SHORTCUT #47: finding the sum of a fibonacci characterstic sequence of 9 terms
1. take the 7th term and multiply by 7 2. subtract the 4th term ex: 3, 4, 7, 11, 18, ... the 7th term is 47 47*7=329-11=318
SHORTCUT #42: finding the positive integers that are less than a number and relatively prime to it
1. take the prime factorization 2. decrease each factor by 1, if there are factors with an exponent greater than one, keep the leftover (ex: 5¹→4, but 5³→4*5²) 3. multiply ex: 30=2¹*3¹*5¹ 1*2*4=8 ex: 40=2³*5¹ 1*2²*4=16
SHORTCUT #40: finding the sum of positive integral divisors
1. take the prime factorization 2. if the exponent is 1, add one to the factor; if the exponent is greater than 1, use: (n^(e+1)-1)/(n-1) 3. multiply ex: 14=2¹*7¹ 3*8=24 ex: 12=2²*3¹ (2^3-1)/1=7; 3+1=4 4*7=28
SHORTCUT #41: finding the number of positive integral divisors
1. take the prime factorization 2. raise each exponent by 1 3. multiply the exponents ex: 48=2⁴*3¹ 5*2=10 ex: 60=2²*3¹*5¹ 3*2*2=12
SHORTCUT #13: multiplying a proper fraction by its numerator
1. the whole number is the numerator minus the difference between the numerator and denominator 2. the numerator of the fraction is the square of the difference between the numerator and denominator ex: (13/14)*13=12 1/14
SHORTCUT #14: multiplying an improper fraction by its numerator
1. the whole number is the numerator plus the difference between the numerator and denominator 2. the numerator of the fraction is the square of the difference between the numerator and denominator ex: (14/13)*14=15 1/13
SHORTCUT #45: multiplying a whole number times a mixed number in the form a*(a+a/(a+1))
1. the whole number will be a*(a+1)-1 ~or~ numerator*denominator-1 2. the fraction will be 1/denominator (1/(a+1)) ex: 3*3¾ = 11¼
SHORTCUT #30: squaring numbers ending in 1
1. write down 1 2. double the digits before the 1 (write down using one place, carry if needed) 3. square the digits before the 1 (write down) ex: 271² write down 1 27*2=54 (41 with 5 carried) 27²=729+5=734 73441
SHORTCUT #29: squaring numbers ending in 15
1. write down 225 2. multiply the number(s) in front of the 15 by 10 and then add 3 3. multiply that by the number(s) in front of the 15 and write down ex: 315² write down 225 3*10+3=33 33*3=99 99225
SHORTCUT #16: multiplying numbers ending in 5 whose digits preceding the 5 are both odd or both even
1. write down 25 2. take the average of the preceding digits and add that to the product of the preceding digits ex: 115*35=4025
SHORTCUT #17: multiplying numbers ending in 5 whose digits preceding the 5 are one odd and one even
1. write down 75 2. take the average of the preceding digits (ignore the ½) and add that to the product of the preceding digits ex: 85*135=11475
SHORTCUT #19: multiplying 101 by a three digit number
1. write down the tens and units digit 2. add the hundreds digit to the number ex: 546*101 write down 46 546+5=551 55146
SHORTCUT #20: multiplying 111 by a three digit number
1. write down the units digit 2. add the tens and units digits (carry if needed) 3. add the hundreds, tens, and units digits (carry if needed) 4. add the hundreds and tens digits (carry if needed) 5. write down the hundreds digit with any carried values ex: 384*111 write down 4 8+4=12 (24 with 1 carried) 3+8+4=15+1=16 (624 with 1 carried) 3+8=11+1=12 (2624 with 1 carried) 3+1=4 42624
SHORTCUT #21: multiplying 111 by a two digit number
1. write down the units digit 2. add the tens and units digits (carry if needed) 3. add the tens and units digits with any carried values 4. write down the tens digit with any carried values ex: 78*111 write down 8 7+8=15 (58 with 1 carried) 7+8=15+1=16 (658 with 1 carried) 7+1=8 8658
SHORTCUT #37: multiplying 14443 by a multiply of 7 that is ≤63
14443*7=101101 14443*14=202202 14443*21=303303 14443*28=404404 14443*63=909909
SHORTCUT #36: multiplying 3367 by a multiply of 3 that is ≤36
3367*3=10101 3367*6=20202 3367*9=30303 3367*12=40404 3367*36=121212
SHORTCUT #39: multiplying certain mixed numbers by percents
4 1/6 * 24% = 1 (1/24 = 4 1/6%) 6¼ * 16% = 1 ( 1/16 = 6½%) 2½ * 40% = 1 (1/40 = 2½%)
SHORTCUT #50: multiplying a 3 digit number by a 3 digit number
ABC*DEF 1. CF 2. FB+EC 3. FA+DC+EB 4. EA+DB 5. DA
SHORTCUT #38: A+B*C-D where A, B, C, and D are consecutive numbers
AD-1 ~or~ BC-3 (use the easier one) ex: 19+20*21-22 20*21=420-3=417
SHORTCUT #51: finding the distance between a point and a line
Ax+By=C; (x, y) since this is number sense, A and B will be the legs of a pythagorean triple 1. to find the numerator of the answer, plug in x and y into the equation and subtract C 2. the denominator is the hypotenuse of the pythagorean triple 3. take the absolute value ex: find the distance between 3x+4y=-2 and (2, 1) numerator= 3(2)+4(1)+2=12 denominator= 5 (PT- 3, 4, 5) the answer is 12/5
SHORTCUT #44: changing yd/min→ft/sec
divide by 20
SHORTCUT #2: multiplying by 25
divide by 4, multiply by 100
SHORTCUT #43: changing ft/sec→yd/min
multiply by 20
SHORTCUT #3: dividing by 25
multiply by 4, divide by 100
SHORTCUT #25: subtracting two squares
multiply the sum and difference of the two numbers (a²-b²=(a+b)(a-b)) ex: 16²-9²=25*7=175
SHORTCUT #28: multiplying two squares
multiply the two numbers and then square ex: 8²*3²=24²=576
SHORTCUT #52: selection from a group with repetition
n= the number to be taken out of r= the number you take (n+r-1)!/r!(n-1)! Ex: A box contains black, red, blue, and green pens. How many different sets of 3 pens can be packaged? (4+3-1)!/3!(4-1)! = 6!/3!3! = 20
SHORTCUT #56: repeating decimals if the bar doesn't go all the way to the decimal
numerator= entire decimal minus non-repeated digits denominator= a nine for every repeated digit and a zero for every non-repeated digit ~reduce~ ex: 0.1[12]=111/990=37/330
SHORTCUT #55: repeating decimals if the bar goes all the way to the decimal
numerator= repeated digit(s) denominator= a 9 for every repeated digit ~reduce~ ex: 0.[15]=15/99=5/33
SHORTCUT #46: finding the sum of a fibonacci characterstic sequence of 10 terms
take the 7th term and multiply by 11 ex: 4, 5, 9, 14, 23, ... the 7th term is 60 60*11=660
SHORTCUT #53: dividing bases
the remainder obtained by dividing an integer in base b by b-1 equals the remainder obtained by dividing the sum of its digits by b-1 ex: 553₅ 553₅/4→R1 5+5+3=13/4→R1
SHORTCUT #23: multiplying by 13-19
use the sme method as multiplying by 12, but instead of doubling, multiply by the ones digit let x= the ones digit in 13-19 1. multiply the ones digit by x (carry if needed) 2. multiply the tens digit by x and add to the ones digit (carry if needed) 3. multiply the hundreds digit by x and add to the tens digit ex: 546*16 6*6=36 (6 with 3 carried) 4*6+6=30+3=33 (36 with 3 carried) 5*6+4=34+3=37 (736 with 3 carried) 0*6+5=5+3=8 8736
SHORTCUT #4: multiplying 101 by a two digit number
write the number down twice ex: 101*74=7474
SHORTCUT #5: multiplying 1001 by a two digit number
write the number down twice with a zero in between ex: 38*1001=38038
