SAT Practice Exam #04
1. A group of students voted on five after-school activities. The bar graph shows the number of students who voted for each of the five activities. How many students chose activity 3?
Explanation: Refer to the bar graph. The height of the bar for activity 3 corresponds to 39 students.
2. What percentage of 300 is 75?
Explanation: Use the formula \( \text{percentage} = \frac{\text{part}}{\text{whole}} \times 100 \). Substituting: \( \frac{75}{300} \times 100 = 25\% \).
3. \( \frac{x^2}{25} = 36 \). What is a solution to the given equation?
Explanation: Multiply both sides by \(25\): \(x^2 = 36 \times 25 = 900\). Taking the square root, \(x = \pm30\).
4. \(3\) more than \(8\) times a number \(x\) is equal to \(83\). Which equation represents this situation?
Explanation: Translate the statement step-by-step: \(8x\) (8 times a number), add 3 (3 more), equals 83.
5. Hana deposited a fixed amount into her bank account each month. The function \(f(t) = 100 + 25t\) gives the amount, in dollars, in Hana’s bank account after \(t\) monthly deposits. What is the best interpretation of \(25\) in this context?
Explanation: In the function \(f(t)\), the coefficient \(25\) represents the rate of change per time period.
6. A customer spent $27 to purchase oranges at $3 per pound. How many pounds of oranges did the customer purchase?
Explanation: Use the equation \( \text{cost} = \text{price per pound} \times \text{pounds} \). Solving \(27 = 3 \times \text{pounds}\), we get \(\text{pounds} = 9\).
7. Nasir bought 9 storage bins that were each the same price. He used a coupon for $63 off the entire purchase. The cost for the entire purchase after using the coupon was $27. What was the original price, in dollars, for 1 storage bin?
Explanation: Let \(p\) be the price per bin. Total cost before the coupon: \(9p\). After the coupon: \(9p - 63 = 27\). Solving \(9p = 90\), we get \(p = 10\).
8. For the linear function \(f\), the table shows three values of \(x\) and their corresponding values of \(f(x)\). Which equation defines \(f(x)\)?
Explanation: Use the slope formula \(m = \frac{\Delta y}{\Delta x}\). Substituting values from the table gives \(m = 3\). Using one point to solve for \(b\), find \(f(x) = 3x + 2\).
9. Right triangles \(PQR\) and \(STU\) are similar, where \(P\) corresponds to \(S\). If the measure of \(\angle Q\) is \(18^\circ\), what is the measure of \(\angle S\)?
Explanation: Since triangles \(PQR\) and \(STU\) are similar, their corresponding angles are equal. \(\angle Q\) corresponds to \(\angle S\), so \(\angle S = 18^\circ\).
10. The scatterplot shows the relationship between two variables, \(x\) and \(y\). Which of the following equations is the most appropriate linear model for the data shown?
Explanation: Based on the scatterplot, as \(x\) increases, \(y\) decreases linearly. This indicates a negative slope. The equation \(y = 9.4 - 0.9x\) accurately models this relationship.
11. The given equation describes the relationship between the number of birds, \(b\), and the number of reptiles, \(r\), that can be cared for at a pet care business on a given day. If the business cares for 16 reptiles on a given day, how many birds can it care for on this day?
Explanation: Substitute \(r = 16\) into the equation \(2.5b + 5r = 80\):
\[ 2.5b + 5(16) = 80 \] Simplify:
\[ 2.5b + 80 = 80 \] Subtract 80 from both sides:
\[ 2.5b = 0 \] Divide by 2.5:
\[ b = 5 \]
12. What is an equation of the graph shown?
Explanation: Observing the graph, the slope is \(-2\), as the line decreases by 2 units in \(y\) for every 1 unit increase in \(x\). The \(y\)-intercept is \(-8\). Therefore, the equation is \(y = -2x - 8\).
13. If \(\frac{x}{8} = 5\), what is the value of \(\frac{8}{x}\)?
Explanation: Start by solving for \(x\) from \(\frac{x}{8} = 5\):
\[ x = 8 \cdot 5 = 40 \] Next, substitute \(x = 40\) into \(\frac{8}{x}\):
\[ \frac{8}{40} = \frac{1}{5} \]
14. The solution to the given system of equations is \((x, y)\). What is the value of \(y\)?
Explanation: Solve the system of equations:
\(24x + y = 48\) (1)
\(6x + y = 72\) (2)
Subtract (2) from (1) to eliminate \(y\):
\[ (24x + y) - (6x + y) = 48 - 72 \] \[ 18x = -24 \implies x = -\frac{24}{18} = -\frac{4}{3} \] Substitute \(x = -\frac{4}{3}\) into \(6x + y = 72\):
\[ 6(-\frac{4}{3}) + y = 72 \] Simplify: \[ -8 + y = 72 \implies y = 72 + 8 = 80
15. Line \(t\) in the \(xy\)-plane has a slope of \(-\frac{1}{3}\) and passes through the point \((9, 10)\). Which equation defines line \(t\)?
Explanation: Start with the point-slope form of the equation:
\[ y - y_1 = m(x - x_1) \] Substitute \(m = -\frac{1}{3}\) and the point \((9, 10)\):
\[ y - 10 = -\frac{1}{3}(x - 9) \] Simplify: \[ y - 10 = -\frac{1}{3}x + 3 \] \[ y = -\frac{1}{3}x + 13 \]
16. The function \(f(x) = 206(1.034)^x\) models the value, in dollars, of a certain bank account by the end of each year from 1957 through 1972, where \(x\) is the number of years after 1957. Which of the following is the best interpretation of \(f(5)\) is approximately equal to 243 in this context?
Explanation: \(f(x)\) gives the estimated value of the account \(x\) years after 1957. \(f(5)\) represents the value of the account in 1962 (1957 + 5). This value is approximately \(243\), indicating that it is 5 times greater than the initial value of \(206\) in 1957: \[ \frac{243}{206} \approx 1.034^5 \approx 5. \]
18. Square \(P\) has a side length of \(x\) inches. Square \(Q\) has a perimeter that is 176 inches greater than the perimeter of square \(P\). The function \(f\) gives the area of square \(Q\) in square inches. Which of the following defines \(f\)?
19. The given equation relates the distinct positive real numbers \(w\), \(x\), and \(y\). Which equation correctly expresses \(w\) in terms of \(x\) and \(y\)?
Explanation: Start by isolating \(w\) in the given equation: \[ \frac{14x}{7y} = \frac{2}{w} + 19. \] Subtract 19 from both sides: \[ \frac{14x}{7y} - 19 = \frac{2}{w}. \] Take the reciprocal and simplify: \[ w = \frac{28x}{14y} - 19. \]
20. Point \(O\) is the center of a circle. The measure of arc \(RS\) on this circle is \(100^\circ\). What is the measure, in degrees, of its associated angle \(ROS\)?
Explanation: The measure of an associated angle at the center of the circle is equal to the measure of the arc. Thus, \(\angle ROS = 100^\circ\).
21. The expression \(6\sqrt[3]{5x^4} \cdot \sqrt{25x}\) is equivalent to \(ax^b\), where \(a\) and \(b\) are positive constants and \(x > 1\). What is the value of \(a + b\)?
Explanation: First, rewrite the expressions in exponential form: \[ 6\sqrt[3]{5x^4} = 6 \cdot 5^{1/3} \cdot x^{4/3}, \quad \sqrt{25x} = 5x^{1/2}. \] Multiply: \[ 6 \cdot 5^{1/3} \cdot x^{4/3} \cdot 5 \cdot x^{1/2} = 30 \cdot 5^{1/3} \cdot x^{(4/3 + 1/2)}. \] Combine exponents: \[ x^{(8/6 + 3/6)} = x^{11/6}. \] Thus, \(a = 30\) and \(b = 11/6\), so \(a + b = 11\).
22. A right triangle has sides of length \(2\sqrt{2}\), \(6\sqrt{2}\), and \(\sqrt{80}\) units. What is the area of the triangle, in square units?
Explanation: The triangle has legs \(2\sqrt{2}\) and \(6\sqrt{2}\). The area is: \[ \text{Area} = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 2\sqrt{2} \cdot 6\sqrt{2}. \] Simplify: \[ \text{Area} = \frac{1}{2} \cdot 24 = 24. \]
23. The expression \(4x^2 + bx - 45\), where \(b\) is a constant, can be rewritten as \((hx + k)(lx + j)\), where \(h\), \(k\), and \(j\) are integer constants. Which of the following must be an integer?
Explanation: In the factored form \((hx + k)(lx + j)\), the coefficients satisfy: \[ h \cdot l = 4, \quad k \cdot j = -45, \quad \text{and} \quad b = hl + kj. \] Since \(k\) and \(j\) are integers, \(\frac{b}{k}\) is guaranteed to be an integer because \(b\) is a linear combination of \(hl\) and \(kj\), both of which involve integers.
24. In the given system of equations, \(a\) is a constant. The graphs of the equations intersect at exactly one point, \((x, y)\), in the \(xy\)-plane. What is the value of \(a\)?
\(y = 2x^2 - 21x + 64\)
\(y = 3x + a\)
Explanation: To find where the graphs intersect at exactly one point, set the equations equal: \[ 2x^2 - 21x + 64 = 3x + a. \] Simplify to: \[ 2x^2 - 24x + (64 - a) = 0. \] For one solution, the discriminant must be zero: \[ b^2 - 4ac = 0. \] Substitute: \[ (-24)^2 - 4(2)(64 - a) = 0 \implies 576 - 8(64 - a) = 0 \implies a = 6. \]
25. An isosceles right triangle has a hypotenuse of length 58 inches. What is the perimeter, in inches, of this triangle?
Explanation: In an isosceles right triangle, the legs are equal in length, and the hypotenuse is: \[ \text{hypotenuse} = \text{leg} \cdot \sqrt{2}. \] Given the hypotenuse is 58, solve for the leg: \[ \text{leg} = \frac{58}{\sqrt{2}} = 29\sqrt{2}. \] The perimeter is: \[ \text{perimeter} = 2 \cdot \text{leg} + \text{hypotenuse} = 2 \cdot 29\sqrt{2} + 58 = 58 + 116\sqrt{2}. \]
26. In the xy-plane, a parabola has vertex \((9, -14)\) and intersects the x-axis at two points. If the equation of the parabola is written in the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, which of the following could be the value of \(a + b + c\)?
Explanation: The vertex form of a parabola is: \[ y = a(x - h)^2 + k, \] where \((h, k)\) is the vertex. Substituting \((h, k) = (9, -14)\): \[ y = a(x - 9)^2 - 14. \] The parabola intersects the x-axis, so set \(y = 0\) to find the roots. Using this information, solve for \(a\), \(b\), and \(c\), and substitute \(a + b + c\) to match one of the provided choices.
27. Function \(f\) is defined by \(f(x) = -ax^2 + b\), where \(a\) and \(b\) are constants. In the xy-plane, the graph of \(y = f(x) - 15\) has a y-intercept at \((0, -\frac{99}{7})\). The product of \(a\) and \(b\) is \(\frac{65}{7}\). What is the value of \(a\)?
Explanation: Substitute the given information into \(f(x) = -ax^2 + b\). For the y-intercept: \[ f(0) - 15 = -\frac{99}{7}. \] Thus: \[ b - 15 = -\frac{99}{7} \implies b = \frac{6}{7}. \] Using \(ab = \frac{65}{7}\): \[ a \cdot \frac{6}{7} = \frac{65}{7} \implies a = 5. \]
Module 2
1. The line graph shows the estimated number of chipmunks in a state park on April 1 of each year from 1989 to 1999.
Based on the line graph, in which year was the estimated number of chipmunks in the state park the greatest?
Explanation: The line graph shows the number of chipmunks each year from 1989 to 1999. The peak value is reached in 1998, where the estimated number of chipmunks is the greatest (approximately 200). Compare the data points visually to confirm this.
2. A fish swam a distance of 5,104 yards. How far did the fish swim, in miles? (1 mile = 1,760 yards)
Explanation: To convert yards to miles, divide the distance in yards by the number of yards in a mile (1,760):
\[ \text{Miles} = \frac{5,104}{1,760} = 2.9 \]
3. Which expression is equivalent to \(12x^3 - 5x^3\)?
Explanation: Combine like terms by subtracting the coefficients of \(x^3\):
\[ 12x^3 - 5x^3 = 7x^3 \]
4. What is the solution \((x, y)\) to the given system of equations?
- \(x + y = 18\)
- \(5y = x\)
Explanation: Solve the system of equations by substitution or elimination. From \(5y = x\), substitute \(x = 5y\) into \(x + y = 18\):
\[ 5y + y = 18 \implies 6y = 18 \implies y = 3 \] Substitute \(y = 3\) into \(x = 5y\):
\[ x = 5(3) = 15 \]
2. A fish swam a distance of 5,104 yards. How far did the fish swim, in miles? (1 mile = 1,760 yards)
Explanation: To convert yards to miles, divide the distance in yards by the number of yards in a mile (1,760):
\[ \text{Miles} = \frac{5,104}{1,760} = 2.9 \]
3. Which expression is equivalent to \(12x^3 - 5x^3\)?
Explanation: Combine like terms by subtracting the coefficients of \(x^3\):
\[ 12x^3 - 5x^3 = 7x^3 \]
4. What is the solution \((x, y)\) to the given system of equations?
- \(x + y = 18\)
- \(5y = x\)
Explanation: Solve the system of equations by substitution or elimination. From \(5y = x\), substitute \(x = 5y\) into \(x + y = 18\):
\[ 5y + y = 18 \implies 6y = 18 \implies y = 3 \] Substitute \(y = 3\) into \(x = 5y\):
\[ x = 5(3) = 15 \]
5. The point \((8, 2)\) in the \(xy\)-plane is a solution to which of the following systems of inequalities?
Explanation: The point \((8, 2)\) has positive \(x\) and \(y\) coordinates, which satisfies \(x > 0\) and \(y > 0\).
6. \(|x - 5| = 10\). What is one possible solution to the given equation?
Explanation: Solve for \(x\):
\[ |x - 5| = 10 \implies x - 5 = 10 \text{ or } x - 5 = -10 \] Add 5 to both equations:
\[ x = 15 \text{ or } x = -5 \]
7. \(f(x) = 7x + 1\). The function gives the total number of people on a company retreat with \(x\) managers. What is the total number of people on a company retreat with 7 managers?
Explanation: Substitute \(x = 7\) into \(f(x) = 7x + 1\):
\[ f(7) = 7(7) + 1 = 49 + 1 = 50 \]
8. A rectangle has a perimeter of 60 units and a width of 15 units. What is the length?
Explanation: The perimeter of a rectangle is given by \(P = 2L + 2W\):
\[ 60 = 2L + 2(15) \implies 60 = 2L + 30 \] Subtract 30 from both sides:
\[ 30 = 2L \implies L = 15 \]
9. What is the slope of the line that passes through the points \((3, 2)\) and \((5, 10)\)?
Explanation: The slope \(m\) is calculated as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 2}{5 - 3} = \frac{8}{2} = 8 \]
10. To estimate the proportion of a population that has a certain characteristic, a random sample was selected from the population. Based on the sample, it is estimated that the proportion of the population that has the characteristic is 0.49, with an associated margin of error of 0.04. Based on this estimate and margin of error, which of the following is the most appropriate conclusion about the proportion of the population that has the characteristic?
Explanation: The margin of error of 0.04 means that the estimated range for the proportion is \(0.49 \pm 0.04\), or 0.45 to 0.53.
11. A moving truck can tow a trailer if the combined weight of the trailer and the boxes it contains is no more than 4,600 pounds. What is the maximum number of boxes this truck can tow in a trailer with a weight of 500 pounds if each box weighs 120 pounds?
Explanation: Subtract the trailer weight from the maximum weight allowed: \(4,600 - 500 = 4,100\). Divide by the weight of each box: \(4,100 \div 120 = 34.16\). Rounding down gives 38.
12. What is the positive solution to the equation \(-4x^2 - 7x = -36\)?
Explanation: Solve the quadratic equation by factoring or using the quadratic formula.
13. The table summarizes the distribution of color and shape for 100 tiles of equal area. If one of these tiles is selected at random, what is the probability of selecting a red tile? (Express your answer as a decimal or fraction, not as a percent.)
| Red | Blue | Yellow | Total | |
|---|---|---|---|---|
| Square | 10 | 20 | 25 | 55 |
| Pentagon | 20 | 10 | 15 | 45 |
| Total | 30 | 30 | 40 | 100 |
Explanation: The total number of red tiles is 30, and the total number of tiles is 100. Thus, the probability is \( \frac{30}{100} = 0.3 \).
14. For the given function \( f(x) = 2x + 3 \), the graph of \( y = f(x) \) in the \( xy \)-plane is parallel to line \( j \). What is the slope of line \( j \)?
Explanation: The slope of a linear function \( f(x) = mx + b \) is \( m \). In this case, the slope is 2, and parallel lines have the same slope.
15. A proposal for a new library was included on an election ballot. A radio show stated that 3 times as many people voted in favor of the proposal as people who voted against it. A social media post reported that 15,000 more people voted in favor of the proposal than voted against it. Based on these data, how many people voted against the proposal?
Explanation: Let \( x \) represent the number of people who voted against the proposal. Then \( 3x - x = 15,000 \), giving \( x = 7,500 \).
16. In the figure, lines \( m \) and \( n \) are parallel. If \( x = 6k + 13 \) and \( y = 8k - 29 \), what is the value of \( z \)?
Explanation: Solve for \( k \) using the equations for parallel lines, then substitute into \( z \).
17. In the given equation, \( p \) is a constant. The equation has no solution. What is the value of \( p \)?
Explanation: For the equation to have no solution, the coefficient ratio must lead to a contradiction.
18. The function \( f(x) = (x - 10)(x + 13) \) is defined by the given equation. For what value of \( x \) does \( f(x) \) reach its minimum?
Explanation: The vertex of a parabola defined by \( f(x) = ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \).
Explanation: Analyze the intersection points of the three lines by solving the system of linear equations graphically or algebraically. The resulting system has two solutions as shown by the graph and equations.
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