Number Sense Section 1 Formulas/Tricks

Formulas/tricks given in Section 1 of Bryant Heath's guide to UIL Number Sense. This involves quickly adding, subtracting, multiplying, and dividing numbers in your head.

Multiplying by 11, 111, 1111, etc.

Add the number to itself with the digits shifted to the left one time for 11, twice for 111, etc.

Dividing by 11, 111, 1111, etc.

Ones digit is the same as dividend

Tens digit is tens of dividend minus ones digit

Hundreds is hundreds of dividend minus tens digit of divisor (for 11) For 111, subtract ones digit of divisor from this number

etc.

Multiplying by 101

Same as 11 trick, but you move the number 2 digits to the left

Multiplying two numbers above 100 (can be extended for 1000, 10000, etc., the first part just gives more digits)

x=100+a y=100+b

Tens/Ones is a*b

Hundreds is x+b or y+a

Multiplying two numbers below 100 (can be extended for 1000, 10000, etc., the first part just gives more digits)

x=100-a y=100-b

Tens/Ones is a*b

Hundreds is x-b or y-a

Multiplying two numbers near 100 (can be extended for 1000, 10000, etc., the first part just gives more digits)

x=100+a y=100-b

Tens/Ones is 100-a*b

Hundreds is x-b-1

Squares ending in 5

(a5)²

Tens/Ones is 25

Hundreds is a(a+1)

41-59 Squares

Tens/Ones is the difference from 50 squared

Hundreds is 25 plus(if larger)/minus(if smaller) the difference from 50

Multiplying Two Numbers Equidistant from a Third

Square the middle number and subtract the square of the difference from the middle number and one of the other numbers

Multiplying Two Digit Reverses (ab*ba)

Ones digit is a*b

Tens digit is a²+b²

Hundreds digit is a*b

If a problem involves adding two products together, it is most likely a

factoring problem

Adding consecutive squares

a²+b² (b is one larger than a)

2a²+2a+1

What should you do for problems like (30²−2²)+(30+2)²?

Expand out each expression and cancel

Adding two squares where the units digit of the first number is one more than the tens digit of the second number and the tens digit of the first number plus the units digit of the second number is 10

Add the squares of the digits of the first number and multiply by 101

Difference of Squares (x²-y²)

(x-y)(x+y)

Multiplying Two Numbers Ending in 5

If a+b is even, the last two digits are 25. If a+b is odd, the last two digits are 75.

The remainder of the answer is a·b plus the greatest integer less than or equal to a+b/2.

Multiplying Mixed Numbers

FOIL if you can quickly

If you can’t, convert to improper fractions and see what cancels

Multiplying two fractions when the integer part (x) is the same and the fractions add to one

Fraction part is the fractions multiplied together

Integer part is x(x+1)

a*(a/b)

Integer: a+(a-b)

Fraction: (a-b)²/b

Remainder when dividing by 4 or 8

Look at last 2 digits for 4

Look at last 3 digits for 8

Remainder when dividing by 3 or 9

Add up all of the digits and find the remainder of that divided by 3 or 9

Remainder when dividing by 11

Add up alternating digits starting with units place then subtract the rest of the digits from that number

Ex.) 13542/11 remainder=2+5+1-4-3=1

Combining remainder tricks

Find the remainder for two factors of the divisor then find a number less than the divisor that fits those requirements

Ex.) If it is 3 mod 4 and 1 mod 3, it is 7 mod 12

Remainders of Expressions

Find the remainders of the individual numbers, then do the operations on the remainders

Converting a/10b to decimals

Find a/b as a decimal then shift the decimal point to the left one

Subtracting Reverses (abc-cba) or (a00c-c00a)

100(a-c)-(a-c)

Multiply by 1000 if it’s a four digit number

a/b(b+1)+a/(b+1)(b+2)+a/(b+2)(b+3)…

Add up numerators and the denominator is the largest denominator times the smallest

a/b+b/a

2+(a-b)²/ab

a/b-(na-1)/(nb+1) or a/b-(na+1)/(nb-1)

(a+b)/b(nb+1) or -(a+b)/b(nb-1)