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SAT Math

Subjects : sat, math

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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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. Finding the Original Value Before it was Increased or Decreased by a Certain Percentage
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.
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
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
Two relatively prime numbers are integers that have no common factor other than 1.

2. Properties of a 30-60-90 Triangle
Do whatever is necessary to both sides to isolate the variable.
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.
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.

3. An Important Property of a Line Tangent to a Circle
Do whatever is necessary to both sides to isolate the variable.
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.
Sum: Average x Number of Terms
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)

4. Properties of Similar Triangles
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
They have the same shape - but not necessarily the same size. Their corresponding angles are equal and their corresponding sides are proportional.
Isolate the radical expression and use the standard rules of algebra.
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.

5. Raising a Power to a Power
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.
Multiply the exponents - (x^3)^4=x^3*4=x^12
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.
Rectangle with four equal sides. Perimeter: 4x the length of 1 side. area of a square is equal to one side squared.

6. Sale Price of a Jacket Marked Down 30% Each Week for 3 Weeks - What % of the Original Price is the Cost of the Jacket After the 3 Week Sale Period
2(pie)r
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
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 an Irrational Number
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.
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
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/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.

8. Simplifying Square Roots
Factor out the perfect squares under the radical - take their square roots - and put the result in front.
To find the prime factorization of an integer - keep breaking it up into factors until all the factors are prime numbers.
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
Four equal acute angles and four equal obtuse angles.

9. Quickly Finding the Average of a Series of Evenly Spaced Numbers
-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.
Average the smallest and largest number
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal
-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.

10. When Given a Series of Percent Increases and Decreases - How do you Determine your ending value?
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds
Get the absolute value equation by itself. Solve.

11. Factoring the Difference of Two Squares
The whole number left over after division.
Check the multiples of the larger number until you find one that's also a multiple of the smaller.
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)

12. Dividing Fractions
Change the division sign to a multiplication sign - invert the second fraction - and multiply.
The value that appears the most often.
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 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.

13. Finding the Length of an Arc in a Circle

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14. Solving a Proportion
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.
Find a common denominator - then add or subtract the numerators.
FOIL: First - Outer - Inner - Last... Combine Like Terms
Cross multiply.

15. Number of Groups - +1 Each 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.
2(pie)r
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
Cancel factors common to the numerator and denominator.

16. What value of x is not in the domain of the function f(x)=x-2/x-3?
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.
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
x(2)+x(2)=(24square root: 2)(2)=576*2 2x(2)=1 -152 - x(2)=576 - Square Root 576= 24
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

17. Finding the Area of a Circle
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.
(pie)r(squared)
Factor out and cancel all factors the numerator and denominator have in common.
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)

18. Finding the Distance Between Two Points on a Coordinate Graph
Use the distance formula: d=(square root over) (x2-x2)2+(y1-y2)2
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
Combine like terms.
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.

19. Finding the Slope When Given an Equation of a Line
(x1+x2)/2 -(y1+y2)/2
Average the smallest and largest number
Put the equation into the slope-intercept form: y=mx+b. The slope is m.
Sum: Average x Number of Terms

20. 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.
FOIL: First - Outer - Inner - Last... Combine Like Terms
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.
Rewrite the number without the negative sign as the bottom of a fraction with 1 as the top.

21. Direct Variation and Inverse Variation
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
Find a common denominator - then add or subtract the numerators.
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
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating decimal.

22. Converting from an Improper Fraction to a Mixed Number
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
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.
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.
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.

23. Solving a Radical Equation
A(2)+b(2)=c(2)
Adjacent angles are supplementary - Vertical angles are equal
Isolate the radical expression and use the standard rules of algebra.
-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.

24. If 4(x(2)-10x+25) - then what is the value of 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.
The value that appears the most often.
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.
Switch the numerator and denominator.

25. Adding and Subtracting Roots
Subtract the exponents and keep the same base.y^13/y^8=y^13-8=y^5
Just add or subtract the coefficients in front of the radicals.
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.
Put each number in the original ratio over the sum of the numbers.

26. Adding and Subtracting Polynomials
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.
Cross multiply.
Combine like terms.

27. If r#t=t(r)-r(t) - then what is 4#3?
Do whatever is necessary to both sides to isolate the variable.
Sum: Average x Number of Terms
Substitute r=4 and t=3 and evaluate the exponents before subtracting the expression.
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides

28. Definition of a Rational Number
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.
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating decimal.
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%

29. Finding the Volume of a Cylinder
VofaC: (pie)r(2)h
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
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
Check the multiples of the larger number until you find one that's also a multiple of the smaller.

30. Converting from part-to-part ratios to part-to-whole ratios
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
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.
Put each number in the original ratio over the sum of the numbers.

31. Counting Consecutive Integers
Find a common denominator - then add or subtract the numerators.
Subtract the smallest from the largest and add 1
Get the absolute value equation by itself. Solve.
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal

32. Formula to Find Average
(pie)r(squared)
Two relatively prime numbers are integers that have no common factor other than 1.
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
Average: Sum of the Terms/Number of the Terms

33. Multiplying Monomials
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.
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
Make sure that the terms to be combined have exactly the same variables. 2a+3a=(2+3)a=5a
Multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying like bases. 2a3a=(23)(a*a)=6a^2

34. Finding the Midpoint Between Two Points
Factor out the perfect squares under the radical - take their square roots - and put the result in front.
-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.
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.
(x1+x2)/2 -(y1+y2)/2

35. Setting Up a Ratio
Four equal acute angles and four equal obtuse angles.
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
Put the number associated with the word of on top and the quantity associated with the word to on the bottom and simplify.
Put each number in the original ratio over the sum of the numbers.

36. Finding the Area of a Sector

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37. If (Square Root: x-1)+5=12 - what is the value of x/2?
Cross multiply.
First - get the square root by itself on one side of the equation. Then - solve for x by squaring each side. Answer: 25
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.
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.

38. Definition of an Integer
Integers are whole numbers and their opposites; they include negatives of whole numbers and zero.
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.
Switch the numerator and denominator.
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.

39. Multiply and Divide Positive and Negative Numbers
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
To solve a quadratic equation - set the equation equal to zero and solve for the numbers of the expression.
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 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.

40. Prime Factorization of a Number
A(2)+b(2)=c(2)
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.
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

41. Finding the Sum of the Interior Angles of a Polygon
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.
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides
Part=Percent x Whole
Plug in the given values for the unknowns and calculate according to PEMDAS.

42. Finding the Surface Area of a Rectangular Solid
The sides of a 45-45-90 triangle are in a ratio of x:x:x(square root)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.
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)

43. Convert From a Fraction to a Decimal - Vice Versa
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
Four equal acute angles and four equal obtuse angles.
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.
-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.

44. What is the length of the shorter leg of a right triangle whose other leg measures 7(square root: 3)?
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.
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.
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.
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.

45. Finding the Missing Number in a Series When You are Given the Average
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I
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.
Factor out and cancel all factors the numerator and denominator have in common.
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.

46. 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?
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 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 units to keep things straight - Snowfall inches/hours
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.

47. 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
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.
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.

48. Dividing Expressions with Exponents that Have a Common Base
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.
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
Slope=change in y/change in x

49. 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
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.
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.
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%

50. 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.
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
Change the division sign to a multiplication sign - invert the second fraction - and multiply.
Switch the numerator and denominator.