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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. Converting from part-to-part ratios to part-to-whole ratios
Probability: Number of Favorable Outcomes/Total Possible Outcomes
Get the absolute value equation by itself. Solve.
Put each number in the original ratio over the sum of the numbers.
Isolate the radical expression and use the standard rules of algebra.

2. When Given a Series of Percent Increases and Decreases - How do you Determine your ending value?
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I
Multiply the numerators and multiply the denominators.

3. Definition of Remainder
The whole number left over after division.
Find a common denominator - then add or subtract the numerators.
Parenthesis - Exponents - Multiplication/Division - Addition/Subtraction
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.

4. Direct Variation and Inverse Variation
Use the distributive property then combine the like terms.
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.
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
Switch the numerator and denominator.

5. Convert From a Fraction to a Decimal - Vice Versa
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.
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.
-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.

6. Definition of an Integer
Integers are whole numbers and their opposites; they include negatives of whole numbers and zero.
Parenthesis - Exponents - Multiplication/Division - Addition/Subtraction
FOIL: First - Outer - Inner - Last... Combine Like Terms
Factor out the perfect squares under the radical - take their square roots - and put the result in front.

7. Solving an Inequality
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.
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.
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides

8. Multiplying Binomials
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.
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.
FOIL: First - Outer - Inner - Last... Combine Like Terms
Find a common denominator - then add or subtract the numerators.

9. Average Rate and How Do You Find It
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.
Average A per B= Total A/Total B so Average Speed = Total Distance/Total Time
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.
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.

10. Finding the Area of a Circle
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.
To find the prime factorization of an integer - keep breaking it up into factors until all the factors are prime numbers.
(pie)r(squared)

11. What is the Triangle Inequality Theorem?
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.
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

12. 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
Use simple numbers like 1 and 2 and see what happens.
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
Probability: Number of Favorable Outcomes/Total Possible Outcomes

13. Determining Absolute Value of a Number
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I
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.
Factor out and cancel all factors the numerator and denominator have in common.

14. Simplifying a Fraction to Lowest Terms
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds
Part=Percent x Whole
Factor out and cancel all factors the numerator and denominator have in common.
Make sure that the terms to be combined have exactly the same variables. 2a+3a=(2+3)a=5a

15. Characteristics of a Parallelogram
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
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.

16. Finding the GCF of Two or More Numbers
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.
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.
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
Change the division sign to a multiplication sign - invert the second fraction - and multiply.

17. Multiplying and Dividing Roots
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.
Express them with a common denominator.
Factor out and cancel all factors the numerator and denominator have in common.
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.

18. 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?
Combine like terms.
Use the formula distance=rate*time and its variations to help in this question.
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.
Two relatively prime numbers are integers that have no common factor other than 1.

19. Finding the Slope When Given an Equation of a Line
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
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.
Use the formula distance=rate*time and its variations to help in this question.
Put the equation into the slope-intercept form: y=mx+b. The slope is m.

20. Solving a Proportion
To solve a quadratic equation - set the equation equal to zero and solve for the numbers of the expression.
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
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.
Cross multiply.

21. Finding the Volume of a Rectangular Solid
-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.
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.
Subtract the exponents and keep the same base.y^13/y^8=y^13-8=y^5
VofaRS=lwh

22. Calculating the Probability that an Event will Take Place
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.
Probability: Number of Favorable Outcomes/Total Possible Outcomes
Subtract the exponents and keep the same base.y^13/y^8=y^13-8=y^5
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.

23. Adding and Subtracting Monomials
Make sure that the terms to be combined have exactly the same variables. 2a+3a=(2+3)a=5a
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.
Factor out and cancel all factors the numerator and denominator have in common.
Put the equation into y=mx+b-- in which case b is the y-intercept.

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

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25. If f(x)=x(1/3)+1/3x - then what is the value of f(27)?
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
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.
First - substitute for the variable. Then evaluate the expression using the correct order of operations.

26. Finding the Rate of Speed
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
Use the formula distance=rate*time and its variations to help in this question.
Part=Percent x Whole
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)

27. Finding the Reciprocal of a Fraction
Switch the numerator and denominator.
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
Slope=change in y/change in x
-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.

28. Numbers that are Relatively Prime
Two relatively prime numbers are integers that have no common factor other than 1.
Just add or subtract the coefficients in front of the radicals.
Use the formula distance=rate*time and its variations to help in this question.
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

29. Properties of a 45-45-90 Triangle
-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.
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.
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 sides of a 45-45-90 triangle are in a ratio of x:x:x(square root)2.

30. Adding and Subtracting Roots
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
Just add or subtract the coefficients in front of the radicals.
Ignore the signs... find the positive difference between the numbers - Then attach the sign of the original number with the larger absolute value
Two relatively prime numbers are integers that have no common factor other than 1.

31. Properties of the Interior and Exterior Angles of a Triangle
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.
Put each number in the original ratio over the sum of the numbers.
Part=Percent x Whole

32. What is the Pythagorean Theorem?
A(2)+b(2)=c(2)
Factor out the perfect squares under the radical - take their square roots - and put the result in front.
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.

33. Finding the Missing Number in a Series When You are Given the Average
Just add or subtract the coefficients in front of the radicals.
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating decimal.
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.

34. Converting From a Mixed Number to an Improper Fraction
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.
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.
Add the exponents and keep the same base - x^3*x^4=x^3+4=x^7
Check the multiples of the larger number until you find one that's also a multiple of the smaller.

35. Formula Used to Find Percent
Part=Percent x Whole
Add the exponents and keep the same base - x^3*x^4=x^3+4=x^7
Put the number associated with the word of on top and the quantity associated with the word to on the bottom and simplify.
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.

36. Finding the Volume of a Cylinder
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)
VofaC: (pie)r(2)h
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds
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.

37. Expressing the Union and Intersection of Sets
(pie)r(squared)
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 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
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.

38. What is the length of a leg of an isosceles rich triangle who's hypotenuse measures 24(square root: 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.
Plug in the given values for the unknowns and calculate according to PEMDAS.
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.

39. Adding a Positive Number to a Negative Number
Ignore the signs... find the positive difference between the numbers - Then attach the sign of the original number with the larger absolute value
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.
Probability: Number of Favorable Outcomes/Total Possible Outcomes
Get the absolute value equation by itself. Solve.

40. Finding the Sum of the Average of a Series of Numbers
Sum: Average x Number of Terms
-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.
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24

41. Simplifying an Algebraic Equation
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds
Cancel factors common to the numerator and 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.
Make sure that the terms to be combined have exactly the same variables. 2a+3a=(2+3)a=5a

42. Finding the Length of an Arc in a Circle

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43. Dividing Fractions
Change the division sign to a multiplication sign - invert the second fraction - and multiply.
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.
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.
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.

44. Finding the Slope of a Line When Given Two Points on the Line
Slope=change in y/change in x
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
Two relatively prime numbers are integers that have no common factor other than 1.
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)

45. What is the greatest of three consecutive odd integers where the sum of the third and twice the first is equal to nine more than twice the second?
Probability: Number of Favorable Outcomes/Total Possible Outcomes
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.
2(pie)r
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)

46. 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
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.
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.
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%

47. Finding the Mode of a Set of Numbers
First - get the square root by itself on one side of the equation. Then - solve for x by squaring each side. Answer: 25
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
The value that appears the most often.
VofaC: (pie)r(2)h

48. Knowing Whether the Sum - Difference - or Product of Several Numbers will be Even/Odd
Use the distributive property then combine the like terms.
Ignore the signs... find the positive difference between the numbers - Then attach the sign of the original number with the larger absolute value
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
Use simple numbers like 1 and 2 and see what happens.

49. Factoring the Difference of Two Squares
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
Adjacent angles are supplementary - Vertical angles are equal
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.
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72

50. Finding the Area of a Sector

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