ACT Basics
For which of the following conditions will the sum of integers m and n always be a positive number?
A) m is an odd number
B) n is an odd number
C) m and n are both odd numbers
D) m and n are both even numbers
E) m is an odd number and n is an even number
What is E?
(Only an odd and an even number can add together to make an odd number. Two of the same number always adds to make an even number.)
If a 3 pound pizza is sliced in half, and each half is sliced into thirds, then what is the weight, in ounces, of each of the slices? (1 pound = 16 ounces)
A) 4
B) 6
C) 8
D) 16
What is C?
3 pounds equals 48 ounces. When splitting in half, you are dividing by 2, leaving 24 ounces. When splitting into thirds, you divide this by 3, leaving 8 ounces per slice.
The inequality 6(x+2) > 7(x-5) is equivalent to which of the following inequalities?
A) x < -23
B) x < 7
C) x < 17
D) x < 37
E) x < 47
What is E?
Distribute to get 6x+12 > 7x-35.
Subtract 6x: 12 > x-35.
Add 35: 47 > x.
Equivalently, x < 47.
f(x) = (3/2)x + b
In the function above, b is a constant. If f(6) = 7, what is the value of f(-2)?
A) -5
B) -2
C) 1
D) 7
What is A?
( f(6)=7 indicates that we can replace x with 6 and f(x) with 7. Then, we have an equation with only b as the unknown. Then we can solve for b.
7 = (3/2)6 + b
7 = 9+b
b = -2
Now that we know that b equals -2, we can replace b with -2. To find f(-2), plug -2 in for x.
(3/2)(-2) - 2 = -3 -2 = -5. )
Xiaoming is baking cookies. Each batch of cookies uses 3 eggs. If Xiaoming has 20 eggs, and assuming he has enough of the other ingredients to make the cookies, what is the greatest number of batches he can make?
A) 3
B) 6
C) 7
D) 17
What is B?
(18 eggs are needed to make 6 batches of cookies. We have more than that, but it takes 21 to make 7 batches, which is more than what we have.
20/3 = 6 remainder 2, or 6 + 2/3. Only wholes count when it comes to batches.)
Source: Next-Generation Accuplacer: Arithmetic, Question 9
Maria ordered a pizza. She ate only 2/9 of it and shared the rest with her brothers. If each brother ate equal parts of the remaining pizza, what fraction of the pizza did each brother eat?
A) 7/9
B) 3/7
C) 1/3
D) 7/27
E) 2/27
What is D?
(When 2/9 of the pizza is eaten, 7/9 remains. Each brother receives 1/3 of that 7/9.
1/3 * 7/9 = 7/27.)
At Lincoln High School, approximately 7 percent of juniors and 5 percent of seniors were inducted into the National Honor Society last year. If there were 562 juniors and 602 seniors enrolled at Lincoln High School that year, which of the following is the closest to the total number of juniors and seniors at Lincoln High School who were inducted into the National Honors Society?
A) 140
B) 69
C) 39
D) 30
What is B?
(7 percent of juniors means (7/100)*562 juniors, which is 39 people (rounded to nearest whole number)
5 percent of seniors means (5/100)*602 seniors, which is 30 people (rounded to nearest whole number)
Adding them together yields 69 inductees.)
(3x3)3 is equivalent to:
A) x
B) 9x6
C) 9x9
D) 27x6
E) 27x9
What is E?
(Three copies of 3x3 are multiplied together. Multiplying coefficients give 3*3*3=27, adding exponents gives 3+3+3=9.)
sqrt(x-a) = x-4
If a = 2, what is the solution set to the equation above?
NOTATION: sqrt(thing) means "the square root of thing"
A) {3, 6}
B) {2}
C) {3}
D) {6}
What is D?
First replace a with 2. Squaring both sides of the equation gives us
x-2 = (x-4)2, which can be rewritten as
x-2 = (x-4)(x-4)
x-2 = x2-8x+16. Set one side equal to 0.
x2-9x+18 = 0. Factoring the left side gives
(x-3)(x-6)=0, which implies that x=3 and x=6 are solutions. Double check to see that there are no extraneous solutions.
sqrt(3-2) = 3-4
sqrt(1) = -1
1=-1. Clearly false, so 3 is extraneous.
sqrt(6-2) = 6-4
sqrt(4) = 2
2=2. 6 is the only valid solution.
Which of the following is equivalent to 5-3?
A) 1/15
B) 1/125
C) -15
D) -125
What is B?
(According to the laws of exponents, x-n = 1/xn,
so this means that 5-3 = 1/53
= 1/125.)
Source: Next-Generation Accuplacer: Quantitative Reasoning, Algebra, and Statistics: Question 15
Which of the following is a rational number? (Notation: sqrt(x) means "The square root of x.")
A) sqrt(2)
B) sqrt(pi)
C) sqrt(7)
D) sqrt(5/25)
E) sqrt(49/64)
What is E?
(sqrt(49/64) can be rewritten as sqrt(49)/sqrt(64), which equal to 7/8, because
sqrt(x) = x1/2, and therefore follows the exponent law (a/b)n=an/bn.)
DAILY DOUBLE #1 -- Check the flipchart.
What is 32?
To add polynomials, you add the like terms together.
-2x2+3x2=x2
x+7x=8x
31-8=23.
The coefficients of each of these terms are added together to get the desired sum. Remember that a term without a number in front has a coefficient of 1. 1+8+23=32.
A group of cells grows in number as described by the equation y = 16(2)t, where t represents the number of days and y represents the number of cells. According to this formula, how many cells will be in the group at the end of the first 5 days?
A) 80
B) 160
C) 400
D) 512
E) 1280
What is D?
(After t days, there will be 16(2)t cells in the group.
After 5 days, there will be 16(2)5 cells in the group.
16(2)5 = 16*32 = 512.)
A group of 202 people went on an overnight camping trip, taking 60 tents with them. Some of the tents held 2 people each, others held 4 people each. Assuming every tent was filled to capacity and that every person got to sleep in a tent, exactly how many tents were 2-person tents?
A) 30
B) 20
C) 19
D) 18
What is C?
We need to set up a system of linear equations to determine how many of each kind of tent we have, and how many slept in each. Let x be the number of 2-person tents, and y be the number of 4-person tents.
2x + 4y = 202. This tells us the number of people on the trip: 2 per 2-person tent, 4 per 4-person tent.
x + y = 60. This tells us that the total number of tents is the sum of 2-person and 4-person tents.
Now we just need to solve the system.
2x+4y = 202.
2x+2y = 120.
Subtracting the bottom from the top yields 2y = 82, so y = 41. Since there were 60 tents total, that means that there were 60-41=19 2-person tents.
The polynomial equation x(x2+4)(x2-x-6) has how many real roots?
A) One only
B) Two only
C) Three only
D) Four only
E) Five
What is C?
(x has 1 real root (zero), x=0.
x2+4 has no real roots (zeros), because it is impossible for this factor to equal 0 if x is real.
x2-x-6 has 2 real roots (zeros), as you can clearly see if you factor it into (x-3)(x+2), x=3 and x=-2.
This gives us 3 real roots. Remember that every polynomial has complex roots equal to its degree (in this case, 5). This fact is called the Fundamental Theorem of Algebra. The question asks only for real roots, however.)
Source: University of Maryland Mathematics Placement Test, Question 10
DAILY DOUBLE #2: Check the Daily Double flipchart. (The other Daily Double and Final Jeopardy are there, too.)
What is 27?
(To get from 6 to 27 by multiplication, you have to multiply by 4.5 (27/6=4.5). Then, multiplying the corresponding parts of the matrix by 4.5 gives 9 for x and 18 for z. x+z = 9+18 = 27.)
In State X, Mr. Camp's eighth grade class consisting of 26 students was surveyed and 34.6 percent of students reported that they had at least two siblings. The average class size in the state is 26. If the students in Mr. Camp's eighth grade class are representative of students in the state's eighth-grade classes and there are 1800 eighth-grade classes in the state, which of the following best estimates the number of eighth-grade students in the state with fewer than two siblings?
A) 16,200
B) 23,400
C) 30,600
D) 46,800
What is C?
Since Mr. Camp's class is representative of other classes in the state, and Mr. Camp's class has a class size equal to the average class size of that of the state, that means that we can assume that the proportion of students with two or more siblings in the state is the same as that of Mr. Camp's class.
1800 students times 26 students per class on average means there are a total of 46,800 eighth grade students in the state. Among them, 34.6 percent have two or more siblings.
(.346)(46800) = 16192.8, or approximately 16,200 do have two or more siblings. The question asks for those with 1 or fewer siblings. We can take all the 8th grade students, subtract those with 2 or more siblings, and we're left with those with 1 or 0 siblings. 46800-16200=30600, our answer.
What is the sum of the first 4 terms of the arithmetic sequence in which the 6th term is 8 and the 10th term is 13?
A) 10.5
B) 14.5
C) 18
D) 21.25
E) 39.5
What is B?
(An arithmetic sequences increases by the same amount as it goes from one term to the next. If it increases by 5 after 4 terms, then it goes up by 1.25 each term.
6th: 8
5th: 6.75
4th: 5.5
3rd: 4.25
2nd: 3
1st: 1.75. 1.75+3+4.25+5.5 = 14.5.)
The expression (5x-2)/(x-3) is equivalent to which of the following?
A) (5-2)/3
B) 5 - 2/3
C) 5 - (2)/(x+3)
D) 5 - (17)/(x+3)
What is D?
A rational expression is expressed as a polynomial divided by another polynomial. We can rewrite it in the form a/(x-h)+k to find the asymptotes by performing polynomial long division. x goes into 5x 5 times, making 5 our quotient. 5 times x-3 is 5x-15, and subtracting this from 5x-2 yields a remainder of -17.
The answer to a division problem is the quotient (5), plus the remainder over the divisor. This means that 5 + (-17)/(x+3) is our equivalent form.
Which of the following is equivalent to 25x=7?
A) x = log2(7/5)
B) x = (log2(7))/5
C) x = (log7(2))/5
D) x = (log7(5))/2
What is B?
25x=7 is an exponential equation. To get x out of the denominator, we must use a base 2 logarithm (since 2 is the base). Taking the base 2 log of both sides,
25x=7
log2(25x)=log2(7)
5x = log2(7). Next, divide both sides by 5.
x = (log2(7))/5, as desired.)
Source: Next-Generation Accuplacer: Advanecd Algebra and Functions, Question 18
An integer from 100 to 999, inclusive, is chosen at random. What is the probability that the number chosen will have 0 as at least 1 digit?
A) 19/900
B) 81/900
C) 90/900
D) 171/900
E) 271/1000
What is D?
(There are 900 numbers between 100 and 999, inclusive. 100, 110, 120,...190 gives 10.
200, 210, 220,... gives 20. Repeat until 900s to get 90 numbers.
Separately, 101, 102, ..., 109 gives 9 numbers. (Don't double-count 100, 200, ..., 900) Repeat for 201, 301, ..., 901 to get 9 sets of 9 for 81.
90+81 gives 171, out of 900 numbers.)
A motor powers a model car so that after starting from rest, the car travels s inches in t seconds, where
s = 16t*sqrt(t). Which of the following gives the average speed of the car, in inches per second, over the first t seconds after it starts?
A) 4sqrt(t)
B) 16sqrt(t)
C) 16/(sqrt(t))
D) 16t
What is B?
Average speed is calculated as distance over time. Therefore, the average speed on an interval is the distance traveled from 0 inches to s inches, divided by the time it takes to travel that distance, from 0 seconds to t seconds. This can be modeled by:
(distance after t seconds - 0 inches) / (t seconds - 0 seconds)
The distance after t seconds is modeled by the equation s = 16t*sqrt(t). All we have to do is divide this by t (because t - 0 = t) in order to get our average speed.
16t*sqrt(t)/t. Cancel out t in the numerator and denominator.
This leaves 16sqrt(t), our answer.
Mai is putting gold foil around the outside of an elliptical picture frame. The formula for the perimeter of an ellipse is (pi/2)sqrt(2(h2+w2)), where h is the height and w is the width. If an elliptical frame has an outside height equal to 4 inches and an outside width equal to 3 inches, what is the outside perimeter, in inches?
(NOTATION: sqrt(x) means "the square root of x.")
A) (5/2)pi*sqrt(2)
B) (7/2)pi*sqrt(2)
C) 5pi*sqrt(2)
D) (pi/2)(4sqrt(2) + 3)
E) (4pi + 3)sqrt(2)
What is A?
(Plug in 4 for h and 3 for w. This results in
(pi/2)sqrt(2(9+16))
=(pi/2)sqrt(2(25))
=(pi/2)sqrt(50).
50 is not a perfect square, but it IS an integer multiple of a perfect square, namely, 2*25. Here's how to simplify radicals:
sqrt(50) = sqrt(2*25) = (2*25)1/2 = 21/2*251/2
= sqrt(25)*sqrt(2) = 5sqrt(2).
So, when fully simplified, our final answer is
(pi/2)*5sqrt(2) = (5/2)pi*sqrt(2). )
Which of the following complex numbers are equivalent to (3-5i)/(8+2i)?
NOTE: i equals the square root of -1.
A) 3/8 - 5i/2
B) 3/8 + 5i/2
C) 7/34 - 23i/34
D) 7/34 + 23i/34
What is C?
The first step is to get i out of the denominator, much like you'd want to get any square root out of the denominator. Multiplying both the top and bottom by the conjugate of the denominator will get rid of the i in the denominator, while keeping the value the same since when you multiply the top and bottom by the same thing, you're really multiplying by 1.
Recall that since i = sqrt(-1), this means i2 = -1. Wherever you see i2, replace it with -1 as you simplify.
((3-5i)(8-2i))/((8+2i)(8-2i)) Multiply by conjugate.
= (24-46i+10i2)/(64-4i2) FOIL.
= (14-46i)/(68) Simplify i2 and add.
= 14/68 - 46i/68 Break up into two fractions.
= 7/34 - 23i/34 Simplify the fraction.
When pi/2 < x < 3pi/4, which of the following could possibly be tan(x)?
A) -8
B) -1/8
C) 0
D) 1/8
E) 8
What is A?
(tan(x) is the equivalent of sin(x)/cos(x), so it is undefined when cos(x)=0. cos(pi/2)=0, so tan(x) has a vertical asymptote at pi/2. The graph of tan(x) reaches 0 again at pi, and 3pi/4 is between pi/2 and pi, which means tan(x) must be negative.
Next, notice that sin(3pi/4)/cos(3pi/4) = -1, since they are opposite signs of the same number. In our desired interval, our answer for tan(x) must be between negative infinity and negative 1. -8 is the only answer that fits that description.)
Source: University of Maryland Mathematics Placement Test, Question 30