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ACT Math Facts1
Vocabulary
Sum = The result of an addition problem. The sum of 3 and 4 is 7.
Product = The result of a multiplication problem. The product of 3 and 4 is 12.
Factor = A number that evenly divides a number. 2, 3 and 4 are factors of 12.
Integer = A number that is not a fraction or decimal. …-2, -1, 0, 1, 2…
Natural number = A nonnegative integer. 0, 1, 2, 3…
Rational number = A number that can be expressed as a fraction of integers. ½, -3, 0…
Irrational number = A number that cannot be expressed as a fraction of integers. √2, π, e...
Real number = A rational or irrational number. For your purposes, all numbers are real.
Consecutive = One after the other.
Consecutive integers
1, 2, 3
15, 16, 17
Consecutive odd integers
-3, -1, 1
5, 7, 9
Prime = A whole number with no factors other than itself and 1. 1 is not prime. 2, 3, 5, 7, 11…
Perimeter = The distance around the sides of a plane figure.
PRectangle = 2(b + h)
PSquare = 4s
Area = The amount of plane covered by a plane figure.
ARectangle = b × h
ATriangle = 1/2 (b × h)
ACircle = πr2
Volume = The amount of space taken up by a solid figure
VCube = s3
VRectangular prism = w×d×h
Congruent = Having the same size and shape.
Similar = Having the same shape, though not necessarily the same size. The sides of similar
polygons will be in the same propotion.
1
This fact sheet is not intended to include all the information on which a student might be tested on the math
sections of the ACT. It is intended to highlight those areas that are typical test material and that students are
particularly likely to have forgotten or not have learned by the time they take the test. Do not rely on this fact sheet
alone in preparing for your test.
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Radius = A line segment from the center of a circle to the edge, or the length of this line.
Chord = A line segment connecting one point on a circle to another point on the circle
Diameter = Twice the radius, a chord passing through the center of a circle.
Circumference = The distance around a circle.
C = 2πr
Average = The sum of the members of a set divided by the number of members of that set.
Avg({2, 7, 9}) = (2+7+9)/3 = 6
Mean = Average
Median = Middle number of a set.
Percent increase/decrease (percent greater/less) = (New value – old value)÷(old value) × 100
Useful terms
Each:
Greater than:
Is:
Of:
Per:
Percent:
Ratio:
What:
×
+
=
×
÷
/100
fraction
x
i.e. “What percent of 12 is 3 greater than 7?”
x /100 × 12 = 3
+
7
Exponents
(xy)z = xy×z
-y
y
x = 1/ x
xy × xz = xy+z
1
y
y
x = x
Angle measurements
A straight line is 180°
A right angle is 90°
The angles of a triangle add up to 180°
The acute angles of a right triangle add up to 90°
Right triangles
a2 + b2 = c2
Perfect triples
3, 4, 5
5, 12, 13
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Trigonometry
opp
sinθ =
hyp
adj
cosθ =
hyp
opp
tanθ =
adj
hyp
opp
hyp
secθ =
adj
adj
cotθ =
opp
cscθ =
Analytic geometry
Quadrant II
x negative; y positive
Quadrant I
x and y both positive
Quadrant III
x and y both negative
Quadrant IV
x positive; y negative
Equation for a line:
‫ ݔ݉ = ݕ‬+ ܾ
Equation for a circle: (‫ ݔ‬− ℎ)ଶ + (‫ ݕ‬− ݇)ଶ = ‫ ݎ‬ଶ
Equation for a parabola: ‫ ݔ(ܽ = ݕ‬− ℎ)ଶ + ݇
m is the slope and b is the y-intercept
(h, k) is the center and r is the radius
(h, k) is the vertex and a is the dilation
General
If xy = 0, then either x = 0 or y = 0.
x y = x2 y
Techniques
Cross multiplying
If two fractions are equal to each other, cross multiply. That is, the product of one
numerator and the other denominator equals the product of the other numerator and the
first denominator.
a c
=
b d
a×d = b×c
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Solving simultaneous equations
If you have two equations and two unknowns, use one of the equations to help you solve
for one of the unknowns in the other equation.
3x – 4y = 11
2x + 2y = 12 x+y=6
x=6–y
Substitute 6 – y for x in the first equation.
3(6 – y) – 4y = 11
7 = 7y
18 – 3y – 4y = 11
y=1
18 – 7y = 11 Now you know y; substitute that into either of the original equations.
3x – 4 = 11
3x = 15
x=5
Alternatively, multiply one of the equations by a constant to get a common term.
3x – 4y = 11
4x + 4y = 24
Then add the two equations together (or subtract one from the other) to eliminate that
term.
3x – 4y = 11
4x + 4y = 24
7x + 0 = 35
x=5
Then substitute the value of the evaluated variable, as in the first method.
15 – 4y = 11
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4 = 4y
y=1
Dimensional analysis
Problems involving rates might ask you to find distance, given a walking speed, or water
used, given a rate of flow. Determine the units that the answer is in and set up the values
given to cancel out all other units.
Speed = 45 miles/hour
Time = 3 hours
Distance = 45m/h × 3h = 45 × 3m h/h = 135m
Flow = 3 gallons/second
Water used = 7 gallons
Time = 7g = 7g×s = 7/3 s
3g/s
3g
Speed = 30 miles/hour
60 minutes/hour
60 seconds/minute
5280 feet/mile
Speed = 30 mi × 5280ft × _1 hr_ × _1 min = 44 ft
hr
mi
60 min 60 sec sec
Matrix multiplication
ቂ
ܽ
ܿ
ሾܽ
ܾ ݁
ቃ൤
݀ ݃
ܾ
ܿ
݂
ܽ݁ + ܾ݃
൨=൤
ℎ
ܿ݁ + ݀݃
݂ܽ + ܾℎ
൨
݂ܿ + ݀ℎ
݁
݂
݀ ሿ ቎݃቏ = ܽ݁ + ܾ݂ + ܿ݃ + ݀ℎ
ℎ
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