passage stringclasses 8
values | question stringlengths 42 1.07k | choices sequencelengths 4 4 | answer stringclasses 4
values | other dict |
|---|---|---|---|---|
If $\frac{x-1}{3}=k$ and $k=3$, what is the value of $x ?$ | [
"(A)2",
"(B)4",
"(C)9",
"(D)10"
] | D | {
"solution": "Choice D is correct. Since $k=3$, one can substitute 3 for $k$ in the equation $\\frac{x-1}{3}=k$, which gives $\\frac{x-1}{3}=3$. Multiplying both sides of $\\frac{x-1}{3}=3$ by 3 gives $x-1=9$ and then adding 1 to both sides of $x-1=9$ gives $x=10$.Choices $\\mathrm{A}, \\mathrm{B}$, and $\\mathrm{C}... | |
For $i=\sqrt{-1}$, what is the sum $(7+3 i)+(-8+9 i) ?$ | [
"(A)$-1+12 i$",
"(B)$-1-6 i$",
"(C)$15+12 i$",
"(D)$15-6 i$ 3"
] | A | {
"solution": "Choice $\\mathbf{A}$ is correct. To calculate $(7+3 i)+(-8+9 i)$, add the real parts of each complex number, $7+(-8)=-1$, and then add the imaginary parts, $3 i+9 i=12 i$. The result is $-1+12 i$.Choices $\\mathrm{B}, \\mathrm{C}$, and $\\mathrm{D}$ are incorrect and likely result from common errors th... | |
On Saturday afternoon, Armand sent $m$ text messages each hour for 5 hours, and Tyrone sent $p$ text messages each hour for 4 hours. Which of the following represents the total number of messages sent by Armand and Tyrone on Saturday afternoon? | [
"(A)$9 m p$",
"(B)$20 m p$",
"(C)$5 m+4 p$",
"(D)$4 m+5 p$"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. The total number of text messages sent by Armand can be found by multiplying his rate of texting, in number of text messages sent per hour, by the total number of hours he spent sending them; that is $m$ texts/hour $\\times 5$ hours $=5 m$ texts. Similarly, the total nu... | |
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 equation $P=108-23 d$, where $P$ is the number of phones left and $d$ is the number of days she has worked that we... | [
"(A)Kathy will complete the repairs within 108 days.",
"(B)Kathy starts each week with 108 phones to fix.",
"(C)Kathy repairs phones at a rate of 108 per hour.",
"(D)Kathy repairs phones at a rate of 108 per day."
] | B | {
"solution": "Choice $\\mathbf{B}$ is correct. The value 108 in the equation is the value of $P$ in $P=108-23 d$ when $d=0$. When $d=0$, Kathy has worked 0 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 equat... | |
$$\left(x^{2} y-3 y^{2}+5 x y^{2}\right)-\left(-x^{2} y+3 x y^{2}-3 y^{2}\right)$$Which of the following is equivalent to the expression above? | [
"(A)$4 x^{2} y^{2}$",
"(B)$8 x y^{2}-6 y^{2}$",
"(C)$2 x^{2} y+2 x y^{2}$",
"(D)$2 x^{2} y+8 x y^{2}-6 y^{2}$"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. Only like terms, with the same variables and exponents, can be combined to determine the answer as shown here:$$\\begin{aligned}& \\left(x^{2} y-3 y^{2}+5 x y^{2}\\right)-\\left(-x^{2} y+3 x y^{2}-3 y^{2}\\right) \\\\= & \\left(x^{2} y-\\left(-x^{2} y\\right)\\right)+\\... | |
$$h=3 a+28.6$$A pediatrician uses the model above to estimate the height $h$ of a boy, in inches, in terms of the boy's age $a$, in years, between the ages of 2 and 5. Based on the model, what is the estimated increase, in inches, of a boy's height each year? | [
"(A)3",
"(B)$\\quad 5.7$",
"(C)9.5",
"(D)14.3"
] | A | {
"solution": "Choice A is correct. In the equation $h=3 a+28.6$, if $a$, the age of the boy, increases by 1 , then $h$ becomes $h=3(a+1)+28.6=3 a+3+28.6=$ $(3 a+28.6)+3$. Therefore, the model estimates that the boy's height increases by 3 inches each year.Alternatively: The height, $h$, is a linear function of the a... | |
$$m=\frac{\left(\frac{r}{1,200}\right)\left(1+\frac{r}{1,200}\right)^{N}}{\left(1+\frac{r}{1,200}\right)^{N}-1} P$$The formula above gives the monthly payment $m$ needed to pay off a loan of $P$ dollars at $r$ percent annual interest over $N$ months. Which of the following gives $P$ in terms of $m, r$, and $N$ ? | [
"(A)$P=\\frac{\\left(\\frac{r}{1,200}\\right)\\left(1+\\frac{r}{1,200}\\right)^{N}}{\\left(1+\\frac{r}{1,200}\\right)^{N}-1} m$",
"(B)$P=\\frac{\\left(1+\\frac{r}{1,200}\\right)^{N}-1}{\\left(\\frac{r}{1,200}\\right)\\left(1+\\frac{r}{1,200}\\right)^{N}} m$",
"(C)$P=\\left(\\frac{r}{1,200}\\right) m$",
"(D)$P... | B | {
"solution": "Choice B is correct. Since the right-hand side of the equation is $P$ times the expression $\\frac{\\left(\\frac{r}{1,200}\\right)\\left(1+\\frac{r}{1,200}\\right)^{N}}{\\left(1+\\frac{r}{1,200}\\right)^{N}-1}$, multiplying both sides of the equation by the reciprocal of this expression results $\\oper... | |
If $\frac{a}{b}=2$, what is the value of $\frac{4 b}{a} ?$ | [
"(A)0",
"(B)1",
"(C)2",
"(D)4"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. Since $\\frac{a}{b}=2$, it follows that $\\frac{b}{a}=\\frac{1}{2}$. Multiplying both sides of the equation by 4 gives $4\\left(\\frac{b}{a}\\right)=4\\left(\\frac{1}{2}\\right)$, or $\\frac{4 b}{a}=2$.Choice A is incorrect because if $\\frac{4 b}{a}=0$, then $\\frac{a}... | |
$$\begin{array}{r}3 x+4 y=-23 \\2 y-x=-19\end{array}$$What is the solution $(x, y)$ to the system of equations above? | [
"(A)$(-5,-2)$",
"(B)$(3,-8)$",
"(C)$(4,-6)$",
"(D)$(9,-6)$"
] | B | {
"solution": "Choice $\\mathbf{B}$ is correct. Adding $x$ and 19 to both sides of $2 y-x=-19$ gives $x=2 y+19$. Then, substituting $2 y+19$ for $x$ in $3 x+4 y=-23$ gives $3(2 y+19)+4 y=-23$. This last equation is equivalent to $10 y+57=-23$. Solving $10 y+57=-23$ gives $y=-8$. Finally, substituting -8 for $y$ in $2... | |
$$g(x)=a x^{2}+24$$For the function $g$ defined above, $a$ is a constant and $g(4)=8$. What is the value of $g(-4)$ ? | [
"(A)8",
"(B)0",
"(C)-1",
"(D)-8"
] | A | {
"solution": "Choice A is correct. Since $g$ is an even function, $g(-4)=g(4)=8$.Alternatively: First find the value of $a$, and then find $g(-4)$.Since $g(4)=8$, substituting 4 for $x$ and 8 for $g(x)$ gives $8=a(4)^{2}+24=16 a+24$. Solving this last equation gives $a=-1$. Thus $g(x)=-x^{2}+24$, from which it follo... | |
$$\begin{aligned}& b=2.35+0.25 x \\& c=1.75+0.40 x\end{aligned}$$In the equations above, $b$ and $c$ represent the price per pound, in dollars, of beef and chicken, respectively, $x$ weeks after July 1 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$"
] | D | {
"solution": "Choice $D$ is correct. To determine the price per pound of beef when it was equal to the price per pound of chicken, determine the value of $x$ (the number of weeks after July 1) when the two prices were equal. The prices were equal when $b=c$; that is, when $2.35+0.25 x=1.75+0.40 x$. This last equatio... | |
A line in the $x y$-plane passes through the origin and has a slope of $\frac{1}{7}$. Which of the following points lies on the line? | [
"(A)$(0,7)$",
"(B)$(1,7)$",
"(C)$(7,7)$",
"(D)$(14,2)$"
] | D | {
"solution": "Choice $\\mathbf{D}$ is correct. In the $x y$-plane, all lines that pass through the origin are of the form $y=m x$, where $m$ is the slope of the line. Therefore, the equation of this line is $y=\\frac{1}{7} x$, or $x=7 y$. A point with coordinates $(a, b)$ will lie on the line if and only if $a=7 b$.... | |
If $x>3$, which of the following is equivalent to $\frac{1}{\frac{1}{x+2}+\frac{1}{x+3}}$ ? | [
"(A)$\\frac{2 x+5}{x^{2}+5 x+6}$",
"(B)$\\frac{x^{2}+5 x+6}{2 x+5}$",
"(C)$2 x+5$",
"(D)$x^{2}+5 x+6$"
] | B | {
"solution": "Choice B is correct. To rewrite $\\frac{1}{\\frac{1}{x+2}+\\frac{1}{x+3}}$, multiplyby $\\frac{(x+2)(x+3)}{(x+2)(x+3)}$. This results in the expression $\\frac{(x+2)(x+3)}{(x+2)+(x+3)}$, which is equivalent to the expression in choice $B$.Choices A, C, and D are incorrect and could be the result of com... | |
If $3 x-y=12$, what is the value of $\frac{8^{x}}{2^{y}} ?$ | [
"(A)$2^{12}$",
"(B)4",
"(C)$8^{2}$",
"(D)The value cannot be determined from the information given."
] | A | {
"solution": "Choice A is correct. One approach is to express $\\frac{8^{x}}{2^{y}}$ so that the numerator and denominator are expressed with the same base. Since 2 and 8 are both powers of 2 , substituting $2^{3}$ for 8 in the numerator of $\\frac{8^{x}}{2^{y}}$ gives $\\frac{\\left(2^{3}\\right)^{x}}{2^{y}}$, whic... | |
If $(a x+2)(b x+7)=15 x^{2}+c x+14$ for all values of $x$, and $a+b=8$, what are the two possible values for $c$ ? | [
"(A)3 and 5",
"(B)6 and 35",
"(C)10 and 21",
"(D)31 and 41"
] | D | {
"solution": "Choice D is correct. One can find the possible values of $a$ and $b$ in $(a x+2)(b x+7)$ by using the given equation $a+b=8$ and finding another equation that relates the variables $a$ and $b$. Since $(a x+2)(b x+7)=15 x^{2}+c x+14$, one can expand the left side of the equation to obtain $a b x^{2}+7 a... | |
If $y=k x$, where $k$ is a constant, and $y=24$ when $x=6$, what is the value of $y$ when $x=5$ ? | [
"(A)6",
"(B)15",
"(C)20",
"(D)23"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. Substituting 6 for $x$ and 24 for $y$ in $y=k x$ gives $24=(k)(6)$, which gives $k=4$. Hence, $y=4 x$. Therefore, when $x=5$, the value of $y$ is $(4)(5)=20$. None of the other choices for $y$ is correct because $y$ is a function of $x$, and so there is only one $y$-val... | |
If $16+4 x$ is 10 more than 14 , what is the value of $8 x$ ? | [
"(A)2",
"(B)6",
"(C)16",
"(D)80 5"
] | C | {
"solution": "Choice C is correct. The description \" $16+4 x$ is 10 more than 14 \" can be written as the equation $16+4 x=10+14$, which is equivalent to $16+4 x=24$. Subtracting 16 from each side of $16+4 x=24$ gives $4 x=8$. Since $8 x$ is 2 times $4 x$, multiplying both sides of $4 x=8$ by 2 gives $8 x=16$. Ther... | |
$$\begin{aligned}1 \text { decagram } & =10 \text { grams } \\1,000 \text { milligrams } & =1 \text { gram }\end{aligned}$$A hospital stores one type of medicine in 2-decagram containers. Based on the information given in the box above, how many 1-milligram doses are there in one 2-decagram container? | [
"(A)$\\quad 0.002$",
"(B)$\\quad 200$",
"(C)$\\quad 2,000$",
"(D)20,000"
] | D | {
"solution": "Choice $\\mathbf{D}$ is correct. Since there are 10 grams in 1 decagram, there are $2 \\times 10=20$ grams in 2 decagrams. Since there are 1,000 milligrams in 1 gram, there are $20 \\times 1,000=20,000$ milligrams in 20 grams.Therefore, 20,000 1-milligram doses of the medicine can be stored in a 2-deca... | |
For what value of $n$ is $|n-1|+1$ equal to 0 ? | [
"(A)0",
"(B)1",
"(C)2",
"(D)There is no such value of $n$."
] | D | {
"solution": "Choice $\\mathbf{D}$ is correct. If the value of $|n-1|+1$ is equal to 0 , then $|n-1|+1=0$. Subtracting 1 from both sides of this equation gives $|n-1|=-1$. The expression $|n-1|$ on the left side of the equation is the absolute value of $n-1$, and the absolute value of a quantity can never be negativ... | |
$$.a=1,052+1.08 t.$$.The speed of a sound wave in air depends on the air temperature. The formula above shows the relationship between $a$, the speed of a sound wave, in feet per second, and $t$, the air temperature, in degrees Fahrenheit $\left({ }^{\circ} \mathrm{F}\right)$. | Which of the following expresses the air temperature in terms of the speed of a sound wave? | [
"(A)$t=\\frac{a-1,052}{1.08}$",
"(B)$t=\\frac{a+1,052}{1.08}$",
"(C)$t=\\frac{1,052-a}{1.08}$",
"(D)$t=\\frac{1.08}{a+1,052}$"
] | A | {
"solution": "Choice A is correct. Subtracting 1,052 from both sides of the equation $a=1,052+1.08 t$ gives $a-1,052=1.08 t$. Then dividing both sides of $a-1,052=1.08 t$ by 1.08 gives $t=\\frac{a-1,052}{1.08}$. Choices B, C, and D are incorrect and could arise from errors in rewriting $a=1,052+1.08 t$. For example,... |
$$.a=1,052+1.08 t.$$.The speed of a sound wave in air depends on the air temperature. The formula above shows the relationship between $a$, the speed of a sound wave, in feet per second, and $t$, the air temperature, in degrees Fahrenheit $\left({ }^{\circ} \mathrm{F}\right)$. | At which of the following air temperatures will the speed of a sound wave be closest to 1,000 feet per second? | [
"(A)$-46^{\\circ} \\mathrm{F}$",
"(B)$-48^{\\circ} \\mathrm{F}$",
"(C)$-49^{\\circ} \\mathrm{F}$",
"(D)$-50^{\\circ} \\mathrm{F}$"
] | B | {
"solution": "Choice B is correct. The air temperature at which the speed of a sound wave is closest to 1,000 feet per second can be found by substituting 1,000 for $a$ and then solving for $t$ in the given formula. Substituting 1,000 for $a$ in the equation $a=1,052+1.08$ t gives $1,000=1,052+1.08 t$. Subtracting 1... |
Which of the following numbers is NOT a solution of the inequality $3 x-5 \geq 4 x-3$ ? | [
"(A)-1",
"(B)-2",
"(C)-3",
"(D)-5"
] | A | {
"solution": "Choice A is correct. Subtracting $3 x$ and adding 3 to both sides of $3 x-5 \\geq 4 x-3$ gives $-2 \\geq x$. Therefore, $x$ is a solution to $3 x-5 \\geq 4 x-3$ if and only if $x$ is less than or equal to -2 and $x$ is NOT a solution to $3 x-5 \\geq 4 x-3$ if and only if $x$ is greater than -2 . Of the... | |
\begin{center}\begin{tabular}{|c|c|c|c|c|c|}\cline { 3 - 5 }\multicolumn{2}{c|}{} & \multicolumn{3}{c|}{Course} & \multicolumn{1}{c|}{} \\\cline { 2 - 5 }\multicolumn{2}{c|}{} & Algebra I & Geometry & $\begin{array}{c}\text { Algebra } \\ \text { II }\end{array}$ & \multirow{2}{*}{Total} \\\hline\multirow{2}{*}{Gender}... | [
"(A)Females taking Geometry",
"(B)Females taking Algebra II",
"(C)Males taking Geometry",
"(D)Males taking Algebra I"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. From the table, there was a total of 310 survey respondents, and $19 \\%$ of all survey respondents is equivalent to $\\frac{19}{100} \\times 310=58.9$ respondents. Of the choices given, 59 , the number of males taking Geometry, is closest to 58.9 respondents.Choices A,... | |
\begin{center}\begin{tabular}{|c|c|c|c|c|c|c|}\hline\multicolumn{7}{|c|}{Lengths of Fish (in inches)} \\\hline8 & 9 & 9 & 9 & 10 & 10 & 11 \\\hline11 & 12 & 12 & 12 & 12 & 13 & 13 \\\hline13 & 14 & 14 & 15 & 15 & 16 & 24 \\\hline\end{tabular}\end{center}The table above lists the lengths, to the nearest inch, of a rando... | [
"(A)Mean",
"(B)Median",
"(C)Range",
"(D)They will all change by the same amount."
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. The range of the lengths of the 21 fish represented in the table is $24-8=16$ inches, and the range of the remaining 20 lengths after the 24 -inch measurement is removed is $16-8=8$ inches. Therefore, after the 24 -inch measurement is removed, the change in range, 8 inc... | |
$$\begin{aligned}& y<-x+a \\& y>x+b\end{aligned}$$In the $x y$-plane, if $(0,0)$ is a solution to the system of inequalities above, which of the following relationships between $a$ and $b$ must be true? | [
"(A)$a>b$",
"(B)$b>a$",
"(C)$|a|>|b|$",
"(D)$a=-b$"
] | A | {
"solution": "Choice A is correct. Since $(0,0)$ is a solution to the system of inequalities, substituting 0 for $x$ and 0 for $y$ in the given system must result in two true inequalities. After this substitution, $y<-x+a$ becomes $0<a$, and $y>x+b$ becomes $0>b$. Hence, $a$ is positive and $b$ is negative. Therefor... | |
A food truck sells salads for $\$ 6.50$ each and drinks for $\$ 2.00$ each. The food truck's revenue from selling a total of 209 salads and drinks in one day was $\$ 836.50$. How many salads were sold that day? | [
"(A)77",
"(B)93",
"(C)99",
"(D)105"
] | B | {
"solution": "Choice B is correct. To determine the number of salads sold, write and solve a system of two equations. Let $x$ equal the number of salads sold and let $y$ equal the number of drinks sold. Since a total of 209 salads and drinks were sold, the equation $x+y=209$ must hold. Since salads cost $\\$ 6.50$ e... | |
Alma bought a laptop computer at a store that gave a 20 percent discount off its original price. The total amount she paid to the cashier was $p$ dollars, including an 8 percent sales tax on the discounted price. Which of the following represents the original price of the computer in terms of $p$ ? | [
"(A)$0.88 p$",
"(B)$\\frac{p}{0.88}$",
"(C)$(0.8)(1.08) p$",
"(D)$\\frac{p}{(0.8)(1.08)}$"
] | D | {
"solution": "Choice $D$ is correct. Let $x$ be the original price of the computer, in dollars. The discounted price is 20 percent off the original price, so $x-0.2 x=0.8 x$ is the discounted price, in dollars. The sales tax is 8 percent of the discounted price, so $0.08(0.8 x)$ represents the sales tax Alma paid. T... | |
Dreams Recalled during One Week\begin{center}\begin{tabular}{|l|c|c|c|c|}\hline& None & 1 to 4 & 5 or more & Total \\\hline\hlineGroup X & 15 & 28 & 57 & 100 \\\hlineGroup Y & 21 & 11 & 68 & 100 \\\hlineTotal & 36 & 39 & 125 & 200 \\\hline\end{tabular}\end{center}The data in the table above were produced by a sleep res... | [
"(A)$\\frac{68}{100}$",
"(B)$\\frac{79}{100}$",
"(C)$\\frac{79}{164}$",
"(D)$\\frac{164}{200}$"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. The probability that a person from Group $Y$ who recalled at least 1 dream was chosen at random from the group of all people who recalled at least 1 dream is equal to the number of people in Group $\\mathrm{Y}$ who recalled at least 1 dream divided by the total number o... | |
Which of the following is an equation of a circle in the $x y$-plane with center $(0,4)$ and a radius with endpoint $\left(\frac{4}{3}, 5\right) ?$ | [
"(A)$x^{2}+(y-4)^{2}=\\frac{25}{9}$",
"(B)$x^{2}+(y+4)^{2}=\\frac{25}{9}$",
"(C)$x^{2}+(y-4)^{2}=\\frac{5}{3}$",
"(D)$x^{2}+(y+4)^{2}=\\frac{3}{5}$"
] | A | {
"solution": "Choice $\\mathbf{A}$ is correct. The equation of a circle can be written as $(x-h)^{2}+(y-k)^{2}=r^{2}$ where $(h, k)$ are the coordinates of the center of the circle and $r$ is the radius of the circle. Since the coordinates of the center of the circle are $(0,4)$, the equation of the circle is $x^{2}... | |
$$h=-4.9 t^{2}+25 t$$The equation above expresses the approximate height $h$, in meters, of a ball $t$ seconds after it is launched vertically upward from the ground with an initial velocity of 25 meters per second. After approximately how many seconds will the ball hit the ground? | [
"(A)3.5",
"(B)4.0",
"(C)4.5",
"(D)5.0"
] | D | {
"solution": "Choice $\\mathbf{D}$ is correct. When the ball hits the ground, its height is 0 meters. Substituting 0 for $h$ in $h=-4.9 t^{2}+25 t$ gives $0=-4.9 t^{2}+25 t$, which can be rewritten as $0=t(-4.9 t+25)$. Thus, the possible values of $t$ are $t=0$ and $t=\\frac{25}{4.9} \\approx 5.1$. The time $t=0$ se... | |
Katarina is a botanist studying the production of pears by two types of pear trees. She noticed that Type A trees produced 20 percent more pears than Type B trees did. Based on Katarina's observation, if the Type A trees produced 144 pears, how many pears did the Type B trees produce? | [
"(A)115",
"(B)120",
"(C)124",
"(D)173"
] | B | {
"solution": "Choice B is correct. Let $x$ represent the number of pears produced by the Type B trees. Type A trees produce 20 percent more pears than Type B trees, or $x$, which can be represented as $x+0.20 x=1.20 x$ pears. Since Type A trees produce 144 pears, it follows that $1.20 x=144$. Thus $x=\\frac{144}{1.2... | |
A square field measures 10 meters by 10 meters. Ten students each mark off a randomly selected region of the field; each region is square and has side lengths of 1 meter, and no two regions overlap. The students count the earthworms contained in the soil to a depth of 5 centimeters beneath the ground's surface in each ... | [
"(A)$\\quad 150$",
"(B)$\\quad 1,500$",
"(C)15,000",
"(D)150,000"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. The area of the field is 100 square meters. Each 1-meter-by-1-meter square has an area of 1 square meter. Thus, on average, the earthworm counts to a depth of 5 centimeters for each of the regions investigated by the students should be about $\\frac{1}{100}$ of the tota... | |
For a polynomial $p(x)$, the value of $p(3)$ is -2 .Which of the following must be true about $p(x)$ ? | [
"(A)$x-5$ is a factor of $p(x)$.",
"(B)$x-2$ is a factor of $p(x)$.",
"(C)$x+2$ is a factor of $p(x)$.",
"(D)The remainder when $p(x)$ is divided by $x-3$ is -2 ."
] | D | {
"solution": "Choice $\\mathbf{D}$ is correct. If the polynomial $p(x)$ is divided by $x-3$, the result can be written as $\\frac{p(x)}{x-3}=q(x)+\\frac{r}{x-3}$, where $q(x)$ is a polynomial and $r$ is the remainder. Since $x-3$ is a degree 1 polynomial, the remainder is a real number. Hence, $p(x)$ can be written ... | |
$$2 z+1=z$$What value of $z$ satisfies the equation above? | [
"(A)-2",
"(B)-1",
"(C)$\\frac{1}{2}$",
"(D)1"
] | B | {
"solution": "Choice B is correct. Subtracting $z$ from both sides of $2 z+1=z$ results in $z+1=0$. Subtracting 1 from both sides of $z+1=0$ results in $z=-1$.Choices $\\mathrm{A}, \\mathrm{C}$, and $\\mathrm{D}$ are incorrect. When each of these values is substituted for $z$ in the given equation, the result is a f... | |
A television with a price of $\$ 300$ is to be purchased with an initial payment of $\$ 60$ and weekly payments of $\$ 30$. Which of the following equations can be used to find the number of weekly payments, $w$, required to complete the purchase, assuming there are no taxes or fees? | [
"(A)$300=30 w-60$",
"(B)$300=30 w$",
"(C)$300=30 w+60$",
"(D)$300=60 w-30$"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. To complete the purchase, the initial payment of $\\$ 60$ plus the $w$ weekly payments of $\\$ 30$ must be equivalent to the $\\$ 300$ price of the television. The total, in dollars, of $w$ weekly payments of $\\$ 30$ can be expressed by $30 w$. It follows that $300=30 ... | |
Shipping Charges\begin{center}\begin{tabular}{|c|c|}\hline$\begin{array}{c}\text { Merchandise weight } \\ \text { (pounds) }\end{array}$ & $\begin{array}{c}\text { Shipping } \\ \text { charge }\end{array}$ \\\hline5 & $\$ 16.94$ \\\hline10 & $\$ 21.89$ \\\hline20 & $\$ 31.79$ \\\hline40 & $\$ 51.59$ \\\hline\end{tabu... | [
"(A)$f(x)=0.99 x$",
"(B)$f(x)=0.99 x+11.99$",
"(C)$f(x)=3.39 x$",
"(D)$f(x)=3.39 x+16.94$"
] | B | {
"solution": "Choice B is correct. Since the relationship between the merchandise weight $x$ and the shipping charge $f(x)$ is linear, a function in the form $f(x)=m x+b$, where $m$ and $b$ are constants, can be used. In this situation, the constant $m$ represents the additional shipping charge, in dollars, for each... | |
$$\sqrt{9 x^{2}}$$If $x>0$, which of the following is equivalent to the given expression? | [
"(A)$3 x$",
"(B)$3 x^{2}$",
"(C)$18 x$",
"(D)$18 x^{4}$"
] | A | {
"solution": "Choice $\\mathbf{A}$ is correct. The expression $\\sqrt{9 x^{2}}$ can be rewritten as $(\\sqrt{9})\\left(\\sqrt{x^{2}}\\right)$. The square root symbol in an expression represents the principal square root, or the positive square root, thus $\\sqrt{9}=3$. Since $x>0$, taking the square root of the seco... | |
$$\frac{x^{2}-1}{x-1}=-2$$What are all values of $x$ that satisfy the equation above? | [
"(A)-3",
"(B)0",
"(C)1",
"(D)-3 and -1"
] | A | {
"solution": "Choice $\\mathbf{A}$ is correct. Factoring the numerator of the rational expression $\\frac{x^{2}-1}{x-1}$ yields $\\frac{(x+1)(x-1)}{x-1}$. The expression $\\frac{(x+1)(x-1)}{x-1}$ can be rewritten as $\\left(\\frac{x+1}{1}\\right)\\left(\\frac{x-1}{x-1}\\right)$. Since $\\frac{x-1}{x-1}=1$, when $x$ ... | |
A circle in the $x y$-plane has center $(5,7)$ and radius 2. Which of the following is an equation of the circle? | [
"(A)$(x-5)^{2}+(y-7)^{2}=4$",
"(B)$(x+5)^{2}+(y+7)^{2}=4$",
"(C)$(x-5)^{2}+(y-7)^{2}=2$",
"(D)$(x+5)^{2}+(y+7)^{2}=2$"
] | A | {
"solution": "Choice A is correct. A circle in the $x y$-plane with center $(h, k)$ and radius $r$ is defined by the equation $(x-h)^{2}+(y-k)^{2}=r^{2}$. Therefore, an equation of a circle with center $(5,7)$ and radius 2 is $(x-5)^{2}+(y-7)^{2}=2^{2}$, or $(x-5)^{2}+(y-7)^{2}=4$.Choice B is incorrect. This equatio... | |
In the $x y$-plane, the graph of the function $f(x)=x^{2}+5 x+4$ has two $x$-intercepts. What is the distance between the $x$-intercepts? | [
"(A)1",
"(B)2",
"(C)3",
"(D)4"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. The $x$-intercepts of the graph of $f(x)=x^{2}+5 x+4$ are the points $(x, f(x))$ on the graph where $f(x)=0$. Substituting 0 for $f(x)$ in the function equation yields $0=x^{2}+5 x+4$. Factoring the right-hand side of $0=x^{2}+5 x+4$ yields $0=(x+4)(x+1)$. If $0=(x+4)(x... | |
$$\sqrt{4 x}=x-3$$What are all values of $x$ that satisfy the given equation?I. 1II. 9 | [
"(A)I only",
"(B)II only",
"(C)I and II",
"(D)Neither I nor II"
] | B | {
"solution": "Choice B is correct. Squaring both sides of the equation $\\sqrt{4 x}=x-3$ yields $4 x=(x-3)^{2}$, or $4 x=(x-3)(x-3)$. Applying the distributive property on the right-hand side of the equation $4 x=(x-3)(x-3)$ yields $4 x=x^{2}-3 x-3 x+9$. Subtracting $4 x$ from both sides of $4 x=x^{2}-3 x-3 x+9$ yie... | |
$$\begin{aligned}& -3 x+y=6 \\& a x+2 y=4\end{aligned}$$In the system of equations above, $a$ is a constant. For which of the following values of $a$ does the system have no solution? | [
"(A)-6",
"(B)-3",
"(C)3",
"(D)6"
] | A | {
"solution": "Choice A is correct. A system of two linear equations has no solution if the graphs of the lines represented by the equations are parallel and are not equivalent. Parallel lines have equal slopes but different $y$-intercepts. The slopes and $y$-intercepts for the two given equations can be found by sol... | |
A helicopter, initially hovering 40 feet above the ground, begins to gain altitude at a rate of 21 feet per second. Which of the following functions represents the helicopter's altitude above the ground $y$, in feet, $t$ seconds after the helicopter begins to gain altitude? | [
"(A)$y=40+21$",
"(B)$y=40+21 t$",
"(C)$y=40-21 t$",
"(D)$y=40 t+21$"
] | B | {
"solution": "Choice B is correct. It's given that the helicopter's initial height is 40 feet above the ground and that when the helicopter's altitude begins to increase, it increases at a rate of 21 feet per second. Therefore, the altitude gain $t$ seconds after the helicopter begins rising is represented by the ex... | |
If $20-x=15$, what is the value of $3 x ?$ | [
"(A)5",
"(B)10",
"(C)15",
"(D)35"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. Subtracting 20 from both sides of the given equation yields $-x=-5$. Dividing both sides of the equation $-x=-5$ by -1 yields $x=5$. Lastly, substituting 5 for $x$ in $3 x$ yields the value of $3 x$, or $3(5)=15$.Choice $\\mathrm{A}$ is incorrect. This is the value of $... | |
$$f(x)=\frac{x+3}{2}$$For the function $f$ defined above, what is the value of $f(-1)$ ? | [
"(A)-2",
"(B)-1",
"(C)1",
"(D)2"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. The value of $f(-1)$ can be found by substituting-1 for $x$ in the given function $f(x)=\\frac{x+3}{2}$, which yields $f(-1)=\\frac{-1+3}{2}$.Rewriting the numerator by adding -1 and 3 yields $\\frac{2}{2}$, which equals 1 .Therefore, $f(-1)=1$.Choice $A$ is incorrect a... | |
Which of the following is equivalent to $2 x\left(x^{2}-3 x\right)$ ? | [
"(A)$-4 x^{2}$",
"(B)$3 x^{3}-x^{2}$",
"(C)$2 x^{3}-3 x$",
"(D)$2 x^{3}-6 x^{2}$"
] | D | {
"solution": "Choice D is correct. To determine which option is equivalent to the given expression, the expression can be rewritten using the distributive property by multiplying each term of the binomial $\\left(x^{2}-3 x\\right)$ by $2 x$, which gives $2 x^{3}-6 x^{2}$.Choices $\\mathrm{A}, \\mathrm{B}$, and $\\ma... | |
A retail company has 50 large stores located in different areas throughout a state. A researcher for the company believes that employee job satisfaction varies greatly from store to store. Which of the following sampling methods is most appropriate to estimate the proportion of all employees of the company who are sati... | [
"(A)Selecting one of the 50 stores at random and then surveying each employee at that store",
"(B)Selecting 10 employees from each store at random and then surveying each employee selected",
"(C)Surveying the 25 highest-paid employees and the 25 lowest-paid employees",
"(D)Creating a website on which employee... | B | {
"solution": "Choice $\\mathbf{B}$ is correct. Selecting employees from each store at random is most appropriate because it's most likely to ensure that the group surveyed will accurately represent each store location and all employees.Choice A is incorrect. Surveying employees at a single store location will only p... | |
$$h(x)=2^{x}$$The function $h$ is defined above. What is $h(5)-h(3) ?$ | [
"(A)2",
"(B)4",
"(C)24",
"(D)28"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. The value of the expression $h(5)-h(3)$ can be found by substituting 5 and 3 for $x$ in the given function. Substituting 5 for $x$ in the function yields $h(5)=2^{5}$, which can be rewritten as $h(5)=32$. Substituting 3 for $x$ in the function yields $h(3)=2^{3}$, which... | |
A researcher surveyed a random sample of students from a large university about how often they see movies. Using the sample data, the researcher estimated that $23 \%$ of the students in the population saw a movie at least once per month. The margin of error for this estimation is $4 \%$. Which of the following is the ... | [
"(A)It is unlikely that less than $23 \\%$ of the students see a movie at least once per month.",
"(B)At least 23\\%, but no more than $25 \\%$, of the students see a movie at least once per month.",
"(C)The researcher is between $19 \\%$ and $27 \\%$ sure that most students see a movie at least once per month.... | D | {
"solution": "Choice D is correct. The margin of error is applied to the sample statistic to create an interval in which the population statistic most likely falls. An estimate of $23 \\%$ with a margin of error of $4 \\%$ creates an interval of $23 \\% \\pm 4 \\%$, or between $19 \\%$ and $27 \\%$. Thus, it's plaus... | |
\begin{center}\begin{tabular}{|c|c|c|c|c|c|c|}\hlineList A & 1 & 2 & 3 & 4 & 5 & 6 \\\hlineList B & 2 & 3 & 3 & 4 & 4 & 5 \\\hline\end{tabular}\end{center}The table above shows two lists of numbers. Which of the following is a true statement comparing list $\mathrm{A}$ and list B ? | [
"(A)The means are the same, and the standard deviations are different.",
"(B)The means are the same, and the standard deviations are the same.",
"(C)The means are different, and the standard deviations are different.",
"(D)The means are different, and the standard deviations are the same."
] | A | {
"solution": "Choice A is correct. The mean number of each list is found by dividing the sum of all the numbers in each list by the count of the numbers in each list. The mean of list $A$ is $\\frac{1+2+3+4+5+6}{6}=3.5$, and the mean of list $B$ is $\\frac{2+3+3+4+4+5}{6}=3.5$. Thus, the means are the same. The stan... | |
A book was on sale for $40 \%$ off its original price. If the sale price of the book was $\$ 18.00$, what was the original price of the book? (Assume there is no sales tax.) | [
"(A)$\\$ 7.20$",
"(B)$\\$ 10.80$",
"(C)$\\$ 30.00$",
"(D)$\\$ 45.00$"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. Let $x$ represent the original price of the book. Then, $40 \\%$ off of $x$ is $(1-0.40) x$, or $0.60 x$. Since the sale price is $\\$ 18.00$, then $0.60 x=18$. Dividing both sides of this equation by 0.60 yields $x=30$. Therefore, the original price of the book was $\\... | |
A right circular cone has a volume of $24 \pi$ cubic inches. If the height of the cone is 2 inches, what is the radius, in inches, of the base of the cone? | [
"(A)$2 \\sqrt{3}$",
"(B)6",
"(C)12",
"(D)36"
] | B | {
"solution": "Choice B is correct. The formula for the volume $V$ of a right circular cone is $V=\\frac{1}{3} \\pi r^{2} h$, where $r$ is the radius of the base and $h$ is the height of the cone. It's given that the cone's volume is $24 \\pi$ cubic inches and its height is 2 inches. Substituting $24 \\pi$ for $V$ an... | |
In 2015 the populations of City $\mathrm{X}$ and City $\mathrm{Y}$ were equal. From 2010 to 2015, the population of City X increased by $20 \%$ and the population of City $\mathrm{Y}$ decreased by $10 \%$. If the population of City $\mathrm{X}$ was 120,000 in 2010, what was the population of City Y in 2010 ? | [
"(A)60,000",
"(B)90,000",
"(C)160,000",
"(D)240,000"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. It's given that the population of City $\\mathrm{X}$ was 120,000 in 2010, and that it increased by $20 \\%$ from 2010 to 2015 . Therefore, the population of City X in 2015 was $120,000(1+0.20)=144,000$. It's also given that the population of City $\\mathrm{Y}$ decreased... | |
The volume of a sphere is given by the formula $V=\frac{4}{3} \pi r^{3}$, where $r$ is the radius of the sphere. Which of the following gives the radius of the sphere in terms of the volume of the sphere? | [
"(A)$\\frac{4 \\pi}{3 V}$",
"(B)$\\frac{3 V}{4 \\pi}$",
"(C)$\\sqrt[3]{\\frac{4 \\pi}{3 V}}$",
"(D)$\\sqrt[3]{\\frac{3 V}{4 \\pi}}$"
] | D | {
"solution": "Choice D is correct. Dividing both sides of the equation $V=\\frac{4}{3} \\pi r^{3}$ by $\\frac{4}{3} \\pi$ results in $\\frac{3 V}{4 \\pi}=r^{3}$. Taking the cube root of both sides produces $\\sqrt[3]{\\frac{3 V}{4 \\pi}}=r$. Therefore, $\\sqrt[3]{\\frac{3 V}{4 \\pi}}$ gives the radius of the sphere ... | |
Survey Results\begin{center}\begin{tabular}{|l|c|}\hlineAnswer & Percent \\\hlineNever & $31.3 \%$ \\\hlineRarely & $24.3 \%$ \\\hlineOften & $13.5 \%$ \\\hlineAlways & $30.9 \%$ \\\hline\end{tabular}\end{center}The table above shows the results of a survey in which tablet users were asked how often they would watch vi... | [
"(A)0.31",
"(B)0.38",
"(C)0.45",
"(D)0.69"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. It's given that the tablet user did not answer \"Never,\" so the tablet user could have answered only \"Rarely,\" \"Often,\" or \"Always.\" These answers make up $24.3 \\%+13.5 \\%+30.9 \\%=68.7 \\%$ of the answers the tablet users gave in the survey. The answer \"Alway... | |
$$y=-(x-3)^{2}+a$$In the equation above, $a$ is a constant. The graph of the equation in the $x y$-plane is a parabola. Which of the following is true about the parabola? | [
"(A)Its minimum occurs at $(-3, a)$.",
"(B)Its minimum occurs at $(3, a)$.",
"(C)Its maximum occurs at $(-3, a)$.",
"(D)Its maximum occurs at $(3, a)$."
] | D | {
"solution": "Choice $D$ is correct. The vertex form of a quadratic equation is $y=n(x-h)^{2}+k$, where $(h, k)$ gives the coordinates of the vertex of the parabola in the $x y$-plane and the sign of the constant $n$ determines whether the parabola opens upward or downward. If $n$ is negative, the parabola opens dow... | |
The maximum value of a data set consisting of 25 positive integers is 84 . A new data set consisting of 26 positive integers is created by including 96 in the original data set. Which of the following measures must be 12 greater for the new data set than for the original data set? | [
"(A)The mean",
"(B)The median",
"(C)The range",
"(D)The standard deviation"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. Let $m$ be the minimum value of the original data set. The range of a data set is the difference between the maximum value and the minimum value. The range of the original data set is therefore $84-m$. The new data set consists of the original set and the positive integ... | |
$$0.10 x+0.20 y=0.18(x+y)$$Clayton will mix $x$ milliliters of a $10 \%$ by mass saline solution with $y$ milliliters of a $20 \%$ by mass saline solution in order to create an $18 \%$ by mass saline solution. The equation above represents this situation. If Clayton uses 100 milliliters of the $20 \%$ by mass saline so... | [
"(A)5",
"(B)25",
"(C)50",
"(D)100"
] | B | {
"solution": "Choice B is correct. It's given that Clayton uses 100 milliliters of the $20 \\%$ by mass solution, so $y=100$. Substituting 100 for $y$ in the given equation yields $0.10 x+0.20(100)=0.18(x+100)$, which can be rewritten as $0.10 x+20=0.18 x+18$. Subtracting $0.10 x$ and 18 from both sides of the equat... | |
The first year Eleanor organized a fund-raising event, she invited 30 people. For each of the next 5 years, she invited double the number of people she had invited the previous year. If $f(n)$ is the number of people invited to the fund-raiser $n$ years after Eleanor began organizing the event, which of the following s... | [
"(A)The function $f$ is a decreasing linear function.",
"(B)The function $f$ is an increasing linear function.",
"(C)The function $f$ is a decreasing exponential function.",
"(D)The function $f$ is an increasing exponential function."
] | D | {
"solution": "Choice D is correct. It's given that the number of people Eleanor invited the first year was 30 and that the number of people invited doubles each of the following years, which is the same as increasing by a constant factor of 2 . Therefore, the function $f$ can be defined by $f(n)=30(2)^{n}$, where $n... | |
\begin{center}\begin{tabular}{|c|c|c|c|}\hline$x$ & $a$ & $3 a$ & $5 a$ \\\hline$y$ & 0 & $-a$ & $-2 a$ \\\hline\end{tabular}\end{center}Some values of $x$ and their corresponding values of $y$ are shown in the table above, where $a$ is a constant. If there is a linear relationship between $x$ and $y$, which of the fol... | [
"(A)$x+2 y=a$",
"(B)$x+2 y=5 a$",
"(C)$2 x-y=-5 a$",
"(D)$2 x-y=7 a$"
] | A | {
"solution": "Choice A is correct. The slope-intercept form of a linear equation in the $x y$-plane is $y=m x+b$, where $m$ is the slope of the graph of the equation and $b$ is the $y$-coordinate of the $y$-intercept of the graph. Any two ordered pairs $\\left(x_{1}, y_{1}\\right)$ and $\\left(x_{2}, y_{2}\\right)$ ... | |
$$\begin{aligned}& 2.4 x-1.5 y=0.3 \\& 1.6 x+0.5 y=-1.3\end{aligned}$$The system of equations above is graphed in the $x y$-plane. What is the $x$-coordinate of the intersection point $(x, y)$ of the system? | [
"(A)-0.5",
"(B)-0.25",
"(C)0.8",
"(D)1.75"
] | A | {
"solution": "Choice A is correct. The intersection point $(x, y)$ of the two graphs can be found by multiplying the second equation in the system $1.6 x+0.5 y=-1.3$ by 3 , which gives $4.8 x+1.5 y=-3.9$. The $y$-terms in the equation $4.8 x+1.5 y=-3.9$ and the first equation in the system $2.4 x-1.5 y=0.3$ have coe... | |
Keith modeled the growth over several hundred years of a tree population by estimating the number of the trees' pollen grains per square centimeter that were deposited each year within layers of a lake's sediment. He estimated there were 310 pollen grains per square centimeter the first year the grains were deposited, ... | [
"(A)$P(t)=310^{t}$",
"(B)$P(t)=310^{1.01 t}$",
"(C)$P(t)=310(0.99)^{t}$",
"(D)$P(t)=310(1.01)^{t}$"
] | D | {
"solution": "Choice D is correct. A model for a quantity that increases by $r \\%$ per time period is an exponential function of the form $P(t)=I\\left(1+\\frac{r}{100}\\right)^{t}$, where $I$ is the initial value at time $t=0$ and each increase of $t$ by 1 represents 1 time period. It's given that $P(t)$ is the nu... | |
$$\frac{2}{3}(9 x-6)-4=9 x-6$$Based on the equation above, what is the value of $3 x-2$ ? | [
"(A)-4",
"(B)$-\\frac{4}{5}$",
"(C)$-\\frac{2}{3}$",
"(D)4"
] | A | {
"solution": "Choice A is correct. Subtracting $\\left(\\frac{2}{3}\\right)(9 x-6)$ from both sides of the given equation yields $-4=\\left(\\frac{1}{3}\\right)(9 x-6)$, which can be rewritten as $-4=3 x-2$. Choices $B$ and $D$ are incorrect and may result from errors made when manipulating the equation. Choice $\\m... | |
$$H=1.88 L+32.01$$The formula above can be used to approximate the height $H$, in inches, of an adult male based on the length $L$, in inches, of his femur. What is the meaning of 1.88 in this context? | [
"(A)The approximate femur length, in inches, for a man with a height of 32.01 inches",
"(B)The approximate increase in a man's femur length, in inches, for each increase of 32.01 inches in his height",
"(C)The approximate increase in a man's femur length, in inches, for each one-inch increase in his height",
... | D | {
"solution": "Choice D is correct. It's given that $L$ is the femur length, in inches, and $H$ is the height, in inches, of an adult male. Because $L$ is multiplied by 1.88 in the equation, for every increase in $L$ by 1 , the value of $H$ increases by 1.88. Therefore, the meaning of 1.88 in this context is that a m... | |
A painter will paint $n$ walls with the same size and shape in a building using a specific brand of paint. The painter's fee can be calculated by the expression $n K \ell h$, where $n$ is the number of walls, $K$ is a constant with units of dollars per square foot, $\ell$ is the length of each wall in feet, and $h$ is ... | [
"(A)$h$",
"(B)$\\ell$",
"(C)$K$",
"(D)$n$"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. The painter's fee is given by $n K \\ell h$, where $n$ is the number of walls, $K$ is a constant with units of dollars per square foot, $\\ell$ is the length of each wall in feet, and $h$ is the height of each wall in feet. Examining this equation shows that $\\ell$ and... | |
If $3 r=18$, what is the value of $6 r+3$ ? | [
"(A)6",
"(B)27",
"(C)36",
"(D)39"
] | D | {
"solution": "Choice D is correct. Dividing each side of the equation $3 r=18$ by 3 gives $r=6$. Substituting 6 for $r$ in the expression $6 r+3$ gives $6(6)+3=39$.Alternatively, the expression $6 r+3$ can be rewritten as $2(3 r)+3$. Substituting 18 for $3 r$ in the expression $2(3 r)+3$ yields $2(18)+3$, or $36+3=3... | |
Which of the following is equal to $a^{\frac{2}{3}}$, for all values of $a$ ? | [
"(A)$\\sqrt{a^{\\frac{1}{3}}}$",
"(B)$\\sqrt{a^{3}}$",
"(C)$\\sqrt[3]{a^{\\frac{1}{2}}}$",
"(D)$\\sqrt[3]{a^{2}}$"
] | D | {
"solution": "Choice $\\mathbf{D}$ is correct. By definition, $a^{\\frac{m}{n}}=\\sqrt[n]{a^{m}}$ for any positive integers $m$ and $n$. It follows, therefore, that $a^{\\frac{2}{3}}=\\sqrt[3]{a^{2}}$.Choice $A$ is incorrect. By definition, $a^{\\frac{1}{n}}=\\sqrt[n]{a}$ for any positive integer $n$. Applying this ... | |
The number of states that joined the United States between 1776 and 1849 is twice the number of states that joined between 1850 and 1900. If 30 states joined the United States between 1776 and 1849 and $x$ states joined between 1850 and 1900 , which of the following equations is true? | [
"(A)$30 x=2$",
"(B)$2 x=30$",
"(C)$\\frac{x}{2}=30$",
"(D)$x+30=2$"
] | B | {
"solution": "Choice B is correct. To fit the scenario described, 30 must be twice as large as $x$. This can be written as $2 x=30$.Choices A, C, and D are incorrect. These equations do not correctly relate the numbers and variables described in the stem. For example, the expression in choice $\\mathrm{C}$ states th... | |
If $\frac{5}{x}=\frac{15}{x+20}$, what is the value of $\frac{x}{5} ?$ | [
"(A)10",
"(B)5",
"(C)2",
"(D)$\\frac{1}{2}$"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. Multiplying each side of $\\frac{5}{x}=\\frac{15}{x+20}$ by $x(x+20)$ gives $5(x+20)=15 x$. Using the distributive property to eliminate the parentheses yields $5 x+100=15 x$, and then subtracting $5 x$ from each side of the equation $5 x+100=15 x$ gives $100=10 x$. Fin... | |
$$\begin{aligned}& 2 x-3 y=-14 \\& 3 x-2 y=-6\end{aligned}$$If $(x, y)$ is a solution to the system of equations above, what is the value of $x-y$ ? | [
"(A)-20",
"(B)$\\quad-8$",
"(C)-4",
"(D)8"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. Multiplying each side of the equation $2 x-3 y=-14$ by 3 gives $6 x-9 y=-42$. Multiplying each side of the equation $3 x-2 y=-6$ by 2 gives $6 x-4 y=-12$. Then, subtracting the sides of $6 x-4 y=-12$ from the corresponding sides of $6 x-9 y=-42$ gives $-5 y=-30$. Dividi... | |
\begin{center}\begin{tabular}{|c|c|}\hline$x$ & $f(x)$ \\\hline0 & 3 \\\hline2 & 1 \\\hline4 & 0 \\\hline5 & -2 \\\hline\end{tabular}\end{center}The function $f$ is defined by a polynomial. Some values of $x$ and $f(x)$ are shown in the table above. Which of the following must be a factor of $f(x)$ ? | [
"(A)$x-2$",
"(B)$x-3$",
"(C)$x-4$",
"(D)$x-5$"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. If $x-b$ is a factor of $f(x)$, then $f(b)$ must equal 0 . Based on the table, $f(4)=0$. Therefore, $x-4$ must be a factor of $f(x)$.Choice A is incorrect because $f(2) \\neq 0$. Choice B is incorrect because no information is given about the value of $f(3)$, so $x-3$ m... | |
The line $y=k x+4$, where $k$ is a constant, is graphed in the $x y$-plane. If the line contains the point $(c, d)$, where $c \neq 0$ and $d \neq 0$, what is the slope of the line in terms of $c$ and $d$ ? | [
"(A)$\\frac{d-4}{c}$",
"(B)$\\frac{c-4}{d}$",
"(C)$\\frac{4-d}{c}$",
"(D)$\\frac{4-c}{d}$"
] | A | {
"solution": "Choice A is correct. The linear equation $y=k x+4$ is in slope-intercept form, and so the slope of the line is $k$. Since the line contains the point $(c, d)$, the coordinates of this point satisfy the equation $y=k x+4$; therefore, $d=k c+4$. Solving this equation for the slope, $k$, gives $k=\\frac{d... | |
$$\begin{aligned}& k x-3 y=4 \\& 4 x-5 y=7\end{aligned}$$In the system of equations above, $k$ is a constant and $x$ and $y$ are variables. For what value of $k$ will the system of equations have no solution? | [
"(A)$\\frac{12}{5}$",
"(B)$\\frac{16}{7}$",
"(C)$-\\frac{16}{7}$",
"(D)$-\\frac{12}{5}$"
] | A | {
"solution": "Choice $\\mathbf{A}$ is correct. If a system of two linear equations has no solution, then the lines represented by the equations in the coordinate plane are parallel. The equation $k x-3 y=4$ can be rewritten as $y=\\frac{k}{3} x-\\frac{4}{3}$, where $\\frac{k}{3}$ is the slope of the line, and the eq... | |
In the $x y$-plane, the parabola with equation $y=(x-11)^{2}$ intersects the line with equation $y=25$ at two points, $A$ and $B$. What is the length of $\overline{A B}$ ? | [
"(A)10",
"(B)12",
"(C)14",
"(D)16"
] | A | {
"solution": "Choice A is correct. Substituting 25 for $y$ in the equation $y=(x-11)^{2}$ gives $25=(x-11)^{2}$. It follows that $x-11=5$ or $x-11=-5$, so the $x$-coordinates of the two points of intersection are $x=16$ and $x=6$, respectively. Since both points of intersection have a $y$-coordinate of 25 , it follo... | |
$$y=a(x-2)(x+4)$$In the quadratic equation above, $a$ is a nonzero constant. The graph of the equation in the $x y$-plane is a parabola with vertex $(c, d)$. Which of the following is equal to $d$ ? | [
"(A)$-9 a$",
"(B)$-8 a$",
"(C)$-5 a$",
"(D)$-2 a$"
] | A | {
"solution": "Choice A is correct. The parabola with equation $y=a(x-2)(x+4)$ crosses the $x$-axis at the points $(-4,0)$ and $(2,0)$. By symmetry, the $x$-coordinate of the vertex of the parabola is halfway between the $x$-coordinates of $(-4,0)$ and $(2,0)$. Thus, the $x$-coordinate of the vertex is $\\frac{-4+2}{... | |
The equation $\frac{24 x^{2}+25 x-47}{a x-2}=-8 x-3-\frac{53}{a x-2}$ is true for all values of $x \neq \frac{2}{a}$, where $a$ is a constant.What is the value of $a$ ? | [
"(A)-16",
"(B)-3",
"(C)3",
"(D)16"
] | B | {
"solution": "Choice B is correct. Since $24 x^{2}+25 x-47$ divided by $a x-2$ is equal to $-8 x-3$ with remainder -53 , it is true that $(-8 x-3)(a x-2)-53=$ $24 x^{2}+25 x-47$. (This can be seen by multiplying each side of the given equation by $a x-2)$. This can be rewritten as $-8 a x^{2}+16 x-3 a x+6-53=$ $24 x... | |
What are the solutions to $3 x^{2}+12 x+6=0 ?$ | [
"(A)$x=-2 \\pm \\sqrt{2}$",
"(B)$x=-2 \\pm \\frac{\\sqrt{30}}{3}$",
"(C)$x=-6 \\pm \\sqrt{2}$",
"(D)$x=-6 \\pm 6 \\sqrt{2}$"
] | A | {
"solution": "Choice A is correct. Dividing each side of the given equation by 3 gives the equivalent equation $x^{2}+4 x+2=0$. Then using the quadratic formula, $\\frac{-b \\pm \\sqrt{b^{2}-4 a c}}{2 a}$ with $a=1, b=4$, and $c=2$, gives the solutions $x=-2 \\pm \\sqrt{2}$.Choices B, C, and D are incorrect and may ... | |
$$C=\frac{5}{9}(F-32)$$The equation above shows how a temperature $F$, measured in degrees Fahrenheit, relates to a temperature $C$, measured in degrees Celsius. Based on the equation, which of the following must be true?I. A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of $\frac{... | [
"(A)I only",
"(B)II only",
"(C)III only",
"(D)I and II only"
] | D | {
"solution": "Choice $\\mathbf{D}$ is correct. If $C$ is graphed against $F$, the slope of the line is equal to $\\frac{5}{9}$ degrees Celsius/degrees Fahrenheit, which means that for an increase of 1 degree Fahrenheit, the increase is $\\frac{5}{9}$ of 1 degree Celsius. Thus, statement I is true. This is the equiva... | |
\begin{center}\begin{tabular}{|l||c|c||c|}\cline { 2 - 4 }\multicolumn{1}{c||}{} & \multicolumn{2}{c||}{Age} & \multirow{2}{c|}{Total} \\\hlineGender & Under 40 & 40 or older \\\hline\hlineMale & 12 & 2 & 14 \\\hlineFemale & 8 & 3 & 11 \\\hline\hlineTotal & 20 & 5 & 25 \\\hline\end{tabular}\end{center}The table above s... | [
"(A)$\\frac{4}{25}$",
"(B)$\\frac{10}{25}$",
"(C)$\\frac{11}{25}$",
"(D)$\\frac{16}{25}$"
] | B | {
"solution": "Choice B is correct. Of the 25 people who entered the contest, there are 8 females under age 40 and 2 males age 40 or older. Because there is no overlap in the categories, the probability that the contest winner will be either a female under age 40 or a male age 40 or older is $\\frac{8}{25}+\\frac{2}{... | |
\begin{center}\begin{tabular}{|c||c|c|c|c|}\hline$n$ & 1 & 2 & 3 & 4 \\\hline$f(n)$ & -2 & 1 & 4 & 7 \\\hline\end{tabular}\end{center}The table above shows some values of the linear function $f$. Which of the following defines $f$ ? | [
"(A)$f(n)=n-3$",
"(B)$f(n)=2 n-4$",
"(C)$f(n)=3 n-5$",
"(D)$f(n)=4 n-6$"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. The graph of $y=f(n)$ in the coordinate plane is a line that passes through each of the points given in the table. From the table, one can see that an increase of 1 unit in $n$ results in an increase of 3 units in $f(n)$; for example, $f 2)-f(1)=1-(-2)=3$. Therefore, th... | |
At Lincoln High School, approximately 7 percent of enrolled juniors and 5 percent of enrolled seniors were inducted into the National Honor Society last year. If there were 562 juniors and 602 seniors enrolled at Lincoln High School last year, which of the following is closest to the total number of juniors and seniors... | [
"(A)140",
"(B)69",
"(C)39",
"(D)30"
] | B | {
"solution": "Choice B is correct. Since 7 percent of the 562 juniors is 0.07(562) and 5 percent of the 602 seniors is $0.05(602)$, the expression $0.07(562)+0.05(602)$ can be evaluated to determine the total number of juniors and seniors inducted into the National Honor Society. Of the given choices, 69 is closest ... | |
$$\begin{aligned}& 3 x^{2}-5 x+2 \\& 5 x^{2}-2 x-6\end{aligned}$$Which of the following is the sum of the two polynomials shown above? | [
"(A)$8 x^{2}-7 x-4$",
"(B)$8 x^{2}+7 x-4$",
"(C)$8 x^{4}-7 x^{2}-4$",
"(D)$8 x^{4}+7 x^{2}-4$"
] | A | {
"solution": "Choice A is correct. The sum of the two polynomials is $\\left(3 x^{2}-5 x+2\\right)+\\left(5 x^{2}-2 x-6\\right)$. This can be rewritten by combining like terms:$$\\left(3 x^{2}-5 x+2\\right)+\\left(5 x^{2}-2 x-6=\\left(3 x^{2}+5 x^{2}\\right)+-5 x-2 x\\right)+(2-6)=8 x^{2}-7 x-4$$Choice $B$ is incorr... | |
If $\frac{3}{5} w=\frac{4}{3}$, what is the value of $w ?$ | [
"(A)$\\frac{9}{20}$",
"(B)$\\frac{4}{5}$",
"(C)$\\frac{5}{4}$",
"(D)$\\frac{20}{9}$"
] | D | {
"solution": "Choice D is correct. To solve the equation for $w$, multiply both sides of the equation by the reciprocal of $\\frac{3}{5}$, which is $\\frac{5}{3}$. This gives $\\left(\\frac{5}{3}\\right) \\cdot \\frac{3}{5} w=\\frac{4}{3} \\cdot\\left(\\frac{5}{3}\\right)$, which simplifies to $w=\\frac{20}{9}$.Choi... | |
The average number of students per classroom at Central High School from 2000 to 2010 can be modeled by the equation $y=0.56 x+27.2$, where $x$ represents the number of years since 2000 , and $y$ represents the average number of students per classroom. Which of the following best describes the meaning of the number 0.5... | [
"(A)The total number of students at the school in 2000",
"(B)The average number of students per classroom in 2000",
"(C)The estimated increase in the average number of students per classroom each year",
"(D)The estimated difference between the average number of students per classroom in 2010 and in 2000"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. In the equation $y=0.56 x+27.2$, the value of $x$ increases by 1 for each year that passes. Each time $x$ increases by $1, y$ increases by 0.56 since 0.56 is the slope of the graph of this equation. Since $y$ represents the average number of students per classroom in th... | |
Nate walks 25 meters in 13.7 seconds. If he walks at this same rate, which of the following is closest to the distance he will walk in 4 minutes? | [
"(A)150 meters",
"(B)450 meters",
"(C)700 meters",
"(D)1,400 meters"
] | B | {
"solution": "Choice B is correct. Because Nate walks 25 meters in 13.7 seconds, and 4 minutes is equal to 240 seconds, the proportion $\\frac{25 \\text { meters }}{13.7 \\mathrm{sec}}=\\frac{x \\text { meters }}{240 \\mathrm{sec}}$ can be used to find out how many meters, $x$, Nate walks in 4 minutes. The proportio... | |
\begin{center}.\begin{tabular}{|l|c|}.\hline.\multicolumn{1}{|c|}{Planet} & Acceleration due to gravity $\left(\frac{\mathrm{m}}{\mathrm{sec}^{2}}\right)$ \\.\hline\hline.Mercury & 3.6 \\.\hline.Venus & 8.9 \\.\hline.Earth & 9.8 \\.\hline.Mars & 3.8 \\.\hline.Jupiter & 26.0 \\.\hline.Saturn & 11.1 \\.\hline.Uranus & 10... | What is the weight, in newtons, of an object on Mercury with a mass of 90 kilograms? | [
"(A)25",
"(B)86",
"(C)101",
"(D)324"
] | D | {
"solution": "Choice D is correct. On Mercury, the acceleration due to gravity is $3.6 \\mathrm{~m} / \\mathrm{sec}^{2}$. Substituting 3.6 for $g$ and 90 for $m$ in the formula $W=m g$ gives $W=90(3.6)=324$ newtons.Choice A is incorrect and may be the result of dividing 90 by 3.6. Choice $B$ is incorrect and may be ... |
\begin{center}.\begin{tabular}{|l|c|}.\hline.\multicolumn{1}{|c|}{Planet} & Acceleration due to gravity $\left(\frac{\mathrm{m}}{\mathrm{sec}^{2}}\right)$ \\.\hline\hline.Mercury & 3.6 \\.\hline.Venus & 8.9 \\.\hline.Earth & 9.8 \\.\hline.Mars & 3.8 \\.\hline.Jupiter & 26.0 \\.\hline.Saturn & 11.1 \\.\hline.Uranus & 10... | An object on Earth has a weight of 150 newtons. On which planet would the same object have an approximate weight of 170 newtons? | [
"(A)Venus",
"(B)Saturn",
"(C)Uranus",
"(D)Neptune"
] | B | {
"solution": "Choice B is correct. On Earth, the acceleration due to gravity is $9.8 \\mathrm{~m} / \\mathrm{sec}^{2}$. Thus, for an object with a weight of 150 newtons, the formula $W=m g$ becomes $150=m(9.8)$, which shows that the mass of an object with a weight of 150 newtons on Earth is about 15.3 kilograms. Sub... |
$$h=-16 t^{2}+v t+k$$The equation above gives the height $h$, in feet, of a ball $t$ seconds after it is thrown straight up with an initial speed of $v$ feet per second from a height of $k$ feet. Which of the following gives $v$ in terms of $h, t$, and $k$ ? | [
"(A)$v=h+k-16 t$",
"(B)$v=\\frac{h-k+16}{t}$",
"(C)$v=\\frac{h+k}{t}-16 t$",
"(D)$v=\\frac{h-k}{t}+16 t$"
] | D | {
"solution": "Choice D is correct. Starting with the original equation, $h=-16 t^{2}+v t+k$, in order to get $v$ in terms of the other variables, $-16 t^{2}$ and $k$ need to be subtracted from each side. This yields $v t=h+16 t^{2}-k$, which when divided by $t$ will give $v$ in terms of the other variables. However,... | |
In order to determine if treatment $\mathrm{X}$ is successful in improving eyesight, a research study was conducted. From a large population of people with poor eyesight, 300 participants were selected at random. Half of the participants were randomly assigned to receive treatment $X$, and the other half did not receiv... | [
"(A)Treatment $\\mathrm{X}$ is likely to improve the eyesight of people who have poor eyesight.",
"(B)Treatment $\\mathrm{X}$ improves eyesight better than all other available treatments.",
"(C)Treatment $X$ will improve the eyesight of anyone who takes it.",
"(D)Treatment $\\mathrm{X}$ will cause a substanti... | A | {
"solution": "Choice $\\mathbf{A}$ is the correct answer. Experimental research is a method used to study a small group of people and generalize the results to a larger population. However, in order to make a generalization involving cause and effect:\\begin{itemize} \\item The population must be well defined. \\i... | |
$$.\begin{aligned}.& S(P)=\frac{1}{2} P+40 \\.& D(P)=220-P.\end{aligned}.$$.The quantity of a product supplied and the quantity of the product demanded in an economic market are functions of the price of the product. The functions above are the estimated supply and demand functions for a certain product. The function $... | How will the quantity of the product supplied to the market change if the price of the product is increased by $\$ 10$ ? | [
"(A)The quantity supplied will decrease by 5 units.",
"(B)The quantity supplied will increase by 5 units.",
"(C)The quantity supplied will increase by 10 units.",
"(D)The quantity supplied will increase by 50 units."
] | B | {
"solution": "Choice $\\mathbf{B}$ is correct. The quantity of the product supplied to the market is given by the function $S(P)=\\frac{1}{2} P+40$. If the price $P$ of the product increases by $\\$ 10$, the effect on the quantity of the product supplied can be determined by substituting $P+10$ for $P$ in the functi... |
$$.\begin{aligned}.& S(P)=\frac{1}{2} P+40 \\.& D(P)=220-P.\end{aligned}.$$.The quantity of a product supplied and the quantity of the product demanded in an economic market are functions of the price of the product. The functions above are the estimated supply and demand functions for a certain product. The function $... | At what price will the quantity of the product supplied to the market equal the quantity of the product demanded by the market? | [
"(A)$\\$ 90$",
"(B)$\\$ 120$",
"(C)$\\$ 133$",
"(D)$\\$ 155$"
] | B | {
"solution": "Choice B is correct. The quantity of the product supplied to the market will equal the quantity of the product demanded by the market if $S(P)$ is equal to $D(P)$, that is, if $\\frac{1}{2} P+40=220-P$. Solving this equation gives $P=120$, and so $\\$ 120$ is the price at which the quantity of the prod... |
$$.\begin{aligned}.& S(P)=\frac{1}{2} P+40 \\.& D(P)=220-P.\end{aligned}.$$.The quantity of a product supplied and the quantity of the product demanded in an economic market are functions of the price of the product. The functions above are the estimated supply and demand functions for a certain product. The function $... | Graphene, which is used in the manufacture of integrated circuits, is so thin that a sheet weighing one ounce can cover up to 7 football fields. If a football field has an area of approximately $1 \frac{1}{3}$ acres, about how many acres could 48 ounces of graphene cover? | [
"(A)250",
"(B)350",
"(C)450",
"(D)1,350"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. It is given that 1 ounce of graphene covers 7 football fields. Therefore, 48 ounces can cover $7 \\times 48=336$ football fields. If each football field has an area of $1 \\frac{1}{3}$ acres, then 336 football fields have a total area of $336 \\times 1 \\frac{1}{3}=448$... |
Of the following four types of savings account plans, which option would yield exponential growth of the money in the account? | [
"(A)Each successive year, $2 \\%$ of the initial savings is added to the value of the account.",
"(B)Each successive year, $1.5 \\%$ of the initial savings and $\\$ 100$ is added to the value of the account.",
"(C)Each successive year, $1 \\%$ of the current value is added to the value of the account.",
"(D)E... | C | {
"solution": "Choice $\\mathbf{C}$ is correct. Linear growth is characterized by an increase of a quantity at a constant rate. Exponential growth is characterized by an increase of a quantity at a relative rate; that is, an increase by the same factor over equal increments of time. In choice $C$, the value of the ac... | |
The sum of three numbers is 855 . One of the numbers, $x$, is $50 \%$ more than the sum of the other two numbers. What is the value of $x$ ? | [
"(A)570",
"(B)513",
"(C)214",
"(D)155"
] | B | {
"solution": "Choice $B$ is correct. One of the three numbers is $x$; let the other two numbers be $y$ and $z$. Since the sum of three numbers is 855 , the equation $x+y+z=855$ is true. The statement that $x$ is $50 \\%$ more than the sum of the other two numbers can be represented as $x=1.5(y+z)$, or $x=\\frac{3}{2... | |
Mr. Kohl has a beaker containing $n$ milliliters of solution to distribute to the students in his chemistry class. If he gives each student 3 milliliters of solution, he will have 5 milliliters left over. In order to give each student 4 milliliters of solution, he will need an additional 21 milliliters. How many studen... | [
"(A)16",
"(B)21",
"(C)23",
"(D)26"
] | D | {
"solution": "Choice D is correct. Let $c$ be the number of students in Mr. Kohl's class. The conditions described in the question can be represented by the equations $n=3 c+5$ and $n+21=4 c$. Substituting $3 c+5$ for $n$ in the second equation gives $3 c+5+21=4 c$, which can be solved to find $c=26$.Choices A, B, a... | |
In the $x y$-plane, the line determined by the points $(2, k)$ and $(k, 32)$ passes through the origin. Which of the following could be the value of $k$ ? | [
"(A)0",
"(B)4",
"(C)8",
"(D)16"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. The line passes through the origin, (2, $k)$, and $(k, 32)$. Any two of these points can be used to find the slope of the line. Since the line passes through $(0,0)$ and $(2, k)$, the slope of the line is equal to $\\frac{k-0}{2-0}=\\frac{k}{2}$. Similarly, since the li... | |
A rectangle was altered by increasing its length by 10 percent and decreasing its width by $p$ percent. If these alterations decreased the area of the rectangle by 12 percent, what is the value of $p$ ? | [
"(A)12",
"(B)15",
"(C)20",
"(D)22"
] | C | {
"solution": "Choice $\\mathbf{C}$ is correct. Let $\\ell$ and $w$ be the length and width, respectively, of the original rectangle. The area of the original rectangle is $A=\\ell w$. The rectangle is altered by increasing its length by 10 percent and decreasing its width by $p$ percent; thus, the length of the alte... | |
In planning maintenance for a city's infrastructure, a civil engineer estimates that, starting from the present, the population of the city will decrease by 10 percent every 20 years. If the present population of the city is 50,000, which of the following expressions represents the engineer's estimate of the population... | [
"(A)$50,000(0.1)^{20 t}$",
"(B)$50,000(0.1)^{\\frac{t}{20}}$",
"(C)$50,000(0.9)^{20 t}$",
"(D)$50,000(0.9)^{\\frac{t}{20}}$"
] | D | {
"solution": "Choice D is correct. For the present population to decrease by 10 percent, it must be multiplied by the factor 0.9. Since the engineer estimates that the population will decrease by 10 percent every 20 years, the present population, 50,000, must be multiplied by $(0.9)^{n}$, where $n$ is the number of ... | |
\begin{center}\begin{tabular}{|l|c|c|}\cline { 2 - 3 }\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{Handedness} \\\hlineGender & Left & Right \\\hline\hlineFemale & & \\\hlineMale & & \\\hline\hlineTotal & 18 & 122 \\\hline\end{tabular}\end{center}The incomplete table above summarizes the number of left-handed studen... | [
"(A)0.410",
"(B)0.357",
"(C)0.333",
"(D)0.250"
] | A | {
"solution": "Choice $\\mathbf{A}$ is correct. Let $x$ be the number of left-handed female students and let $y$ be the number of left-handed male students. Then the number of right-handed female students will be $5 x$ and the number of right-handed male students will be $9 y$. Since the total number of lefthanded st... | |
$$\begin{aligned}& 3 x+b=5 x-7 \\& 3 y+c=5 y-7\end{aligned}$$In the equations above, $b$ and $c$ are constants.If $b$ is $c$ minus $\frac{1}{2}$, which of the following is true? | [
"(A)$x$ is $y$ minus $\\frac{1}{4}$.",
"(B)$x$ is $y$ minus $\\frac{1}{2}$.",
"(C)$x$ is $y$ minus 1 .",
"(D)$x$ is $y$ plus $\\frac{1}{2}$."
] | A | {
"solution": "Choice A is correct. Subtracting the sides of $3 y+c=5 y-7$from the corresponding sides of $3 x+b=5 x-7$ gives$(3 x-3 y)+(b-c)=\\left(5 x-5 y+(-7-(-7))\\right.$. Since $b=c-\\frac{1}{2}$, or $b-c=-\\frac{1}{2}$,it follows that $(3 x-3 y)+\\left(-\\frac{1}{2}\\right)=(5 x-5 y)$. Solving this equation fo... |
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