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SAT Math

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1. Direct Variation and Inverse Variation
Express them with a common denominator.
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
Direct Variation- y=kx - where k is constant. As x gets larger - y gets larger. If the number of units of B were to double - the number of units of A would double - Inverse Variation- xy=k - where k is a constant. As x gets larger - y gets smaller. A

2. Definition of Remainder
Standard function notation is written f(x) and read 'f of x.' To evaluate the function f(x)=2x+3 for f(4) - replace every x with 4 and simplify. f(4)=2(4)+3=11.
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
The whole number left over after division.
VofaC: (pie)r(2)h

3. If x varies directly with y - and the value of x is 12 when y is 11 - what is the value of y when x is 66?
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.
Ignore the signs... find the positive difference between the numbers - Then attach the sign of the original number with the larger absolute value
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides
Slope=change in y/change in x

4. Knowing if an Integer is a Multiple of 2 or 4
Ue the formula y=mx+b - and remember that the value of m is the slope of the line. The slopes of two perpendicular lines are negative reciprocals of one another.
Put the equation into y=mx+b-- in which case b is the y-intercept.
-An integer is divisible by 2 if the last digit is even - An integer is divisible by 4 if the last two digits form a multiple of 4.
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.

5. PEMDAS
Cross multiply.
Parenthesis - Exponents - Multiplication/Division - Addition/Subtraction
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
VofaC: (pie)r(2)h

6. Finding a Term in a Geometric Sequence
Integers are whole numbers and their opposites; they include negatives of whole numbers and zero.
The 3 interior angles of a any triangle add up to 180 degrees. An exterior angle of a triangle is equal to the sum of the remote - that is - non-adjacent - interior angles.
Use the distributive property then combine the like terms.
Use the formula - a(n)=a(1)r(n-1) - where a(n) is the nth term - a1 is the first term - and r is the ratio between the terms.

7. Finding the Area of a Circle
(pie)r(squared)
Use the formula - a(n)=a(1)r(n-1) - where a(n) is the nth term - a1 is the first term - and r is the ratio between the terms.
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
When a line is tangent to a circle - or touches the circle at just one point - the radius of the circle is perpendicular to the line at the point of contact.

8. Finding the Distance Between Two Points on a Coordinate Graph
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating decimal.
Factor out and cancel all factors the numerator and denominator have in common.
Use the distance formula: d=(square root over) (x2-x2)2+(y1-y2)2
Multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying like bases. 2a3a=(23)(a*a)=6a^2

9. Simplifying Square Roots
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.
If there are m ways one event can happen and n ways a second event can happen - then there are m*n ways for the 2 events to happen.
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
Factor out the perfect squares under the radical - take their square roots - and put the result in front.

10. Solving a Proportion
Cross multiply.
Subtract the exponents and keep the same base.y^13/y^8=y^13-8=y^5
Combine like terms.
Isolate the radical expression and use the standard rules of algebra.

11. Identifying Which Number of a Fraction is the Part and Which Number is the Whole

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12. Finding the Missing Number in a Series When You are Given the Average
Part=Percent x Whole
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
Use the sum... If the average of 4 numbers is 7 - then the sum of those 4 numbers is 4*7 - or 28. Suppose that 3 of the numbers are 3 -5 - and 9. These 3 numbers add up to 16 of that 28 - 12 is the fourth number.
Area of a Triangle=1/29(base)(height) - The height is the perpendicular distance between the side that's chosen as the base and the opposite vertex.

13. Multiply and Divide Positive and Negative Numbers
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds
(x1+x2)/2 -(y1+y2)/2
Parenthesis - Exponents - Multiplication/Division - Addition/Subtraction
Multiply and/or divide positives and negatives treat the number parts as usual - and attach a negative sign if there is an odd number of negatives

14. Finding the Rate of Speed
Rewrite the number without the negative sign as the bottom of a fraction with 1 as the top.
(pie)r(squared)
Use the formula distance=rate*time and its variations to help in this question.
Four equal acute angles and four equal obtuse angles.

15. Finding the Slope When Given an Equation of a Line
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24
-The Factors of integer n are the positive integers that divide into n with no remainder - The Multiples of n are the positive integers that n divides into with no remainder.

16. If f(x)=x(1/3)+1/3x - then what is the value of f(27)?
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
The value that falls in the middle of the set.
Four equal acute angles and four equal obtuse angles.
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%

17. Calculating the Probability that an Event will Take Place
When a line is tangent to a circle - or touches the circle at just one point - the radius of the circle is perpendicular to the line at the point of contact.
An integer is divisible by 3 if the sum of its digits is divisible by 3 - - An integer is divisible by 9 is the sum of its digits is divisible by 9.
Isolate the radical expression and use the standard rules of algebra.
Probability: Number of Favorable Outcomes/Total Possible Outcomes

18. Setting Up a Ratio
Put the number associated with the word of on top and the quantity associated with the word to on the bottom and simplify.
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.
Values of x the take the denominator of a faction equal to zero - thereby dividing by zero which is not possible - are not in the domain of a function. Set the denominator equal to zero and solve for x. Ans: 3
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides

19. Quickly Finding the Average of a Series of Evenly Spaced Numbers
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.
Area of a Triangle=1/29(base)(height) - The height is the perpendicular distance between the side that's chosen as the base and the opposite vertex.
Average the smallest and largest number
-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.

20. Finding the GCF of Two or More Numbers
Average the smallest and largest number
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
Use the formula distance=rate*time and its variations to help in this question.
Sum: Average x Number of Terms

21. Simplifying a Fraction to Lowest Terms
-An integer is divisible by 2 if the last digit is even - An integer is divisible by 4 if the last two digits form a multiple of 4.
Cross multiply.
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.
Factor out and cancel all factors the numerator and denominator have in common.

22. Formula to Find Average
If a right triangle's leg-to-leg ratio is 3:4 - or if the leg-to-hypotenuse ratio is 3:5 - it's a 3-4-5 triangle and you don't need to use the Pythagorean theorem to find the third side.
Average: Sum of the Terms/Number of the Terms
Two relatively prime numbers are integers that have no common factor other than 1.
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24

23. Adding a Positive Number to a Negative Number
(x1+x2)/2 -(y1+y2)/2
The union of Set A and Set B is the set of elements that are in either or both Set A and Set B - The intersection of Set A and Set B is the set of elements common to both Set A and Set B
If there are m ways one event can happen and n ways a second event can happen - then there are m*n ways for the 2 events to happen.
Ignore the signs... find the positive difference between the numbers - Then attach the sign of the original number with the larger absolute value

24. Multiplying and Dividing Roots
Factor out and cancel all factors the numerator and denominator have in common.
Scalene- No sides or angles are equal - Isosceles- Have two equal sides. The angles opposite the equal sides are also equal - Equilateral- All three sides and angles are equal.
FOIL: First - Outer - Inner - Last... Combine Like Terms
Multiplying: Multiply the numbers under the different radicals. The product goes under one radical - Dividing: The quotient of more than one square root is equal to the square root of their total quotient.

25. Difference Between a Factor and a Multiple
-The Factors of integer n are the positive integers that divide into n with no remainder - The Multiples of n are the positive integers that n divides into with no remainder.
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)
The sides of a 45-45-90 triangle are in a ratio of x:x:x(square root)2.

26. Multiplying Fractions
Express them with a common denominator.
Use the formula - a(n)=a(1)r(n-1) - where a(n) is the nth term - a1 is the first term - and r is the ratio between the terms.
Multiply the numerators and multiply the denominators.
Multiply the exponents - (x^3)^4=x^3*4=x^12

27. Dividing Fractions
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides
Change the division sign to a multiplication sign - invert the second fraction - and multiply.
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.
Average A per B= Total A/Total B so Average Speed = Total Distance/Total Time

28. If the average of a - b - and 48 is 48 - what is the value of a + b?
Put the equation into y=mx+b-- in which case b is the y-intercept.
Set the quadratic equation equal to zero with the terms in order of decreasing exponents - and solve for the roots by factoring. Don't forget to then find the difference in the roots by subtracting them. Ans: 2
VofaC: (pie)r(2)h
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96

29. Adding and Subtracting Polynomials
Combine like terms.
Change the division sign to a multiplication sign - invert the second fraction - and multiply.
The 3 interior angles of a any triangle add up to 180 degrees. An exterior angle of a triangle is equal to the sum of the remote - that is - non-adjacent - interior angles.
Use the units to keep things straight - Snowfall inches/hours

30. Multiplying Monomials
Use the distance formula: d=(square root over) (x2-x2)2+(y1-y2)2
Slope=change in y/change in x
Sector is a piece of the area of a circle. Area of a Sector: (n/360)((pie)r2) - n is the degree measure of the sector's central angle
Multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying like bases. 2a3a=(23)(a*a)=6a^2

31. Definition of an Integer
Put the number associated with the word of on top and the quantity associated with the word to on the bottom and simplify.
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal
Multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying like bases. 2a3a=(23)(a*a)=6a^2
Integers are whole numbers and their opposites; they include negatives of whole numbers and zero.

32. How do you know if an integer is a multiple of 3 or 9?
Probability: Number of Favorable Outcomes/Total Possible Outcomes
Combine like terms.
Set the quadratic equation equal to zero with the terms in order of decreasing exponents - and solve for the roots by factoring. Don't forget to then find the difference in the roots by subtracting them. Ans: 2
An integer is divisible by 3 if the sum of its digits is divisible by 3 - - An integer is divisible by 9 is the sum of its digits is divisible by 9.

33. Formula Used to Find Percent
Integers are whole numbers and their opposites; they include negatives of whole numbers and zero.
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
Part=Percent x Whole
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal

34. What value of x is not in the domain of the function f(x)=x-2/x-3?
Values of x the take the denominator of a faction equal to zero - thereby dividing by zero which is not possible - are not in the domain of a function. Set the denominator equal to zero and solve for x. Ans: 3
Divide the denominator into the numerator to get a whole number quotient with a remainder. The quotient becomes the whole number of the mixed number. Remainder becomes the numerator.
The whole number left over after division.
The 3 interior angles of a any triangle add up to 180 degrees. An exterior angle of a triangle is equal to the sum of the remote - that is - non-adjacent - interior angles.

35. General Procedure for Multiplying Polynomials
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
-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.
Average the smallest and largest number
Use the distributive property then combine the like terms.

36. Prime Factorization of a Number
Use the formula distance=rate*time and its variations to help in this question.
Don't be confused by all the variables. Recall that (x-y)(x+y)=x(2)-y(2). Then - x(2)-y(2)=5*7=35. x(2)+1-y(2)=x(2)-y(2)+1=35+1=36.
To find the prime factorization of an integer - keep breaking it up into factors until all the factors are prime numbers.
Rectangle with four equal sides. Perimeter: 4x the length of 1 side. area of a square is equal to one side squared.

37. Solving a Radical Equation
Isolate the radical expression and use the standard rules of algebra.
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I
Integers are whole numbers and their opposites; they include negatives of whole numbers and zero.

38. If y varies inversely with x - and y is 3 when x is 10 - what is the value of x when y is 6?
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
Variables that have an inverse relationship will always have the same product. As one value gets larger - the other gets smaller. Set up and solve the equation 310=6x. Answer: 5
When a line is tangent to a circle - or touches the circle at just one point - the radius of the circle is perpendicular to the line at the point of contact.
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.

39. Finding the Sum of the Average of a Series of Numbers
Divide the denominator into the numerator to get a whole number quotient with a remainder. The quotient becomes the whole number of the mixed number. Remainder becomes the numerator.
Sum: Average x Number of Terms
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
Multiply the numerators and multiply the denominators.

40. Finding the Surface Area of a Rectangular Solid
Multiplying: Multiply the numbers under the different radicals. The product goes under one radical - Dividing: The quotient of more than one square root is equal to the square root of their total quotient.
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24
-Increasing: Add the percent to 100 percent - convert to a decimal - and multiply - Decrease: Subtract the percent from 100 percent - convert to a decimal - and multiply.

41. Counting the Total Number of Possibilities for Several Events to Occur
If there are m ways one event can happen and n ways a second event can happen - then there are m*n ways for the 2 events to happen.
Scalene- No sides or angles are equal - Isosceles- Have two equal sides. The angles opposite the equal sides are also equal - Equilateral- All three sides and angles are equal.
Remember that any non-zero base to the power of zero is equal to one. Set the exponent equal to zero and solve for x.
Express them with a common denominator.

42. If 3(3x)=9(x+2) - what is the value of x?
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
Multiply the numerators and multiply the denominators.
Divide the denominator into the numerator to get a whole number quotient with a remainder. The quotient becomes the whole number of the mixed number. Remainder becomes the numerator.
Multiplying: Multiply the numbers under the different radicals. The product goes under one radical - Dividing: The quotient of more than one square root is equal to the square root of their total quotient.

43. Solving a Linear Equation
Find a common denominator - then add or subtract the numerators.
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24
Do whatever is necessary to both sides to isolate the variable.
Use the formula distance=rate*time and its variations to help in this question.

44. Finding the Reciprocal of a Fraction
They have the same shape - but not necessarily the same size. Their corresponding angles are equal and their corresponding sides are proportional.
Multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying like bases. 2a3a=(23)(a*a)=6a^2
When a line is tangent to a circle - or touches the circle at just one point - the radius of the circle is perpendicular to the line at the point of contact.
Switch the numerator and denominator.

45. Properties of a 30-60-90 Triangle
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.
Average: Sum of the Terms/Number of the Terms

46. Finding the Volume of a Cylinder
Sector is a piece of the area of a circle. Area of a Sector: (n/360)((pie)r2) - n is the degree measure of the sector's central angle
Direct Variation- y=kx - where k is constant. As x gets larger - y gets larger. If the number of units of B were to double - the number of units of A would double - Inverse Variation- xy=k - where k is a constant. As x gets larger - y gets smaller. A
VofaC: (pie)r(2)h
Isolate the radical expression and use the standard rules of algebra.

47. What Does Solving In Terms of Mean?
The whole number left over after division.
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.
Change the division sign to a multiplication sign - invert the second fraction - and multiply.
Sector is a piece of the area of a circle. Area of a Sector: (n/360)((pie)r2) - n is the degree measure of the sector's central angle

48. An Important Property of a Line Tangent to a Circle
Ue the formula y=mx+b - and remember that the value of m is the slope of the line. The slopes of two perpendicular lines are negative reciprocals of one another.
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
When a line is tangent to a circle - or touches the circle at just one point - the radius of the circle is perpendicular to the line at the point of contact.
The whole number left over after division.

49. Finding the y-intercept When Given an Equation of a Line
Put the equation into y=mx+b-- in which case b is the y-intercept.
Plug in the given values for the unknowns and calculate according to PEMDAS.
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)
The sides of a 45-45-90 triangle are in a ratio of x:x:x(square root)2.

50. Factoring a Polynomial ('FOIL in Reverse')
Adjacent angles are supplementary - Vertical angles are equal
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)
Sum: Average x Number of Terms
Use simple numbers like 1 and 2 and see what happens.

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