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SAT Math

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Answer 50 questions in 15 minutes.
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1. Finding the Length of an Arc in a Circle

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2. General Procedure for Multiplying Polynomials
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.
Two relatively prime numbers are integers that have no common factor other than 1.
Use the distributive property then combine the like terms.
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.

3. If (Square Root: x-1)+5=12 - what is the value of x/2?
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
Cancel factors common to the numerator and denominator.
Average the smallest and largest number
First - get the square root by itself on one side of the equation. Then - solve for x by squaring each side. Answer: 25

4. Convert From a Fraction to a Decimal - Vice Versa
The value that appears the most often.
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.
Cross multiply.
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%

5. Raising a Power to a Power
Put each number in the original ratio over the sum of the numbers.
Multiply the exponents - (x^3)^4=x^3*4=x^12
Average: Sum of the Terms/Number of the Terms
Cross multiply.

6. Solving a Radical Equation
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.
Isolate the radical expression and use the standard rules of algebra.
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.

7. If f(x)=x(1/3)+1/3x - then what is the value of f(27)?
The sides of a 45-45-90 triangle are in a ratio of x:x:x(square root)2.
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
Plug in the given values for the unknowns and calculate according to PEMDAS.

8. Finding the y-intercept When Given an Equation of a Line
Put the equation into y=mx+b-- in which case b is the y-intercept.
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.
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.

9. Definition of an Irrational Number
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal
The sides of a 45-45-90 triangle are in a ratio of x:x:x(square root)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.
Rectangle with four equal sides. Perimeter: 4x the length of 1 side. area of a square is equal to one side squared.

10. Finding the Volume of a Rectangular Solid
VofaRS=lwh
To find the prime factorization of an integer - keep breaking it up into factors until all the factors are prime numbers.
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.
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.

11. Calculating the Probability that an Event will Take Place
Probability: Number of Favorable Outcomes/Total Possible Outcomes
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.
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I
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.

12. 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.
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
VofaRS=lwh

13. Characteristics of a Rectangle
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 distance formula: d=(square root over) (x2-x2)2+(y1-y2)2
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.
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.

14. Determining Absolute Value of a Number
Adjacent angles are supplementary - Vertical angles are equal
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
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
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I

15. Solving a Problem Involving Rates
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.
Use the formula distance=rate*time and its variations to help in this question.
Factor out and cancel all factors the numerator and denominator have in common.
Use the units to keep things straight - Snowfall inches/hours

16. Finding the Slope When Given an Equation of a Line
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.
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
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

17. Dividing Expressions with Exponents that Have a Common Base
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.
Subtract the exponents and keep the same base.y^13/y^8=y^13-8=y^5
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.
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

18. If 3(3x)=9(x+2) - what is the value of x?
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
Slope=change in y/change in x
(x1+x2)/2 -(y1+y2)/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.

19. If 4(x(2)-10x+25) - then what is the value of x?
Put the equation into y=mx+b-- in which case b is the y-intercept.
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.
VofaRS=lwh
Parenthesis - Exponents - Multiplication/Division - Addition/Subtraction

20. What is the length of the shorter leg of a right triangle whose other leg measures 7(square root: 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.
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.
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.
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

21. Finding the Reciprocal of a Fraction
Change the division sign to a multiplication sign - invert the second fraction - and multiply.
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.
Switch the numerator and denominator.
FOIL: First - Outer - Inner - Last... Combine Like Terms

22. Characteristics of a Square
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.
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
Rectangle with four equal sides. Perimeter: 4x the length of 1 side. area of a square is equal to one side squared.
VofaRS=lwh

23. What is the value of x(2)+1-y(2) - if x-y=5 and x+y=7?

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24. Average Rate and How Do You Find It
Average A per B= Total A/Total B so Average Speed = Total Distance/Total Time
(pie)r(squared)
Two relatively prime numbers are integers that have no common factor other than 1.
Put the equation into the slope-intercept form: y=mx+b. The slope is m.

25. Definition of Remainder
Combine like terms.
Factor out and cancel all factors the numerator and denominator have in common.
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds
The whole number left over after division.

26. Definition of a Rational 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.
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating decimal.
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal
Multiply the numerators and multiply the denominators.

27. Simplifying a Fraction to Lowest 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
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.
Factor out and cancel all factors the numerator and denominator have in common.

28. Finding the GCF of Two or More Numbers
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
FOIL: First - Outer - Inner - Last... Combine Like Terms
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides
VofaC: (pie)r(2)h

29. Finding the Surface Area of a Rectangular Solid
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
Subtract the smallest from the largest and add 1
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating decimal.
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)

30. Converting From a Mixed Number to an Improper Fraction
2(pie)r
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)

31. Finding the Sum of the Interior Angles of a Polygon
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
Use the units to keep things straight - Snowfall inches/hours
Put each number in the original ratio over the sum of the numbers.
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides

32. Prime Factorization of a Number
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
Put the equation into y=mx+b-- in which case b is the y-intercept.
To find the prime factorization of an integer - keep breaking it up into factors until all the factors are prime numbers.
Two relatively prime numbers are integers that have no common factor other than 1.

33. What is the slope of the line perpendicular to the line with linear equation 4x+2y=12?
Ignore the signs... find the positive difference between the numbers - Then attach the sign of the original number with the larger absolute value
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I
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.
Put each number in the original ratio over the sum of the numbers.

34. Properties of the Interior and Exterior Angles of a Triangle
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
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.
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.
Plug in the given values for the unknowns and calculate according to PEMDAS.

35. Numbers that are Relatively Prime
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds
Two relatively prime numbers are integers that have no common factor other than 1.
The whole number left over after division.
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.

36. Counting Consecutive Integers
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
Subtract the smallest from the largest and add 1
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
Do whatever is necessary to both sides to isolate the variable.

37. What types of angles are formed when two lines intersect?
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds
2(pie)r
Factor out and cancel all factors the numerator and denominator have in common.
Adjacent angles are supplementary - Vertical angles are equal

38. Finding the Rate of Speed
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
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
Use the formula distance=rate*time and its variations to help in this question.
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.

39. 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 common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72
Use simple numbers like 1 and 2 and see what happens.
Do whatever is necessary to both sides to isolate the variable.

40. Finding the Original Value Before it was Increased or Decreased by a Certain Percentage
Use simple numbers like 1 and 2 and see what happens.
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.
Put each number in the original ratio over the sum of the numbers.

41. If 2+l6-xl=10 and x>0 - what is the value of 2x?
Multiply the exponents - (x^3)^4=x^3*4=x^12
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.
Use the distributive property then combine the like terms.
Get the absolute value equation by itself. Solve.

42. Multiplying Binomials
FOIL: First - Outer - Inner - Last... Combine Like Terms
The value that appears the most often.
Put each number in the original ratio over the sum of the numbers.
Check the multiples of the larger number until you find one that's also a multiple of the smaller.

43. Formula Used to Find Percent
Part=Percent x Whole
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.

44. What is the length of a leg of an isosceles rich triangle who's hypotenuse measures 24(square root: 2)?
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
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.
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24
Two relatively prime numbers are integers that have no common factor other than 1.

45. Multiplying and Dividing Roots
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24
(pie)r(squared)
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
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.

46. 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?
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
Integers are whole numbers and their opposites; they include negatives of whole numbers and zero.
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.

47. Finding the Mode of a Set of Numbers
The value that appears the most often.
Factor out and cancel all factors the numerator and denominator have in common.
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
Do whatever is necessary to both sides to isolate the variable.

48. Factoring a Polynomial ('FOIL in Reverse')
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
Get the absolute value equation by itself. Solve.
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)
Use the units to keep things straight - Snowfall inches/hours

49. Simplifying an Algebraic Equation
Rewrite the number without the negative sign as the bottom of a fraction with 1 as the top.
Switch the numerator and denominator.
Average: Sum of the Terms/Number of the Terms
Cancel factors common to the numerator and denominator.

50. Counting the Total Number of Possibilities for Several Events to Occur
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
-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.
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.

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