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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. Factoring a Polynomial ('FOIL in Reverse')
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
Use the units to keep things straight - Snowfall inches/hours
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
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)

2. Characteristics of a Rectangle
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.
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.
(pie)r(squared)
Combine like terms.

3. Formula Used to Find Percent
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides
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.
Part=Percent x Whole
Factor out the perfect squares under the radical - take their square roots - and put the result in front.

4. Definition of a Rational Number
Factor out and cancel all factors the numerator and denominator have in common.
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating decimal.
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle

5. Calculating Negative Exponents and Radical Exponents
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
Rewrite the number without the negative sign as the bottom of a fraction with 1 as the top.
(pie)r(squared)
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%

6. Properties of a 30-60-90 Triangle
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 30-60-90 triangle are in a ratio of x:x(square root)3:2x.
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 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. Solving an Inequality
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides
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.
Cross multiply.

8. Difference Between a Factor and a Multiple
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
-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.
Sum: Average x Number of Terms
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96

9. Factoring the Difference of Two Squares
Sum: Average x Number of Terms
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
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)

10. Finding the Reciprocal of a Fraction
Switch the numerator and denominator.
The value that falls in the middle of the set.
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)

11. Increasing and Decreasing a Number by a Certain Percentage
-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.
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.
Sum: Average x Number of Terms
Add the exponents and keep the same base - x^3*x^4=x^3+4=x^7

12. Multiplying and Dividing Roots
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.
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.
VofaC: (pie)r(2)h
The whole number left over after division.

13. Dividing Fractions
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 45-45-90 triangle are in a ratio of x:x:x(square root)2.
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
Change the division sign to a multiplication sign - invert the second fraction - and multiply.

14. What is the length of a leg of an isosceles rich triangle who's hypotenuse measures 24(square root: 2)?
Subtract the smallest from the largest and add 1
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.
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24
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

15. Finding the Surface Area of a Rectangular Solid
-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.
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
2(pie)r
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72

16. Calculating the Probability that an Event will Take Place
Probability: Number of Favorable Outcomes/Total Possible Outcomes
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
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.
Slope=change in y/change in x

17. Knowing if an Integer is a Multiple of 2 or 4
-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.
Use the formula distance=rate*time and its variations to help in this question.
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.
First - substitute for the variable. Then evaluate the expression using the correct order of operations.

18. Expressing the Union and Intersection of Sets
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds
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
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

19. Finding the Missing Number in a Series When You are Given the Average
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 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.
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.

20. Finding the Domain and Range of a Function
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.
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides
The sides of a 45-45-90 triangle are in a ratio of x:x:x(square root)2.
FOIL: First - Outer - Inner - Last... Combine Like Terms

21. Finding the Area of a Circle
(pie)r(squared)
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.
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

22. Knowing if an Integer is a Multiple of 5 or 10
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds
-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.
They have the same shape - but not necessarily the same size. Their corresponding angles are equal and their corresponding sides are proportional.
Check the multiples of the larger number until you find one that's also a multiple of the smaller.

23. If (Square Root: x-1)+5=12 - what is the value of x/2?
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
First - get the square root by itself on one side of the equation. Then - solve for x by squaring each side. Answer: 25
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.

24. Simplifying a Fraction to Lowest Terms
FOIL: First - Outer - Inner - Last... Combine Like Terms
Factor out and cancel all factors the numerator and denominator have in common.
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.
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72

25. Numbers that are Relatively Prime
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.
Rectangle with four equal sides. Perimeter: 4x the length of 1 side. area of a square is equal to one side squared.
Two relatively prime numbers are integers that have no common factor other than 1.

26. Knowing Whether the Sum - Difference - or Product of Several Numbers will be Even/Odd
Use simple numbers like 1 and 2 and see what happens.
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.
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96

27. Finding the Slope of a Line When Given Two Points on the Line
Use the distance formula: d=(square root over) (x2-x2)2+(y1-y2)2
Use the units to keep things straight - Snowfall inches/hours
Multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying like bases. 2a3a=(23)(a*a)=6a^2
Slope=change in y/change in x

28. If the average of a - b - and 48 is 48 - what is the value of a + b?
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
The whole number left over after division.
-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 distance=rate*time and its variations to help in this question.

29. Converting from an Improper Fraction to a Mixed Number
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
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.
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
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.

30. Counting Consecutive Integers
Subtract the smallest from the largest and add 1
Integers are whole numbers and their opposites; they include negatives of whole numbers and zero.
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.

31. Converting from part-to-part ratios to part-to-whole ratios
The value that falls in the middle of the set.
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(pie)r

32. Finding the LCM of Two or More Numbers

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33. Number of Groups - +1 Each Time
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.
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.
Get the absolute value equation by itself. Solve.
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72

34. Finding a Term in a Geometric Sequence
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.
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.
Express them with a common denominator.
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.

35. Finding the Area of a Sector

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36. What is the Pythagorean Theorem?
A(2)+b(2)=c(2)
Change the division sign to a multiplication sign - invert the second fraction - and multiply.
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.

37. Multiplying Binomials
Subtract the smallest from the largest and add 1
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.
FOIL: First - Outer - Inner - Last... Combine Like Terms
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.

38. 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.
VofaRS=lwh
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
Adjacent angles are supplementary - Vertical angles are equal

39. What types of angles are formed when a transversal cross parallel lines?
Four equal acute angles and four equal obtuse angles.
Part=Percent x Whole
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
Get the absolute value equation by itself. Solve.

40. Adding and Subtracting Monomials
Make sure that the terms to be combined have exactly the same variables. 2a+3a=(2+3)a=5a
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.
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.

41. Definition of Remainder
The whole number left over after division.
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 distributive property then combine the like terms.
The value that falls in the middle of the set.

42. Direct Variation and Inverse Variation
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.
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 the exponents - (x^3)^4=x^3*4=x^12
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.

43. Finding the Original Value Before it was Increased or Decreased by a Certain Percentage
The whole number left over after division.
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
Isolate the radical expression and use the standard rules of algebra.

44. If 4(x(2)-10x+25) - then what is the value of x?
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
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.
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.
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72

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

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46. 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.
Find a common denominator - then add or subtract the numerators.
Add the exponents and keep the same base - x^3*x^4=x^3+4=x^7
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.

47. If r#t=t(r)-r(t) - then what is 4#3?
Part=Percent x Whole
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.
To find the prime factorization of an integer - keep breaking it up into factors until all the factors are prime numbers.
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.

48. If 2+l6-xl=10 and x>0 - what is the value of 2x?
Get the absolute value equation by itself. Solve.
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.
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
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.

49. When Given a Series of Percent Increases and Decreases - How do you Determine your ending value?
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.
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
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.
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)

50. Adding and Subtracting Polynomials
Plug in the given values for the unknowns and calculate according to PEMDAS.
Combine like terms.
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
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.

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