Below is our free SAT Math practice test that is fully updated for the new 2016 SAT. The new SAT Math test focuses on real world math problems, including the types of problems you may need to know for college courses, careers, and your personal life. There is also an emphasis on multistep problems. The math test is divided into two sections, the first of which must be completed without a calculator.
Directions for Questions 115: Solve each problem and then select the best answer from the choices provided. Directions for Questions 1620: Solve each problem and then type in the correct answer. Use the forward slash symbol ( / ) for fractions.
SAT Math Practice Test  No Calculator
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Question 1 of 20
1. Question
In a certain coffee shop, X lattes were sold each hour for 6 hours on Monday, and Y americanos were sold each hour for 7 hours on Monday. What expression represents the total number of lattes and americanos that were sold by the coffee shop on Monday?
CorrectConvert the word problem into an expression representing the total coffees sold:
This solution can be verified by testing values. Let’s say X = 2 and Y = 3. If 2 lattes were sold each hour for 6 hours, then 2 x 6 = 12 lattes total were sold. If 3 americanos were sold each hour for 7 hours, then 3 x 7 = 21 americanos were sold. The total number of lattes and americanos would therefore be 12 + 21 = 33. Plug in X = 2 and Y = 3 into the answer choices. The one that yields 33 is correct.Alternatively, notice that we multiplied the number of each drink by the number of hours that drink was sold, then we added the two totals together: 6X + 7Y. This matches the expression in answer choice (D).
IncorrectConvert the word problem into an expression representing the total coffees sold:
This solution can be verified by testing values. Let’s say X = 2 and Y = 3. If 2 lattes were sold each hour for 6 hours, then 2 x 6 = 12 lattes total were sold. If 3 americanos were sold each hour for 7 hours, then 3 x 7 = 21 americanos were sold. The total number of lattes and americanos would therefore be 12 + 21 = 33. Plug in X = 2 and Y = 3 into the answer choices. The one that yields 33 is correct.Alternatively, notice that we multiplied the number of each drink by the number of hours that drink was sold, then we added the two totals together: 6X + 7Y. This matches the expression in answer choice (D).

Question 2 of 20
2. Question
CorrectStart by substituting 12 for “p” in the original equation:
Because the unknown is in the denominator, and the equation represents a simple proportion, we can simplify the problem by cross multiplying and then dividing:
IncorrectStart by substituting 12 for “p” in the original equation:
Because the unknown is in the denominator, and the equation represents a simple proportion, we can simplify the problem by cross multiplying and then dividing:

Question 3 of 20
3. Question
If the function ƒ(x) is defined for all real numbers by the following equation:
Then ƒ(ƒ(2)) =CorrectTo solve this function question, start by substituting 2 for x:
IncorrectTo solve this function question, start by substituting 2 for x:

Question 4 of 20
4. Question
If a car averages 36.8 miles per gallon of gasoline, approximately how many kilometers per liter of gasoline does the car average if 1 gallon = 3.8 liters and 1 mile = 1.6 kilometers?
CorrectThis problem is asking you to do two conversions: miles to kilometers, and gallons to liters. Both conversions can be done simultaneously to ensure correct dimensional analysis. Notice that because 1 gallon = 3.8 liters, both sides can be divided by either 1 gallon or 3.8 liters to produce 1:
IncorrectThis problem is asking you to do two conversions: miles to kilometers, and gallons to liters. Both conversions can be done simultaneously to ensure correct dimensional analysis. Notice that because 1 gallon = 3.8 liters, both sides can be divided by either 1 gallon or 3.8 liters to produce 1:

Question 5 of 20
5. Question
If the product of x and y does not equal zero, which of the following could be true based on the figure above?I. (−x, y) lies above the xaxis and to the right of the yaxis.
II. (x, −y) lies below the xaxis and to the left of the yaxis.
III. (x, y) lies on the xaxis to the right of the yaxis.CorrectIn Roman numeral I, if (−x, y) lies above the xaxis and to the right of the yaxis (that is, in the first quadrant), then x must equal a negative number, since all xvalues to the right of the yaxis are positive, and the only way −x = positive is if x = negative. So, y must be positive since it is above the xaxis. This is possible.
In Roman numeral II, if (x, −y) is in the third quadrant, then x = negative, and −y = negative. This is only true if x = negative and y = positive. This could be true.
In Roman numeral III, if a point lies on the xaxis, then its value is 0, that would make the product of x and y zero, which contradicts the given information. Only the first two Roman numerals are possible given the constraints of the problem.
IncorrectIn Roman numeral I, if (−x, y) lies above the xaxis and to the right of the yaxis (that is, in the first quadrant), then x must equal a negative number, since all xvalues to the right of the yaxis are positive, and the only way −x = positive is if x = negative. So, y must be positive since it is above the xaxis. This is possible.
In Roman numeral II, if (x, −y) is in the third quadrant, then x = negative, and −y = negative. This is only true if x = negative and y = positive. This could be true.
In Roman numeral III, if a point lies on the xaxis, then its value is 0, that would make the product of x and y zero, which contradicts the given information. Only the first two Roman numerals are possible given the constraints of the problem.

Question 6 of 20
6. Question
2x + 4y = 16
3x − 6y = 12
If (x,y) is a solution to the system of equations above, what is the value of x + y?
CorrectStart by simplifying each equation. Do the terms have any common factors? Notice how “2”, “4,” and “16,” in the first equation can all be divided by 2:
x + 2y = 8
In the second equation, “3,” “6,” and “12” can all be divided by 3:
x − 2y = 4
Since “2y” and “−2y” will cancel out if added together, let’s use the method of Elimination/Combination to solve for x:
x + 2y = 8
+ (x − 2y = 4)
——————
2x = 12
x = 6Plugging x = 6 back into either equation:
(6) + 2y = 8, tells us that y = 1. The question is asking for x + y, so:
x + y =
(6) + (1) = 7IncorrectStart by simplifying each equation. Do the terms have any common factors? Notice how “2”, “4,” and “16,” in the first equation can all be divided by 2:
x + 2y = 8
In the second equation, “3,” “6,” and “12” can all be divided by 3:
x − 2y = 4
Since “2y” and “−2y” will cancel out if added together, let’s use the method of Elimination/Combination to solve for x:
x + 2y = 8
+ (x − 2y = 4)
——————
2x = 12
x = 6Plugging x = 6 back into either equation:
(6) + 2y = 8, tells us that y = 1. The question is asking for x + y, so:
x + y =
(6) + (1) = 7 
Question 7 of 20
7. Question
If Eric was 22 years old x years ago and Shelley will be 24 years old in y years, what was the average of their ages 4 years ago?
CorrectTo do this problem, choose values for x and y. If you are choosing numbers, try to use low numbers that are easy to work with (such as 2, 3, 4, etc.).
Let’s say x = 2. If Eric was 22 years old 2 years ago, then today he is 24 years old. Four years ago, he would have been 20.
Let’s say y = 3. If Shelley will be 24 in 3 years, then she is 21 years old now. Four years ago, she was 17 years old.
Remember, we are looking for the average for their ages four years ago, NOT today!
Find the answer that gives you 18.5 when you plug in x = 2 and y = 3.IncorrectTo do this problem, choose values for x and y. If you are choosing numbers, try to use low numbers that are easy to work with (such as 2, 3, 4, etc.).
Let’s say x = 2. If Eric was 22 years old 2 years ago, then today he is 24 years old. Four years ago, he would have been 20.
Let’s say y = 3. If Shelley will be 24 in 3 years, then she is 21 years old now. Four years ago, she was 17 years old.
Remember, we are looking for the average for their ages four years ago, NOT today!
Find the answer that gives you 18.5 when you plug in x = 2 and y = 3. 
Question 8 of 20
8. Question
In the coordinate plane, line m passes through the origin and has a slope of 3. If points (6,y) and (x,12) are on line m, then y − x = ?
CorrectThe standard equation of a line is y = mx + b, where m is the slope, b is the yintercept, and (x, y) represent any coordinate pair on the line. Let’s fill in the equation of line m based on what we know:
y = 3x + b
Since the line passes through the origin: (0,0), we know the yintercept is 0:
y = 3x
To find what “y” is when x = 6, plug in x = 6. (6,18) is a point on the line, and y = 18. To find x, plug in 12 for y:
12 = 3x
4 = x
y − x = 18 − 4 = 14IncorrectThe standard equation of a line is y = mx + b, where m is the slope, b is the yintercept, and (x, y) represent any coordinate pair on the line. Let’s fill in the equation of line m based on what we know:
y = 3x + b
Since the line passes through the origin: (0,0), we know the yintercept is 0:
y = 3x
To find what “y” is when x = 6, plug in x = 6. (6,18) is a point on the line, and y = 18. To find x, plug in 12 for y:
12 = 3x
4 = x
y − x = 18 − 4 = 14 
Question 9 of 20
9. Question
CorrectIn this case, multiplying each term by the least common multiple (LCM) of the denominators will best clarify the problem:
An alternate approach is to work backwards and try each of the answer choices. However, this is best used as a method of verification, as it is likely that attempting every answer choice will require more time than algebraically evaluating the original statement.IncorrectIn this case, multiplying each term by the least common multiple (LCM) of the denominators will best clarify the problem:
An alternate approach is to work backwards and try each of the answer choices. However, this is best used as a method of verification, as it is likely that attempting every answer choice will require more time than algebraically evaluating the original statement. 
Question 10 of 20
10. Question
CorrectThe expression abcd can be rewritten to:
Only answer choice (2) correctly places abcd in the range: −2 < −1 < 0.IncorrectThe expression abcd can be rewritten to:
Only answer choice (2) correctly places abcd in the range: −2 < −1 < 0. 
Question 11 of 20
11. Question
The average of several test scores is 80. One makeup exam was given. Included with the other scores, the new average was 84. If the score on the makeup exam was 92, how many total exams were given?
CorrectRecall that average is calculated by adding a group of numbers and then dividing by the count of those numbers. Plugging in what we’re given to the formula for average:
Multiplying both sides by x + 1:
84x + 84 = Sum + MakeUp ScoreLet’s substitute “80x” for the “Sum”:
84x + 84 = 80x + Makeup Score
4x + 84 = Makeup ScoreWe are told that the makeup score was 92:
4x + 84 = 92
4x = 8
x = 2IncorrectRecall that average is calculated by adding a group of numbers and then dividing by the count of those numbers. Plugging in what we’re given to the formula for average:
Multiplying both sides by x + 1:
84x + 84 = Sum + MakeUp ScoreLet’s substitute “80x” for the “Sum”:
84x + 84 = 80x + Makeup Score
4x + 84 = Makeup ScoreWe are told that the makeup score was 92:
4x + 84 = 92
4x = 8
x = 2 
Question 12 of 20
12. Question
CorrectA negative exponent is another way of writing a fraction:
IncorrectA negative exponent is another way of writing a fraction:

Question 13 of 20
13. Question
In quadrilateral ABCD, what is the value of angle ADC if Angle BAD + Angle ABC + Angle BCD = 280?
CorrectFor this type of Geometry question, it’s simpler to draw the figure if none is provided. Remember that just because it’s a quadrilateral, this does not necessarily mean it is a square or rectangle, so let’s draw an irregular quadrilateral.
Let’s label the angles with other letters of the alphabet so it’s easier to understand which angles we’re discussing.The value we are looking for is angle ADC, here called “z”. The interior angles of any quadrilateral must sum to 360 degrees:
W + X + Y + Z = 360
Z = 360 − (W + X + Y)
280 + Z = 360, so Z = 80 degrees.IncorrectFor this type of Geometry question, it’s simpler to draw the figure if none is provided. Remember that just because it’s a quadrilateral, this does not necessarily mean it is a square or rectangle, so let’s draw an irregular quadrilateral.
Let’s label the angles with other letters of the alphabet so it’s easier to understand which angles we’re discussing.The value we are looking for is angle ADC, here called “z”. The interior angles of any quadrilateral must sum to 360 degrees:
W + X + Y + Z = 360
Z = 360 − (W + X + Y)
280 + Z = 360, so Z = 80 degrees. 
Question 14 of 20
14. Question
CorrectSimplify the given expression by rewriting 9 and 27 as exponents of base 3, also, recall that:
IncorrectSimplify the given expression by rewriting 9 and 27 as exponents of base 3, also, recall that:

Question 15 of 20
15. Question
Point A lies between X and Y. Point B lies on YZ and Point C lies on XZ. BZ is congruent to CZ, and angle XYZ = 90. XZ – CZ = XA. What is the value of angle ACB?CorrectThis is a Geometry question, so you’ll want to begin by drawing the figure on your own and filling in the given information. Let’s call angle ACB “z” to keep track of it.
We know a, z, and b are supplementary, so a + b + z = 180. We also know that (180 − 2b) + (180 − 2a) = 90, this is because the angle at X and the angle at Z must combine with the angle Y to make the sum of the interior angles of a triangle: 180°. Simplifying this second equation:360 − 2b − 2a = 90. Subtracting 360:
−2b − 2a = −270 → 270 = 2a + 2b
Let’s factor out the 2 from the variables: 270 = 2(a + b)
If we manipulate the first equation we can see that a + b = 180 − z. Let’s substitute “180 − z” for “a + b”: 270 = 2(180 − z). Angle ACB (or “z”) = 45. This is a difficulty level 5 question.
IncorrectThis is a Geometry question, so you’ll want to begin by drawing the figure on your own and filling in the given information. Let’s call angle ACB “z” to keep track of it.
We know a, z, and b are supplementary, so a + b + z = 180. We also know that (180 − 2b) + (180 − 2a) = 90, this is because the angle at X and the angle at Z must combine with the angle Y to make the sum of the interior angles of a triangle: 180°. Simplifying this second equation:360 − 2b − 2a = 90. Subtracting 360:
−2b − 2a = −270 → 270 = 2a + 2b
Let’s factor out the 2 from the variables: 270 = 2(a + b)
If we manipulate the first equation we can see that a + b = 180 − z. Let’s substitute “180 − z” for “a + b”: 270 = 2(180 − z). Angle ACB (or “z”) = 45. This is a difficulty level 5 question.

Question 16 of 20
16. Question
If x > 0, and x^{2} − 2x = 80, what is the value of x?
CorrectThe correct answer is 10.
First, subtract 80 from both sides of the equation to make the quadratic format clear. It should be recognized that the equation is a quadratic, and that the easiest method for solving quadratic equations is by factoring (when possible). Rearranging to find:
x^{2} − 2x − 80 = 0
Now that the quadratic is in standard form (ax^{2} + bx + c = 0), look for factors of the c value −80, that sum to the b value, −2. In this case, the factors 8 and 10 multiply to 80 and also have a difference of 2. Because the b value is negative, the larger number must be negative:
(x – 10) (x + 8) = 0
Recall that the zero product rule states that the product of 0 and anything is 0. So, in this case, if (x − 10) = 0, then it doesn’t matter what the other factor is, as the product will be 0. Set both terms equal to 0 and solve for x:
x − 10 = 0 → x = 10 and x + 8 = 0 → x = −8
Since x must be greater than zero, the correct answer is 10.
IncorrectThe correct answer is 10.
First, subtract 80 from both sides of the equation to make the quadratic format clear. It should be recognized that the equation is a quadratic, and that the easiest method for solving quadratic equations is by factoring (when possible). Rearranging to find:
x^{2} − 2x − 80 = 0
Now that the quadratic is in standard form (ax^{2} + bx + c = 0), look for factors of the c value −80, that sum to the b value, −2. In this case, the factors 8 and 10 multiply to 80 and also have a difference of 2. Because the b value is negative, the larger number must be negative:
(x – 10) (x + 8) = 0
Recall that the zero product rule states that the product of 0 and anything is 0. So, in this case, if (x − 10) = 0, then it doesn’t matter what the other factor is, as the product will be 0. Set both terms equal to 0 and solve for x:
x − 10 = 0 → x = 10 and x + 8 = 0 → x = −8
Since x must be greater than zero, the correct answer is 10.

Question 17 of 20
17. Question
CorrectThe correct answer is 5.
An often optimal approach to problems of this type, which contain unique denominators, is to eliminate the denominators as the first step of the solution. In this case, each of the denominators is a prime number, and the least common multiple will be their product: 3 * 2 * 7 = 42. By multiplying each term by 42, the denominators will cancel out, and the ratio of y to z can be determined:
IncorrectThe correct answer is 5.
An often optimal approach to problems of this type, which contain unique denominators, is to eliminate the denominators as the first step of the solution. In this case, each of the denominators is a prime number, and the least common multiple will be their product: 3 * 2 * 7 = 42. By multiplying each term by 42, the denominators will cancel out, and the ratio of y to z can be determined:

Question 18 of 20
18. Question
CorrectThe correct answer is 12/13
In a right triangle, the tangent is the ratio of the opposite side to the adjacent side. If one side is 10 and the other side is 24, we can tell that the third side, the hypotenuse, will be 26. This follows the common 5: 12: 13 right triangle ratio.
The figure above shows the triangle. Notice that the angle (90 − x)° is represented with the variable y. The sine of y is equal to the side opposite y divided by the hypotenuse:
IncorrectThe correct answer is 12/13
In a right triangle, the tangent is the ratio of the opposite side to the adjacent side. If one side is 10 and the other side is 24, we can tell that the third side, the hypotenuse, will be 26. This follows the common 5: 12: 13 right triangle ratio.
The figure above shows the triangle. Notice that the angle (90 − x)° is represented with the variable y. The sine of y is equal to the side opposite y divided by the hypotenuse:

Question 19 of 20
19. Question
What is the value of x?CorrectThe correct answer is 15.
Use your knowledge of supplemental angles, vertical angles, and the fact that the sum of the interior angles of a triangle sum to 180 degrees, to fill in the unknown angles and determine the value of x.
Starting from the upperright angle, and the fact that straight lines are 180°: 180 − 145 = 35.
Use the 2 known angles of the triangle to solve for the third angle: 35 + 80 + the unknown angle = 180.
The unknown angle = 65°.
Vertical angles are congruent, so the top angle in the adjacent triangle is 65°. Again using the known angles of a triangle to find the third angle: 65 + 35 + the unknown angle = 180.
The unknown angle = 80Repeating the process from the bottom of the figure: 180 − 140 = 40°.
Solving the triangle: 40 + 75 + the unknown angle = 180
The unknown angle = 65°.
Solving the next triangle: 65 + 30 + the unknown angle = 180.
The unknown angle = 85°.
Using the values of the other two angles in the triangle with x, we can solve for x:
85 + 80 + x = 180.
x = 15°.IncorrectThe correct answer is 15.
Use your knowledge of supplemental angles, vertical angles, and the fact that the sum of the interior angles of a triangle sum to 180 degrees, to fill in the unknown angles and determine the value of x.
Starting from the upperright angle, and the fact that straight lines are 180°: 180 − 145 = 35.
Use the 2 known angles of the triangle to solve for the third angle: 35 + 80 + the unknown angle = 180.
The unknown angle = 65°.
Vertical angles are congruent, so the top angle in the adjacent triangle is 65°. Again using the known angles of a triangle to find the third angle: 65 + 35 + the unknown angle = 180.
The unknown angle = 80Repeating the process from the bottom of the figure: 180 − 140 = 40°.
Solving the triangle: 40 + 75 + the unknown angle = 180
The unknown angle = 65°.
Solving the next triangle: 65 + 30 + the unknown angle = 180.
The unknown angle = 85°.
Using the values of the other two angles in the triangle with x, we can solve for x:
85 + 80 + x = 180.
x = 15°. 
Question 20 of 20
20. Question
An English professor gave the same exam to three groups of students. The average (arithmetic mean) scores for the three groups were 36, 60, and 84. The ratio of the numbers of students in each group who took the test was 2 to 3 to 5. What was the average score for the three groups combined?
CorrectThe correct answer is 67.2.
An average is calculated by summing the data values and dividing the calculated sum by the number of data values:
Average = Sum ÷ Number of Data
In this case we know the averages and the corresponding number of data points for each. The number of data points is given as a ratio— 2 to 3 to 5, which can be represented as 2x:3x:5x in order to accommodate the unknown number of students.
The average score for the three groups is the sum of the three groups divided by the number of data in the three groups. Begin by finding the sums of each group:
IncorrectThe correct answer is 67.2.
An average is calculated by summing the data values and dividing the calculated sum by the number of data values:
Average = Sum ÷ Number of Data
In this case we know the averages and the corresponding number of data points for each. The number of data points is given as a ratio— 2 to 3 to 5, which can be represented as 2x:3x:5x in order to accommodate the unknown number of students.
The average score for the three groups is the sum of the three groups divided by the number of data in the three groups. Begin by finding the sums of each group: