SAT Math No Calculator practice tests.

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Question 1 |

**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?**

$13Y + 13X$ | |

$13XY$ | |

$6Y + 7X$ | |

$6X + 7Y$ |

Question 1 Explanation:

The correct answer is (D).

Notice that we multiplied the number of each drink by the number of hours that drink was sold, and then added the two totals together: $(6*X) + (7*Y).$

This matches the expression in answer choice (D).

Alternatively, 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 $2 * 6 = 12$ lattes total were sold.

Similarly, if 3 americanos were sold each hour for 7 hours, then $3 * 7 = 21$ 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.

Notice that we multiplied the number of each drink by the number of hours that drink was sold, and then added the two totals together: $(6*X) + (7*Y).$

This matches the expression in answer choice (D).

Alternatively, 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 $2 * 6 = 12$ lattes total were sold.

Similarly, if 3 americanos were sold each hour for 7 hours, then $3 * 7 = 21$ 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.

Question 2 |

**If $\dfrac{6}{z+4} = p$, and $p = 12$, what is the value of $z $?**

−1.5 | |

−3.5 | |

1.5 | |

3.5 |

Question 2 Explanation:

The correct answer is (B).

Start by substituting $12$ for $p$ in the original equation:

$\dfrac{6}{z + 4} = 12$

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:

$6 = 12(z + 4)$

Divide both sides by 12:

$\dfrac{6}{12} = z + 4$

Simplify the fraction:

$\dfrac{1}{2} = z + 4$

Subtract 4:

$\dfrac{1}{2} - 4 = z$

$-3.5 = z$

Start by substituting $12$ for $p$ in the original equation:

$\dfrac{6}{z + 4} = 12$

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:

$6 = 12(z + 4)$

Divide both sides by 12:

$\dfrac{6}{12} = z + 4$

Simplify the fraction:

$\dfrac{1}{2} = z + 4$

Subtract 4:

$\dfrac{1}{2} - 4 = z$

$-3.5 = z$

Question 3 |

**If the function ƒ(x) is defined for all real numbers by the following equation: $f(x) = \dfrac{x^2 + 2}{2}$**

**Then $ƒ(ƒ(2)) = $**

1.5 | |

3.0 | |

5.5 | |

7.0 |

Question 3 Explanation:

The correct answer is (C).

To solve this function question, start by substituting $2$ for $x$:

$f(2)$ = $\dfrac{2$

To solve this function question, start by substituting $2$ for $x$:

$f(2)$ = $\dfrac{2$

^{2$ + 2}{2}$ $f(2)$ = $\dfrac{4 + 2}{2}$ $f(2)$ = $\dfrac{6}{2}$ $f(2)$ = $3$ But we can't stop there! Notice the question asks for the value of $f(f(2))$ $f(2)$ = $3$, so $f(f(2))$ = $f(3)$ Substitute $3$ for $x$ in the function to get the the final answer: $f(3) $= $\dfrac{3$2 + 2}{2}$ $f(3)$ = $\dfrac{9 + 2}{2}$ $f(3)$ = $\dfrac{11}{2}$ $f(2)$ = $5.5$ }Question 4 |

**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?**

16 kpl | |

21 kpl | |

36 kpl | |

49 kpl |

Question 4 Explanation:

This 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:
M8rev

Question 5 |

**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 x-axis and to the right of the y-axis. II. (x, −y) lies below the x-axis and to the left of the y-axis. III. (x, y) lies on the x-axis to the right of the y-axis.**

I only | |

II only | |

I and II only | |

II and III |

Question 5 Explanation:

In Roman numeral I, if (−x, y) lies above the x-axis and to the right of the y-axis (that is, in the first quadrant), then x must equal a negative number, since all x-values to the right of the y-axis are positive, and the only way −x = positive is if x = negative. So, y must be positive since it is above the x-axis. 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 x-axis, 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 |

**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?**

5 | |

6 | |

7 | |

8 |

Question 6 Explanation:

Start 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 = 6
Plugging 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 |

**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?**

\dfrac{x + y}{2} | |

\dfrac{y + 38}{3} | |

\dfrac{x - y + 28}{4} | |

\dfrac{x - y + 38}{2} |

Question 7 Explanation:

To 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.
M5x1
Remember, we are looking for the average for their ages four years ago, NOT today!
M5x2
Find the answer that gives you 18.5 when you plug in x = 2 and y = 3.

Question 8 |

**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 = ?**

14 | |

18 | |

22 | |

26 |

Question 8 Explanation:

The standard equation of a line is y = mx + b, where m is the slope, b is the y-intercept, 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 y-intercept 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 |

**If $\dfrac{1}{y} - \dfrac{1}{y + 1}$ = $\dfrac{1}{y + 3}$, then $y$ could be**

2 | |

0 | |

-2 | |

√3 |

Question 9 Explanation:

In this case, multiplying each term by the least common multiple (LCM) of the denominators will best clarify the problem:
M6x
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 |

**If a = \dfrac{-1}{b} and c = \dfrac{1}{d} for integers a, b, c and d, in which of the following ranges does the product**

*abcd*fall?

−4 < abcd < -2 | |

−2 < abcd < 0 | |

0 < abcd < 2 | |

2 < abcd < 4 |

Question 10 Explanation:

The expression abcd can be rewritten to:
M11x
Only answer choice (2) correctly places abcd in the range: −2 < −1 < 0.

Question 11 |

**The average of several test scores is 80. One make-up exam was given. Included with the other scores, the new average was 84. If the score on the make-up exam was 92, how many total exams were given?**

2 | |

3 | |

4 | |

5 |

Question 11 Explanation:

Recall 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:
M15
Multiplying both sides by x + 1:
84x + 84 = Sum + Make-Up Score
Let’s substitute “80x” for the “Sum”:
84x + 84 = 80x + Make-up Score
4x + 84 = Make-up Score
We are told that the make-up score was 92:
4x + 84 = 92
4x = 8
x = 2

Question 12 |

**If (x)**

^{-1}= \dfrac{-1}{2} then (x)^{-2}equals which of the following?

-4 | |

-1 | |

\dfrac{-1}{4} | |

\dfrac{1}{4} |

Question 12 Explanation:

A negative exponent is another way of writing a fraction:
M9x

Question 13 |

**In quadrilateral ABCD, what is the value of angle ADC if Angle BAD + Angle ABC + Angle BCD = 280°?**

80° | |

120° | |

160° | |

180° |

Question 13 Explanation:

For 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 |

**For all positive integers**

*x*, if 3^{2x + 1}< $\dfrac{1}{2}$, what is a possible value for \dfrac{9^{x + 4}}{27}?

1474 | |

1648 | |

1896 | |

2187 |

Question 14 Explanation:

Simplify the given expression by rewriting 9 and 27 as exponents of base 3, also, recall that:
M13x

Question 15 |

**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?**

45° | |

60° | |

75° | |

90° |

Question 15 Explanation:

This 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.

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