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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 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.
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
Sum: Average x Number of Terms
Find a common denominator - then add or subtract the numerators.

2. If 4(x(2)-10x+25) - then what is the value of x?
-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.
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 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 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

3. Finding the Mode of a Set of Numbers
Average A per B= Total A/Total B so Average Speed = Total Distance/Total Time
Check the multiples of the larger number until you find one that's also a multiple of the smaller.
The value that appears the most often.
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72

4. Quickly Finding the Average of a Series of Evenly Spaced Numbers
The whole number left over after division.
Average the smallest and largest number
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal
Use the units to keep things straight - Snowfall inches/hours

5. Evaluating an Algebraic Expression
Plug in the given values for the unknowns and calculate according to PEMDAS.
Put the number associated with the word of on top and the quantity associated with the word to on the bottom and simplify.
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.
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.

6. Definition of Remainder
The whole number left over after division.
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
Cross multiply.
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.

7. Adding and Subtracting Roots
Just add or subtract the coefficients in front of the radicals.
Set up an equation. Think of the result of a 15 percent increase over x as 1.15x.
A rational number is a number that can be expressed as a ratio of two numbers or as a terminating or repeating decimal.
Find a common denominator - then add or subtract the numerators.

8. 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?
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
Average the smallest and largest number
Use common multiples of 6 and 8 - and divide by 7 to find a remainder of 2. Answer: 72
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.

9. 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
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.
Parenthesis - Exponents - Multiplication/Division - Addition/Subtraction
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.

10. Finding the Slope of a Line When Given Two Points on the Line
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
Cancel factors common to the numerator and denominator.
Just add or subtract the coefficients in front of the radicals.
Slope=change in y/change in x

11. Finding the Domain and Range of a Function
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.
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.
Rectangle with four equal sides. Perimeter: 4x the length of 1 side. area of a square is equal to one side squared.
Express them with a common denominator.

12. Characteristics of a Parallelogram
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
Multiply the whole number part by the denominator - then add the numerator. Keep the same denominator.
Use simple numbers like 1 and 2 and see what happens.

13. Finding the Volume of a Rectangular Solid
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.
VofaRS=lwh
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.
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

14. Scalene - Isosceles - and Equilateral Triangles
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.
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.
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.
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.

15. If the average of a - b - and 48 is 48 - what is the value of a + b?
Put each number in the original ratio over the sum of the numbers.
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.
Since the average of the three numbers is 48 - then a+b+48=48*3. A+B=96
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.

16. Solving a Proportion
The whole number left over after division.
x(2)-5x+6 - What factors of +6 also have a sum of -5? (x-2)(x-3)
Cross multiply.
Make sure that the terms to be combined have exactly the same variables. 2a+3a=(2+3)a=5a

17. PEMDAS
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
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.
Parenthesis - Exponents - Multiplication/Division - Addition/Subtraction
Use the distributive property then combine the like terms.

18. Simplifying Square Roots
Factor out the perfect squares under the radical - take their square roots - and put the result in front.
Combine like terms.
Multiply the numerators and multiply the denominators.
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.

19. Finding the Sum of the Average of a Series of Numbers
Multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying like bases. 2a3a=(23)(a*a)=6a^2
Sum: Average x Number of Terms
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
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

20. Factoring the Difference of Two Squares
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
Cancel factors common to the numerator and denominator.
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.
Multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying like bases. 2a3a=(23)(a*a)=6a^2

21. 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.
Average the smallest and largest number
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.
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.

22. Expressing the Union and Intersection of Sets
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
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.
Probability: Number of Favorable Outcomes/Total Possible Outcomes
(pie)r(squared)

23. Solving a Problem Involving Rates
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
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.
Factor out the perfect squares under the radical - take their square roots - and put the result in front.
Use the units to keep things straight - Snowfall inches/hours

24. What is the Triangle Inequality Theorem?
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.
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.
-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.
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

25. Calculating Negative Exponents and Radical Exponents
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.
Rewrite the number without the negative sign as the bottom of a fraction with 1 as the top.
-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.
Use simple numbers like 1 and 2 and see what happens.

26. 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?
First - get the base numbers to be the same. Then - set the exponents equal to each other and solve for x. Answer: 4
If two values vary directly - the values increase at the same rate. Express the relationship as x/y - and use a proportion to solve.
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.

27. Solving a System of Equations
Length of an Arc: (n/360)(2pier) - n is the degree measure of the arc's central angle
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.
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
Factor out the perfect squares under the radical - take their square roots - and put the result in front.

28. Finding the Area of a Sector

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29. Comparing the values of two or more Fractions
Express them with a common denominator.
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
SA=2lw+2wh+2lh (Length: l - Width: w - Height: h)
Use the distance formula: d=(square root over) (x2-x2)2+(y1-y2)2

30. Finding the Surface Area of a Rectangular Solid
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)
First - substitute for the variable. Then evaluate the expression using the correct order of operations.
VofaC: (pie)r(2)h

31. Finding the Distance Between Two Points on a Coordinate Graph
Use the formula distance=rate*time and its variations to help in this question.
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
Use the distance formula: d=(square root over) (x2-x2)2+(y1-y2)2
Average A per B= Total A/Total B so Average Speed = Total Distance/Total Time

32. 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.
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
Just add or subtract the coefficients in front of the radicals.

33. Finding the Length of an Arc in a Circle

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34. If y varies inversely with x - and y is 3 when x is 10 - what is the value of x when y is 6?
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
-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.
Factor out the perfect squares under the radical - take their square roots - and put the result in front.
Two pairs of parallel sides. Opposite sides/angles are equal. Any two consecutive angles add up to 180. Area: base*height.

35. Finding the Midpoint Between Two Points
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
Distance of the Number from Zero on the Number Line. AV is always positive. AV of 7: I7I
(x1+x2)/2 -(y1+y2)/2
'In a group of 260 boys - half of the boys are blonds.' 1/2 of 260 boys are blondes--> 130= # of blonds

36. Knowing Whether the Sum - Difference - or Product of Several Numbers will be Even/Odd
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.
Use simple numbers like 1 and 2 and see what happens.
Factor out and cancel all factors the numerator and denominator 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

37. Characteristics of a Rectangle
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
VofaRS=lwh
The value that appears the most often.
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.

38. 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.
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
Adjacent angles are supplementary - Vertical angles are equal
Express them with a common denominator.

39. Multiply and Divide Positive and Negative Numbers
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 formula distance=rate*time and its variations to help in this question.
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.

40. Prime Factorization of a Number
(pie)r(squared)
To find the prime factorization of an integer - keep breaking it up into factors until all the factors are prime numbers.
Four-sided figure with four right angles. Perimeter: 2(length+width). Area: length*width.
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. Number of Groups - +1 Each Time
Combine like terms.
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
The least common multiple of 2 -3 - and 5 is 30. Therefore - the least number of students in the class is 30+1=31.
Do whatever is necessary to both sides to isolate the variable.

42. Formula to Find 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.
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.
Make sure that the terms to be combined have exactly the same variables. 2a+3a=(2+3)a=5a
Average: Sum of the Terms/Number of the Terms

43. Formula Used to Find Percent
Sum: Average x Number of Terms
Cancel factors common to the numerator and denominator.
Part=Percent x Whole
Put each number in the original ratio over the sum of the numbers.

44. Properties of a 30-60-90 Triangle
The sides of a 30-60-90 triangle are in a ratio of x:x(square root)3:2x.
-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.
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.

45. Definition of an Irrational Number
Real numbers - Have locations on the number line - Cannot be expressed precisely as a fraction/decimal
Break down the integers into their prime factorizations - and multiply all prime factors they have in common.
Factor the expressions to factor is the difference of squares. a^2-b^2=(a+b)(a-b)
Find a common denominator - then add or subtract the numerators.

46. 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
Multiply 100 by .30 - then that # by .30 - then that # by .30=34.3. The answer is 34.3%
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.
Find a common denominator - then add or subtract the numerators.
Isolate the one variable on one side of the equation - leaving an expression containing the other variable on the other side of the equation.

47. When Given a Series of Percent Increases and Decreases - How do you Determine your ending value?
The whole number left over after division.
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.
Start with 100%--> 100% + (10 percent of 100) = 110%... 110% + (20 percent of 110) = 132%.
Isolate the variable. When you multiply both sides by a negative number - you must reverse the inequality symbol.

48. Numbers that are Relatively Prime
(x1+x2)/2 -(y1+y2)/2
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.
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
Two relatively prime numbers are integers that have no common factor other than 1.

49. An Important Property of a Line Tangent to a Circle
Put the number associated with the word of on top and the quantity associated with the word to on the bottom and simplify.
Combine like terms.
Plug in the given values for the unknowns and calculate according to PEMDAS.
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.

50. Multiplying Expressions with Exponents that Have a Common Base
Sum of the Angles of a Polygon: (n-2)*180 - n is the number of sides
Add the exponents and keep the same base - x^3*x^4=x^3+4=x^7
Multiply the coefficients and the variables separately. Be sure to add the exponents when multiplying like bases. 2a3a=(23)(a*a)=6a^2
Average: Sum of the Terms/Number of the Terms