Thea practice test_4 pptx

The number line is a graphical representation of the order of numbers.. If we need a number line to reflect certain rational or irrational numbers, we can estimate where they should be..

■ Integers include the whole numbers and their opposites Remember, the opposite of zero is

zero: –3, –2, –1, 0, 1, 2, 3,

■ Rational numbers are all numbers that can be written as fractions, where the numerator and

denomina-tor are both integers, but the denominadenomina-tor is not zero For example,23is a rational number, as is56 Thedecimal form of these numbers is either a terminating (ending) decimal, such as the decimal form of34which is 0.75; or a repeating decimal, such as the decimal form of13which is 0.3333333

■ Irrational numbers are numbers that cannot be expressed as terminating or repeating decimals (i.e

non-repeating, non-terminating decimals such as π, 2 , 12)

The number line is a graphical representation of the order of numbers As you move to the right, the valueincreases As you move to the left, the value decreases

If we need a number line to reflect certain rational or irrational numbers, we can estimate where they should be

(x can be 5 or any number > 5)

(x can be 3 or any number < 3)

When two or more numbers are being multiplied, they are called factors The answer that results is called the

prod-uct In the following example, 5 and 6 are factors and 30 is their product:

5  6 = 30

There are several ways to represent multiplication in the above mathematical statement

■ A dot between factors indicates multiplication:

In division, the number being divided BY is called the divisor The number being divided INTO is called the

div-idend The answer to a division problem is called the quotient.

There are a few different ways to represent division with symbols In each of the following equivalentexpressions, 3 is the divisor and 8 is the dividend:

8 ÷ 3, 8/3, 83, 38

P RIME AND C OMPOSITE N UMBERS

A positive integer that is greater than the number 1 is either prime or composite, but not both

■ A prime number is a number that has exactly two factors: 1 and itself.

1 Align the addends in the ones column Since it is necessary to work from right to left, begin to add

start-ing with the ones column Since the ones column totals 13, and 13 equals 1 ten and 3 ones, write the 3 inthe ones column of the answer, and regroup or “carry” the 1 ten to the next column as a 1 over the tenscolumn so it gets added with the other tens:

1

40129+ 243

2 Add the tens column, including the regrouped 1.

1

40129+ 2493

3 Then add the hundreds column Since there is only one value, write the 1 in the answer.

1

40129+ 24193

If Becky has 52 clients, and Claire has 36, how many more clients does Becky have?

1 Find the difference between their client numbers by subtracting Start with the ones column Since 2 is

less than the number being subtracted (6), regroup or “borrow” a ten from the tens column Add theregrouped amount to the ones column Now subtract 12 – 6 in the ones column

54

21– 366

2 Regrouping 1 ten from the tens column left 4 tens Subtract 4 – 3 and write the result in the tens column

of the answer Becky has 16 more clients than Claire Check by addition: 16 + 36 = 52

54

21– 3616

M ULTIPLICATION

In multiplication, the same amount is combined multiple times For example, instead of adding 30 three times,

30 + 30 + 30, it is easier to simply multiply 30 by 3 If a problem asks for the product of two or more numbers,the numbers should be multiplied to arrive at the answer

Example

A school auditorium contains 54 rows, each containing 34 seats How many seats are there in total?

1 In order to solve this problem, you could add 34 to itself 54 times, but we can solve this problem easier

with multiplication Line up the place values vertically, writing the problem in columns Multiply thenumber in the ones place of the top factor (4) by the number in the ones place of the bottom factor (4): 4

 4 = 16 Since 16 = 1 ten and 6 ones, write the 6 in the ones place in the first partial product Regroup orcarry the ten by writing a 1 above the tens place of the top factor

1

34

 546

2 Multiply the number in the tens place in the top factor (3) by the number in the ones place of the bottom

factor (4); 4  3 = 12 Then add the regrouped amount 12 + 1 = 13 Write the 3 in the tens column andthe one in the hundreds column of the partial product

1

34

 54136

3 The last calculations to be done require multiplying by the tens place of the bottom factor Multiply 5

(tens from bottom factor) by 4 (ones from top factor); 5  4 = 20, but since the 5 really represents anumber of tens, the actual value of the answer is 200 (50  4 = 200) Therefore, write the two zeros underthe ones and tens columns of the second partial product and regroup or carry the 2 hundreds by writing

a 2 above the tens place of the top factor

2

34

 5413600

4 Multiply 5 (tens from bottom factor) by 3 (tens from top factor); 5  3 = 15, but since the 5 and the 3each represent a number of tens, the actual value of the answer is 1,500 (50  30 = 1,500) Add the twoadditional hundreds carried over from the last multiplication: 15 + 2 = 17 (hundreds) Write the 17 infront of the zeros in the second partial product.

2

34

 541361,700

5 Add the partial products to find the total product:

2

34

 54136+ 1,700

1,836

Note: It is easier to perform multiplication if you write the factor with the greater number of digits in the top row.

In this example, both factors have an equal number of digits, so it does not matter which is written on top

1 Divide the total amount ($54) by the number of ways the money is to be split (3) Work from left to right.

How many times does 3 divide 5? Write the answer, 1, directly above the 5 in the dividend, since both the

5 and the 1 represent a number of tens Now multiply: since 1(ten)  3(ones) = 3(tens), write the 3under the 5, and subtract; 5(tens) – 3(tens) = 2(tens)

1354

–3

2

2 Continue dividing Bring down the 4 from the ones place in the dividend How many times does 3 divide

24? Write the answer, 8, directly above the 4 in the dividend Since 3  8 = 24, write 24 below the other 24and subtract 24 – 24 = 0

18354

–3↓

24 –24

Working with Integers

Remember, an integer is a whole number or its opposite Here are some rules for working with integers:

A DDING

Adding numbers with the same sign results in a sum of the same sign:

(positive) + (positive) = positive and (negative) + (negative) = negative

When adding numbers of different signs, follow this two-step process:

1 Subtract the positive values of the numbers Positive values are the values of the numbers without any

signs

2 Keep the sign of the number with the larger positive value.

–2 + 3 =

1 Subtract the positive values of the numbers: 3 – 2 = 1.

2 The number 3 is the larger of the two positive values Its sign in the original example was positive, so the

sign of the answer is positive The answer is positive 1

Example

8 + –11 =

1 Subtract the positive values of the numbers: 11 – 8 = 3.

2 The number 11 is the larger of the two positive values Its sign in the original example was negative, so

the sign of the answer is negative The answer is negative 3

M ULTIPLYING AND D IVIDING

A simple method for remembering the rules of multiplying and dividing is that if the signs are the same when tiplying or dividing two quantities, the answer will be positive If the signs are different, the answer will be nega-tive

mul-(positive)  mul-(positive) = positive = positive

(positive)  (negative) = negative = negative

(negative)  (negative) = positive = positive

Sequence of Mathematical Operations

There is an order in which a sequence of mathematical operations must be performed:

P: Parentheses/Grouping Symbols Perform all operations within parentheses first If there is more than

one set of parentheses, begin to work with the innermost set and work toward the outside If morethan one operation is present within the parentheses, use the remaining rules of order to determinewhich operation to perform first

E: Exponents Evaluate exponents.

M/D: Multiply/Divide Work from left to right in the expression.

A/S: Add/Subtract Work from left to right in the expression.

This order is illustrated by the following acronym PEMDAS, which can be remembered by using the first

let-ter of each of the words in the phrase: Please Excuse My Dear Aunt Sally.

Listed below are several properties of mathematics:

■ Commutative Property: This property states that the result of an arithmetic operation is not affected by

reversing the order of the numbers Multiplication and addition are operations that satisfy the tive property

■ Associative Property: If parentheses can be moved to group different numbers in an arithmetic

problem without changing the result, then the operation is associative Addition and multiplication are associative

Examples

2 + (3 + 4) = (2 + 3) + 4

2(ab) = (2a)b

■ Distributive Property: When a value is being multiplied by a sum or difference, multiply that value by

each quantity within the parentheses Then, take the sum or difference to yield an equivalent result

A DDITIVE AND M ULTIPLICATIVE I DENTITIES AND I NVERSES

■ The additive identity is the value which, when added to a number, does not change the number For all

of the sets of numbers defined above (counting numbers, integers, rational numbers, etc.), the additiveidentity is 0

Examples

5 + 0 = 5

–3 + 0 = –3

Adding 0 does not change the values of 5 and –3, so 0 is the additive identity

■ The additive inverse of a number is the number which, when added to the number, gives you the

addi-tive identity

Example

What is the additive inverse of –3?

This means, “what number can I add to –3 to give me the additive identity (0)?”

–3 + _ = 0

–3 + 3 = 0

The answer is 3

■ The multiplicative identity is the value which, when multiplied by a number, does not change the

number For all of the sets of numbers defined previously (counting numbers, integers, rational numbers,etc.) the multiplicative identity is 1

Examples

5  1 = 5

–3  1 = –3

Multiplying by 1 does not change the values of 5 and –3, so 1 is the multiplicative identity

■ The multiplicative inverse of a number is the number which, when multiplied by the number, gives you

the multiplicative identity

Example

What is the multiplicative inverse of 5?

This means, “what number can I multiply 5 by to give me the multiplicative identity (1)?”

5  _ = 1

5 1

5 = 1

The answer is 15

There is an easy way to find the multiplicative inverse It is the reciprocal, which is obtained by reversing

the numerator and denominator of a fraction In the above example, the answer is the reciprocal of 5; 5 can bewritten as 51, so the reciprocal is 15

Some numbers and their reciprocals:

Factors and Multiples

The factors of 24 = 1, 2, 3, 4, 6, 8, 12, and 24

The factors of 18 = 1, 2, 3, 6, 9, and 18

From the examples above, you can see that the common factors of 24 and 18 are 1, 2, 3, and 6 From this list

it can also be determined that the greatest common factor of 24 and 18 is 6 Determining the greatest common

factor (GCF) is useful for simplifying fractions.

Example

Simplify 1260

The factors of 16 are 1, 2, 4, 8, and 16 The factors of 20 are 1, 2, 4, 5, and 20 The common factors of 16 and

20 are 1, 2, and 4 The greatest of these, the GCF, is 4 Therefore, to simplify the fraction, both numerator anddenominator should be divided by 4

The number 35 is, therefore, a multiple of the number 5 and of the number 7 Other multiples of 5 are 5,

10, 15, 20, etc Other multiples of 7 are 7, 14, 21, 28, etc

The common multiples of two numbers are the multiples that both numbers share.

Example

Some multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36

Some multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48

Some common multiples are 12, 24, and 36 From the above it can also be determined that the least

com-mon multiple of the numbers 4 and 6 is 12, since this number is the smallest number that appeared in both lists

The least common multiple, or LCM, is used when performing addition and subtraction of fractions to find the

least common denominator

Example (using denominators 4 and 6 and LCM of 12)

) )

+ 56

( (

2 2

) )

The most important thing to remember about decimals is that the first place value to the right of the decimal point

is the tenths place The place values are as follows:

In expanded form, this number can also be expressed as:

1,268.3457 = (1  1,000) + (2  100) + (6  10) + (8  1) + (3  1) + (4  01) + (5  001) + (7

 0001)

1 T H O U S A N D S

2 H U N D R E D S

6 T E N S

8 O N E S

• D E C I M A L

3 T E N T H S

4 H U N D R E D T H S

5 T H O U S A N D T H S

7 T E N T H O U S A N D T H SPOINT

A DDING AND S UBTRACTING D ECIMALS

Adding and subtracting decimals is very similar to adding and subtracting whole numbers The most importantthing to remember is to line up the decimal points Zeros may be filled in as placeholders when all numbers donot have the same number of decimal places

Example

What is the sum of 0.45, 0.8, and 1.36?

1 1

0.450.80+ 1.362.61

Take away 0.35 from 1.06

10

.016–0.350.71

M ULTIPLICATION OF D ECIMALS

Multiplication of decimals is exactly the same as multiplication of integers, except one must make note of the totalnumber of decimal places in the factors

Example

What is the product of 0.14 and 4.3?

First, multiply as usual (do not line up the decimal points):

4.3 .14172+ 430602

Now, to figure out the answer, 4.3 has one decimal place and 14 has two decimal places Add in order to mine the total number of decimal places the answer must have to the right of the decimal point In this problem,there are a total of 3 (1 + 2) decimal places When finished multiplying, start from the right side of the answer,and move to the left the number of decimal places previously calculated

deter-In this example, 602 turns into 602 since there have to be 3 decimal places in the answer If there are notenough digits in the answer, add zeros in front of the answer until there are enough.

Example

Multiply 0.03  0.2

.03

 26

There are three total decimal places in the problem; therefore, the answer must contain three decimalplaces Starting to the right of 6, move left three places The answer becomes 0.006

D IVIDING D ECIMALS

Dividing decimals is a little different from integers for the set-up, and then the regular rules of division apply It

is easier to divide if the divisor does not have any decimals In order to accomplish that, simply move the decimalplace to the right as many places as necessary to make the divisor a whole number If the decimal point is moved

in the divisor, it must also be moved in the dividend in order to keep the answer the same as the original lem; 4 ÷ 2 has the same solution as its multiples 8 ÷ 4 and 28 ÷ 14, etc Moving a decimal point in a division prob-lem is equivalent to multiplying a numerator and denominator of a fraction by the same quantity, which is thereason the answer will remain the same

prob-If there are not enough decimal places in the answer to accommodate the required move, simply add zerosuntil the desired placement is achieved Add zeros after the decimal point to continue the division until the dec-imal terminates, or until a repeating pattern is recognized The decimal point in the quotient belongs directly abovethe decimal point in the dividend

Example

What is 4251.53 ?

First, to make 425 a whole number, move the decimal point 3 places to the right: 425

Now move the decimal point 3 places to the right for 1.53: 1,530

The problem is now a simple long division problem

3.6425.1,530.0

–1,275↓

2,550–2,5500