Arithmetic & Fractions Flashcards
Real Number
ANY number on the number line (positive, negative, fractions, decimals, INCLUDING zero)
Remember that zero is not positive nor negative
Integer
All positive and negative whole numbers, including zero.
Does NOT include fractions or decimals
Positive Integer
All positive whole numbers
Does NOT include fractions, decimals, or zero
Sum
Result of addition
Difference
Result of subtraction
Product
Result of multiplication
Quotient
Result of division
Sign Rules for Multiplication
positive x positive = positive
negative x negative = positive
negative x positive = negative
Sign Rules for Division
positive / positive = positive
negative / negative = positive
negative / positive = negative
positive / negative = negative
Absolute Value
Gives the distance of the number from the origin
|x| = the distance of x from the origin
|x - 5| = the distance of x from +5
|x + 3| = the distance of x from -3
Order of Operations
G - Grouping Symbols (parentheses, brackets, square root signs, long fraction bar, exponent slot)
E - Exponents
M/D - Multiplication & Division
A/S - Addition & Subtraction
Goblins Eat Many Doughnuts A Second
Decimals as Powers
- 1 = 10^ -1
- 01 = 10^ -2
- 001 = 10^ -3
- 0001 = 10^ -4
For negative powers of 10, the power equals how many decimal places to the right of the decimal point.
Adding and Subtracting Decimals
Line up decimal points and add or subtract vertically
Multiplying Decimals
1) Count the number of digits to the right of the decimal point for each.
For example: 6.25 x 0.048
- 6.25 has 2 decimal places to the right of the decimal point
- 0.048 has 3 decimal places to right of the decimal point
2) Add those two numbers together.
- 2 + 3 = 5 SO the product will have 5 decimal places
3) Ignore the decimal points completely and multiply as if they were two positive integers.
- 625 x 48 = 30,000
4) with 5 decimal places to the left, the answer is 0.3
Note that powers are a special case of multiplying
(0.03)^3 = 0.03 x 0.03 x 0.03
2 + 2 + 2 = 6 SO the product will have 6 decimal places to the right of the decimal point.
3^3 = 27
The 7 must fall 6 places to the right of the decimal point
The answer is 0.000027
Dividing Decimals
1) Move the decimals of the numerator and denominator/divisor to the right until they are both whole numbers.
For example: 0.56/0.0007
0.56/0.0007 = 5.6/0.007 = 56/0.07 = 560/0.7 = 5600/7 = 800
Numerator
The top number of a fraction
Denominator
The bottom number of a fraction
Multiples of 10 greater than 1
For multiples of 10 greater than 1, the number of zeroes after the one equals the factors of 10 the multiple contains
For example:
10,000 = 10^4 = 10x10x10x10
1,000,000 = 10^6 = 10x10x10x10x10x10
Multiples of 10 less than 1
For powers of 10 less than 1, the number of factors of ten in the denominator equals the number of places to the right of the decimal point.
For example:
- 01 = 10^-2 = 1/10x10
- 0001 = 10^-4 = 1/10x10x10x10
Multiplying by Positive Powers of 10
When multiplying a number by positive powers of 10, we move the decimal point the number of spaces to the right equal to the power, that is, equal to the number of factors of 10
350x100
100 = 10^2
So we have to move the decimal point two to the right.
= 35,000
Dividing by Positive Powers of 10 OR Multiplying by Negative Powers of 10
When we divide any number by any positive power of 10 OR multiply any number by a negative power of 10, we move the decimal point a number of spaces to the left equal to the absolute value of the power, that is, equal to the factors of 10.
Dividing by a Negative Power of 10
This is the equivalent of multiplying by a positive power of 10 and will ultimately make the product bigger
2.35/0.01 = 2.35 x 100 = 235
Scientific Notation
N = A x 10^p
Where A is greater than or equal to 1 and is less than or equal to 10
AND
Where p is an integer
Multiplying/Dividing in Scientific Notation
We can multiply/divide numbers in scientific notation by multiplying/dividing the numbers in front and then adding (for multiplication) or subtracting (for dividing) the exponents of 10.
Adding and Subtracting in Scientific Notation
We can add or subtract two numbers in scientific notation, but FIRST we have to make the powers of 10 the same.
Equivalent Fractions
Fractions are equivalent if they have equal numerical value despite having different numerators and denominators.
We can always find an equivalent fraction by multiplying the numerator and denominator by the sane integer.
For example:
3/8 = 3x4/8x4 = 12/32
We can also factor out the same positive integer from both the numerator and denominator.
For example:
6/42 = 6x1/6x7
The sixes cancel and we’re left with 1/7
Lowest Term Fraction/Simplest Form
An equivalent fraction is in its simplest form when the lowest possible integer values are in the numerator and denominator
Improper Fraction
When a fraction is greater than one and expressed by a numerator that is greater than the denominator
Ex: 5/3
Mixed Numeral
A fraction greater than one that is expressed by an integer part plus a fraction part
Ex: 1 2/3
Note that this is addition! 1 + 2/3
1/2
0.5
1/4
0.25
3/4
0.75
2/5
0.4
4/5
0.8
1/3
0.33333 repeating
2/3
0.666666 repeating ~ 0.667
1/5
0.2
3/5
0.6
1/6
0.1666666 repeating ~ 0.1667
5/6
0.833333 repeating
1/7
0.142857 repeating ~ 0.143
1/8
0.125
3/8
0.375
5/8
0.625
7/8
0.875
1/9
0.11111 repeating
This means that any numerator on top of 9 will just be that number repeating as a decimal.
2/9 = 0.22222 repeating
3/9 = 0.33333 repeating
4/9 = 0.44444 repeating
Etc.
Fractions with denominators that are multiples of 10
Break up the fraction to simplify to a fraction you already know as a decimal and then move the decimal place based on the factor of 10.
1/40 = 1/4 x 1/10 = (0.25)(10^-1) = 0.025
1/600 = 1/6 x 1/100 = (0.1667)(10^-2)
= 0.001667
n/1
n/1 = n
Fractions with 0 in the denominator are UNDEFINED
Fractions with 0 in the numerator always equal 0
Note: 0/0 is also illegal
n/n
n/n = 1
Anything divided by itself equals 1 and we can always multiply a fraction by n/n because it doesn’t change the value of the product.
Reciprocal Fraction
For a/b, the reciprocal fraction is flipped over and equals b/a.
The product of a fraction and its reciprocal always equals 1.
Ex: 2/3 x 3/2 = 1
1 divided by a fraction will always equal its reciprocal.
Ex: 1/(3/7) = 7/3
Changing the numerator but keeping the same denominator
If a > b, then a/c > b/c
Ex: 4/13 < 6/14
Changing the denominator but keeping the same numerator
If p>q, then s/p < s/q
If we’re dividing by more, the quotient is less…bigger denominators make smaller fractions.
Ex: 2/5 > 2/7
Adding the same number to the numerator and denominator
The fraction will move closer to 1.
If the original fraction is LESS THAN 1 to begin with, adding the same number to the numerator and denominator will result in a BIGGER fraction
Ex: 2/5 (less than 1)
If you add 6 to numerator and denominator, the resulting fraction is 8/11, which is closer to 1 on the number line and GREATER THAN 2/5
8/11 > 2/5
If the original fraction is MORE THAN 1 to begin with, adding the same number to the numerator and denominator will result in a SMALLER fraction
Ex: 7/4 (greater than 1)
If you add 2 to the numerator and denominator, the resulting fraction is 9/6, which is closer to 1 on the number line and LESS THAN 7/4.
Adding different numbers to the numerator and denominator
Suppose we add 2 to the numerator and 5 to the denominator…the resulting fraction will be closer to 2/5 on the number line than the original fraction was.
SO if the original fraction was LESS THAN 2/5, adding 2 to the numerator and 5 to the denominator would result in a BIGGER fraction.
BUT if the original fraction was MORE THAN 2/5, adding 2 to the numerator and 5 to the denominator would result in a SMALLER fraction.
Ex: 1/8 (smaller than 2/5)
(1+2)/(8+5) = 3/13
1/8 < 3/13
Ex: 3/4 (greater than 2/5)
(3+2)/(4+5) = 5/9
3/4 > 5/9
Add to numerator and subtract from denominator (or vice versa!)
Whichever fraction has a larger numerator AND a smaller denominator is bigger
Cross Multiplying Fractions
Used when we have an equation in the form fraction = fraction.
For a/b = c/d, we can multiply both sides by the denominator of the opposite side to eliminate all fractions. The numerator stays on the appropriate side!
a x d = c x b
This also works if we want to quickly compare two relatively close fractions.
Instead of an equal sign, out a ? in between the two fractions. This could represent greater than or less than. But if only works if we keep the inequality pointing in the same direction. If we have to multiply or divide by a negative, the inequality sign will flip and this won’t work!
Ex: 7/11 and 5/8
7/11 ?? 5/8
7 x 8 ?? 5 x 11
56 ?? 55
56 > 55 and because we didn’t change the direction of the inequality, it’s true of the original fractions as well and 7/11 > 5/8
Adding/Subtracting Fractions
We can only perform addition and subtraction on fractions when they share a common denominator — that is, they must have the same denominator.
If you’re given a fraction that does not have a common denominator, find an equivalent fraction with the same denominator. You must multiply fractions to ensure there’s a common denominator and once you have the common denominator you can add/subtract across the numerator.
Ex: 1/3 + 1/7
(1/3 x 7/7) + (1/7 x 3/3) = 7/21 + 3/21
= 10/21
Multiplying Fractions
You simply multiply across numerators and denominators.
Remember: when multiplying two or more fractions, you can cancel any numerator with any denominator. Always cancel before you multiply!
Dividing Fractions
To divide by a fraction, we simply multiply by its reciprocal.
Ex: 1/4 / 3/2 = 1/4 x 2/3
= 1/6
To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction.
Ex: 6/(3/4) = 6/1 x 4/3 = 8/1
= 8
To divide a fraction by a whole number, multiply the fraction by the reciprocal or the whole number, which will be in the form 1/n.
Ex: (3/5) / 2 = 3/5 x 1/2
= 3/10
Addition or Subtraction in Numerator
We CAN separate a fraction into two fractions by addition or subtraction in the numerator.
a + b/c = a/c + b/c
Addition or Subtraction in the Denominator
We CANNOT separate a fraction into two fractions by addition or Subtraction in the denominator.
Illegal move!
Addition or Subtraction in Numerator and Denominator
If we have addition or Subtraction in both the numerator and denominator, we can split up the numerator but the denominator remains unchanged.
Ex: a + b /c + d = a/c + d + b/c + d
Multiplying a Fraction by its Denominator
Fraction x Denominator = Numerator
Can be very useful in solving equations.
Simplifying complex fractions
Strategy: multiply numerator and denominator of the big fraction by each denominator of the inner fractions
Cancellation in Proportions
For proportion
a/b = c/d
You’re allowed to cancel numerators and denominators vertically (a & b and c & d)
You’re also allowed to cancel horizontally across numerators and across denominators (a & c and b & d)
You are NOT ALLOWED to cancel diagonally - this is only allowed in multiplication, it’s completely illegal to use in proportions.
Word Problem Secret
In most word problems:
- is = equals
- of = multiply
