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. Formula to Find Average
Average: Sum of the Terms/Number of the Terms
Make sure that the terms to be combined have exactly the same variables. 2a+3a=(2+3)a=5a
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I
Part=Percent x Whole


2. Number of Groups - +1 Each Time
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
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.


3. What value of x is not in the domain of the function f(x)=x-2/x-3?
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
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
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
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


4. If f(x)=x(1/3)+1/3x - then what is the value of f(27)?
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
(pie)r(squared)


5. Finding the Sum of the Interior Angles of a Polygon
Check the multiples of the larger number until you find one that's also a multiple of the smaller.
FOIL: First - Outer - Inner - Last... Combine Like Terms
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides
Cross multiply.


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

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7. Calculating the Probability that an Event will Take Place
Probability: Number of Favorable Outcomes/Total Possible Outcomes
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.
Rectangle with four equal sides. Perimeter: 4x the length of 1 side. area of a square is equal to one side squared.
Subtract the smallest from the largest and add 1


8. If 4(x(2)-10x+25) - then what is the value of x?
Four equal acute angles and four equal obtuse angles.
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.
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.
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.


9. Properties of the Interior and Exterior Angles of a Triangle
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.
Find a common denominator - then add or subtract the numerators.
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.
Average the smallest and largest number


10. General Procedure for Multiplying Polynomials
Use the distributive property then combine the like terms.
Express them with a common denominator.
Cancel factors common to the numerator and denominator.
Factor out and cancel all factors the numerator and denominator have in common.


11. Factoring a Polynomial ('FOIL in Reverse')
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)
2(pie)r
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72


12. If 3(3x)=9(x+2) - what is the value of x?
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
Factor out the perfect squares under the radical - take their square roots - and put the result in front.
A(2)+b(2)=c(2)
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4


13. Finding the Area of a Triangle

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14. What is the slope of the line perpendicular to the line with linear equation 4x+2y=12?
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.
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.
To solve a quadratic equation - set the equation equal to zero and solve for the numbers of the expression.
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.


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

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16. Finding the Area of a Circle
The sides of a 45-45-90 triangle are in a ratio of x:x:x(square root)2.
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.
-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.
(pie)r(squared)


17. Finding the Domain and Range of a Function
Fraction to a Decimal: Divide the Denominator into the numerator - 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.
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
Switch the numerator and denominator.
Domain- The domain of a a function is the set of values for which the function is defined - Range- The range of a function is the set of outputs or results of the function.


18. Finding the Reciprocal of a Fraction
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.
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.
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
Switch the numerator and denominator.


19. Converting from 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 of the mixed number. Remainder becomes the numerator.
Put the equation into y=mx+b-- in which case b is the y-intercept.
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.


20. Solving a System of Equations
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.
Change the division sign to a multiplication sign - invert the second fraction - and multiply.
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.
Fraction to a Decimal: Divide the Denominator into the numerator - 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.


21. If the average of a - b - and 48 is 48 - what is the value of a + b?
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
Adjacent angles are supplementary - Vertical angles are equal
The value that falls in the middle of the set.


22. Characteristics of a Square
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.
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.
Rectangle with four equal sides. Perimeter: 4x the length of 1 side. area of a square is equal to one side squared.
Factor out the perfect squares under the radical - take their square roots - and put the result in front.


23. What is the length of the shorter leg of a right triangle whose other leg measures 7(square root: 3)?
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.
The sides of a 45-45-90 triangle are in a ratio of x:x:x(square root)2.
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
In a 30-60-90 triangle - the measure of the longer leg is 7(square root: 3) - then the length of the shorter side or shorter leg must be 7.


24. Finding the GCF of Two or More Numbers
The whole number left over after division.
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
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.
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating decimal.


25. Finding the Mode of a Set of Numbers
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
Isolate the radical expression and use the standard rules of algebra.
The value that appears the most often.
Use simple numbers like 1 and 2 and see what happens.


26. Counting Consecutive Integers
They have the same shape - but not necessarily the same size. Their corresponding angles are equal and their corresponding sides are proportional.
-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.
Subtract the smallest from the largest and add 1
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)


27. Finding the Slope of a Line When Given Two Points on the Line
Slope=change in y/change in x
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.
Adjacent angles are supplementary - Vertical angles are equal
-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.


28. Finding the Area of a Sector

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29. What types of angles are formed when a transversal cross parallel lines?
Four equal acute angles and four equal obtuse angles.
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
Plug in the given values for the unknowns and calculate according to PEMDAS.
Combine like terms.


30. Finding the Length of an Arc in a Circle

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31. When Given a Series of Percent Increases and Decreases - How do you Determine your ending value?
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
Do whatever is necessary to both sides to isolate the variable.
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
To solve a quadratic equation - set the equation equal to zero and solve for the numbers of the expression.


32. Difference Between a Factor and a Multiple
Fraction to a Decimal: Divide the Denominator into the numerator - 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.
-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.
VofaC: (pie)r(2)h
Use the formula distance=rate*time and its variations to help in this question.


33. Finding the Median of a Set of Numbers
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds
The value that falls in the middle of the set.
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.
Rectangle with four equal sides. Perimeter: 4x the length of 1 side. area of a square is equal to one side squared.


34. 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
Cross multiply.
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
In a 30-60-90 triangle - the measure of the longer leg is 7(square root: 3) - then the length of the shorter side or shorter leg must be 7.


35. Finding the Slope When Given an Equation of a Line
Ignore the signs... find the positive difference between the numbers - Then attach the sign of the original number with the larger absolute value
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
Factor out and cancel all factors the numerator and denominator have in common.


36. Multiplying Expressions with Exponents that Have a Common Base
Add the exponents and keep the same base - x^3*x^4=x^3+4=x^7
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.


37. Dividing Expressions with Exponents that Have a Common Base
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
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.
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24
Subtract the exponents and keep the same base.y^13/y^8=y^13-8=y^5


38. Definition of Remainder
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
Multiply the exponents - (x^3)^4=x^3*4=x^12
The whole number left over after division.
Rewrite the number without the negative sign as the bottom of a fraction with 1 as the top.


39. Calculating Negative Exponents and Radical Exponents
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.
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
Rewrite the number without the negative sign as the bottom of a fraction with 1 as the top.
-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.


40. Characteristics of a Parallelogram
The value that falls in the middle of the set.
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.
Get the absolute value equation by itself. Solve.
Subtract the smallest from the largest and add 1


41. Counting the Total Number of Possibilities for Several Events to Occur
The whole number left over after division.
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.
Express them with a common denominator.
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.


42. Average Rate and How Do You Find It
Express them with a common denominator.
Average A per B= Total A/Total B so Average Speed = Total Distance/Total Time
Make sure that the terms to be combined have exactly the same variables. 2a+3a=(2+3)a=5a
Do whatever is necessary to both sides to isolate the variable.


43. 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
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.
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.
Use the formula distance=rate*time and its variations to help in this question.


44. Finding the Missing Number in a Series When You are Given the Average
FOIL: First - Outer - Inner - Last... Combine Like Terms
Do whatever is necessary to both sides to isolate the variable.
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.


45. Finding the Surface Area of a Rectangular Solid
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.
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)


46. Simplifying Square Roots
Express them with a common denominator.
-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.
Factor out the perfect squares under the radical - take their square roots - and put the result in front.
Fraction to a Decimal: Divide the Denominator into the numerator - 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.


47. If r#t=t(r)-r(t) - then what is 4#3?
Do whatever is necessary to both sides to isolate the variable.
Factor out and cancel all factors the numerator and denominator have in common.
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4


48. Comparing the values of two or more Fractions
Adjacent angles are supplementary - Vertical angles are equal
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
Express them with a common denominator.
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.


49. Definition of an Irrational Number
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
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
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.


50. Factoring the Difference of Two Squares
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal
In a 30-60-90 triangle - the measure of the longer leg is 7(square root: 3) - then the length of the shorter side or shorter leg must be 7.
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 the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)