ACT Maths

If the volume of a cube is 64, what is the shortest distance from the center of the cube to the base of the cube? A. 2 B. 4 C. 2√4 D. √32 E. 16

A. 2

It is estimated that, from the beginning of 1993 to the end of 1997, the average number of CDs bought by teenagers increased from 7 per year to 15 per year. During the same time period, the average number of video games purchased by teenagers increased from 6 per year to 18 per year. Assuming that in each case the rates or purchase are the same, in what year did teenagers buy the same average number of CDs and video games? A. 1993 B. 1994 C. 1995 D. 1996 E. 1997

The correct answer is B. The best way to solve this problem is to set up a table indicating the time period in years and the number of both CDs and video games purchased during the years given. The rate of purchase is the same, so, based on information in the problem, you can fill in the table below. Time CDs Video games 1993 7 6 1994 9 9 1995 11 12 1996 13 15 1997 15 18 Teenagers bought the same average number of CDs and video games in 1994.

If 5 times a number x is subtracted from 15, the result is negative. Which of the following gives the possible value(s) for x? A. All x < 3 B. All x > 3 C. 10 only D. 3 only E. 0 only

The correct answer is B. To solve this problem, first convert the information in the question into its mathematical equivalent, as follows: 5 times a number x = 5x Subtracting 5x from 15 yields a negative result, so 15 − 5x < 0. Now, solve the inequality for x: 15 − 5x < 0 − 5x < −15 Divide both sides by −5 and reverse the inequality to get x > 3.

If a board 9 feet 6 inches in length is cut into 2 equal parts, what will be the length of each part? A. 3 feet 8 inches B. 4 feet 5 inches C. 4 feet 8 inches D. 4 feet 9 inches E. 5 feet 2 inches

The correct answer is D. To find half of 9 feet 6 inches, recall that there are 12 inches in a foot. Therefore, 9 feet 6 inches is equal to 9(12) + 6 = 114 inches. Half of 114 inches is 57 inches, which is equivalent to 4 feet 9 inches [4(12) + 9 = 57]. 1 foot =12 inches 1 meter =3.2808399 feet (3 feet 3⅜ inches)

The larger of two numbers exceeds twice the smaller number by 9. The sum of twice the larger and 5 times the smaller number is 74. If a is the smaller number, which equation below determines the correct value of a? A. 5(2a + 9) + 2a = 74 B. 5(2a − 9) + 2a = 74 C. (4a + 9) + 5a = 74 D. 2(2a + 9) + 5a = 74 E. 2(2a − 9) + 5a = 74

The correct answer is D. To solve this problem, first convert the information given into its mathematical equivalent, as follows (use b to represent the larger number): The larger of two numbers exceeds twice the smaller number by nine: b = 2a + 9 The sum of twice the larger and five times the smaller number is 74: 2b + 5a = 74 Now, simply substitute the value of b into the second equation in order to solve for a: 2(2a + 9) + 5a = 74

Rebecca is trying to schedule volunteers to help at a school carnival. There are 5 one-hour shifts to be filled by 5 different volunteers. If each shift must have one and only one volunteer, how many different arrangements can the schedule have? F. 5 G. 20 H. 25 J. 50 K. 120

The correct answer is K. Think of each shift as a spot to be filled: __×__×__×__×__. In each spot there are only a certain number of available workers. For example, the first spot can have any of the 5 volunteers working, leaving 4 possible workers for the next shift, and so on. Therefore, the number of possible arrangements of volunteers is 5 × 4 × 3 × 2 × 1 = 120.

isosceles

adj. (三角形)二等边的；等腰的

When the choir is arranged in rows of 5 people each, the last row is one person short. When the choir is arranged in rows of 6 people each, the last row is still one person short. What is the least possible number of people in the choir? A. 29 B. 30 C. 56 D. 60 E. 99

A. 29 题意：最小的，+1之后既可以被5整除，也可以被6整除的，正整数

Which of the following is the set of all real numbers x such that x − 3 < x − 5? A. The empty set B. The set containing only zero C. The set containing all nonnegative real numbers D. The set containing all negative real numbers E. The set containing all real numbers

A. The empty set

In the (x, y) coordinate plane, what is the y-intercept of the line −9x − 3y = 15? A. −9 B. −5 C. −3 D. 3 E. 15 intercept

B. −5 slope 斜率 intercept 截距

Let n equal 3a + 2b − 7. What happens to the value of n if the value of a increases by 2 and the value of b decreases by 1? A. It is unchanged. B. It decreases by 1. C. It increases by 4. D. It decreases by 4. E. It decreases by 2.

C. It increases by 4.

Which of the following numbers has the digit 5 in the thousandths place? A. 5,000.00 B. 50.0 C. 0.05 D. 0.005 E. 0.0005

D. 0.005 thousandths place, 小数点后第三位，注意系thousandth，而不是thousand

An oil refinery produces gasoline from crude oil. For every 10,000 barrels of crude oil supplied, the refinery can produce 6,500 barrels of gasoline. How many barrels of gasoline can be produced from 3,500 barrels of crude oil? A. 1,265 B. 1,750 C. 2,125 D. 2,275 E. 5,385

D. 2,275

The average of 7 consecutive numbers is 16. What is the sum of the least and greatest of the 7 integers? A. 13 B. 14 C. 16 D. 19 E. 32

E. 32 求和公式=（首项+末项)*项数/2 等差数列之和=平均数*个数 （首项+末项)*7/2=16*7 （首项+末项)=32

If mn = k and k = x2n, and nk not equal to 0, which of the following is equal to m? (2 is exponent) A. 1 B. 1/x C. √ x D. x E. x2

E. x2

The product of two integers is between 137 and 149. Which of the following CANNOT be one of the integers? F. 15 G. 13 H. 11 J. 10 K. 7

F. 15 用139和149分别除以每个选项

What is the y-coordinate of the point in the standard (x,y) coordinate plane at which the 2 lines y = x/2 + 3 and y = 3x − 2 intersect? F. 5 G. 4 H. 3 J. 2 K. 1

G. 4

(When written in symbols, "the product of r and s, raised to the fourth power," is represented as: (4 is exponent) F. r^4s^4 G. (r + s)4 H. (rs)4 J. r^4/s^4 K. rs^4

H. (rs)^4

For what value of n would the following system of equations have an infinite number of solutions? 3a + b = 12 12a + 4b = 3n F. 4 G. 9 H. 16 J. 36 K. 48

H. 16

Gary has turtles, cats, and birds for pets. The number of birds he has is 4 more than the number of turtles, and the number of cats is 2 times the number of birds. Of the following, which could be the total number of Gary's pets? F. 14 G. 18 H. 20 J. 22 K. 26

H. 20

What is the slope of a line that passes through the origin and the point (−6, 2) ? F. 3 G. 1/3 H. −1/3 J. −3 K. −6

H. −1/3

Mandy and Jordan each bought some of the same notebooks and the same three-ring binder. Mandy paid $5.85 for 3 notebooks and 1 binder. Jordan paid $4.65 for 2 notebooks and 1 binder. What is the price of one of the notebooks? F. $2.70 G. $2.25 H. $1.80 J. $1.20 K. $0.75

J. $1.20

Kate rode her bicycle to visit her grandmother. The trip to Kate's grandmother's house was mostly uphill, and took m minutes. On the way home, Kate rode mostly downhill and was able to travel at an average speed twice that of her trip to her grandmother's house. Which of the following expresses the total number of minutes that Kate bicycled on her entire trip? F. 3m G. 2m H. (m + 1)/2 J. (3m)/2 K. m/2

J. 3m/2

What is the slope of any line parallel to the y-axis in the (x, y) coordinate plane? F. −1 G. 0 H. 1 J. Undefined K. Cannot be determined from the given information

J. Undefined 平行于y轴的直线的倾斜角是90度,斜率是不存在.

What is the slope of a line that is perpendicular to the line determined by the equation 7x + 4y = 11? F. −4 G. −(7/4) H. 11/4 J. 4 K. 4/7

K. 4/7 妈蛋，睁开你的狗眼好好审题，"perpendicular to"，两直线垂直时,其中一条直线的倾斜角等于另一条直线倾斜角加九十度,由此可得两直线斜率的乘积为-1.

A partial deck of cards was found sitting out on a table. If the partial deck consists of 6 spades, 3 hearts, and 7 diamonds, what is the probability of randomly selecting a red card from this partial deck? (Note: diamonds and hearts are considered "red," while spades and clubs are considered "black.") F. 9/6 G. 13/16 H. 7/16 J. 3/8 K. 5/8

K. 5/8

Three years ago, the population of a certain species of bird was calculated at 20 birds per acre. This year, a biologist recorded a total of 47 birds in an area equal to 3.25 acres. By about what percentage has the bird population in the biologist's sample decreased over the last 3 years, to the nearest tenth? A. 14.7% B. 27.7% C. 38.3% D. 42.6% E. 72.3%

The correct answer is B. Numbers of birds this year/acre: 47÷3.25=14.46 Numbers of birds 3 years ago/acre: 20÷1=20 Numbers of birds decreased over 3 years: 20-14.46=5.54 Decreased percentage: 5.54÷20=0.277=27.7%

What is the solution set of |3a − 3| ≥ 12? A. a ≥ 5 and a ≤ −5 B. a ≥ 5 and a ≤ −3 C. a ≥ −5 and a ≤ 5 D. a ≥ −5 and a ≤ 3 E. a ≤ 5 and a ≥ −5

The correct answer is B. To solve this problem, remember that |3a − 3| ≥ 12 is equivalent to 3a − 3 ≥ 12 or 3a − 3 ≤ −12. Adding 3 to both sides and dividing by 3 yields a ≥ 5 or a ≤ −3.

The balance of Juan's savings account quadrupled during the year. At the end of the year, Juan withdrew $300, and the resulting balance was $400. What was the balance in the account before it quadrupled? A. $100 B. $175 C. $300 D. $350 E. $700

The correct answer is B. When Juan withdrew $300, the resulting balance was $400, making the balance of the account before withdrawal $400 + $300, or $700. Since the balance of Juan's savings account quadrupled during the year, the balance, b, in the account before it quadrupled is represented by 4b = 700. Therefore b = 700/4, or $175.

A classroom has 10 tables that will seat up to 4 students each. If 20 students are seated at tables, and NO tables are empty, what is the greatest possible number of tables that could be filled with students? A. 5 B. 3 C. 2 D. 1 E. 0

The correct answer is B. You are given that no table can be empty, which means that there must be at least one student seated at each table. There are 20 students in all and 10 tables, so if one student is seated at each table, you have accounted for 10 students. Now simply fill tables with the remaining 10 students until no students remain: Table 1 already has 1 student; add 3 more so that it is filled. You now have 7 more students to seat. Table 2 already has 1 student; add 3 more so that it is filled. You now have 3 more students to seat. Table 3 already has 1 student; add the 3 remaining students so that it is filled. You have accounted for all of the students and have filled 3 tables. You cannot fill more than 3 tables with students while leaving no tables empty.

In a certain music store, CDs were put on display and assigned prices for May. Each month after that, the price was 20% less than the price for the previous month. If the price of a CD was d dollars in May, what was the price in August? A. 0.2d B. 0.3d C. 0.512d D. 0.64d E. 0.8d

The correct answer is C. Between May and August, there were 3 price decreases (in June, July, and August). If the price was decreased by 20%, then the resulting price was 80% of the previous month's price. Thus, in June the price was 0.8d; in July the price was 0.8(0.8d); in August the price was 0.8(0.8(0.8d)), which is equivalent to (0.8)3d, or 0.512d.

Which of the following is the slope of a line parallel to the line y = (2/5)x + 7 in the standard (x, y) coordinate plane? A. −7 B. −5/2 C.2/5 D. 2 E.5/2

The correct answer is C. In the equation of a line, y = mx + b, m is the slope. Recall that parallel lines have the same slope. Therefore, because the slope of the given line is 2/5, the slope of any line parallel to that line will also have a slope of2/5.

For what value of b would the following system of equations have an infinite number of solutions? 3x + 5y = 27 12x + 20y = 3b A. 9 B. 27 C. 36 D. 81 E. 126

The correct answer is C. Systems of equations have an infinite number of solutions when the equations are equivalent (i.e. they graph the same lane). In order for the two equations to be equivalent, the constants and coefficients must be proportional. If the entire equation 3x + 5y = 27 is multiplied by 4, the result is 4(3x + 5y) = 4(27), or 12x + 20y = 108. Thus, in order for the two equations to be equivalent, 3b = 108, or b = 36.

What is the least common multiple of 40, 70, and 60? A. 240 B. 420 C. 840 D. 1,680 E. 168,000

The correct answer is C. To solve this problem, start by finding the least common multiple of 60 and 70, which is 420. However, 420 is not a multiple of 40. Next try 2 × 420, which is 840. The number 840 is still a multiple of 60 and 70 and is also a multiple of 40. Another possible strategy for this problem would be to start with the smallest answer choice (you are asked for the least common multiple) and stop when you find an answer choice that is evenly divisible by 40, 60, and 70.

Which of the following is (are) equivalent to the mathematical operation a(b − c) for all real numbers a, b, and c? I. ca − ba II. ab − ac III. (b − c)a A. II only B. I and II only C. I and III only D. II and III only E. I, II and III

The correct answer is D. This question tests your ability to recognize and apply the distributive property of multiplication. According to the distributive property, for any numbers a, b, and c, a(b − c) = ab − ac. a(b − c) = ab − ac, which is NOT equivalent to ca − ba, so Roman Numeral I is incorrect; eliminate answer choices B, C, and E. ab - ac, so Roman Numeral II is correct. (b − c)a, so Roman Numeral III is also correct. Since II and III are equivalent to a(b − c), answer choice D is correct.

If√(2x) + 5 = 9, then x = ? A. −4 B. 2 C. 4 D. 8 E. 16

The correct answer is D. To solve this problem, first subtract 5 from both sides of √(2x) + 5 = 9, to get √(2x) = 4. Squaring both sides of √(2x) = 4 yields 2x = 16, or x = 8.

If a is inversely proportional to b and a = 36 when b = 12, what is the value of a when b = 48? A. 0 B. 1 3 C. 1 4 D. 4 E. 9

The correct answer is E. By definition, if a and b are inversely proportional, then āb1 = áb2. Therefore, (36)(12) = (48)a. Solve for a as follows: (36)(12) = (48)a 432 = 48a 9 = a inversely proportional directly proportional The symbol used to denote the proportionality is '∝'. For example, if we say, a is proportional to b, then it is represented as 'a∝b' and if we say, a is inversely proportional to b, then it is denoted as 'a∝1/b'.

For all real integers, which of the following is always an even number? I. x^3 + 4 II. 2x + 4 III. 2x^2 + 4 A. I only B. II only C. III only D. I and II only E. II and III only

The correct answer is E. Integers can be even or odd, and positive or negative. Pick real numbers to substitute into the expressions in each Roman Numeral, then eliminate any Roman Numerals that do not always yield and odd number. Whenever a number is multiplied by 2, the result is always even. Thus, II and III are true; answer choice E is correct.

In a 3-dimensional (x, y, z) space, the set of all points 5 units from the x-axis is: A. a line. B. 2 parallel lines. C. a circle. D. a sphere. E. a cylinder.

The correct answer is E. The set of all points 5 units from the x-axis in 3-dimensional space can be likened to the set of all points 5 units from the x-axis in 2-dimensional space, which are two lines. Revolving such lines about the x-axis creates a tube-like shape that is a cylinder.

If the function f satisfies the equation f (x+y) = f (x)+ f (y) for every pair of real numbers x and y, what is (are) the possible value(s) of f (1)? F. Any real number G. Any positive real number H. 0 and 1 only J. 0 only K. 1 only

The correct answer is F. If the function f satisfies the equation f (x+y) = f (x)+f (y) for every pair of real numbers x and y, then f is a linear function. For some unknown linear function f , the value of f (1) can be any real number (think of all the possible lines that can be drawn).

For what nonzero whole number k does the quadratic equation y^2+2ky+4k = 0 have exactly one real solution for y? F. 8 G. 4 H. 2 J. −4 K. −8

The correct answer is G. To solve this problem, recall that a quadratic equation has exactly one real solution when the solution is a "double zero." Such an equation would have the factored form of (y + b)^2 = 0, where b is a real number. When you expand the equation, it becomes (y + b)^2 is y^2 + 2by + b^2. The equation y^2 + 2ky + 4k = 0 has a similar form; however, in order for the term 4k to represent b^2, k must equal 4.

One traffic light flashes every 6 seconds. Another traffic light flashes every 9 seconds. If they flash together and you begin counting seconds, how many seconds after they flash together will they next flash together? F. 6 G. 9 H. 18 J. 36 K. 54

The correct answer is H. To solve this problem, find the least common multiple of 6 and 9. The correct answer is 18 because 6 × 3 = 18 and 9 × 2 = 18.

If r and s are constants and x^2 + rx + 12 is equivalent to (x + 3)(x + s), what is the value of r? F. 3 G. 4 H. 7 J. 12 K. Cannot be determined from the given information

The correct answer is H. 方程式先化为相似形式 To solve this problem, multiply the expression (x + 3) by (x + s) to get x^2 +3x +sx +3s. You are given that x^2 +rx +12 is equivalent to x^2 + 3x + sx + 3s. Therefore, 3s is equal to 12, making s equal to 4. It is also apparent that 3x + sx is equivalent to rx. Set the quantities equal and solve for r, as follows: rx = 3x + sx rx = x(3 + s) r = 3 + s Because s = 4, r must equal 7.

The speed of a car exceeds twice the speed of a truck by 15 mph. If t is the speed of the truck, which of the following expresses the speed, in miles per hour, of the car? F. t + 15 G. t + 30 H. t − 30 J. 2t + 15 K. 2t + 30

The correct answer is J. If t is the speed of the truck, then twice the speed of the truck can be written as 2t. If the speed of the car exceeded twice the speed of the truck by 15 mph, the car's speed was 2t + 15.

A house painter charges $24.00 per hour for a painting job that requires more than 5 hours to complete. For any job requiring 5 hours or less, the house painter charges a flat fee of $100. If n represents the number of hours the job requires, which of the following expressions gives the charge, in dollars, for a job requiring more than 5 hours to complete? F. 124.0 G. −24n + 100 H. 24n − 100 J. 24n K. 24n + 100

The correct answer is J. Since the house painter charges $24.00 per hour for a painting job that requires more than 5 hours to complete and n represents the number of hours the job requires, the charge, in dollars, for a job requiring more than 5 hours to complete can be expressed as (24.00)(n), or 24n. This problem tested your ability to disregard irrelevant information.

For every positive 2-digit number, a, with units digit x and tens digit y, let b be the 2-digit number formed by reversing the digits of a. Which of the following expressions is equivalent to a − b? F. 0 G. 9x − y H. 9y − x J. 9(x − y) K. 9(y − x)

The correct answer is J. You are given that a is a number with units digit x and tens digit y. Therefore, x is equivalent to 10 times y, and a = xy = 10x+y. You are given that b is formed by reversing the digits of a. Therefore, b = yx = 10y + x. Set up an equation and solve for a − b as follows: a − b = (10x + y) − (10y + x) = 10x + y − 10y − x = 9x − 9y = 9(x − y)

Which of the following is a factor of the polynomial x^2 + 3x − 18? F. x − 6 G. x − 12 H. x − 18 J. x + 3 K. x + 6

The correct answer is K. To solve this problem, factor the polynomial x2+3x−18. To do so, think of x2+3x−18 as (x +?)(x −?). To fill in the question marks, find two numbers that multiply to equal −18 and add up to 3. One such pair of numbers is 6 and −3. To check, make sure that (x + 6)(x − 3) = x2+3x−18, which it does. Of the answer choices, only (x + 6) is a factor.

quadrupled

adj. 四倍的；四部分的；[音]四节拍的 n. 四倍 v. 成四倍

hypotenuse

n. (直角三角形的)斜边

even integer odd integer

偶整数 奇整数

coordinate

坐标

Complex number

复数，为实数的延伸，它使任一多项式方程都有根。