SAT Math

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• Answer 50 questions in 15 minutes.
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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. Prime Factorization of a Number
Average: Sum of the Terms/Number of the Terms
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.
To find the prime factorization of an integer - keep breaking it up into factors until all the factors are prime numbers.


2. Solving a System of Equations
To find the prime factorization of an integer - keep breaking it up into factors until all the factors are prime numbers.
Use the distance formula: d=(square root over) (x2-x2)2+(y1-y2)2
Combine the equations so that one of the variables cancels out. 4x+3y=8 and x+y=3. Multiply equation 2 by -3... then add the two equations.
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.


3. 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.
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.
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I


4. An Important Property of a Line Tangent to a Circle
Put each number in the original ratio over the sum of the numbers.
Average A per B= Total A/Total B so Average Speed = Total Distance/Total Time
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
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.


5. Setting Up a Ratio
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.
Rectangle with four equal sides. Perimeter: 4x the length of 1 side. area of a square is equal to one side squared.
Put the number associated with the word of on top and the quantity associated with the word to on the bottom and simplify.
Use the units to keep things straight - Snowfall inches/hours


6. Characteristics of a Parallelogram
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.
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.
Check the multiples of the larger number until you find one that's also a multiple of the smaller.
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.


7. Finding the Length of an Arc in a Circle

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8. Multiplying Binomials
FOIL: First - Outer - Inner - Last... Combine Like Terms
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.
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.
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.


9. Multiplying and Dividing Roots
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
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.
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
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.


10. Finding the Sum of the Interior Angles of a Polygon
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
2(pie)r
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides
Express them with a common denominator.


11. Finding the LCM of Two or More Numbers

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12. Adding and Subtracting Roots
Factor out the perfect squares under the radical - take their square roots - and put the result in front.
Just add or subtract the coefficients in front of the radicals.
They have the same shape - but not necessarily the same size. Their corresponding angles are equal and their corresponding sides are proportional.
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.


13. What is the positive difference between the answers to the equation 35+x(2)=12x?

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14. Calculating Negative Exponents and Radical Exponents
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides
Use the units to keep things straight - Snowfall inches/hours
Rewrite the number without the negative sign as the bottom of a fraction with 1 as the top.
Part=Percent x Whole


15. What is a Function and how do you Evaluate one?

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16. Properties of a 45-45-90 Triangle
Cross multiply.
The sides of a 45-45-90 triangle are in a ratio of x:x:x(square root)2.
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.
Express them with a common denominator.


17. Finding the Distance Between Two Points on a Coordinate Graph
Cross multiply.
Use the distance formula: d=(square root over) (x2-x2)2+(y1-y2)2
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.
Multiply the numerators and multiply the denominators.


18. Comparing the values of two or more Fractions
Adjacent angles are supplementary - Vertical angles are equal
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.
Express them with a common denominator.
Cross multiply.


19. Simplifying Square Roots
Use the distance formula: d=(square root over) (x2-x2)2+(y1-y2)2
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
Factor out the perfect squares under the radical - take their square roots - and put the result in front.
Combine the equations so that one of the variables cancels out. 4x+3y=8 and x+y=3. Multiply equation 2 by -3... then add the two equations.


20. If f(x)=x(1/3)+1/3x - then what is the value of f(27)?
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 out the perfect squares under the radical - take their square roots - and put the result in front.
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.
First - substitute for the variable. Then evaluate the expression using the correct order of operations.


21. What is the length of a leg of an isosceles rich triangle who's hypotenuse measures 24(square root: 2)?
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
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24
Cross multiply.


22. Dividing Expressions with Exponents that Have a Common Base
A(2)+b(2)=c(2)
They have the same shape - but not necessarily the same size. Their corresponding angles are equal and their corresponding sides are proportional.
Subtract the exponents and keep the same base.y^13/y^8=y^13-8=y^5
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24


23. PEMDAS
The value that appears the most often.
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.
Parenthesis - Exponents - Multiplication/Division - Addition/Subtraction
2(pie)r


24. Finding the Surface Area of a Rectangular Solid
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
Subtract the exponents and keep the same base.y^13/y^8=y^13-8=y^5
Use simple numbers like 1 and 2 and see what happens.


25. Formula Used to Find Percent
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.
Part=Percent x Whole
The length of one side of a triangle must be greater than the difference and less of the sum of the lengths of the other two sides.
The sides of a 45-45-90 triangle are in a ratio of x:x:x(square root)2.


26. Increasing and Decreasing a Number by a Certain Percentage
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.
-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.
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.


27. Numbers that are Relatively Prime
To find the prime factorization of an integer - keep breaking it up into factors until all the factors are prime numbers.
Two relatively prime numbers are integers that have no common factor other than 1.
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.
Get the absolute value equation by itself. Solve.


28. Solving a Proportion
Ignore the signs... find the positive difference between the numbers - Then attach the sign of the original number with the larger absolute value
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24
Cross multiply.
Probability: Number of Favorable Outcomes/Total Possible Outcomes


29. What is the Pythagorean Theorem?
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.
To solve a quadratic equation - set the equation equal to zero and solve for the numbers of the expression.
A(2)+b(2)=c(2)
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating decimal.


30. What Does Solving In Terms of Mean?
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.
The length of one side of a triangle must be greater than the difference and less of the sum of the lengths of the other two sides.
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.


31. 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
To solve a quadratic equation - set the equation equal to zero and solve for the numbers of the expression.
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
Average A per B= Total A/Total B so Average Speed = Total Distance/Total Time


32. Dividing Fractions
The value that appears the most often.
(pie)r(squared)
Change the division sign to a multiplication sign - invert the second fraction - and multiply.
Slope=change in y/change in x


33. Solving an Inequality
Average the smallest and largest number
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.
Use simple numbers like 1 and 2 and see what happens.
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.


34. Number of Groups - +1 Each Time
The length of one side of a triangle must be greater than the difference and less of the sum of the lengths of the other two sides.
Do whatever is necessary to both sides to isolate the variable.
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.
-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.


35. Properties of Similar Triangles
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
They have the same shape - but not necessarily the same size. Their corresponding angles are equal and their corresponding sides are proportional.
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
Sum: Average x Number of Terms


36. Multiplying Monomials
-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.
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.
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
Multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying like bases. 2a3a=(23)(a*a)=6a^2


37. Properties of the Interior and Exterior Angles of a Triangle
Two relatively prime numbers are integers that have no common factor other than 1.
VofaC: (pie)r(2)h
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.
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.


38. Knowing 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.
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.
Use simple numbers like 1 and 2 and see what happens.
The value that appears the most often.


39. 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.
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
The whole number left over after division.
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


40. What is the value of x(2)+1-y(2) - if x-y=5 and x+y=7?

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41. Determining Absolute Value of a Number
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
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.


42. Solving a Radical Equation
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
Isolate the radical expression and use the standard rules of algebra.
Get the absolute value equation by itself. Solve.
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


43. Sale Price of a Jacket Marked Down 30% Each Week for 3 Weeks - What % of the Original Price is the Cost of the Jacket After the 3 Week Sale Period
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.


44. Adding and Subtracting Monomials
Get the absolute value equation by itself. Solve.
2(pie)r
Make sure that the terms to be combined have exactly the same variables. 2a+3a=(2+3)a=5a
Average: Sum of the Terms/Number of the Terms


45. If (Square Root: x-1)+5=12 - what is the value of x/2?
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.
First - get the square root by itself on one side of the equation. Then - solve for x by squaring each side. Answer: 25
Cancel factors common to the numerator and denominator.
Use the distributive property then combine the like terms.


46. General Procedure for Multiplying Polynomials
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.
Find a common denominator - then add or subtract the numerators.
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
Use the distributive property then combine the like terms.


47. Let N represent the smallest positive integer that is a multiple of 6 and 8 - but leaves a remainder of 2 when divided by 7. What is the value of N?
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72
Find a common denominator - then add or subtract the numerators.
To solve a quadratic equation - set the equation equal to zero and solve for the numbers of the expression.


48. Direct Variation and Inverse Variation
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
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
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.
Cross multiply.


49. 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 is the sum of its digits is divisible by 9.
-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.
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
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.


50. What types of angles are formed when a transversal cross parallel lines?
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
Four equal acute angles and four equal obtuse angles.
Use the units to keep things straight - Snowfall inches/hours
The whole number left over after division.