1. Fractions & Decimals Flashcards
1
Q
Number on top of a fraction
A
Numerator
2
Q
Number on bottom of a fraction
A
Denominator
3
Q
0 / 3
A
0
4
Q
3 / 0
A
undefined
5
Q
How are fractions properly expressed?
A
in their lowest or simplest terms
6
Q
What is a mixed numeral
A
The combination of a whole number and a faction
Ex: 6⅔
7
Q
How to convert a mixed numeral into a fraction?
A
- Multiply the denominator by the whole number
- Add the numerator
- Place the sum over the denominator
8
Q
How to convert a fraction into a mixed numeral?
A
- Divide the numerator by the denominator
2. Place the remainder over the denominator
9
Q
Before adding and subtracting fractions
A
be sure to simplify the factions first if possible
10
Q
To add whole numbers and fractions
A
- combine the two into a mixed numeral
2. convert the mixed numeral
11
Q
When multiplying large or complex number
A
- rip up the numerators and denominators
- cancel out the common terms
- express what remains as a fraction
12
Q
If all the terms in the numerator or denominator cancel out
A
be sure to leave a 1 in their place
13
Q
To divide two fractions
A
- Flip the second fraction
2. multiply the two together
14
Q
When dividing large or complex numbers
A
- rip up the numerators and denominators
- cancel out the common terms
- express what remains as a fraction
15
Q
What is a complex fraction?
A
Any fraction that contains addition or subtraction in either the numerator or denominator
16
Q
What is a simple fraction?
A
No addition or subtraction in numerator or denominator?
17
Q
Can terms within complex fractions be cancelled?
A
No, NEVER.
Only within simple fractions
18
Q
What is the Complex Numerator Shortcut?
A
If the numerator of a fraction contains addition or subtraction, the fraction can be split to simplify the arithmetic
19
Q
Can the denominator of a complex fraction be split?
A
No, NEVER.
20
Q
When adding, subtracting, multiplying or dividing mixed numerals
A
always convert the mixed numerals into fractions before performing the operation
21
Q
Reciprocals (definition)
A
Any two numbers who product equals 1
22
Q
Numbers that are not “flips” mays also be reciprocals
A
Especially w/ square roots
23
Q
How many reciprocals can fractions have?
A
More than one
24
Q
Fractions within fractions
A
Multiply the numerator by the “flip” of the denominator
25
Fractions within the Denominator
Put the numerator over one and multiplying the numerator by the "flip" of the denominator
26
Fractions within the Nominator
Put the denominator over one and multiplying the numerator by the "flip" of the denominator
27
What are fractions with multiple "levels"?
Complex fractions that contain fraction sin either the numerator or the denominator
28
How to solve complex fractions with multiple "levels"?
Work on "level" at a time, starting with the bottom "level" and progressing upwards
29
What are the properties of fractions (2)
1. If the num = 0, that fraction = 0
| 2. If the den = 0, that faction is undefined
30
Proper fractions (definition)
Any fraction whose numerator is smaller than its denominator is known
0 < x/y < 1
31
0 < x/y < 1
Value of positive proper fractions
32
Multiplying any number by a whole number yields
a product equal or larger than the original number
33
Squaring any number by a whole number yields
a product equal to or larger than the original number
34
Dividing any number by a whole number yields
a quotient equal to or smaller than the original number
35
Taking the square root of any number by a whole number yields
a root equal to or smaller than the original number
36
Multiplying any number by a proper fraction yields
a product smaller than the original number
37
Squaring any number by a proper fraction yields
a product smaller than the original number
38
Taking the square root of any number by a proper fraction yields
a root larger than the original number
39
Dividing any number by a proper fraction yields
a quotient larger than the original number
40
k = -3/7
| Which one is bigger? k^2 or k^4
k^2 > k^4
The more we multiply a fraction between 0 and 1, the smaller it gets
41
What are the different ways to compare any two fractions? (3)
1. Cross-multiplying
2. Approximating
3. Picking a Litmus
42
Cross-multiplying
Cross-multiply the num and den.
Multiply from the bottom up, and write each product over the corresponding numerator.
The fraction under the larger product will always be larger
43
Approximating
If num and den too large to multiply easily, identify approximate equivalents that are easy to work with
44
How to rank fractions?
1. Pick a Litmus
¼, ⅓, ½, ⅔, ¾
2. Determine which fractions are bigger than the litmus and which are smaller
45
FUQ
Fractions with Unspecified Quantities
46
How to solve FUGs?
1. Pick numbers
| 2. Let the unspecified quantity equal the product of the den of the fractions involved
47
Properties of Num and Den
1. Increasing the num increases the value of the fraction, while increasing the den decreases the value of the fraction
2. Increasing the num and the den by the same amount brings the value of a fraction closer to 1, while decreasing the num and the den by the same amount moves the value of a fraction away from 1
48
Zeroes can be added and removed from the end of a decimal without altering the value of the decimal
Not true for whole numbers
4.2 = 4.20 = 4.200
23 ≠ 230 ≠ 2300
49
Difference thousands and thousandths?
1000 vs 0,001
50
Rounding decimals
Limiting a value of a decimal to a specified place value
51
How to round digit
1. Find rounding digit
2. Look at the digit immediately to its right
3. IF less than 5, leave rounding digit unchanged and drop all the digits to its right
IF 5 or greater, add one to the rounding digit and drop all digits to its right
52
Power of 10
Any number formed by multiplying one or more tens together
53
Multiplying by a power of 10 shifts the decimal point of any number to the right
Multiplying by a number greater than 1 makes a number larger.
The number of shifts to the right will equal the number of the zeroes in the power of 10
54
Multiplying by a power of 10 shifts the decimal point of any number to the ______
right
Multiplying by a number greater than 1 makes a number larger.
The number of shifts to the right will equal the number of the zeroes in the power of 10
55
Dividing by a power of 10 shifts the decimal point of any number to the _____
left
as dividing by a number created than 1 makes a number smaller
The number of shifts to the left will equal the number of zeroes in the power of 10
56
Powers of 10 written in the form of 10^-n work in the opposite manner of regular powers of 10
Multiplying by 10^-n will shift the decimal point to the left, and dividing by 10^-n will shift the decimal point to the right
57
Approximation - know when to round
Decimal problems that contain words such as "approximately", "nearest" and "closest" can usually be solved through approximation
58
Know when not to round
1. numbers can easily be reduced
| 2. doesn't not contain the word approximately
59
How to divide decimals (2)
1. Long division
| 2. Slide approach
60
Slide approach (3)
1. Put the division in fraction form
2. Slide the decimal point an equal number of times in both the num and the den
3. until neither the num nor the den contains a decimal
61
Decimal property 0 < x < 1
Numbers 0 < x < 1 always behave in the opposite manner of numbers larger than 1 when it comes to multiplying/dividing
x > 1 : 2*5 = 10
x < 1 : 2*0.5 = 1
Idem for squaring, square root, mult., div.
62
Converting decimals to fractions
To convert a decimal into a fraction, place the digits of that decimal over a 1 with a number of zeroes equal to the number of decimal places in the decimal
63
Ways to convert fractions to decimals (3)
1. Long division
2. Conversion Chart
3. Set the Den equal to a power of 10
64
1/6
0.16
65
1/7
0.14
66
11/9
0.11
67
1/11
0.09
68
1/99
0.01
69
Setting the Den equal to a power of 10
See them equal to a power of 10 such as 10, 100 or 1000
Ex: 17/50 * 2/2 = 34/100 = 0.34
70
A repeating decimal
Any decimals whose digits repeat in a FIXED pattern without end
Represented by horizontal line
71
Converting repeating decimals to fractions
Set the repeating decimal to x and multiply both sides of the equation by a power of 10
! Let the number of zeroes in the power of 10 equal the length of the repeating sequence (provide two equations that can be subtracted to determine your fraction)
72
Fractional equivalent of 0.5757
1. Since 0.57 2 digits -> multiply by 100
2. 100x = 57.57
3. Subtract both :
100x = 57.57
- x = 0.57
__________
99x = 57
4. Solve
x = 57/99
x = 3*19 / 3*33
x = 19/33
73
Terminating decimals
Decimals that do end
! Adding more zeroes to the end does NOT make it non-terminating, since 0 can always be removed
74
Non-terminating decimals
Any decimal that never ends
! Adding more zeroes to the end does NOT make it non-terminating, since 0 can always be removed
75
Which prime factors need to be included in the denominator to terminate?
2 and 5
3/10 ; 21/50 ; 7/100
76
How to determine the number of consecutive zeroes that an integer has to the LEFT of its decimal point
count the number of "10"s it contains
| 2 and 5
77
How to determine the number of consecutive zeroes that an integer has to the RIGHT of its decimal point
1. Simplify the faction and count the number of 10s left in the denominator
2. Determine the product of any numbers that do not produce a "10"
_ less than 10 - no 10
_ less than 100 - one 10
_ less than 1000 -two 10
