Mathematics Test—No Calculator
20 Questions
Turn to Section 3 of your answer sheet to answer the questions in this section.
Directions
For questions 1 through 15, solve each problem, choose the best answer from the choices provided, and fill in the corresponding circle on your answer sheet. For questions 16through 20, solve the problem and enter your answer in the grid on the answer sheet. You may use any available space in your test booklet for scratch work.
Notes
1.The use of a calculator is not permitted.
2.All variables and expressions used represent real numbers unless otherwise indicated.
3.Figures provided in this test are drawn to scale unless otherwise indicated.
4.All figures lie in a plane unless otherwise indicated.
5.Unless otherwise indicated, the domain of a given functionfis the set of all real numbersxfor which f of xis a real number.
Reference
Begin skippable figure descriptions.
The figure presents information for your reference in solving some of theproblems.
Reference figure1 is a circle with radiusr. Two equations are presented below reference figure1.
Aequals pi times the square ofr.
Cequals 2 pir.
Reference figure2 is a rectangle with lengthℓand widthw. An equation is presented below reference figure2.
A equals ℓw.
Reference figure3 is a triangle with baseband heighth. An equation is presented below reference figure3.
Aequals onehalfbh.
Reference figure4 is a right triangle. The two sides that form the right angle are labeledaandb, and the side opposite the right angle is labeledc. An equation is presented below reference figure4.
csquared equalsasquared plusbsquared.
Special Right Triangles
Reference figure5 is a right triangle with a 30degree angle and a 60degree angle. The side opposite the 30degree angle is labeledx. The side opposite the 60degree angle is labeledxtimes the squareroot of3. The side opposite the right angle is labeled2x.
Reference figure6 is a right triangle with two 45degree angles. Two sides are each labeleds. The side opposite the rightangle is labeledstimes the squareroot of2.
Reference figure7 is a rectangular solid whose base has lengthℓand widthwand whose height ish. An equation is presented below reference figure7.
Vequalsℓwh.
Reference figure8 is a rightcircularcylinder whose base has radiusrand whose height ish. An equation is presented below reference figure8.
Vequalspitimes the square ofrtimesh.
Reference figure9 is a sphere with radiusr. An equation is presented below reference figure9.
Vequalsfourthirds pi times the cube ofr.
Reference figure10 is a cone whose base has radiusrand whose height ish. Anequation is presented below reference figure10.
Vequals onethird times pi times the square ofrtimesh.
Reference figure11 is an asymmetrical pyramid whose base has lengthℓand widthwand whose height ish. An equation is presented below reference figure11.
V equalsonethirdℓwh.
End skippable figure descriptions.
Additional Reference Information
The number of degrees of arc in a circle is360.
The number of radians of arc in a circle is 2pi.
The sum of the measures in degrees of the angles of a triangle is180.
The SAT®Page 1
Copyright 2015 by the College BoardWF-5KSA09
Question 1.
Ifthe fraction whose numerator is xminus 1, and whose denominator is 3, equalsk,andkequals 3,what is the value ofx?
A.12
B.14
C.19
D.10
Answer and explanation for question 1.
Question 2.
Fori equals the square root of negative 1,what is the sum parenthesis, 7 plus 3i, close parenthesis, plus, parenthesis, negative 8 plus 9i, close parenthesis ?
A.negative 1 plus 12i
B.negative 1 minus 6i
C.15 plus 12i
D.15 minus 6i
Answer and explanation for question 2.
Question 3.
On Saturday afternoon, Armand sentmtext messages each hour for 5hours, and Tyrone sentptext messages each hour for 4hours. Which of the following represents the total number of messages sent by Armand and Tyrone on Saturday afternoon?
A.9mp
B.20mp
C.5m plus 4p
D.4m plus 5p
Answer and explanation for question 3.
Question 4.
Kathy is a repair technician for a phone company. Each week, she receives a batch of phones that need repairs. The number of phones that she has left to fix at the end of each day can be estimated with the equationP equals 108 minus 23d,where Pis the number of phones left anddis the number of days she has worked that week. What is the meaning of the value108 in this equation?
A.Kathy will complete the repairs within 108days.
B.Kathy starts each week with 108phones to fix.
C.Kathy repairs phones at a rate of 108 per hour.
D.Kathy repairs phones at a rate of 108 per day.
Answer and explanation for question 4.
Question 5.
parenthesis, x squared, y, minus 3ysquared, plus 5xy squared, close parenthesis, minus, parenthesis, negative xsquared, y, plus 3xy squared, minus 3y squared, close parenthesis
Which of the following is equivalent to the preceding expression?
A.4x squared, y squared
B.8xy squared, minus 6y squared
C.2x squared, y, plus, 2xy squared
D.2x squared, y, plus 8xy squared, minus 6y squared
Answer and explanation for question 5.
Question 6.
h equals 3a, plus 28.6
A pediatrician uses the model above to estimate the heighthof a boy, in inches, in terms of the boy’s agea, in years, between the ages of 2 and 5. Based on the model, what is the estimated increase, in inches, of a boy’s height each year?
A.13
B.15.7
C.19.5
D.14.3
Explanation for question 6.
Question 7.
m equals an expression times P, where the expression is the fraction whose numerator is parenthesis, the fraction r over 1,200, close parenthesis, times parenthesis, 1 plus the fractionr over 1,200, close parenthesis, to the power N, and whose denominator is parenthesis, 1 plus the fractionr over 1,200, close parenthesis, to the power N, end power, minus 1, end fraction.
The preceding formula gives the monthly paymentmneeded to pay off a loan ofPdollars atrpercent annual interest overNmonths. Which of the following givesPin terms ofm,r, andN?
A.P equals an expression timesm, where the expression is the fraction whose numerator is parenthesis, the fractionr over 1,200, close parenthesis, times, parenthesis, 1 plus the fraction rover 1,200, close parenthesis, to the power N, and whose denominator is parenthesis, 1plus the fractionr over 1,200, close parenthesis, to the powerN, minus 1.
B.Pequals an expression timesm, where the expression is the fraction whose numerator is parenthesis, 1 plus the fraction rover 1,200, close parenthesis, to the power N, minus 1, and whose denominator is parenthesis, the fraction r over 1,200, close parenthesis, times, parenthesis, 1 plus the fraction r over 1,200, close parenthesis, to the power N.
C.P equals, parenthesis, the fraction r over 1,200, close parenthesis, timesm.
D.P equals, parenthesis, the fraction 1,200 over r, close parenthesis, timesm.
Answer and explanation for question 7.
Question 8.
Ifthe fraction,a over b, equals 2,what is the value ofthe fraction 4b overa?
A.0
B.1
C.2
D.4
Answer and explanation for question 8.
Question 9.
3x plus 4y equals negative 23
2y minus x equals negative 19
What is the solutionparenthesis, x comma y, close parenthesis,to the preceding system of equations?
A.parenthesis, negative 5, comma negative 2, close parenthesis
B.parenthesis, 3 comma, negative 8, close parenthesis
C.parenthesis, 4 comma, negative 6, close parenthesis
D.parenthesis, 9 comma, negative 6, close parenthesis
Answer and explanation for question 9.
Question 10.
g of x equals a, x squared, plus 24.
For the functiongdefined,ais a constant andgof 4 equals 8.What is the value ofg of negative 4?
A.8
B.0
C.negative 1
D.negative 8
Answer and explanation for question 10.
Question 11.
b equals 2.35 plus 0.25x
c equals 1.75 plus 0.40x
In the preceding equations,bandcrepresent the price per pound, in dollars, of beef and chicken, respectively,xweeks after July1 during last summer. What was the price per pound of beef when it was equal to the price per pound of chicken?
A.$2.60
B.$2.85
C.$2.95
D.$3.35
Answer and explanation for question 11.
Question 12.
A line in the xy-plane passes through the origin and has a slope ofone seventh.Which of the following points lies on the line?
A.parenthesis, 0 comma 7, close parenthesis
B.parenthesis, 1 comma 7, close parenthesis
C.parenthesis, 7 comma 7, close parenthesis
D.parenthesis, 14 comma 2, close parenthesis
Answer and explanation for question 12.
Question 13.
Ifx is greater than 3,which of the following is equivalent tothe fraction whose numerator is 1, and whose denominator is the sum of fraction,1 over x plus 2, end fraction, and the fraction1 over x plus 3, end fraction, end expression?
A.the fraction whose numerator is 2x plus 5, and whose denominator is xsquared, plus 5x, plus 6.
B.the fraction whose numerator is x squared plus 5x plus 6, and whose denominator is 2x plus 5.
C.2x plus 5
D.x squared plus 5x plus 6
Answer and explanation for question 13.
Question 14.
If3x minus y equals 12,what is the value ofthe fraction 8 to the powerx, over 2 to the power y?
A.2 to the power 12
B.4 to the power 4
C.8 to the power 2
D.The value cannot be determined from the information given.
Answer and explanation for question 14.
Question 15.
Ifparenthesis, a, x plus 2, close parenthesis, times, parenthesis, bx plus 7, close parenthesis, equals 15x squared, plus cx, plus 14,for all values ofx, anda, plus b equals 8,what are the two possible valuesforc?
A.3 and 5
B.6 and 35
C.10 and 21
D.31 and 41
Answer and explanation for question 15.
Question 16.
Ift is greater than 0andt squared minus 4 equals 0,what is the valueoft?
Answer and explanation for question 16.
Directions
For questions 16 through 20, solve the problem and enter your answer in the grid, as described below, on the answer sheet.
1.Although not required, it is suggested that you write your answer in the boxes at the top of the columns to help you fill in the circles accurately. You will receive credit only if the circles are filled in correctly.
2.Mark no more than one circle in any column.
3.No question has a negative answer.
4.Some problems may have more than one correct answer. In such cases, grid only one answer.
5.Mixed numbers such asthree and one half must be gridded as 3.5 orsevenslashtwo. (Ifthree,one,slash,two, is entered into the grid, it will be interpreted as thirty one halves, notthree and one half.)
6.Decimal answers: If you obtain a decimal answer with more digits than the grid can accommodate, it may be either rounded or truncated, but it must fill the entire grid.
The following are four examples of how to record your answer in the spaces provided. Keep in mind that there are four spaces provided to record each answer.
Examples 1 and 2
Begin skippable figure description.
Example 1: If your answer is a fraction such as seventwelfths, it should be recorded as follows. Enter 7 in the first space, the fractionbar (aslash) in the second space, 1 in the third space, and 2 in the fourth space. All four spaces would be used in this example.
Example 2: If your answer is a decimal value such as 2.5, it could be recorded as follows. Enter 2 in the secondspace, the decimal point in the third space, and 5 in the fourth space. Only three spaces would be used in thisexample.
End skippable figure description.
Example 3
Begin skippable figure description.
Example 3: Acceptable ways to record twothirds are: 2 slash 3, .666, and .667.
End skippable figure description.
Example 4
Note: You may start your answers in any column, spacepermitting. Columns you don’t need to use should be left blank.
Begin skippable figure description.
Example 4: It is not necessary to begin recording answers in the first space unless all four spaces are needed. For example, if your answer is 201, you may record 2 in the second space, 0 in the third space, and 1 in the fourth space. Alternatively, you may record 2 in the first space, 0 in the second space, and 1 in the third space. Spaces not needed should be left blank.
End skippable figure description.
Question 17 is based on the following graphic.
Begin skippable figure description.
The figure presents the outline of a lake and some geometric figures with measurements. In the figure, from pointA, which is on the top side of the lake, to point E, which is on the bottom side of the lake, the length of the lake,AE, is labeledxfeet. To the right of the lake, line segmentsAC andED are drawn such thatAC slants downward,ED slants upward, and both line segments intersect at pointBthat is to the right of the lake. In triangleAEB and triangleCDB, angleAEB and angleCDB are both marked with an angle symbol.
End skippable figure description.
Question 17.
A summer camp counselor wants to find a length,x, in feet, across a lake as represented in the preceding sketch. The lengths represented byAB,EB,BD, andCD on the sketch were determined to be 1800feet, 1400feet, 700feet, and 800feet, respectively. SegmentsAC andDE intersect atB, andangle AEBandangle CDBhave the same measure. What is the value ofx?
Answer and explanation for question 17.
Question 18 is based on the following system of equations.
x plus y equals negative 9
x plus 2y equals negative 25
Question 18.
According to the preceding system of equations, what is the value ofx?
Answer and explanation for question 18.
Question 19.
In a right triangle, one angle measuresx degrees, wheresine of xdegrees equals four fifths. What iscosine of, parenthesis, 90 degrees minus x degrees, close parenthesis ?
Answer and explanation for question 19.
Question 20.
Ifa, equals 5 times the square root of 2and2a, equals the square root of 2x,what is the value ofx ?
Answer and explanation for question 20.
Stop.
If you finish before time is called, you may check your work on this section only. Do not turn to any other section.
Answers and explanations for questions1 through20 are provided in the nextsection of this document.
Answers and Explanations for Questions1 through20.
Explanation for question 1.
Choice D is correct. Sincek equals 3, one can substitute 3 forkin the equation the fraction whose numerator is x minus 1, and whose denominator is 3, equalsk which givesthe fraction whose numerator is x minus 1, and whose denominator is 3, equals 3. Multiplying both sides ofthe fraction whose numerator is xminus 1 and whose denominator is 3, equals 3, by 3 givesxminus 1, equals 9 and then adding 1 to both sides ofxminus 1, equals 9 givesx equals 10.
Choices A, B, and C are incorrect because the result of subtracting 1 from the value and dividing by 3 is not the given value ofk, which is 3.
Explanation for question 2.
Choice A is correct. To calculateparenthesis, 7 plus 3i, close parenthesis, plus, parenthesis, negative 8 plus 9i, close parenthesis, add the real parts of each complex number,7 plus, parenthesis, negative 8, close parenthesis, equals negative 1, and then add the imaginary parts,3i plus 9i, equals 12i. The result isnegative 1 plus 12i.
Choices B, C, and D are incorrect and likely result from common errors that arise when adding complex numbers. For example, Choice B is the result of adding3i andnegative 9i Choice C is the result of adding 7 and 8.
Explanation for question 3.
Choice C is correct. The total number of messages sent by Armand is the 5hours he spent texting multiplied by his rate of texting:mtexts per hour, times 5hours, equals 5m texts. Similarly, the total number of messages sent by Tyrone is the 4hours he spent texting multiplied by his rate of texting: ptexts per hour, times 4hours, equals 4ptexts. The total number of messages sent by Armand and Tyrone is the sum of the total number of messages sent by Armand and the total number of messages sent by Tyrone:5m plus 4p.
Choice A is incorrect and arises from adding the coefficients and multiplying the variables of5mand4p. Choice B is incorrect and is the result of multiplying5m and4p. The total number of messages sent by Armand and Tyrone should be the sum of 5m and 4p, not the product of these terms. Choice D is incorrect because it multiplies Armand’s number of hours spent texting by Tyrone’s rate of texting, and vice versa. This mix-up results in an expression that does not equal the total number of messages sent by Armand and Tyrone.
Explanation for question 4.
Choice B is correct. The value 108 in the equation is the value ofPinPequals 108 minus 23d, whend equals 0. Whend equals 0, Kathy has worked0 days that week. In other words, 108 is the number of phones left before Kathy has started work for the week. Therefore, the meaning of the value 108 in the equation is that Kathy starts each week with 108 phones to fix because she has worked 0 days and has 108 phones left to fix.
Choice A is incorrect, because Kathy will complete the repairs whenP equals 0. SinceP equals 108 minus 23d, this will occur when0equals 108 minus 23d, or whend equals the fraction 108 over 23, not when d equals 108. Therefore, the value 108 in the equation does not represent the number of days it will take Kathy to complete the repairs. Choices C and D are incorrect because the number23 inP equals 108 minus 23, P equals 108 indicates that the number of phones left will decrease by 23 for each increase in the value ofdby 1; in other words, that Kathy is repairing phones at a rate of 23 per day, not 108per hour (choice C) or 108 per day (choice D).
Explanation for question 5.
Choice C is correct. Only like terms, with the same variables and exponents, can be combined to determine the answer as shown here:
Parenthesis,x squared, y, minus 3y squared, plus 5xy squared, close parenthesis, minus, parenthesis, negative x squared, y, plus 3xy squared, minus 3y squared, close parenthesis. Equals, parenthesis,x squared, y, minus, parenthesis, negative x squared, y, close double parentheses, plus, parenthesis, negative 3y squared, minus, parenthesis, negative 3y squared, close double parentheses, plus, parenthesis, 5xy squared, minus 3xy squared, close parenthesis. Equals, 2x squared, y, plus 0, plus 2xy squared. Equals, 2x squared, y, plus 2xy squared
Choices A, B, and D are incorrect and are the result of common calculation errors or of incorrectly combining like and unlike terms.
Explanation for question 6.
Choice A is correct. In the equationh equals 3a, plus 28.6ifa, the age of the boy, increases by 1, thenhbecomesh equals 3, parenthesis,a, plus1, close parenthesis, plus 28.6, equals, 3a, plus 3, plus 28.6, equals, parenthesis, 3a, plus 28.6, close parenthesis, plus 3. Therefore, the model estimates that the boy’s height increases by 3inches each year.