Math Flashcards
What is the fraction representation of 105/377?
105/377
This represents a ratio of two integers.
How can 1025.0749 be expressed in sums of powers of tens?
1.0 × 10^3 + 0.0 × 10^2 + 2.0 × 10^1 + 5.0 × 10^0 + 0.7 × 10^-1 + 4.9 × 10^-2
This breaks down the number into its constituent parts based on decimal placement.
What is the decimal form of 113,076/10,000?
11.3076
This result is obtained by moving the decimal point four places to the left.
What is the decimal result of dividing 7 by 8?
0.875
This division results in a repeating decimal.
What is the fraction form of 0.0053?
53/10,000
This fraction is simplified from the decimal by recognizing the place value.
Convert 74% to a fraction.
74/100 Simplified = 3/50
This conversion involves expressing the percentage as a fraction out of 100 and simplifying.
Convert 125% to decimal.
1.25
This conversion is done by dividing the percentage by 100.
How do you convert 0.65 to a percentage?
Move decimal two places to the right and add percent symbol
This method converts a decimal to a percentage.
What is the fraction form of 0.876?
7/8
This fraction represents the decimal value precisely.
What is the prime factorization of 144?
2 x 2 x 2 x 2 x 3
The prime factorization indicates that 144 can be expressed as the product of prime numbers.
What is the lowest prime number that divides 144 with zero remainder?
2
The factorization process starts with the lowest prime number, which in this case is 2.
What do we call numbers that have factors other than one and itself?
Composite numbers
144 is classified as a composite number because it has multiple prime factors.
What is the next prime number after 2 that divides 9?
3
9 is not divisible by 2, but it can be divided by 3.
True or False: The prime factorization process can end with a prime number.
True
In this case, after dividing by 3, the quotient is 3, which is a prime number.
Fill in the blank: 144 is a _______ number.
composite
This indicates that 144 has factors other than just 1 and itself.
How do you multiply two fractions?
Multiply the numerators and multiply the denominators (multiply ‘straight across,’ top and bottom).
What is the product of the numerators and denominators in the example 3/5 x 5/9?
15/45
What is the simplified form of 15/45?
⅓
What is the reciprocal of a nonzero fraction a/b?
b/a
What is the product of a fraction and its reciprocal?
1
Fill in the blank: The reciprocal of 3/5 is _______.
5/3
True or False: The product of 3/5 and its reciprocal is 1.
True
What is the reciprocal of ¾?
½
The product of a number and its reciprocal is always 1.
What is the product of ¾ and its reciprocal?
1
The product is calculated as ¾ × ½ = 3 × 4 = 12.
Define the least common multiple (LCM).
The smallest number that is a multiple of both numbers
For example, the LCM of 2 and 3 is 6.
How do you find the LCM of two numbers?
Find the prime factorization of each number
The LCM will be a product of the different factors of both numbers, taking the greatest number of occurrences.
What is the LCM of 2 and 3?
6
6 is the smallest number that is a multiple of both 2 and 3.
Fill in the blank: The LCM will occur the ______ number of times it appears in either one of the factorizations.
greatest
This ensures that all factors are accounted for in the LCM.
What is the first step to add or subtract fractions with the same denominator?
Add or subtract their numerators
The denominator remains the same.
What must you find to add or subtract fractions with different denominators?
The least common denominator (LCD)
The least common denominator is the LCM of all the denominators.
To compute 4 + 1 - 7, what is the answer?
19 + 3 - 7 - 1 + 2 - 7 - 3
This illustrates the addition and subtraction of fractions.
Fill in the blank: The least common denominator is the _______ of all the denominators.
LCM
True or False: To add fractions with the same denominator, you must change the denominators.
False
The denominator remains the same when adding or subtracting fractions with the same denominator.
What is the process for adding or subtracting fractions with different denominators?
Find the least common denominator, change the fractions, then add or subtract
This ensures that the fractions are compatible for addition or subtraction.
What does LCM stand for in the context of adding fractions?
Least Common Multiple
What is the LCM of 15, 21, and 70?
210
The least common multiple (LCM) can be verified by multiplying factors: 15 × 14 = 210, 21 × 10 = 210, and 70 × 3 = 210.
Fill in the blank: The LCM of 15, 21, and 70 is _______.
210
What operation is used to find the LCM of a set of numbers?
Multiplication of their factors
True or False: The LCM of 15, 21, and 70 is a product of their least factors.
True
What does LCM stand for?
Least Common Multiple
What is the first step in performing arithmetic operations with rational numbers?
Identify the rational numbers involved
How can the task of adding a whole number and a fraction be simplified?
Multiply the whole number by the denominator and add the numerator
For example, to add 5 and 2/3, multiply 5 by 3 and then add 2.
What is the method to convert a fraction back to mixed-number form?
Divide the numerator by the denominator and write the quotient next to the remainder over the denominator
For example, converting 17/3 gives a quotient of 5 and a remainder of 2, resulting in 5 2/3.
What is the process for adding, subtracting, multiplying, and dividing mixed numbers?
Convert them to single improper fractions and then perform the operations as usual
Fill in the blank: To convert a mixed number to an improper fraction, you multiply the whole number by the _______ and add the numerator.
denominator
True or False: Mixed numbers can be added directly without conversion to improper fractions.
False
What is the additive inverse of a number?
The negative of a number
The additive inverse of a number is a value that, when added to the original number, results in zero.
What is the additive inverse of 7?
-7
The additive inverse of 7 is -7 because 7 + (-7) = 0.
What is the additive inverse of -7?
7
The additive inverse of -7 is 7 because -7 + 7 = 0.
What is the result of multiplying two negative integers, -3 and -5?
15
-3 × (-5) = -1 × 15 = 15
What happens when you multiply two negative numbers?
The result will always be positive
This is because multiplying or dividing two negative numbers involves their absolute values.
What is the result of dividing 27 by -3?
-9
27 ÷ (-3) = -9.
To multiply or divide two negative numbers, you multiply or divide their _______.
absolute values
The result will always be positive.
What is the result of multiplying -7 x (-8)?
56
The multiplication of two negative numbers yields a positive result.
True or False: The additive inverse of a number is always positive.
False
The additive inverse is the opposite of the number, which can be negative.
What is a fraction bar also known as?
Vinculum
What must be done before dividing in a fraction problem?
Perform all calculations above and below the fraction bar
In the example (36 - 6) / (12 + 3), what is the final answer?
2
What is often required when dealing with mixed numbers in arithmetic?
Convert to a fraction first
Fill in the blank: The TEAS preparation can be supported by searching the internet for _______.
order of operations practice
What is the result of the operation -3 x (2) + 2 x 8.23?
0.61
The calculation involves performing arithmetic operations with rational numbers.
Fill in the blank: The operation 0.75 - 6.6 + 1646 results in _______.
1640.15
What is the first step in solving the expression -3 x (2) + 2 x 8.23?
Perform multiplication operations first.
True or False: The expression -5.85 + 1646 can be simplified by adding the two numbers.
True
What is the final answer after performing all arithmetic operations in the expression 0.75 - 3 x 2 + 2 x 8.23?
0.61
List the arithmetic operations performed in the expression: -3 x (2) + 2 x 8.23.
- Multiplication
- Subtraction
- Addition
Fill in the blank: The expression 075 - 3 x 22 + 2 x 8.23 simplifies to _______.
-5.85 + 1646
What type of numbers are involved in the operation -3 x (2) + 2 x 8.23?
Rational numbers
What is the importance of performing arithmetic operations in the correct order?
It ensures accurate results.
What does the expression 0.75 - 6.6 + 1646 equal to?
1640.15
What is the arithmetic operation performed in the example 14.12 × (3.03 - 1.051) + 2.2 × (4.5 + 9.63)?
Perform arithmetic operations with rational numbers
The example illustrates the use of multiplication and addition involving rational numbers.
What is the first step in the calculation of 14.12 × (3.03 - 1.051)?
Calculate the expression inside the parentheses: 3.03 - 1.051
The result of this calculation is essential for the subsequent multiplication.
What is the result of the expression 3.03 - 1.051?
1.979
This value is used in the multiplication with 14.12.
What is the next calculation after finding 1.979 in the expression 14.12 × 1.979?
Multiply 14.12 by 1.979
This yields the first part of the overall expression.
What is the calculation performed in 2.2 × (4.5 + 9.63)?
Calculate the expression inside the parentheses: 4.5 + 9.63
This is followed by multiplying the result by 2.2.
What is the result of the expression 4.5 + 9.63?
14.13
This value is essential for the multiplication by 2.2.
What is the next step after calculating 14.13 in 2.2 × 14.13?
Multiply 2.2 by 14.13
This provides the second part of the overall expression.
What are the final calculations in the expression 14.12 × 1.979 + 2.2 × 14.13?
Add the results of the two multiplications
The final step combines both parts to yield the overall result.
What is the final result of the expression 14.12 × 1.979 + 2.2 × 14.13?
59.02948
This is the total after performing all calculations.
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EXAMPLE 38
₴ - 3 x(2})+2• x 8.23
Answer:
3 -3 x(2=) +2 x 823
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= 0.75 - 6.6 + 2 × 8.23
= 0.75 - 6.6 + 16.46
= -5.85 + 16.46
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What is the purpose of comparing and ordering rational numbers?
To understand the relative sizes of positive and negative numbers.
What are some examples of true inequalities?
0 < 5, 20 > -50, 12 < 13, 10 > 3, -4 < -2, 7 = 7
The last two inequalities are true because the inequality symbols used allow for equality.
Why is 7 < 7 or 7 > 7 not true?
Because both statements imply that 7 is not equal to itself.
How can you visualize the relationship between numbers?
By drawing a number line and noting the relative positions of the numbers.
What is a helpful way to remember the use of inequality symbols?
The large, open end of the symbol faces the larger number, and the small, pointed end points to the smaller number.
What is the main technique used to compare rational numbers?
We can use the techniques for multiplying by 1 to compare two or more rational numbers.
How do you determine which of two fractions is larger?
If the fractions have the same denominator, compare the numerators.
What is a common denominator?
A common denominator is a shared multiple of the denominators of two or more fractions.
How do you find the least common denominator?
List the multiples of each denominator and find the smallest number that appears in both lists.
What are the first few multiples of 7?
14, 21, 28, 35, 42, 49, 56, …
What are the first few multiples of 8?
16, 24, 32, 40, 48, 56, 64, …
What is the main technique used to compare rational numbers?
We can use the techniques for multiplying by 1 to compare two or more rational numbers.
How do you determine which of two fractions is larger?
If the fractions have the same denominator, compare the numerators.
What is a common denominator?
A common denominator is a shared multiple of the denominators of two or more fractions.
How do you find the least common denominator?
List the multiples of each denominator and find the smallest number that appears in both lists.
What are the first few multiples of 7?
14, 21, 28, 35, 42, 49, 56, …
What are the first few multiples of 8?
16, 24, 32, 40, 48, 56, 64, …
What is the focus of ATI TEAS SmartPrep 2.0?
Math
This includes comparing and ordering rational numbers.
What are the multiples of 8?
16, 24, 32, 40, 48, 56, 64, …
These are examples of multiples of 8.
What is the least common denominator in the example given?
56
It is not necessary to find the least common multiple when comparing rational numbers.
What is the quickest way to compare rational numbers?
Multiply the denominators and use the result as a common denominator.
What can be concluded from 49/56 > 48/56?
718 > 61.
How can you compare three or more rational numbers?
Multiply all the denominators to get a common denominator.
Order the following from least to greatest: 2/3, 3/4, 5/8.
5/8, 2/3, 3/4.
What is the objective of M.1.3?
Compare and order rational numbers (including positive and negative numbers).
How can decimal numbers be ordered?
Write the numbers vertically, lining up the decimals.
What is the first step in analyzing decimal numbers for order?
Analyze the digit with the highest place value.
If two numbers start with the same digit in the ones place, what should you compare next?
Compare digits in the tenths place.
What is the decreasing order of the numbers 3.245, 3.524, and 0.3245?
3.524, 3.245, 0.3245.
What is the increasing order of the numbers 3.245, 3.524, and 0.3245?
0.3245, 3.245, 3.524.
How can inequalities be expressed for the numbers 0.3245 and 3.245?
0.3245 < 3.245 or 0.3245 ≤ 3.245.
What should you do if the numbers are not in decimal form?
Divide the fractions or fraction parts to convert them into decimal form.
How do you convert 5 ⅖ into decimal form?
Divide 2 by 7 to get approximately 0.285714.
What is the purpose of algebra?
Algebra allows us to deal with unknown quantities using variables, which stand for unknown numbers.
What symbol is commonly used for multiplication in algebra?
In algebra, multiplication is represented by a dot (•), parentheses, or by placing factors next to each other.
What symbols are used for division in algebra?
Standard symbols for division include the vinculum (/), either oblique or horizontal, and the obelus (÷).
What are the components of an algebraic equation?
An algebraic equation consists of terms and mathematical operations.
What is a term in algebra?
A term is a number, variable, or product of a number and variables, separated by addition and subtraction signs.
What is a coefficient?
A coefficient is the numeric part of a variable term that is being multiplied by the variable.
What is the coefficient of a variable written without a numerical part?
A variable written without a numerical part has a coefficient of one.
Identify the parts of the algebraic equation 8 - 10x + 2 = 11.
Variables: x; Coefficients: -10; Constants: 8, 2, 11.
What is a coefficient?
A coefficient is the numeric part of a variable term that is being multiplied by the variable.
What is the coefficient of x in the term -10x?
The coefficient of x in the term -10x is -10.
What is the coefficient of a variable that is written without a numerical part?
A variable written without a numerical part has a coefficient of one.
Identify the parts of the algebraic equation 8 - 10x + 2 = 11.
The variable terms are -10x, the coefficients are -10, and the constants are 8, 2, 11.
What is a solution in an algebraic equation?
A solution is any number(s) that can be substituted for the variable that makes the equation true.
What is the solution for the equation 5x = 20?
The solution is x = 4, because 5 • 4 = 20.
What are the variable term, coefficient, and constant in the equation 5x = 20?
The variable term is 5x, the coefficient is 5, and the constant is 20.
What is the addition principle in algebra?
The addition principle states that in a true equation where a = b, adding any number c to both sides results in a true equation: a + c = b + c.
What happens if c is less than 0 in the addition principle?
If c < 0, it is equivalent to subtracting c from both sides of the equation.
What are the inverse operations of addition?
The inverse operation of addition is subtraction.
How do you isolate a variable using the addition principle?
To isolate a variable, you can add or subtract the same number from both sides of the equation.
Solve the equation x + 10 = -4.
To isolate x, subtract 10 from both sides: x + 10 - 10 = -4 - 10, resulting in x = -14.
What is the goal of M.1.4 in ATI TEAS SmartPrep 2.0?
Solve equations with one variable.
What is the multiplication principle?
In a true equation where a = b, multiplying both sides by any number c results in a true equation: a * c = b * c. This also applies to division by a nonzero number.
Division by c is the same as multiplication by the reciprocal 1/c.
How can you solve the equation 2x = 8?
You can isolate x by either dividing both sides by 2 or multiplying both sides by the reciprocal of 2 (1/2).
