Page 1
Question ID: ac472881
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Hard
Question
In the given equation, and are constants, and . If the equation has infinitely many solutions, what is the value of ?
Correct Answer: 403
Rationale
The correct answer is . For a linear equation in one variable to have infinitely many solutions, the coefficients of the variable must be equal
on both sides of the equation and the constant terms must also be equal on both sides of the equation. The given equation can be rewritten as
, or . Applying the distributive property to the right-hand side of this equation yields
. For this equation to have infinitely many solutions, the coefficients of must be equal, so it follows that . Additionally,
the constant terms must be equal, which means . Substituting for in this equation yields , or
. Adding to both sides of this equation yields . Adding to both sides of this equation yields . Multiplying both
sides of this equation by yields . Therefore, if the equation has infinitely many solutions, the value of is .
Page 2
Question ID: 2937ef4f
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Hard
Question
Hector used a tool called an auger to remove corn from a storagebin at a constant rate. The bin contained 24,000bushels of corn when Hector
began to use theauger. After 5hours of using the auger, 19,350bushels of corn remained in the bin. If the auger continues to remove corn at this
rate, what is the total number of hours Hector will have been using the auger when 12,840bushels of corn remain in thebin?
Answer
A. 3
B. 7
C. 8
D. 12
Correct Answer: D
Rationale
Choice D is correct. After using the auger for 5 hours, Hector had removed 24,000 – 19,350 = 4,650 bushels of corn from the storage bin. During
the 5-hour period, the auger removed corn from the bin at a constant rate of bushels per hour. Assuming the auger continues
to remove corn at this rate, after x hours it will have removed 930x bushels of corn. Because the bin contained 24,000 bushels of corn when
Hector started using the auger, the equation 24,000 – 930x = 12,840 can be used to find the number of hours, x, Hector will have been using the
auger when 12,840 bushels of corn remain in the bin. Subtracting 12,840 from both sides of this equation and adding 930x to both sides of the
equation yields 11,160 = 930x. Dividing both sides of this equation by 930 yields x = 12. Therefore, Hector will have been using the auger for 12
hourswhen 12,840 bushels of corn remain in the storage bin.
Choice A is incorrect. Three hours after Hector began using the auger, 24,000 – 3(930) = 21,210 bushels of corn remained, not 12,840. Choice B
is incorrect. Seven hours after Hector began using the auger, 24,000 – 7(930) = 17,490 bushels of corn will remain, not 12,840. Choice C is
incorrect. Eight hours after Hector began using the auger, 24,000 – 8(930) = 16,560 bushels of corn will remain, not 12,840.
Page 3
Question ID: 9b886541
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
If , what is the value of ?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. It's given that . Adding to both sides of this equation yields . Adding to both sides of this equation
yields . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the value of , not .
Choice C is incorrect and may result from conceptual or calculation errors.
Page 4
Question ID: f14484a5
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Hard
Question
A manufacturing plant makes -inch, -inch, and -inch frying pans. During a certain day, the number of -inch frying pans that the
manufacturing plant makes is times the number of -inch frying pans it makes, and the number of -inch frying pans it makes is . During
this day, the manufacturing plant makes frying pans total. Which equation represents this situation?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. It's given that during a certain day, the number of -inch frying pans the manufacturing plant makes is and the number of
-inch frying pans it makes is . It's also given that during this day the number of -inch frying pans that the manufacturing plant makes is
times the number of -inch frying pans, or . Therefore, the total number of -inch, -inch, and -inch frying pans the manufacturing plant
makes is , or . It's given that during this day the manufacturing plant makes frying pans total. Thus, the equation
represents this situation.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Page 5
Question ID: 9d4270fe
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
A company that creates and sells tape dispensers calculates its monthly profit, in dollars, by subtracting its fixed monthly costs, in dollars, from
its monthly sales revenue, in dollars. The equation represents this situation for a month where tape dispensers are
created and sold. Which statement is the best interpretation of in this context?
Answer
A. The monthly sales revenue, in dollars, from selling tape dispensers
B. The monthly sales revenue, in dollars, from each tape dispenser sold
C. The monthly cost, in dollars, of creating each tape dispenser
D. The monthly cost, in dollars, of creating tape dispensers
Correct Answer: A
Rationale
Choice A is correct. It’s given that the equation represents this situation for a month where tape dispensers are
created and sold. It’s also given that the company calculates its monthly profit, in dollars, by subtracting its fixed monthly costs, in dollars, from
its monthly sales revenue, in dollars. It follows that represents the monthly sales revenue, in dollars. Therefore, the best interpretation of
in this context is the monthly sales revenue from selling tape dispensers.
Choice B is incorrect. This is the best interpretation of , not .
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect. This is the best interpretation of , not .
Page 6
Question ID: 7a5a74a6
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
If x is the solution to the equation above, what is the value of ?
Answer
A.
B.
C.
D.
Correct Answer: B
Rationale
Choice B is correct. Because 2 is a factor of both and 6, the expression can be rewritten as . Substituting for
on the left-hand side of the given equation yields , or .
Subtracting from both sides of this equation yields . Adding 11 to both sides of this equation yields
. Dividing both sides of this equation by 2 yields .
Alternate approach: Distributing 3 to the quantity on the left-hand side of the given equation and distributing 4 to the quantity
on the right-hand side yields , or . Subtracting from both sides of this equation yields
. Adding 29 to both sides of this equation yields . Dividing both sides of this equation by 2 yields .
Therefore, the value of is , or .
Choice A is incorrect. This is the value of x, not . Choices C and D are incorrect. If the value of is or , it follows that
the value of x is or , respectively. However, solving the given equation for x yields . Therefore, the value of can’t be
or .
Page 7
Page 8
Question ID: b7e6394d
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Hard
Question
Alan drives an average of 100miles each week. His car can travel an average of 25miles per gallon of gasoline. Alan would like to reduce his
weekly expenditure on gasoline by $5. Assuming gasoline costs $4per gallon, which equation can Alan use to determine how many fewer
average miles, m, he should drive each week?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. Since gasoline costs $4 per gallon, and since Alan’s car travels an average of 25 miles per gallon, the expression gives
the cost, in dollars per mile, to drive the car. Multiplying by m gives the cost for Alan to drive m miles in his car. Alan wants to reduce his
weekly spending by $5, so setting m equal to 5 gives the number of miles, m, by which he must reduce his driving.
Choices A, B, and C are incorrect. Choices A and B transpose the numerator and the denominator in the fraction. The fraction would result
in the unit miles per dollar, but the question requires a unit of dollars per mile. Choices A and C set the expression equal to 95 instead of 5, a
mistake that may result from a misconception that Alan wants to reduce his driving by 5 miles each week; instead, the question says he wants to
reduce his weekly expenditure by $5.
Page 9
Question ID: aa85b138
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
A tree had a height of 6 feet when it was planted. The equation above can be used to find how many yearsn it took the tree to reach a height of
14 feet. Which of the following is the best interpretation of the number 2 in this context?
Answer
A. The number of years it took the tree to double its height
B. The average number of feet that the tree grew per year
C. The height, in feet, of the tree when the tree was 1yearold
D. The average number of years it takes similar trees to grow 14 feet
Correct Answer: B
Rationale
Choice B is correct. The height of the tree at a given time is equal to its height when it was planted plus the number of feet that the tree grew. In
the given equation, 14 represents the height of the tree at the given time, and 6 represents the height of the tree when it was planted. It follows
that represents the number of feet the tree grew from the time it was planted until the time it reached a height of 14 feet. Since n represents
the number of years between the given time and the time the tree was planted, 2 must represent the average number of feet the tree grew each
year.
Choice A is incorrect and may result from interpreting the coefficient 2 as doubling instead of as increasing by 2 each year. Choice C is incorrect.
The height of the tree when it was 1 year old was feet, not 2 feet. Choice D is incorrect. No information is given to connect the
growth of one particular tree to the growth of similar trees.
Page 10
Question ID: 15daa8d6
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
In the given equation, is a constant. If the equation has infinitely many solutions, what is the value of ?
Correct Answer: 2
Rationale
The correct answer is . An equation with one variable, , has infinitely many solutions only when both sides of the equation are equal for any
defined value of . It's given that , where is a constant. This equation can be rewritten as . If this
equation has infinitely many solutions, then both sides of this equation are equal for any defined value of . Both sides of this equation are equal
for any defined value of when . Therefore, if the equation has infinitely many solutions, the value of is .
Alternate approach: If the given equation,
, has infinitely many solutions, then both sides of this equation are equal for any
value of . If , then substituting for in yields , or . Dividing both sides of this
equation by yields .
Page 11
Question ID: 12ee1edc
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
In the given equation, b is a constant. If the equation has no solution, what is the value of b ?
Answer
A. 2
B. 4
C. 6
D. 10
Correct Answer: A
Rationale
Choice A is correct. This equation has no solution when there is no value of x that produces a true statement. Solving the given equation for x by
dividing both sides by gives . When , the right-hand side of this equation will be undefined, and the equation
will have no solution. Therefore, when , there is no value of x that satisfies the given equation.
Choices B, C, and D are incorrect. Substituting 4, 6, and 10 for b in the given equation yields exactly one solution, rather than no solution, for x.
For example, substituting 4 for b in the given equation yields , or . Dividing both sides of by 2 yields .
Similarly, if or , and , respectively.
Page 12
Question ID: 25e1cfed
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Hard
Question
How many solutions does the equation have?
Answer
A. Exactly one
B. Exactly two
C. Infinitely many
D. Zero
Correct Answer: C
Rationale
Choice C is correct. Applying the distributive property to each side of the given equation yields . Applying the
commutative property of addition to the right-hand side of this equation yields . Since the two sides of the equation
are equivalent, this equation is true for any value of . Therefore, the given equation has infinitely many solutions.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Page 13
Question ID: 6ac23de7
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
In the equation above, what is the value of x ?
Answer
A. 25
B. 24
C. 16
D. 15
Correct Answer: A
Rationale
Choice A is correct. Multiplying both sides of the equation by 5 results in . Dividing both sides of the resulting equation by 4 results in
.
Choice B is incorrect and may result from adding 20 and 4. Choice C is incorrect and may result from dividing 20 by 5 and then multiplying the
result by 4. Choice D is incorrect and may result from subtracting 5 from 20.
Page 14
Question ID: e6cb2402
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Hard
Question
In the given equation, is a constant. The equation has no solution. What is the value of ?
Correct Answer: .9411, .9412, 16/17
Rationale
The correct answer is . It's given that the equation has no solution. A linear equation in the form
,
where , , , and are constants, has no solution only when the coefficients of on each side of the equation are equal and the constant terms
aren't equal. Dividing both sides of the given equation by yields , or
. Since the coefficients of
on each side of the equation must be equal, it follows that the value of is . Note that 16/17, .9411, .9412, and 0.941 are examples of ways to
enter a correct answer.
Page 15
Question ID: 7392dfc1
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
Which of the following is equivalent to ?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. Dividingeach side of the original equation by yields
, which simplifies to
.
Choice A is incorrect. Dividing each side of the original equation by gives , which is not equivalent to . Choice B is
incorrect. Dividing each side of the original equation by gives , which is not equivalent to . Choice C is incorrect.
Dividing each side of the original equation by gives
, which is not equivalent to .
Page 16
Question ID: 93954cfa
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
One pound of grapes costs$2. At this rate, how many dollars will cpounds of grapes cost?
Answer
A.
B.
C.
D.
Correct Answer: A
Rationale
Choice A is correct. If one pound of grapes costs $2, two pounds of grapes will cost 2 times $2, three pounds of grapes will cost 3 times $2, and
so on. Therefore, c pounds of grapes will cost c times $2, which is 2c dollars.
Choice B is incorrect and may result from incorrectly adding instead of multiplying. Choice C is incorrect and may result from assuming that c
pounds cost $2, and then finding the cost per pound. Choice D is incorrect and could result from incorrectly assuming that 2 pounds cost $c, and
then finding the cost per pound.
Page 17
Question ID: 018a2704
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
If , what is the value of ?
Answer
A.
B.
C.
D.
Correct Answer: C
Rationale
Choice C is correct. Subtracting from both sides of the given equationyields . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Page 18
Question ID: 3d04de9c
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
A principal used a total of flags that were either blue or yellow for field day. The principal used blue flags. How many yellow flags were
used?
Answer
A.
B.
C.
D.
Correct Answer: A
Rationale
Choice A is correct. It's given that a principal used a total of blue flags and yellow flags. It's also given that of the flags used, flags were
blue. Subtracting the number of blue flags used from the total number of flags used results in the number of yellow flags used. It follows that the
number of yellow flags used is , or .
Choice B is incorrect. This is the number of blue flags used.
Choice C is incorrect. This is the total number of flags used.
Choice D is incorrect and may result from conceptual or calculation errors.
Page 19
Question ID: 60f71697
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
What value of is the solution to the given equation?
Answer
A.
B.
C.
D.
Correct Answer: A
Rationale
Choice A is correct. Dividing both sides of the given equation by yields . Therefore, is the solution to the given equation.
Choice B is incorrect. This is the solution to the equation .
Choice C is incorrect. This is the solution to the equation .
Choice D is incorrect. This is the solution to the equation .
Page 20
Question ID: 70e29454
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
In the equation above, a and b are constants. If the equation has infinitely many solutions, what are the values of a and b ?
Answer
A. and
B. and
C. and
D. and
Correct Answer: B
Rationale
Choice B is correct. Distributing the a on the left-hand side of the equation gives 3a– b–
ax = –1 – 2x. Rearranging the terms in each side of the
equation yields –ax + 3a – b = –2x –1. Since the equation has infinitely many solutions, it follows that the coefficients of x and the free terms on
both sides must be equal. That is, –a= –2, or a = 2, and 3a – b = –1. Substituting 2 for a in the equation 3a–
b =–
1 gives 3(2) – b = –1, so b =
7.
Choice A is incorrect and may be the result of a conceptual error when finding the value of b. Choices C and D are incorrect and may result from
making a sign error when simplifying.
Page 21
Question ID: f09097b1
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
An agricultural scientist studying the growth of corn plants recorded the height of a corn plant at the beginning of a study and the height of the
plant each day for the next 12days. The scientist found that the height of the plant increased by an average of 1.20centimeters per day for the
12days. If the height of the plant on the last day of the study was 36.8centimeters, what was the height, in centimeters, of the corn plant at the
beginning of the study?
Rationale
The correct answer is 22.4. If the height of the plant increased by an average of 1.20 centimeters per day for 12 days, then its total growth over
the 12 days was centimeters. The plant was 36.8 centimeters tall after 12 days, so at the beginning of the studyits height
was centimeters. Note that 22.4 and 112/5 are examples of ways to enter a correct answer.
Alternate approach:The equation can be used to represent this situation, where h is the height of the plant, in
centimeters, at the beginning of the study. Solving this equation for h yields 22.4 centimeters.
Page 22
Question ID: 550b352c
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
How many solutions exist to the equation shown above?
Answer
A. None
B. Exactly 1
C. Exactly 3
D. Infinitely many
Correct Answer: B
Rationale
Choice B is correct. Subtracting 4 from each side of the given equation yields , or , so the equation has a unique solution of
.
Choice A is incorrect. Since 3 is a value of x that satisfies the given equation, the equation has at least 1 solution. Choice C is incorrect. Linear
equations can have 0, 1, or infinitely many solutions; no linear equation has exactly 3 solutions. Choice D is incorrect. If a linear equation has
infinitely many solutions, it can be reduced to . This equation reduces to , so there is only 1 solution.
Page 23
Question ID: 87071893
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
What value of is the solution to the given equation?
Correct Answer: 55
Rationale
The correct answer is . Subtracting from both sides of the given equation yields . Therefore, the value of is .
Page 24
Question ID: 771bd0ca
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Hard
Question
What value of is the solution to the given equation?
Correct Answer: -22
Rationale
The correct answer is . The given equation can be rewritten as . Dividing both sides of this equation by yields
. Subtracting from both sides of this equation yields . Therefore, is the value of that is the solution to the given equation.
Page 25
Question ID: 620abf36
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
If , what is the value of ?
Answer
A.
B.
C.
D.
Correct Answer: C
Rationale
Choice C is correct. Subtracting from both sides of the given equation yields , or . Therefore, the value of
is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the value of , not .
Choice D is incorrect and may result from conceptual or calculation errors.
Page 26
Question ID: ed18c4f7
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
Cathy has n CDs. Gerry has 3 more than twice the number of CDs that Cathy has. In terms of n, how many CDs does Gerry have?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. The term 2n represents twice the number of CDs that Cathy has, and adding 3 represents 3 more than that amount.
Choices A and B are incorrect. The expression 3n represents three times the number of CDs that Cathy has. Choice C is incorrect. Subtracting 3
represents 3 fewer than twice the number of CDs that Cathy has.
Page 27
Question ID: d9d83c02
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
For what value of w
does
?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. To solve the equation, use the distributive property to multiply on the right-hand side of the equation which gives w – 10 = 2w
+ 10. Subtract w from both sides of the equation, which gives –10 = w + 10. Finally, subtract 10 from both sides of the equation, which gives –20
= w.
Choices A and B are incorrect and may result from making sign errors. Choice C is incorrect and may result from incompletely distributing the 2
in the expression 2(w + 5).
Page 28
Question ID: 4f669597
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
What value of p is the solution of the equation above?
Rationale
The correct answer is 1.2. One way to solve the equation is to first distribute the terms outside the parentheses to
the terms inside the parentheses: . Next, combine like terms on the left side of the equal sign: .
Subtracting 10p from both sides yields . Finally, dividing both sides by gives , which is equivalent to . Note
that 1.2 and 6/5 are examples of ways to enter a correct answer.
Page 29
Question ID: 3c4ce699
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
If , what is the value of ?
Correct Answer: 27
Rationale
The correct answer is . Multiplying both sides of the given equation by yields
, or
. Therefore, the value of
is .
Page 30
Question ID: ce314070
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
If , what is the value of ?
Answer
A. 2
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. Multiplying the given equation by 2 on each side yields . Applying the distributive property, this
equation can be rewritten as , or .
Choices A, B, and C are incorrect and may result from calculation errors in solving the given equation for and then substituting that value of x
in the expression .
Page 31
Question ID: feb78194
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
A museum rents tablets to visitors. The museum earns revenue of for each tablet rented for the day. On Wednesday, the museum earned
in profit from renting tablets after paying daily expenses of . How many tablets did the museum rent on Wednesday?
Correct Answer: 37
Rationale
The correct answer is . It's given that the museum earns revenue of for each tablet rented for the day. It's also given that on Wednesday,
the museum earned in profit from renting tablets after paying daily expenses of . Let represent the number of tablets the museum
rented on Wednesday. It follows that the total revenue can be represented by the expression . Because
, the equation
represents this situation. Adding to both sides of this equation yields .
Dividing both sides of this equation by yields . Therefore, the museum rented tablets on Wednesday.
Page 32
Question ID: ce6b52d8
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
If , what is the value of ?
Correct Answer: 60
Rationale
The correct answer is . Subtracting from both sides of the given equation yields . Applying the distributive property to
the left-hand side of this equation yields . Adding to both sides of this equation yields . Subtracting from
both sides of this equation yields . Therefore, the value of is .
Page 33
Question ID: aee9fd2d
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Hard
Question
If , the value of is between which of the following pairs of values?
Answer
A. and
B. and
C. and
D. and
Correct Answer: B
Rationale
Choice B is correct. Multiplying both sides of the given equation by , or , yields , or
. Subtracting
from both sides of this equation yields
. Dividing both sides of this equation by yields
. Therefore, if
, then the value of is . It follows that of the given choices, the value of is between and .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Page 34
Question ID: 7a987ae4
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
If , what is the value of ?
Answer
A. 24
B. 49
C. 50
D. 99
Correct Answer: B
Rationale
Choice B is correct. Multiplying both sides of the given equation by 5 yields . Substituting 50 for in the expression yields
.
Alternate approach: Dividing both sides of by 2 yields . Evaluating the expression for yields
.
Choice A is incorrect and may result from finding the value of instead of . Choice C is incorrect and may result from finding the
value of instead of . Choice D is incorrect and may result from finding the value of instead of .
Page 35
Question ID: 76f29fa5
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
The cost to rent a commercial fishing boat from a certain company is for the first hours and an additional per hour for each hour
after the first hours. If the total cost to rent the commercial fishing boat from the company for hours, where , is , which equation
represents this situation?
Answer
A.
B.
C.
D.
Correct Answer: C
Rationale
Choice C is correct. It’s given that the cost to rent a commercial fishing boat is for the first hours and an additional per hour for each
hour after the first hours. It’s also given that represents the total number of hours and . Therefore, the number of additional hours after
the first hours can be represented with the expression . The cost for these additional hours is per hour, so the cost for the additional
hours can be represented by the expression . The total cost can be calculated by adding the cost for the first hours to the cost for
the additional hours and can be represented by the expression . It’s also given that the total cost to rent the commercial fishing
boat from the company for hours is . Thus, the equation that represents this situation is .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
Page 36
Question ID: 4f8bd093
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
What value of is the solution to the equation ?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. Subtracting from both sides of the given equation yields . Dividing both sides of this equation by yields
. Therefore, the solution to the given equationis .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Page 37
Question ID: 9ff10b3b
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
If , what is the value of x ?
Answer
A.
B.
C. 3
D. 6
Correct Answer: C
Rationale
Choice C is correct. To make it easier to add like terms on the left-hand side of the given equation, both sides of the equation can be multiplied
by 6, which is the lowest common denominator of and . This yields , which can be rewritten as . Dividing both sides
of this equation by 2 yields .
Choice A is incorrect and may result from subtracting the denominators instead of numerators with common denominators to get , rather
than , on the left-hand side of the equation. Choice B is incorrect and may result from rewriting the given equation as instead of
. Choice D is incorrect and may result from conceptual or computational errors.
Page 38
Question ID: 36ab4122
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
Megan’s regular wage at her job is p
dollars per hour for the first 8 hours of work in a day plus 1.5times her regular hourly wage for work in
excess of 8hours that day. On a given day, Megan worked for 10hours, and her total earnings for that day were$137.50. What is Megan’s regular
hourly wage?
Answer
A. $11.75
B. $12.50
C. $13.25
D. $13.75
Rationale
Choice B is correct. Since p represents Megan’s regular pay per hour, 1.5p represents the pay per hour in excess of 8 hours. Since Megan worked
for 10 hours, she must have been paid p dollars per hour for 8 of the hours plus 1.5p dollars per hour for the remaining 2 hours. Therefore, since
Megan earned $137.50 for the 10 hours, the situation can be represented by the equation 137.5 = 8p + 2(1.5)p.Distributing the 2 in the equation
gives 137.5 = 8p + 3p, and combining like terms gives 137.5 = 11p. Dividing both sides by 11 gives p = 12.5. Therefore, Megan’s regular wage is
$12.50.
Choices A and C are incorrect and may be the result ofcalculation errors. Choice D is incorrect and may result from finding the average hourly
wage that Megan earned for the 10 hours of work.
Page 39
Question ID: 4f7981a0
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
If , what is the value of ?
Rationale
The correct answer is 24. Multiplying both sides of the given equation by 3 yields . Using the distributive property to rewrite the
left-hand side of this equation yields .
Page 40
Question ID: 3f8a701b
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Hard
Question
The equation , where a and b are constants, has no solutions. Which of the following must be true?
I.
II.
III.
Answer
A. None
B. I only
C. I and II only
D. I and III only
Correct Answer: D
Rationale
Choice D is correct. For a linear equation in a form to have no solutions, the x-terms must have equal coefficients and the
remaining terms must not be equal. Expanding the right-hand side of the given equation yields . Inspecting the x-terms, 9
must equal a, so statement I must be true. Inspecting the remaining terms, 5 can’t equal . Dividing both of these quantities by 9 yields that b
can’t equal . Therefore, statement III must be true. Since b can have any value other than , statement II may or may not be true.
Choice A is incorrect. For the given equation to have no solution, both and must be true. Choice B is incorrect because it must
also be true that . Choice C is incorrect because when , there are many values of b that lead to an equation having no solution.
That is, b might be 5, but b isn’t required to be 5.
Page 41
Question ID: 429fb7c0
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Hard
Question
What value of is the solution to the equation
?
Correct Answer: -.3266, -.3267, -49/150
Rationale
The correct answer is . Applying the distributive property to the right-hand side of the given equation yields
, or . Subtracting from both sides of this equation yields . Subtracting from
both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, thevalue of
is
approximately . Note that -.3267, -.3266, -0.326, and -0.327 are examples of ways to enter a correct answer.
Page 42
Question ID: 5ad9eff0
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
The width of a rectangular dance floor is w feet. The length of the floor is 6 feet longer than its width. Which of the following expresses the
perimeter, in feet, of the dance floor in terms of w ?
Answer
A.
B.
C.
D.
Correct Answer: B
Rationale
Choice B is correct. It is given that the width of the dance floor is w feet. The length is 6 feet longer than the width; therefore, the length of the
dance floor is . So the perimeter is .
Choice A is incorrect because it is the sum of one length and one width, which is only half the perimeter. Choice C is incorrect and may result
from using the formula for the area instead of the formula for the perimeter and making a calculation error. Choice D is incorrect because this is
the area, not the perimeter, of the dance floor.
Page 43
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
A librarian has 43 books to distribute to a group of children. If he gives each child 2 books, he will have 7books left over. How many children are
in the group?
Answer
A. 15
B. 18
C. 25
D. 29
Rationale
Choice B is correct. Subtracting the number of books left over from the total number of books results in , which is the number of
books distributed. Dividing the number of books distributed by the number of books given to each child results in .
Choice A is incorrect and results from dividing the total number of books by the number of books given to each child, , then
subtracting the number of books left over from the result, . Choice C is incorrect and results from adding the number of books left
over to the total number of books, , then dividing the result by the number of books given to each child, . Choice D is
incorrect and results from dividing the total number of books by the number of books given to each child, , then adding the number of
books left over, .
Page 44
Question ID: e53870b6
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
In the given equation, k is a constant. If the equation has infinitely many solutions, what is the value of k ?
Rationale
The correct answer is 5. Subtracting from both sides of the given equation gives , so for any value of x, if and only
if . Therefore, if the given equation has infinitely many solutions, the value of k is 5.
Page 45
Question ID: f5ff91b2
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Hard
Question
If , the value of is between which of the following pairs of values?
Answer
A. and
B. and
C. and
D. and
Correct Answer: B
Rationale
Choice B is correct. Multiplying both sides of the given equationby , or , yields
, or .
Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields .
Therefore, if , then the value of is . It follows that of the given choices, the value of is between and .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Page 46
Question ID: 628300a9
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Hard
Question
A science teacher is preparing the 5stations of a science laboratory. Each station will have either ExperimentA materials or ExperimentB
materials, but not both. ExperimentA requires 6teaspoons of salt, and ExperimentB requires 4teaspoons of salt. If x is the number of stations
that will be set up for ExperimentA and the remaining stations will be set up for ExperimentB, which of the following expressions represents the
total number of teaspoons of salt required?
Answer
A.
B.
C.
D.
Correct Answer: C
Rationale
Choice C is correct. It is given that x represents the number of stations that will be set up for Experiment A and that there will be 5 stations total,
so it follows that 5 – x is the number of stations that will be set up for Experiment B. It is also given that Experiment A requires 6 teaspoons of
salt and that Experiment B requires 4 teaspoons of salt, so the total number of teaspoons of salt required is 6x + 4(5 – x), which simplifies to 2x
+ 20.
Choices A, B, and D are incorrect and may be the result of not understanding the description of the context.
Page 47
Question ID: 45bba652
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
If , what is the value of ?
Answer
A. 2
B. 5
C. 7
D. 12
Correct Answer: A
Rationale
Choice A is correct. Adding the like terms on the left-hand side of the given equation yields . Dividing both sides of this equation
by 5 yields .
Choice B is incorrect and may result from subtracting 5, not dividing by 5, on both sides of the equation . Choice C is incorrect.
This is the value of x, not the value of . Choice D is incorrect. This is the value of , not the value of .
Page 48
Question ID: a9c04a21
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Easy
Question
What is the solution to the equation ?
Answer
A. 1
B. 1.5
C. 2
D. 4
Correct Answer: C
Rationale
Choice C is correct. Subtracting 3 from both sides of the given equation yields . Dividing both sides by 2 results in .
Choices A and B are incorrect and may result from computational errors. Choice D is incorrect. This is the value of .
Page 49
Question ID: 8c515062
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in one
variable
Medium
Question
A candle is made of ounces of wax. When the candle is burning, the amount of wax in the candle decreases by ounce every hours. If
ounces of wax remain in this candle, for how many hours has it been burning?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. It’s given that the candle starts with ounces of wax and has ounces of wax remaining after a period of time has passed.
The amount of wax the candle has lost during the time period can be found by subtracting the remaining amount of wax from the amount of wax
the candle was made of, which yields ounces, or ounces. This means the candle loses ounces of wax during that period of time.
It’s given that the amount of wax decreases by ounce every hours. If represents the number of hours the candle has been burning, it follows
that . Multiplying both sides of this equation by yields . Therefore, the candle has been burning for hours.
Choice A is incorrect and may result from using the equation rather than
to represent the situation, and then rounding to the
nearest whole number.
Choice B is incorrect. This is the amount of wax, in ounces, remaining in the candle, not the number of hours it has been burning.
Choice C is incorrect and may result from using the equation
rather than
to represent the situation.
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