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SAT Math

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Answer 50 questions in 15 minutes.
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1. Evaluating an Algebraic Expression
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.
Subtract the smallest from the largest and add 1
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.
Plug in the given values for the unknowns and calculate according to PEMDAS.

2. Adding and Subtracting Monomials
Average the smallest and largest number
Do whatever is necessary to both sides to isolate the variable.
Make sure that the terms to be combined have exactly the same variables. 2a+3a=(2+3)a=5a
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.

3. 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?
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
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.
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.
(x1+x2)/2 -(y1+y2)/2

4. Adding and Subtracting Roots
Do whatever is necessary to both sides to isolate the variable.
Just add or subtract the coefficients in front of the radicals.
Multiply the numerators and multiply the denominators.
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

5. Adding and Subtracting Polynomials
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
Combine like terms.
Integers are whole numbers and their opposites; they include negatives of whole numbers and zero.
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

6. Definition of an Integer
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
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.
Integers are whole numbers and their opposites; they include negatives of whole numbers and zero.

7. Finding the Area of a Triangle

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8. If f(x)=x(1/3)+1/3x - then what is the value of f(27)?
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
Subtract the smallest from the largest and add 1
2(pie)r
Get the absolute value equation by itself. Solve.

9. 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
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
Check the multiples of the larger number until you find one that's also a multiple of the smaller.
Express them with a common denominator.

10. What are Two Special Pythagorean Triples?

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11. Multiplying Monomials
Multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying like bases. 2a3a=(23)(a*a)=6a^2
Factor out and cancel all factors the numerator and denominator have in common.
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
Average A per B= Total A/Total B so Average Speed = Total Distance/Total Time

12. Solving a System of Equations
Subtract the smallest from the largest and add 1
Use the distributive property then combine the like terms.
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.
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.

13. Finding the Midpoint Between Two Points
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
Four equal acute angles and four equal obtuse angles.
Two relatively prime numbers are integers that have no common factor other than 1.
(x1+x2)/2 -(y1+y2)/2

14. Finding the Rate of Speed
Use the formula distance=rate*time and its variations to help in this question.
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.
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.
Factor out the perfect squares under the radical - take their square roots - and put the result in front.

15. Calculating Negative Exponents and Radical Exponents
Rewrite the number without the negative sign as the bottom of a fraction with 1 as the top.
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
To solve a quadratic equation - set the equation equal to zero and solve for the numbers of the expression.

16. Converting from an Improper Fraction to a Mixed Number
Part=Percent x Whole
Use the distance formula: d=(square root over) (x2-x2)2+(y1-y2)2
Sum: Average x Number of Terms
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.

17. Finding a Term in a Geometric Sequence
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.
Part=Percent x Whole
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.
Plug in the given values for the unknowns and calculate according to PEMDAS.

18. If x(2)=7x+18 - what is the positive value of x?
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal
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.
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

19. Convert From a Fraction to a Decimal - Vice Versa
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.
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.
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I
Use the distributive property then combine the like terms.

20. What is the Triangle Inequality Theorem?
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
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.
Sum: Average x Number of Terms
Adjacent angles are supplementary - Vertical angles are equal

21. Finding the Volume of a Cylinder
VofaC: (pie)r(2)h
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
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
Subtract the exponents and keep the same base.y^13/y^8=y^13-8=y^5

22. Properties of the Interior and Exterior Angles of a Triangle
Part=Percent x Whole
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)
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.

23. Characteristics of a Square
Make sure that the terms to be combined have exactly the same variables. 2a+3a=(2+3)a=5a
(x1+x2)/2 -(y1+y2)/2
Combine like terms.
Rectangle with four equal sides. Perimeter: 4x the length of 1 side. area of a square is equal to one side squared.

24. Definition of an Irrational Number
Integers are whole numbers and their opposites; they include negatives of whole numbers and zero.
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: Sum of the Terms/Number of the Terms
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal

25. If the average of a - b - and 48 is 48 - what is the value of a + b?
First - get the square root by itself on one side of the equation. Then - solve for x by squaring each side. Answer: 25
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
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.

26. Finding the Area of a Circle
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.
VofaRS=lwh
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.
(pie)r(squared)

27. Setting Up a Ratio
Plug in the given values for the unknowns and calculate according to PEMDAS.
They have the same shape - but not necessarily the same size. Their corresponding angles are equal and their corresponding sides are proportional.
Put the number associated with the word of on top and the quantity associated with the word to on the bottom and simplify.
Cross multiply.

28. How do you know if an integer is a multiple of 3 or 9?
(x1+x2)/2 -(y1+y2)/2
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.
Find a common denominator - then add or subtract the numerators.
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

29. Knowing Whether the Sum - Difference - or Product of Several Numbers will be Even/Odd
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.
Use simple numbers like 1 and 2 and see what happens.
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.
Plug in the given values for the unknowns and calculate according to PEMDAS.

30. If r#t=t(r)-r(t) - then what is 4#3?
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.
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.
Plug in the given values for the unknowns and calculate according to PEMDAS.
Sum: Average x Number of Terms

31. Prime Factorization of a Number
To find the prime factorization of an integer - keep breaking it up into factors until all the factors are prime numbers.
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.
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.
Put the number associated with the word of on top and the quantity associated with the word to on the bottom and simplify.

32. Formula Used to Find Percent
Isolate the radical expression and use the standard rules of algebra.
Parenthesis - Exponents - Multiplication/Division - Addition/Subtraction
Average the smallest and largest number
Part=Percent x Whole

33. If 4(x(2)-10x+25) - then what is the value of x?
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.
The whole number left over after division.
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 - get the square root by itself on one side of the equation. Then - solve for x by squaring each side. Answer: 25

34. What is the length of the shorter leg of a right triangle whose other leg measures 7(square root: 3)?
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating decimal.
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.
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.
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. What is the length of a leg of an isosceles rich triangle who's hypotenuse measures 24(square root: 2)?
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24
Use the distributive property then combine the like terms.
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
Average A per B= Total A/Total B so Average Speed = Total Distance/Total Time

36. What Does Solving In Terms of Mean?
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.
Use simple numbers like 1 and 2 and see what happens.
Plug in the given values for the unknowns and calculate according to PEMDAS.
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.

37. Average Rate and How Do You Find It
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
Average A per B= Total A/Total B so Average Speed = Total Distance/Total Time
To find the prime factorization of an integer - keep breaking it up into factors until all the factors are prime numbers.

38. Converting From a Mixed Number to an Improper Fraction
-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.
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
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.

39. Number of Groups - +1 Each Time
To find the prime factorization of an integer - keep breaking it up into factors until all the factors are prime numbers.
-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.
Cancel factors common to the numerator and denominator.
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.

40. Finding the Missing Number in a Series When You are Given the Average
Use simple numbers like 1 and 2 and see what happens.
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
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 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.

41. Properties of a 30-60-90 Triangle
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.
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.
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.
Average: Sum of the Terms/Number of the Terms

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

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43. What types of angles are formed when a transversal cross parallel lines?
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.
Four equal acute angles and four equal obtuse angles.
The whole number left over after division.
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.

44. Simplifying a Fraction to Lowest Terms
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
Factor out and cancel all factors the numerator and denominator have in common.
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
Subtract the smallest from the largest and add 1

45. What is the Pythagorean Theorem?
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal
Just add or subtract the coefficients in front of the radicals.
A(2)+b(2)=c(2)
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.

46. Adding and Subtracting Fractions
Multiply the numerators and multiply the denominators.
Find a common denominator - then add or subtract the numerators.
Parenthesis - Exponents - Multiplication/Division - Addition/Subtraction
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.

47. Multiplying Binomials
-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.
FOIL: First - Outer - Inner - Last... Combine Like Terms
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.
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.

48. If y varies inversely with x - and y is 3 when x is 10 - what is the value of x when y is 6?
Add the exponents and keep the same base - x^3*x^4=x^3+4=x^7
-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.
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
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

49. Characteristics of a Rectangle
(x1+x2)/2 -(y1+y2)/2
Put the number associated with the word of on top and the quantity associated with the word to on the bottom and simplify.
Rewrite the number without the negative sign as the bottom of a fraction with 1 as the top.
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.

50. Finding the Sum of the Interior Angles of a Polygon
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.
FOIL: First - Outer - Inner - Last... Combine Like Terms
Sum: Average x Number of Terms
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides

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