Fractions, Decimals, and Probability Flashcards
This deck covers different types of fractions and decimals as well as rules of operations with them. We will review the counting principle and the topic of probability, providing you with clues for decoding any probability problem on the SAT.
1
Q
What is a fraction?
A
A fraction is a part of a whole.
Example: 3/4
2
Q
What is a numerator?
A
The top number of a fraction is called the numerator.
The numerator shows how many equal parts of a whole are in the fraction.
3
Q
What is a denominator?
A
The bottom number of a fraction is called the denominator.
The denominator indicates how many parts make up a whole.
The denominator cannot equal zero.
4
Q
What is a simple fraction?
A
In simple (common) fractions, both the numerator and the denominator are integers expressed as a ratio.
Examples: 3/4, 1/3
These are the types of fractions you’ll see most often on the SAT.
5
Q
What is a simplified fraction?
A
In simplified fractions, the greatest common factor of both the numerator and the denominator is 1.
Examples: 1/2, 3/4
6
Q
What fractions are called proper fractions?
A
In proper fractions, the numerator is less than the denominator.
Examples: 1/4, 3/4
7
Q
What fractions are called improper fractions?
A
In improper fractions, the numerator is equal to or greater than the denominator.
Examples: 23/11, 9/7
8
Q
What fractions are called equivalent fractions?
A
Equivalent fractions have the same value, but are expressed differently.
Examples: 1/2, 2/4, 3/6
9
Q
What are similar fractions?
A
Similar fractions have the same denominator.
Examples: 1/4, 3/4
10
Q
When the numerator of a fraction is zero, the fraction is….?
When the denominator of a fraction is zero, the fraction is…?
A
When the numerator of a fraction is zero, it’s called a zero fraction. It equals 0.
When the denominator of a fraction is zero, the fraction is undefined.
Remember, nothing can be divided by 0.
11
Q
What are reciprocal fractions?
A
In reciprocal fractions, the numerators and the denominators are switched.
Examples: 5 is a reciprocal of 1/5.
3/8 is a reciprocal of 8/3.
12
Q
What are complex fractions?
A
In complex fractions, the numerator and the denominator are fractions themselves.
Example:
13
Q
What are mixed fractions?
A
Mixed fractions combine an integer and a proper fraction.
Example: 6 2/3
Mixed fractions are also called mixed numerals or mixed numbers.
14
Q
How do you simplify a fraction by using the GCF (greatest common factor)?
A
Divide both the numerator and the denominator by their GCF.
Example: Simplify 24/36
The GCF of 24 and 36 is 12. Simplify the fraction by dividing by 12. The lowest term of this fraction is 2/3.
15
Q
How do you use prime factorization to simplify a fraction?
A
Write out the prime factorization of both the numerator and the denominator and cancel out any common prime factors.
Example: Simplify 76/84
16
Q
How do you reduce a fraction to its lowest terms?
A
To reduce a fraction to the lowest terms, factor out both the numerator and the denominator. Then, cancel out all common factors.
Example: 16/24 = 4 x 2 x 2/4 x 2 x 3 = 2/3
17
Q
How do you add or subtract fractions that have the same denominator?
A
When fractions are similar (i.e. have like denominators), work with the numerators (add or subtract) and leave the denominators alone.
Example: 5/22 + 7/22 = 13/22
18
Q
What is the Least Common Denominator (LCD) of two or more fractions?
A
The Least Common Denominator (LCD) of two or more fractions is the smallest whole number that is divisible by each of the denominators.
19
Q
How do you find the LCD of two or more fractions?
A
To find the LCD, use the same methods used to find the Least Common Multiple (LCM) of two numbers:
Successive multiplication Prime factorization See “Factors and Multiples” deck for detailed explanation.
20
Q
How do you add or subtract fractions with different denominators?
A
To add or subtract fractions with different denominators:
Find the LCD
Write equivalent fractions with the LCD
Add or subtract the numerators
Example: 1/5 + 2/7 = 7/35 + 10/35 = 17/35
21
Q
How do you multiply fractions?
A
- To multiply fractions, simply multiply the numerators, then multiply the denominators.
- It doesn’t matter whether fractions have same or different denominators.
- Use cancelling whenever possible as a shortcut. Cancelling can take place only between numerators and denominators.
22
Q
How do you divide fractions?
A
Rewrite the second fraction as its reciprocal and multiply by the first fraction.
Example: 5/6 ÷ 2/3 = 5/6 x 3/2 = 15/12 = 5/4
23
Q
How do you turn an improper fraction into a mixed number?
A
To convert an improper fraction into a mixed number, divide the numerator of the improper fraction by its denominator. If there is a remainder, it will become the numerator of the mixed number.
Example: 18/14 = 1 4/14 = 1 2/7
24
Q
How do you turn a mixed number into an improper fraction?
A
To convert a mixed number into an improper fraction, multiply the denominator by the whole number and add the numerator. This number becomes the numerator of your improper fraction.
Example: 1 2/3 = 5/3
3 * 1 + 2 = 5 It’s the numerator of the improper fraction.
25
How do you add or subtract mixed numbers containing similar fraction parts?
Add or subtract the fraction parts first, then work with the whole numbers. Simplify if possible.
## Footnote
*Examples:* 53/7 + 82/7 = 135/7
53/7 + 86/7 = 139/7 = 142/7
If the answer contains an improper fraction, you must turn it into a mixed number.
26
How do you subtract mixed numbers with similar fraction parts when the numerator of the second number is greater than the numerator of the first number?
83/7 - 56/7 = ?
"Regroup," or remove 1 (or 7*/*7) from the whole part of the first mixed number by adding it to the fraction part:
83*/*7 - 5 6*/*7= 710/7 - 56/7
Perform the subtraction, dealing first with the fractions, then the whole numbers:
710/7 - 56/7 = 24/7
27
How do you add or subtract mixed numbers with different denominators?
53/5 + 81/4 = ?
53/5 + 81/4 = ?
Write equivalent fractions using the LCD of 4 and 5.
53/5 = 512/20 ; 81/4 = 85/20
Add the fraction parts, then the whole numbers:
53/5 + 81/4 = 512/20 + 85/20 = 1317/20
28
How do you subtract mixed numbers with different denominators?
83/7 - 52/3 = ?
Using the LCD of 3 and 7, write new equivalent fractions :
83/7 - 52/3 = 89/21 - 514/21
Make a new fraction equal to 1 and "regroup," or remove, that fraction from the whole part of the first mixed number by adding it to the fraction part:
89/21 = 730/21
The fraction in the first mixed number is now greater than the fraction in the second number. Subtract the fractions first, then the whole numbers:
730/21 - 514/21 = 216/21
29
How do you multiply or divide mixed numbers?
Change mixed numbers into improper fractions and follow the rules explained in "Multiplying and Dividing Fractions" cards.
30
What is **decimal notation**?
**Decimal notation** is the most common way to write the base 10 numeral system, which uses digits from 0 to 9 together with the decimal point (.)
31
What is a **decimal fraction**?
A decimal or **decimal** **fraction** is a fraction whose denominator is 10 or a power of 10.
Adding zeros to the right side of a decimal fraction does not change its value.
## Footnote
*Examples:* 0.62; 0.5; 0.009
The above numbers can be written as proper fractions:
62/100; 5/10; 9/1000
32
What is a **mixed decimal**?
A **mixed decimal** is a number consisting of an integer plus a decimal fraction.
## Footnote
*Examples:* 5.34; 654.006; 1.1
33
What decimals are called **similar decimals**?
**Similar decimals** are decimals that use the same number of places to the right of the decimal point.
## Footnote
*Examples:* 7.345 and 2.012
34
What is a **periodic decimal**?
A **periodic** (or **repeating**) **decimal** is a non-terminating decimal with repeating digits or group of digits.
## Footnote
*Example*: 1/3 = 0.33333333.......
One way to indicate a repeating decimal is to put a horizontal line above the repeated numerals.
35
What is a **non-repeating, non-terminating** **decimal**?
A decimal that neither terminates nor repeats is called a **non-repeating, non-terminating** **decimal**.
## Footnote
*Example:* 1.41421356237........
36
How do you convert a fraction into a decimal?
Just divide the numerator by the denominator (if the denominator doesn't equal 10, 100, 1000, etc)
## Footnote
*Example*: 5/8 = 0.625
37
Besides dividing the numerator by the denominator, what is another way to convert fractions into decimals?
Multiply both the numerator and the denominator by such number that makes the denominator 10, 100, 1000, etc.
38
How do you convert a decimal number into a fraction?
The key is to understand the place value of the decimal.
Read the decimal: the 1st place to the right of the decimal point indicates tenths, 2nd place indicates hundredths, 3rd place indicates thousandths.
## Footnote
*Example*: 0.145 = 145*/*1000 = 29*/*200
39
How do you mutiply a decimal number by a power of 10?
To multiply a decimal by a power of 10, move the decimal point to the *right*, one place for each zero in the power of 10.
## Footnote
*Example*: 15.8567 x 100 = 1585.67
40
How do you divide a decimal number by a power of 10?
To divide a decimal number by a power of 10, move the decimal point to the *left*, one place for each zero in the power of 10.
## Footnote
*Example:* 15.86 ÷ 1000 = 0.01586
41
How do you add or subtract decimals with different number of decimal places?
To add or subtract decimal numbers, *align* them first.
## Footnote
Tenths have to be under tenths, hundredths - under hundredths, etc. The decimal point in the answer goes right under the decimal points of the numbers you are calculating.
If one number has more decimal places than the other, add zeros to the right end.
42
When you multiply decimal numbers, what is the number of decimal places in the product?
The number of decimal places in the product is the *sum* of the decimal places in the numbers you are multiplying.
## Footnote
*Example*: 2.3 x 1.1 = 2.53
Multiply the numbers as if they were whole numbers. Each factor has one decimal place so the answer would have two decimal places.
43
How do you divide by a decimal number?
In order to divide by a decimal:
* Convert the divisor to a whole number by multiplying by a power of 10
* Multiply the dividend by the same power of 10
## Footnote
*Example:* 34 ÷ 0.17 = 3400 ÷ 17 = 200
Turn 0.17 into a whole number by multiplying by 100. Then, multiply 34 by 100 as well.
44
# Define:
**probability**
**Probability** is the likelihood or a chance that a random event will occur.
## Footnote
The probability that *one* event will or will not happen is a **simple** probability.
The probability that *two* or *more* events happen is a **compound** probability.
It's denoted as P (A) - probability of an event A.
45
How do you express probability as a number?
Probability can be expressed as a number between 0 and 1.
## Footnote
"0" probability means that the event is impossible. "1" probability means that the event will certainly take place.
Probability can also be expressed as a percentage between 0% and 100%.
46
How do you calculate the probability that an event will take place?
The probability of an event happenning is the number of *favorable* outcomes over the *total* number of *possible* outcomes.
## Footnote
*Example*:
Out of 24 tops in your closet, 8 have long sleeves and the rest have short sleeves. What is the probability that at random you pick a short sleeve top?
16 short sleeve tops over 24 total tops
16/24 = 2/3
47
What is the probability that an event A will *not* happen?
The opposite of an event A is the event *not* A. Its probability can be expressed as:
**1 - P (A)**.
## Footnote
*Example*:
What is the probability of *not* rolling two on a six-sided die?
1 - (chance of rolling two) = 1 - 1/6 = 5/6
48
# Define:
**independent** **events**
Two events are **independent** when the outcome of the 1st event doesn't affect the outcome of the 2nd event.
## Footnote
*Example*:
Two decks of cards.... pulling a king from each one.
P1(K) = 4*/*52 = 1*/*13
P2(K) = 4*/*52 = 1*/*13
49
# Define:
**dependent** **events**
Two events are **dependent** when the outcome of the 1st event affects the outcome of the 2nd event.
## Footnote
*Example*:
Pulling a king from a deck of cards, then, without replacement, pulling a queen.
P (K) = 4*/***52** = 1*/*13
P (Q) = 4*/***51**
50
If two events A and B are *independent*, what is the probability of *both* of them happening?
The probability of two *independent* events A and B happening is a *product* of a probability of event A and a probability of event B.
**P (A and B) = P (A) x P (B)**
## Footnote
*Example*:
What is a probability of having two tails when you flip two coins?
1/2 x 1/2 = 1/4
51
When two events A and B are *dependent*, what is the probability of *both* of them happening?
The probability of two *dependent* events A and B happening is a *product* of a probability of A and a probability of B after A occurs.
**P (A and B) = P (A) x P (B given A)**
## Footnote
*Example*:
A drawer contains 3 white tee-shirts and 4 black tee-shirts. Find a probability of first pulling a white tee-shirt and then, without replacing, pulling a black tee-shirt.
3*/***7** x 4*/***6** = 2*/*7
The total number of possible outcomes of the 2nd event is smaller than the number of possible outcomes of the 1st event.
52
# Define:
**mutually exclusive** **events**
**Mutually exclusive events** are events that *cannot* occur at the same time.
## Footnote
*Example*: Probability of rolling 2 *or* 4 on a six-sided die....
53
If two events A and B are *mutually exclusive*, what is the probability of either of them occuring?
The probability of either of *mutually exclusive* events A or B happening is a *sum* of the probability of event A and the probability of event B.
**P (A or B) = P (A) + P (B)**
## Footnote
*Example*:
What is a probability of rolling 2 *or* 4 on a six-sided die?
1/6 + 1/6 = 1/3
54
What do the words "without replacement" mean in the context of a probability problem?
The words "*without replacement*" mean the number of possible outcomes of an event following the 1st event is reduced.
## Footnote
*Example*:
There are 6 apples in a fruit bowl; 4 green and 2 red. What is the probability of picking a red apple after you have picked a green apple from the bowl, *without replacement*?
There are 2 red apples out of 5 apples left in the bowl so, it's 2/5. Had you put the green apple back in the bowl, the probability of picking a red apple would have been 2 out of 6.
55
How do you know when to multiply probabilities of different events or when to add them?
What words can give you the clues?
When you see the words *"both", "all", "together", "combined"*, _multiply_ the probabilities of different events.
When you see *"either.... or"*, _add_ them.
## Footnote
*Example*:
What is the probability of picking *BOTH* 2 and 4 out of a set of numbers from 0 to 9 (with replacement)?
1/10 x 1/10 = 1/100
*Example*:
What is the probabiltiy of picking 2 *OR* 4 from the set of numbers from 0 to 9 (with replacement)?
1/10 + 1/10 = 2/10
56
There are 6 apples in a fruit bowl; 4 green and 2 red. What is the probability that a green apple is picked and then a red apple is picked, assuming *no replacement* between turns?
Certain words in this problem give you clues how to solve it. What are they?
The "clues" words are *no replacement.*
4*/*6 is the probability of picking a green apple. Since you didn't replace the apple, the probability of picking a red apple *after* you had pick the green apple is 2/5. The number of possible outcomes in the probability number of the 2nd event is reduced by 1.
## Footnote
Since you are looking for the probability of *both* events happening, *multiply* probabilities of each event.
4*/*6 x 2*/*5 = 4*/*15
57
What is the essense of the **Fundamental Counting Principle**?
How do you count the total number of possibilities for several events to occur?
The **Fundamental Counting Principle:** If there are *"a" ways* for one activity to occur, and *"b" ways* for a second activity to occur, there are *a x b* ways for both to occur.
Simply multiply the number of ways each activity can occur to count the total number of possibilities.
## Footnote
*Example*:
There are 6 pairs of jeans and 8 tops in your closet. How many outfits can you build out of those?
6 x 8 = 48
58
What are the two ways you can simplify a fraction?
Simplify 36*/*90 using both methods.
* Divide by GCF
The GCF of 36 and 90 is 18. Divide both number by 18 to reduce the fraction to its lowest term 2*/*5.
* Cancel out common prime factors
Factor both 36 and 90 into primes and cancel out common factors.
59
What is the reciprocal fraction of 13*/*25?
What type of fraction is the answer?
The reciprocal of 13*/*25 is 25*/*13.
It's an improper fraction.
## Footnote
Remember, you need to convert it into a mixed number.
25*/*13 = 1 12/13
60
A prime number multiplied by its reciprocal will *always* be:
(a) even
(b) odd
(c) prime
(d) 0
(b) odd
## Footnote
Note: The product of any number and its reciprocal equals 1.
61
If you divide a prime number by its reciprocal, the result will *never* be:
(a) even
(b) odd
(c) prime
(d) 4
(c) prime
## Footnote
Dividing a number by its reciprocal is equivalent to multiplying the number by itself. Therefore, the result can never be prime.
62
What is the reciprocal of
(1 + 6/7):
(a) 1 + 7/6
(b) 13/7
(c) 6/13
(d) 7/13
(d) 7*/*13
## Footnote
The expression in the parenthesis equals 13/7. The reciprocal of this fraction is 7/13.
63
What is the reciprocal of the quotient of
(5 ÷ 1*/*25)?
(a) 125
(b) 1*/*5
(c) 5
(d) 1*/*125
(d) 1/125
## Footnote
Take the reciprocal of 1*/*25 and multiply by 5 to find the quotient. The quotient is 125. Then, flip that number to find the reciprocal.
64
What operation in arithmetic requires finding the LCD?
Adding or subtracting fractions with the different denominators requires you to find the Least Common Denominator.
65
Of the following, which is the closest to 0.6?
(a) 4*/*5
(b) 19*/*30
(c) 8*/*15
(d) 11*/*18
(d) 11/18
## Footnote
The LCD of these 4 fractions is 90. Write equivalent fractions with 90 in the denominator and compare.
(a) 72/90
(b) 57/90
(c) 48/90
(d) 55/90
0.6 equals 54/90. Evaluate 57/90 and 55/90, because the other two choices are clearly out. 57/90 is 3/90 (or 1/30) removed from 0.6. 55/90 is 1/90 removed from 0.60.
66
What is the value of
3/4 x 7/8 x 16/21
rounded to the nearest whole number?
The product of the three rounded numbers is 1.
## Footnote
Always look for ways to simplify fractions before operating with them. Notice that you can cancel out common factors before multiplying numbers in the numerators and the denominators. So, the simplified fraction is 1/2. Rounded to the nearest whole number it's 1.
Remember, cancelling takes place only between the numerator and the denominator.
67
Is the quotient of a mixed number and a proper fraction greater than or less than 1?
The quotient is greater than 1.
## Footnote
Any mixed number can be expressed as an improper fraction, and is greater than 1. Dividing by a fraction means multiplying by a reciprocal of that fraction. A reciprocal of a proper fraction is an improper fraction, which is greater than 1. The product of two numbers greater than 1 is itself greater than 1.
68
What type of number is the quotient
(2 ÷ 3) ÷ (4 ÷ 5) = ?
(a) mixed number
(b) odd number
(c) even number
(d) proper fraction
(d) proper fraction
## Footnote
2/3 **÷** 4/5 = 2/3 **x** 5/4 = 5/6
69
(2 ÷ 3) ÷ (4 ÷ 5) = 5*/*6
Would it affect the result if the parentheses are removed?
2 ÷ 3 ÷ 4 ÷ 5 = ?
Yes, the removal of the parentheses affects the result.
## Footnote
According to the order of operations rule, when the operations have the same rank, work from left to right.
2/3 ÷ 4 = 2/3 x 1/4 = 1/61/6 ÷ 5 = 1*/*6 x 1*/*5 = 1*/*30
70
Is the product of ten proper fractions less than or greater than 1?
The product of ten proper fractions is always less than 1.
## Footnote
In a proper fraction, the numerator is always smaller than the denominator. Multiplying ten proper fractions will give you a fraction with the numerator that is smaller than the denominator.
71
Which number is greater:
the product of 3*/*4 and 8*/*3
or
the sum of 3*/*4 and 4*/*3?
The sum is greater than the product.
## Footnote
3*/*4 x 8*/*3 = 2
3*/*4 + 4*/*3 = 9*/*12 + 16*/*12 = 25*/*12 = 21*/*12
Find the LCD when you add fractions with the different denominators.
Remember to change improper fractions to mixed numbers and reduce fractions to lowest terms when possible.
72
Total population of towns A, B and C in France is a certain whole number. The population of town B is one-third of the population of town A. The population of town C is four times the population of town B. What fraction of the total is the population of town B?
The population of town B is 1/8 of the total.
## Footnote
The population of town A is not related directly to the population of town C. The common element between them is town B's population. So, express their respective populations in terms of B.
Total population (T) = A + B + C.
B = 1/3 A ⇒
A = 3B; C = 4B
Substitute the equivalents of A and C in the equation and solve for B.
T = 3B + B + 4B = 8B ⇒ B = 1/8T
73
# Complete the statement:
If you write a decimal number as a fraction, the denominator of that fraction is always....?
The denominator is a power of 10.
74
What are the tens and tenths digits of 182*/*8 written in decimal form?
The tens digit is 2. The tenths digit is 7.
## Footnote
182*/*8 = 22.75
75
How many decimal places does the product of numbers below have?
0.5 x 0.5 x 0.5
The result, 0.125, has three decimal places. The number of decimal places in a product is equal to the sum of the number of decimal places in factors.
76
Convert 4*/*125 into a decimal number.
0.032
## Footnote
Convert 4*/*125 to a fraction with the denominator of 1000 by multiplying both the numerator and the denominator by 8.
77
Which pair consists of two *unequal* numbers?
(a) 3.5, 7*/*2
(b) 0.032, 4*/*125
(c) 90*/*38, 5*/*2
(d) 201*/*3, 67
(c) 90*/*38 and 5*/*2 are not equal.
78
Which of these numbers *cannot* represent the probability of an event?
(a) - 0.5
(b) 0.5
(c) 0
(d) 1.1
(e) 104%
(f) 20%
(a), (d), (e)
## Footnote
cannot express the probability because the probability is always greater than or equal to 0 and less than or equal to 1. In percents, the probability lies between 0% and 100%.
79
If a card is drawn from a deck of cards at random, what is the probablity of getting a king?
4*/*52 = 1*/*13
## Footnote
There are 52 cards in a deck. There are 4 kings among them. Therefore, there are 4 possible choices out of 52 total choices.
80
There are 5 cans of Coke, 3 cans of Sprite, 4 bottles of spring water and 8 bottles of ice tea in a cooler. What is the probability, in percent, of getting a drink other than a soda can?
60%
## Footnote
12*/*20 = 6*/*10
There are 12 non-soda drinks in a cooler that contains a total of 20 cans and bottles.
81
If two dice are rolled, what is the probability that the sum is equal to 5?
4*/*36 = 1*/*9
## Footnote
There are 4 possible choices that give you the sum of 5. {1, 4}; {2, 3}, {4, 1} and {3, 2}.
By using the fundamental counting principle, you can find the total number of choices. Each dice has 6 sides; i.e. 6 possible choices. 6 x 6 = 36
82
There are 5 white, 8 black and 4 multicolor t-shirts in your closet. In three days, you've worn one of each color and removed them from the closet.
What is the probability of you picking a multicolor t-shirt on the 4th day?
3*/*14
## Footnote
4th day: There are 14 tee-shirts left in the closet. Among them there are 3 multicolor t-shirts.
83
There are 5 white, 8 black and 4 multicolor t-shirts in your closet. In three days, you've worn one of each color and removed them from the closet.
What is the probability of picking a multicolor t-shirt on the 4th day *and* (without replacing the multicolor t-shirt) picking a white t-shirt on the 5th day?
The probability of *both* events happening equals a product of probabilities of two events.
## Footnote
3*/*14 x 4*/*13 = 6/91
These events are *dependent* since the t-shirts were pulled without replacement.
4th day: The probability of picking a multicolor t-shirt is 3*/*14.
5th day: The probability of picking a white t-shirt is 4*/*13.
84
How many different ways can three students occupy four seats in a classroom?
There are 12 different ways for 3 students to occupy 4 chairs.
## Footnote
Each student has *four* choices to select a seat. There are *three* students; therefore, alltogether, they have 12 possible choices of selecting a seat in a classroom.
Simply multiply the number of ways each activity can occur to count the total number of possibilities.
