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How do you know if an integer is a multiple of 3 or 9?
An integer is divisible by 3 if the sum of its digits is divisible by 3.
An integer is divisible by 9 if the sum of its digits is divisible by 9.
What is the definition of an integer?
Integers are whole numbers and their opposites; they include negative whole numbers and zero.
What is the definition of a rational number?
A rational number is a number that can be expressed as a ratio of two integers or as a terminating or repeating decimal.
What is the definition of an irrational number?
Irrational numbers are real numbers—they have locations on the number line—but they can't be expressed precisely as a fraction or decimal. They are nonrepeating, nonterminating decimals.
How do you add a positive number to a negative number?
First, ignore the signs and find the positive difference between the number parts. Then attach the sign of the original number with the larger absolute value.
How do you multiply and divide positive and negative numbers?
Treat the number parts as you usually would, then attach a negative sign if there is an odd number of negatives.
Explain the memory device PEMDAS.
PEMDAS means Parentheses first, then Exponents, then Multiplication and Division (left to right), and lastly Addition and Subtraction (left to right).
How do you count consecutive integers?
Subtract the smallest consecutive integer from the largest and add 1.
How do you express the union of a set?
The union of Set A and Set B, sometimes expressed as A χ B, is the set of elements that are in either or both Set A and Set B.
How do you express the intersection of a set?
The intersection of Set A and Set B, sometimes expressed as , is the set of elements common to both Set A and Set B.
What is the difference between a factor and a multiple?
The factors of integer are the positive integers that divide into with no remainder. The multiples of are the positive integers that divides into with no remainder.
What is the prime factorization of a number?
To find the prime factorization of an integer, just keep breaking it up into factors until all the factors are prime numbers.
What are numbers that are relatively prime?
Two relatively prime numbers are integers that have no common factor other than 1. To determine this, break down both numbers to their prime factorizations.
How do you find the least common multiple (LCM) of two or more numbers?
To find the LCM, check out the multiples of the larger integer until you find one that's also a multiple of the smaller.
How do you find the greatest common factor (GCF) of two or more numbers?
To find the GCF, break down the integers into their prime factorizations and multiply all the prime factors they have in common.
How do you know if the sum, difference, or product of several numbers will be even or odd?
There are rules—"odd times even is even," for example—but there's no need to memorize them. Instead, just perform the operation on simple numbers, like 1 and 2, and see what happens. What happens with one set of numbers generally happens with all similar sets.
How do you know if an integer is a multiple of 2 or 4?
An integer is divisible by 2 (even) if the last digit is even.
An integer is divisible by 4 if the last two digits form a multiple of 4.
How do you know if an integer is a multiple of 5 or 10?
An integer is divisible by 5 if the last digit is 5 or 0. An integer is divisible by 10 if the last digit is 0.
What is the definition of a remainder?
The remainder is the whole number left over after division.
How do you simplify a fraction to lowest terms?
To reduce a fraction to lowest terms, factor it out and then cancel all factors the numerator and denominator have in common.
How do you add and subtract fractions?
To add or subtract fractions, first find a common denominator, then add or subtract the numerators.
How do you multiply fractions?
To multiply fractions, simply multiply the numerators and multiply the denominators.
How do you divide fractions?
To divide fractions, change the division sign to a multiplication sign, invert the second fraction, and multiply.
How do you convert a mixed number to an improper fraction?
Multiply the whole number part by the denominator, then add the numerator. The result is a new numerator over the same denominator.
How do you convert an improper fraction to a mixed number?
Divide the denominator into the numerator to get a whole number quotient with a remainder. The quotient becomes the whole number part of the mixed number, and the remainder becomes the new numerator (with the same denominator).
How do you find the reciprocal of a fraction?
To find the reciprocal of a fraction, switch the numerator and the denominator.
How do you compare the values of two or more fractions?
One way to compare fractions is to express them with a common denominator.
How do you convert a fraction to a decimal?
To convert a fraction to a decimal, divide the bottom into the top.
How do you convert a decimal to a fraction?
To convert a decimal to a fraction, set the decimal over 1 and multiply the numerator and denominator by 10 raised to the number of digits to the right of the decimal point.
What formula is used to find a percent?
Whether you need to find the part, the whole, or the percent, use the same formula:
part = percent * whole
How do you increase or decrease a number by a certain percentage?
To increase a number by a percent, add the percent to 100 percent, convert the result to a decimal, and multiply. To decrease a number by a percent, subtract the percent from 100 percent, convert the result to a decimal, and multiply.
How do you find an original value, that is, a value before it was increased or decreased by a certain percentage?
To find the original whole before a percent increase or decrease, set up an equation that multiplies the percent change (as a decimal) by the original number. For example, think of the result of a 15 percent increase of x as 115x .
When given a series of percent increases and decreases, how do you determine the ending value?
To determine the combined effect of multiple percent increases and/or decreases, start with 100 and see what happens.
How do you set up a ratio?
To find a ratio, put the number associated with the word of on top and the number associated with the word to on the bottom and simplify.
How do you convert part-to-part ratios to part-to-whole ratios?
If the parts add up to the whole, a part-to-part ratio can be turned into two part-to-whole ratios by putting each number in the original ratio over the sum of the numbers.
How do you solve a proportion?
To solve a proportion, cross multiply.
How do you solve a problem involving rates?
Use the units in the problem to keep things straight.
How do you find the average rate?
Average rate is the change in the value of a quantity divided by the elapsed time.
What formula is used to find average?
average = sum of the terms / number of the terms
How do you quickly find the average of a series of evenly spaced numbers?
To find the average of evenly spaced numbers, just average the smallest and the largest.
If you have an average of a series of numbers, how can you find their sum?
Use the formula:
sum = (average) x (number of terms)
If the average of ten numbers is 50, then they add up to , or 500.
How do you find the missing number in a series when you are given the average?
To find a missing number when you're given the average, use the sum.
How do you find the median of a set of numbers?
The median of a set of numbers is the value that falls in the middle of the set.
How do you find the mode of a set of numbers?
The mode of a set of numbers is the value that appears most often.
How do you count the total number of possible outcomes of more than one event?
Use the fundamental counting principle: if there are m ways one event can happen and n ways a second event can happen, then there are (m) x (n) ways ways for the two events to happen in combination. For example, with five shirts and seven pairs of pants to choose from, you can have (5) x (7) = 35 different outfits.
How do you calculate the probability that an event will take place?
probability = number of favorable outcomes / total possible outcomes
How do you multiply expressions with exponents that have a common base?
To multiply powers with the same base, add the exponents and keep the same base.
How do you divide expressions with exponents that have a common base?
To divide powers with the same base, subtract the exponents and keep the same base.
How do you raise a power to a power?
To raise a power to a power, multiply the exponents.
How do you simplify square roots?
To simplify a square root, factor out the perfect squares under the radical, take their square roots, and put the result in front.
How do you add and subtract roots?
You can add or subtract radical expressions when the part under the radicals is the same. Just add or subtract the coefficients in front of the radicals.
How do you multiply and divide roots?
The product of more than one square root is equal to the square root of their total product.
The quotient of more than one square root is equal to the square root of their total quotient:
How do you determine the absolute value of a number?
The absolute value of a number is the distance of the number from zero on the number line. Because absolute value is a distance, it is always positive.
How do you evaluate an algebraic expression?
To evaluate an algebraic expression, plug in the given values for the unknowns and calculate according to the correct order of operations, or PEMDAS.
How do you add and subtract monomials?
First, make sure that the terms to be combined have exactly the same variable(s). This makes them like terms. Then, to combine like terms, keep the variable part unchanged while adding or subtracting the coefficients.
How do you add and subtract polynomials?
Combine like terms.
How do you multiply monomials?
Multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying like bases.
How do you multiply binomials?
Use FOIL: First terms, Outer terms, Inner terms, Last terms.
How do you factor the difference of two squares?
Use the following formula:
a2 - b2 = (a + b) (a - b)
How do you simplify an algebraic fraction?
Find factors common to the numerator and denominator and then cancel them. Thus, simplifying an algebraic fraction begins with factoring.
How do you solve a linear equation?
Do whatever is necessary to both sides to isolate the variable.
What does solving in terms of mean?
To solve an equation for one variable in terms of another means to isolate the one variable on one side of the equation, leaving an expression containing the other variable on the other side of the equation.
How do you solve a quadratic equation?
Put the quadratic equation in the form, factor the left side (if you can), and set each factor equal to 0 separately to get the two solutions.
How do you solve an inequality?
To solve an inequality, do whatever is necessary to both sides to isolate the variable. Remember that when you multiply or divide both sides by a negative number, you must reverse the inequality symbol.
How do you solve a radical equation?
A radical equation contains at least one radical expression. Solve radical equations by isolating the radical expression and using standard rules of algebra.
What is a function, and how do you evaluate one?
Standard function notation is written f(x) and read as "f of x."
Describe direct variation.
In direct variation, , where is a nonzero constant. The variable changes directly as does.
Describe inverse variation.
In inverse variation, xy = k, where x and y are variables and k is a constant. The larger x is, the smaller y is, and vice versa.
How do you find the domain of a function?
The domain of a function is the set of values for which the function is defined.
How do you find the range of a function?
The range of a function is the set of outputs or results of the function.
How do you find the distance between two points on a coordinate graph?
Use the distance formula:
d = square root of (x1 - x2)2 + (y1 - y2)2
How do you find the slope of a line when given two points on the line?
slope = (change in y) / (change in x) = (rise) / (rise)
If given an equation of a line, how do you find its slope?
Put the equation into the slope-intercept form: y = mx + b
If given an equation of a line, how do you find its -intercept?
Either put the equation into y = mx + b (slope-intercept) form—in which case b is the y-intercept—or just plug x = 0 into the equation and solve for y = 0 into the equation and solve for x.
How do you find the midpoint between two points?
The midpoint is the average of the x-coordinates of the endpoints and the average of the y-coordinates of the endpoints. If the endpoints are (x1, y1) and (x2, y2), the midpoint is (x1 + x2) / 2 (y1 + y2) / 2.
What types of angles form when two lines intersect?
When two lines intersect, adjacent angles are supplementary, and vertical angles are equal.
What are the properties of similar triangles?
Similar triangles have the same shape. Their corresponding angles are equal, and their corresponding sides are proportional.
How do you find the area of a triangle?
area of triangle = 1/2 (base)(height)
The height is the perpendicular distance between the side that's chosen as the base and the opposite vertex.
What is the Triangle Inequality Theorem?
The Triangle Inequality Theorem says that the length of one side of a triangle must be greater than the difference and less than the sum of the lengths of the other two sides.
What are isosceles triangles?
Isosceles triangles are triangles with two equal sides. Not only are two sides equal, but the angles opposite the equal sides, called base angles, are also equal.
What are equilateral triangles?
Equilateral triangles are triangles in which all three sides are equal. Because all the sides are equal, all the angles are also equal. All three angles in an equilateral triangle measure 60 degrees, regardless of the lengths of sides.
What is the Pythagorean theorem?
a2 + b2 = c2
What are two special Pythagorean triples?
3:4:5 and 5:12:13
If a right triangle's leg-to-leg ratio is 3:4, or if the leg-to-hypotenuse ratio is 3:5 or 4:5, it's a 3-4-5 triangle and you don't need to use the Pythagorean theorem to find the third side. Just figure out what multiple of 3-4-5 it is. Same goes for the 5:12:13 ratio.
What are the properties of a 30-60-90 triangle?
The sides of a 30-60-90 triangle are in a ratio of x, x square root of 3, and 2x. You don't need the Pythagorean theorem to solve for the length of a side of a 30-60-90 triangle.
What are the properties of a 45-45-90 triangle?
The sides of a 45-45-90 triangle are in a ratio of x, x, x square root of 2.
What are the characteristics of a rectangle?
A rectangle is a four-sided figure with four right angles. Opposite sides are equal. Diagonals are equal. The perimeter of a rectangle is equal to the sum of the lengths of the four sides, which is equivalent to 2 (length + width).
The area of a rectangle = (length) x (width)
What are the characteristics of a parallelogram?
A parallelogram has two pairs of parallel sides. Opposite sides are equal. Opposite angles are equal. Consecutive angles add up to 180 degrees.
The area of a parallelogram = base x height
What are the characteristics of a square?
A square is a rectangle with four equal sides.
The perimeter of a square is equal to four times the length of one side.
The area of a square is equal to one side squared.
How do you find the sum of the interior angles of a polygon?
sum of the angles = (n - 2) x 180
The sum of the measures of the interior angles of a polygon = (n - 2) x 180, where is the number of sides.
How do you find the circumference of a circle?
circumference = 2 (3.14159...) r
How do you find the length of an arc in a circle?
An arc is a piece of the circumference. If n is the degree measure of the arc's central angle, then the formula is:
length of an arc = n / 360 x 2 (3.14159...) r
How do you find the area of a circle?
area of a circle = (3.14159...) r2
How do you find the area of a sector?
A sector is a piece of the area of a circle. If is the degree measure of the sector's central angle, then the formula is:
area of a sector = n / 360 x (3.14159...) r2
What is an important property of a line tangent to a circle?
When a line is tangent to a circle, or touches the circle at one point, the radius of the circle is perpendicular to the line at the point of contact.
How do you find the surface area of a rectangular solid?
The surface of a rectangular solid consists of three pairs of identical faces. To find the surface area, find the area of each face and add them up. If the length is , the width is , and the height is , the formula is:
surface area = 2lw + 2wh +2lh
How do you find the volume of a rectangular solid?
volume of a rectangular solid = lwh
How do you find the volume of a cube?
A cube is a rectangular solid with length, width, and height all equal. If e is the length of an edge of a cube, the volume formula is:
volume of a cube = e3
How do you find the volume of a cylinder?
volume of a cylinder = (3.14159...) r2 h
How do you factor a polynomial (or perform FOIL "in reverse")?
Think about what binomials you could use FOIL on to get that quadratic expression.
How do you identify which number of a fraction is the part and which number is the whole?
The key to solving most story problems involving fractions and percents is to identify the part and the whole. Usually, you'll find the part associated with the verb is/are and the whole associated with the word of.
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