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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. Characteristics of a Square
Rectangle with four equal sides. Perimeter: 4x the length of 1 side. area of a square is equal to one side squared.
(x1+x2)/2 -(y1+y2)/2
Multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying like bases. 2a3a=(23)(a*a)=6a^2
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24

2. Simplifying Square Roots
VofaC: (pie)r(2)h
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
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.
Factor out the perfect squares under the radical - take their square roots - and put the result in front.

3. Finding the Area of a Triangle

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4. Number of Groups - +1 Each Time
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
Average A per B= Total A/Total B so Average Speed = Total Distance/Total 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.
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

5. Dividing Fractions
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.
Change the division sign to a multiplication sign - invert the second fraction - and multiply.
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I

6. Calculating the Probability that an Event will Take Place
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.
Sum: Average x Number of Terms
Probability: Number of Favorable Outcomes/Total Possible Outcomes
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.

7. Definition of a Rational Number
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating 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.
Plug in the given values for the unknowns and calculate according to PEMDAS.

8. Solving a Problem Involving Rates
Use the units to keep things straight - Snowfall inches/hours
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.
FOIL: First - Outer - Inner - Last... Combine Like Terms
Factor out and cancel all factors the numerator and denominator have in common.

9. Finding the Area of a Circle
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.
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.
To solve a quadratic equation - set the equation equal to zero and solve for the numbers of the expression.
(pie)r(squared)

10. Knowing Whether the Sum - Difference - or Product of Several Numbers will be Even/Odd
Factor out and cancel all factors the numerator and denominator have in common.
Use simple numbers like 1 and 2 and see what happens.
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24
The value that appears the most often.

11. Finding the Midpoint Between Two Points
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.
Average: Sum of the Terms/Number of the Terms
(x1+x2)/2 -(y1+y2)/2
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.

12. What is the length of the shorter leg of a right triangle whose other leg measures 7(square root: 3)?
-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.
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.
Four equal acute angles and four equal obtuse angles.

13. PEMDAS
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.
Slope=change in y/change in x
Parenthesis - Exponents - Multiplication/Division - Addition/Subtraction

14. If (Square Root: x-1)+5=12 - what is the value of x/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.
First - get the square root by itself on one side of the equation. Then - solve for x by squaring each side. Answer: 25
Use the units to keep things straight - Snowfall inches/hours
Sum: Average x Number of Terms

15. Definition of Remainder
Subtract the exponents and keep the same base.y^13/y^8=y^13-8=y^5
Put the equation into y=mx+b-- in which case b is the y-intercept.
The whole number left over after division.
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

16. Finding the Sum of the Average of a Series of Numbers
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
Sum: Average x Number of 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.

17. Finding the Slope of a Line When Given Two Points on the Line
Slope=change in y/change in x
Sum: Average x Number of Terms
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.

18. Evaluating an Algebraic Expression
VofaRS=lwh
(pie)r(squared)
Plug in the given values for the unknowns and calculate according to PEMDAS.
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.

19. Finding the Domain and Range of a Function
Check the multiples of the larger number until you find one that's also a multiple of the smaller.
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.
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.
2(pie)r

20. Expressing the Union and Intersection of Sets
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.
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.
VofaC: (pie)r(2)h
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

21. Finding the GCF of Two or More Numbers
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.
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
Part=Percent x Whole
Plug in the given values for the unknowns and calculate according to PEMDAS.

22. 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?
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.
Sum: Average x Number of Terms
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

23. If x(2)=7x+18 - what is the positive value of x?
To solve a quadratic equation - set the equation equal to zero and solve for the numbers of the expression.
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
Rectangle with four equal sides. Perimeter: 4x the length of 1 side. area of a square is equal to one side squared.
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.

24. Calculating Negative Exponents and Radical Exponents
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
Part=Percent x Whole
Rewrite the number without the negative sign as the bottom of a fraction with 1 as the top.

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

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26. 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.
Subtract the smallest from the largest and add 1
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 2 if the last digit is even - An integer is divisible by 4 if the last two digits form a multiple of 4.

27. Adding a Positive Number to a Negative Number
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.
Ignore the signs... find the positive difference between the numbers - Then attach the sign of the original number with the larger absolute value
Express them with a common denominator.
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.

28. Finding the Missing Number in a Series When You are Given the Average
Change the division sign to a multiplication sign - invert the second fraction - and multiply.
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.
VofaC: (pie)r(2)h

29. Adding and Subtracting Roots
VofaC: (pie)r(2)h
Subtract the smallest from the largest and add 1
Just add or subtract the coefficients in front of the radicals.
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.

30. Solving a Proportion
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
Cross multiply.
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.

31. Multiplying Fractions
A(2)+b(2)=c(2)
Put each number in the original ratio over the sum of the numbers.
Multiply the numerators and multiply the denominators.
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96

32. 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?
Adjacent angles are supplementary - Vertical angles are equal
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.
Put each number in the original ratio over the sum of the numbers.
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72

33. Finding the Length of an Arc in a Circle

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34. What Does Solving In Terms of Mean?
Four equal acute angles and four equal obtuse angles.
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.
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides

35. Finding the Volume of a Rectangular Solid
Plug in the given values for the unknowns and calculate according to PEMDAS.
VofaRS=lwh
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

36. If 4(x(2)-10x+25) - then what is the value of x?
2(pie)r
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.
Average: Sum of the Terms/Number of the Terms
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.

37. Factoring the Difference of Two Squares
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.
Cross multiply.
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
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.

38. Difference Between a Factor and a Multiple
The value that appears the most often.
VofaC: (pie)r(2)h
-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.
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.

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

40. How do you know if an integer is a multiple of 3 or 9?
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72
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.
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.

41. Increasing and Decreasing a Number by a Certain Percentage
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
-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.
Use the distance formula: d=(square root over) (x2-x2)2+(y1-y2)2
Put each number in the original ratio over the sum of the numbers.

42. Multiply and Divide Positive and Negative Numbers
Just add or subtract the coefficients in front of the radicals.
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
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating decimal.
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

43. Simplifying an Algebraic Equation
FOIL: First - Outer - Inner - Last... Combine Like Terms
Four equal acute angles and four equal obtuse angles.
Cancel factors common to the numerator and denominator.
-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.

44. Converting from part-to-part ratios to part-to-whole ratios
Put each number in the original ratio over the sum of the numbers.
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
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
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.

45. General Procedure for Multiplying Polynomials
Use the distributive property then combine the like terms.
Use the distance formula: d=(square root over) (x2-x2)2+(y1-y2)2
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.
Average A per B= Total A/Total B so Average Speed = Total Distance/Total Time

46. Multiplying Monomials
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.
-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.
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
Multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying like bases. 2a3a=(23)(a*a)=6a^2

47. Solving an Inequality
Sum: Average x Number of Terms
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.
Parenthesis - Exponents - Multiplication/Division - Addition/Subtraction
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.

48. 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.
Change the division sign to a multiplication sign - invert the second fraction - and multiply.
Ignore the signs... find the positive difference between the numbers - Then attach the sign of the original number with the larger absolute value
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.

49. If the average of a - b - and 48 is 48 - what is the value of 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.
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.
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
Plug in the given values for the unknowns and calculate according to PEMDAS.

50. Simplifying a Fraction to Lowest 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.
Factor out and cancel all factors the numerator and denominator have in common.
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.

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