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English Test
Set 1
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Set 2
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Set 3
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Set 4
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Set 5
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Mathematics Test
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Set 2
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Set 3
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Set 4
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Set 5
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Reading Test
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Set 2
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Set 3
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Set 4
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Science Test
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Set 2
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Set 3
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Set 4
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Set 5
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Writing Test
Sample Essay 1
Sample Essay 2
Sample Essay 3
Sample Essay 4
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ACT Assessment Sample Question Answer Key and
Question Explanations
Set 1: Mathematics
1. The correct answer is B.
By definition, x4 means the product of 4 factors of x so x4 is equivalent to (x)(x)(x)(x).
A. 4x = x + x + x + x, which is not the same as the product of 4 factors of x.
C. x + 4 is not the same as the product of 4 factors of x.
D. 4 x is x factors of 4, which is not the same as the product of 4 factors of x.
E. 2x2 is 2 x x, which is not the same as the product of 4 factors of x.
2. The correct answer is K.
The rectangle is 3 inches by 6 inches, so the area is 3(6) = 18 square inches.
F. If the area were 6, then l =
= 2, but this length isn't twice the width.
G. If the area were 9, then l = = 3, but this length isn't twice the width.
H. If the area were 12, then l =
= 4, but this length isn't twice the width.
J. If the area were 15, then l =
= 5, but this length isn't twice the width.
3. The correct answer is A.
There are 14 balloons, of which 9 are yellow. If a yellow balloon is sold, there are 13
balloons left, of which 8 are yellow. The probability of selecting a yellow balloon is .
B.
cannot be the probability of selecting a yellow balloon;
because 9(13)
C.
cannot be the probability of selecting a yellow balloon;
because 5(13) = 65, 14
(8) = 112, and 65 112.
D. cannot be the probability of selecting a yellow balloon;
E.
cannot be the probability of selecting a yellow balloon;
14(8) = 112, and 117
112.
4. The correct answer is J.
because 8(13)
8(13).
8(14).
because 9(13) = 117,
Test Descriptions
3 10- 4 = 3
=3
= 0.0003
F. -30,000 in scientific notation is -3 104, not 3 10- 4.
G. -120 in scientific notation is -1.2 102, not 3 10- 4.
H. 0.00003 in scientific notation is 3 10- 5, not 3 10- 4.
K. 0.12 in scientific notation is 1.2 10-1, not 3 10- 4.
5. The correct answer is C.
=
=
= 2(x + 3)
simplifies to x + 3, then both should have the same value for all x > 0. But
A. If
if x = 1,
=
= 8 and x + 3 = 1 + 3 = 4.
simplifies to x + 4, then both should have the same value for all x > 0. But
B. If
if x = 1,
=
D. If
= 8 and x + 4 = 1 + 4 = 5.
simplifies to 2(x + 4), then both should have the same value for all x > 0.
But if x = 1,
=
= 8 and 2(x + 4) = 2(1 + 4) = 10.
simplifies to 2(x + 3)(x + 4), then both should have the same value for all
E. If
> 0. But if x = 1,
=
= 8 and 2(x + 3)(x + 4) = 2(4)(5) = 40.
6. The correct answer is F.
By the Pythagorean theorem, the length of the hypotenuse is
.
G.
is the hypotenuse of a right triangle with legs 12 and 4, since
; so
can't be the hypotenuse of a right triangle with
legs 12 and 8.
H.
can't be the hypotenuse because is would be the shortest
side of the triangle.
J.
one of the legs.
can't be the hypotenuse because it would be shorter than
K. 4 < 8, so 4 can't be the hypotenuse because it would be the shortest side of the triangle.
7. The correct answer is D.
Since circles have equations of the form (x - h)2 + (y - k)2 = r2, for the given circle, r2 = 9,
so r = 3.
Then A = r2 = (3)2 = 9 .
A. A circle with area 3 has radius , since ( )2 = 3 ;
but the radius of the given circle is 3, not .
B. A circle with area 4 has radius 2, since (2)2 = 4 ; but the radius of the given circle is
3, not 2.
C. A circle with area 6 has radius , since ( )2 = 6 ; but the radius of the given circle
is 3, not .
E. A circle with area 16 has radius 4, since (4)2 = 16 ; but the radius of the given circle
is 3, not 4.
8. The correct answer is H.
x2 - 15 = 0, so x2 = 15, x = ±
.
2
2
) - 15 = 15 - 15 = 0.
If x = - , x - 15 = (2
2
) - 15 = 15 - 15 = 0.
If x =
, x - 15 = (
Thus both and
satisfy the equation, so there are two solutions.
F and G are not correct because two numbers have been shown to satisfy the equation.
J and K are not correct because a quadratic equation has at most two solutions.
9. The correct answer is B.
If the circle with center (-3,4) is tangent to the x-axis, the point of tangency is (-3,0) and
the distance between these points is 4, so the radius is 4.
A. If r = 3, the equation of the circle is (x + 3)2 + (y - 4)2 = 9.
This circle is tangent to the x-axis so it contains a point for which y = 0.
But then (x + 3)2 + (0 - 4)2 = 9, (x + 3)2 + 16 = 9, (x + 3)2 = -7, which is impossible since
squares can't be negative.
C. If r = 5, the equation of the circle is (x + 3)2 + (y - 4)2 = 25.
When y = 0, (x + 3)2 + (0 - 4)2 = 25, (x + 3)2 + 16 = 25, (x + 3)2 = 9, x + 3 = ±3, x = 0, So this circle crosses the x-axis at (0,0) and (-6,0) and is not tangent to the x-axis.
D. If r = 9, the equation of the circle is (x + 3)2 + (y - 4)2 = 81.
When y = 0, (x + 3)2 + (0 - 4)2 = 81, (x + 3)2 + 16 = 81, (x + 3)2 = 65, x + 3 = ±
,x=
±
.
,0) and is not tangent to the
So this circle crosses the x-axis at (-3 ,0) and (-3 +
axis.
E. If r = 16, the equation of the circle is (x + 3)2 + (y - 4)2 = 256.
When y = 0, (x + 3)2 + (0 - 4)2 = 256, (x + 3)2 + 16 = 256, (x + 3)2 = 240, x + 3 = ±
= -3 ±
.
So this circle crosses the x-axis at (-3 x-axis.
,0) and (-3 +
,0) and is not tangent to the
10. The correct answer is F.
Since sin2A + cos2A = 1, ( )2 + cos2A = 1, cos2A = 1 ±
.
But since A is acute, cosA =
.
G. ( )2 + (
1, so cosA
)2 =
+
=
+
=
H. ( )2 + ( )2 =
2
2
) =
+
=
J. ( ) + (
)2 + (
K. (
)2
=
+
=
1, so cosA
.
1, so cosA
=
, cos2A =
, so cosA = ±
.
.
1, so cosA
.
11. The correct answer is A.
If a|b - 2| < 0, one of a or |b - 2| must be negative and one must be positive.
Since |b - 2| can't be negative, a must be negative. Then |b - 2| must be positive, so b
2.
B. If b = 2, |b - 2| = 0 and a|b - 2| = 0, but 0 0.
C. The values a = 1 and b = 3 meet the conditions a 0 and b > 2,
but a|b - 2| = 1|3 - 2| = 1 1 = 1, and 1 0.
D. The values a = 1 and b = 1 meet the conditions a > 0 and b < 2,
but a|b - 2| = 1|1 - 2| = 1 1 = 1, and 1 0.
E. Values of a and b do exist for which a|b - 2| < 0.
For example, if a = -1 and b = 3, a|b - 2| = (-1)|3 - 2| = -1 1 = -1, and -1 < 0.
12. The correct answer is G.
of 1,000 light bulbs were defective, so
(1,000) or 25 bulbs were defective.
The rest, 1,000 - 25 or 975, were nondefective. The ratio of defective to nondefective is
=
.
F. If the ratio of defective bulbs to nondefective bulbs were , then out of every 26 bulbs,
1 would be defective, so
of the bulbs would be defective. But the fraction of defective
bulbs is .
H. If the ratio of defective bulbs to nondefective bulbs were , then out of every 41 bulbs,
1 would be defective, so
of the bulbs would be defective. But the fraction of defective
bulbs is .
J. If the ratio of defective bulbs to nondefective bulbs were , then out of every 40 bulbs,
39 would be defective, so
of the bulbs would be defective. But the fraction of defective
bulbs is .
K. If the ratio of defective bulbs to nondefective bulbs were , then out of every 41 bulbs,
40 would be defective, so
of the bulbs would be defective. But the fraction of defective
bulbs is .
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