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

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2. Finding the Reciprocal of a Fraction
Probability: Number of Favorable Outcomes/Total Possible Outcomes
Switch the numerator and denominator.
2(pie)r
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)


3. Finding the Length of an Arc in a Circle

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4. Dividing Fractions
Change the division sign to a multiplication sign - invert the second fraction - and multiply.
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.
Two relatively prime numbers are integers that have no common factor other than 1.
-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.


5. Solving a Linear Equation
Subtract the exponents and keep the same base.y^13/y^8=y^13-8=y^5
Switch the numerator and denominator.
Do whatever is necessary to both sides to isolate the variable.
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


6. Dividing Expressions with Exponents that Have a Common Base
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.
Subtract the exponents and keep the same base.y^13/y^8=y^13-8=y^5
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.
-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.


7. Finding the Rate of Speed
Subtract the smallest from the largest and add 1
Use the formula distance=rate*time and its variations to help in this question.
Use the distance formula: d=(square root over) (x2-x2)2+(y1-y2)2
Put the equation into y=mx+b-- in which case b is the y-intercept.


8. Finding the Domain and Range of a Function
Ignore the signs... find the positive difference between the numbers - Then attach the sign of the original number with the larger absolute value
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.
Express them with a common denominator.
-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.


9. Number of Groups - +1 Each Time
Get the absolute value equation by itself. Solve.
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.
Four equal acute angles and four equal obtuse 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.


10. What is the length of the shorter leg of a right triangle whose other leg measures 7(square root: 3)?
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
Switch the numerator and denominator.
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.
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.


11. Solving a Proportion
Cancel factors common to the numerator and denominator.
Use simple numbers like 1 and 2 and see what happens.
Cross multiply.
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


12. Finding the Sum of the Average of a Series of Numbers
Get the absolute value equation by itself. Solve.
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.
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.
Sum: Average x Number of Terms


13. General Procedure for Multiplying Polynomials
Use the distributive property then combine the like terms.
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
Isolate the radical expression and use the standard rules of algebra.
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating decimal.


14. What Does Solving In Terms of Mean?
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.
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.
Cross multiply.
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.


15. Properties of Similar Triangles
Put the number associated with the word of on top and the quantity associated with the word to on the bottom and simplify.
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
VofaC: (pie)r(2)h
They have the same shape - but not necessarily the same size. Their corresponding angles are equal and their corresponding sides are proportional.


16. What is the Triangle Inequality Theorem?
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.
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.
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
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.


17. Calculating the Probability that an Event will Take Place
First - get the square root by itself on one side of the equation. Then - solve for x by squaring each side. Answer: 25
Find a common denominator - then add or subtract the numerators.
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.
Probability: Number of Favorable Outcomes/Total Possible Outcomes


18. Characteristics of a Rectangle
Multiply the exponents - (x^3)^4=x^3*4=x^12
To find the prime factorization of an integer - keep breaking it up into factors until all the factors are prime numbers.
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.
Four equal acute angles and four equal obtuse angles.


19. 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?
-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.
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
Factor out the perfect squares under the radical - take their square roots - and put the result in front.
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72


20. What is the slope of the line perpendicular to the line with linear equation 4x+2y=12?
Adjacent angles are supplementary - Vertical angles are equal
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.
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.
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


21. Adding and Subtracting Polynomials
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
Combine like terms.
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)


22. Multiplying Binomials
Adjacent angles are supplementary - Vertical angles are equal
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.
FOIL: First - Outer - Inner - Last... Combine Like Terms
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal


23. 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.
The sides of a 45-45-90 triangle are in a ratio of x:x:x(square root)2.
Put the equation into y=mx+b-- in which case b is the y-intercept.
Adjacent angles are supplementary - Vertical angles are equal


24. Direct Variation and Inverse Variation
Plug in the given values for the unknowns and calculate according to PEMDAS.
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
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72
(pie)r(squared)


25. Increasing and Decreasing a Number by a Certain Percentage
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 sides of a 45-45-90 triangle are in a ratio of x:x:x(square root)2.
Ignore the signs... find the positive difference between the numbers - Then attach the sign of the original number with the larger absolute value
-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.


26. Raising a Power to a Power
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.
-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.
Multiply the exponents - (x^3)^4=x^3*4=x^12


27. Scalene - Isosceles - and Equilateral Triangles
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.
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.
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.


28. What types of angles are formed when a transversal cross parallel lines?
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.
Part=Percent x Whole
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.
Four equal acute angles and four equal obtuse angles.


29. Finding a Term in a Geometric Sequence
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
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
Subtract the smallest from the largest and add 1
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.


30. Knowing if an Integer is a Multiple of 2 or 4
Slope=change in y/change in x
Use simple numbers like 1 and 2 and see what happens.
-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.
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.


31. Finding the Original Value Before it was Increased or Decreased by a Certain Percentage
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.
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
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.
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.


32. 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.
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.
Rectangle with four equal sides. Perimeter: 4x the length of 1 side. area of a square is equal to one side 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.


33. Finding the Area of a Sector

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

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35. If 3(3x)=9(x+2) - what is the value of x?
Find a common denominator - then add or subtract the numerators.
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
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.
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.


36. Simplifying a Fraction to Lowest Terms
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.
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.
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
Factor out and cancel all factors the numerator and denominator have in common.


37. Finding the Median of a Set of Numbers
A(2)+b(2)=c(2)
Multiply the exponents - (x^3)^4=x^3*4=x^12
The value that falls in the middle of the set.
2(pie)r


38. Average Rate and How Do You Find It
Average A per B= Total A/Total B so Average Speed = Total Distance/Total Time
Add the exponents and keep the same base - x^3*x^4=x^3+4=x^7
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.


39. Knowing Whether the Sum - Difference - or Product of Several Numbers will be Even/Odd
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.
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I
Use simple numbers like 1 and 2 and see what happens.
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.


40. Counting the Total Number of Possibilities for Several Events to Occur
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
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.
Just add or subtract the coefficients in front of the radicals.
A(2)+b(2)=c(2)


41. Comparing the values of two or more Fractions
Consecutive odd integers are odd integers in order in a row - such as 1 - 3 - and 5. The difference between any two consecutive odd integers is 2. Translate the question into an equation that expresses the relationship between the integers.
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
Express them with a common denominator.


42. Simplifying an Algebraic Equation
Cancel factors common to the numerator and denominator.
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72
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.
Isolate the radical expression and use the standard rules of algebra.


43. Properties of the Interior and Exterior Angles of a Triangle
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating decimal.
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
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 sides of a 45-45-90 triangle are in a ratio of x:x:x(square root)2.


44. If (Square Root: x-1)+5=12 - what is the value of x/2?
First - get the square root by itself on one side of the equation. Then - solve for x by squaring each side. Answer: 25
-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.
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.
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


45. Multiply and Divide Positive and Negative Numbers
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.
Check the multiples of the larger number until you find one that's also a multiple of the smaller.
Consecutive odd integers are odd integers in order in a row - such as 1 - 3 - and 5. The difference between any two consecutive odd integers is 2. Translate the question into an equation that expresses the relationship between the integers.
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


46. If y varies inversely with x - and y is 3 when x is 10 - what is the value of x when y is 6?
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
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.
The whole number left over after division.
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.


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

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48. Properties of a 30-60-90 Triangle
Factor out and cancel all factors the numerator and denominator have in common.
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)
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.


49. Multiplying Fractions
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.
Multiply the numerators and multiply the denominators.
Use the units to keep things straight - Snowfall inches/hours
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.


50. Factoring a Polynomial ('FOIL in Reverse')
Just add or subtract the coefficients in front of the radicals.
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
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.