ACT Math - Full-Length Practice Test 1
Calculator is allowed on all questions. Figures are not necessarily drawn to scale.
Question 1
The mean of the numbers \(12, 15, 18, 21,\) and \(x\) is \(18\). What is the value of \(x\)?
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The total must be \(5(18)=90\).
The known values total \(66\): \(x=90-66=24\).
Answer: A
Question 2
The Venn diagram represents 50 students. How many students are in the art group?
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The art group includes the 15 students in art only and the 12 in both groups: \(15+12=27\).
Answer: G
Question 3
When \(a=-3\) and \(b=5\), what is the value of \(2a^2-3b\)?
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\(2(-3)^2-3(5)=18-15=3\).
Answer: C
Question 4
A jacket originally costs \(80\). It is discounted by \(25\%\). What is the sale price?
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The discount is \(0.25(80)=20\).
The sale price is \(80-20=60\).
Answer: J
Question 5
In the right triangle shown, what is the value of \(x\)?
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\(x=\sqrt{6^2+8^2}=\sqrt{100}=10\).
Answer: A
Question 6
Which expression is equivalent to \(4(2x-3)-5x\)?
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\(4(2x-3)-5x=8x-12-5x=3x-12\).
Answer: G
Question 7
An arithmetic sequence begins \(7,11,15,19,\ldots\). What is the 10th term?
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For an arithmetic sequence, use \(a_n=a_1+(n-1)d\).
Here \(a_1=7\) and the common difference is \(d=4\).
The 10th term is \(a_{10}=7+(10-1)(4)=7+36=43\).
Answer: C
Question 8
If \(3x+7=25\), what is the value of \(x\)?
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\(3x=18\), so \(x=6\).
Answer: J
Question 9
The line shown passes through \((0,-1)\) and \((2,3)\). What is its slope?
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Use the slope formula \(m=\frac{y_2-y_1}{x_2-x_1}\).
Substitute the two points: \(m=\frac{3-(-1)}{2-0}=\frac42=2\).
Answer: A
Question 10
A recipe uses \(3\) cups of flour for \(8\) servings. At the same rate, how many cups are needed for \(20\) servings?
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\(\frac{3}{8}(20)=7.5\) cups.
Answer: G
Question 11
Which of the following is a factor of \(x^2-9x+20\)?
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\(x^2-9x+20=(x-4)(x-5)\): \(x-4\) is a factor.
Answer: C
Question 12
The circle shown has radius \(7\). What is its circumference?
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\(C=2\pi r=2\pi(7)=14\pi\).
Answer: J
Question 13
For \(f(x)=3x-4\), what is \(f(6)\)?
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\(f(6)=3(6)-4=14\).
Answer: A
Question 14
A rectangle has length \(12\) and width \(5\). What is its area?
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\(A=lw=12(5)=60\).
Answer: G
Question 15
Which best describes the association shown in the scatterplot?
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As \(x\) increases, the plotted \(y\)-values generally increase, indicating a positive association.
Answer: C
Question 16
Solve \(5(2x-1)=3x+23\) for \(x\).
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\(10x-5=3x+23\), so \(7x=28\) and \(x=4\).
Answer: J
Question 17
If \(g(x)=x^2-6x+5\), what is the minimum value of \(g(x)\)?
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Complete the square: \(g(x)=(x-3)^2-4\).
The minimum value is \(-4\).
Answer: A
Question 18
The graph shown has \(x\)-intercepts at \(0\) and \(4\). Which equation could represent the graph?
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The graph opens upward and has zeros 0 and 4: \(y=x(x-4)\) fits.
Answer: G
Question 19
A bag contains 5 red, 3 blue, and 2 green marbles. One marble is chosen at random. What is the probability that it is not blue?
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Use \(P(\text{event})=\frac{\text{favorable outcomes}}{\text{total outcomes}}\).
There are 10 marbles total and 7 are not blue, so \(P(\text{not blue})=\frac{7}{10}\).
Answer: C
Question 20
If \(2^{x+1}=32\), what is the value of \(x\)?
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\(32=2^5\), so \(x+1=5\) and \(x=4\).
Answer: J
Question 21
Lines \(m\) and \(n\) are parallel. In the figure, one acute angle is \(68^\circ\). What is the measure of any obtuse angle formed?
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An obtuse angle is supplementary to \(68^\circ\): \(180^\circ-68^\circ=112^\circ\).
Answer: A
Question 22
The system \(y=2x+1\) and \(y=-x+10\) has solution \((x,y)\). What is \(x+y\)?
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Set the equations equal: \(2x+1=-x+10\), so \(x=3\) and \(y=7\).
Thus \(x+y=10\).
Answer: G
Question 23
Which expression is equivalent to \(\frac{x^2-16}{x+4}\) for \(x\ne -4\)?
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\(x^2-16=(x-4)(x+4)\).
The expression simplifies to \(x-4\).
Answer: C
Question 24
The table shows the number of books read by 20 students. What is the mean number of books read?
| Books | Students |
|---|---|
| 1 | 2 |
| 2 | 5 |
| 3 | 8 |
| 4 | 5 |
Show solution
The weighted total is \(1(2)+2(5)+3(8)+4(5)=56\).
Divide by 20: \(56/20=2.8\).
Answer: J
Question 25
Two similar right triangles are shown. What is the value of \(x\)?
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The scale factor from 6 to 9 is \(\frac{3}{2}\).
Thus \(x=8\left(\frac32\right)=12\).
Answer: A
Question 26
For \(h(x)=|x-3|+2\), what is \(h(-1)\)?
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\(h(-1)=|-1-3|+2=4+2=6\).
Answer: G
Question 27
If \(\sin\theta=\frac{5}{13}\) for an acute angle \(\theta\), what is \(\cos\theta\)?
Show solution
A \(5\)-\(12\)-\(13\) right triangle has opposite 5 and hypotenuse 13.
The adjacent side is 12.
Thus \(\cos\theta=\frac{12}{13}\).
Answer: C
Question 28
If \(f(x)=2x+3\) and \(g(x)=x^2\), what is \(g(f(2))\)?
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\(f(2)=7\), then \(g(7)=49\).
Answer: J
Question 29
A sector of a circle has radius \(12\) and central angle \(60^\circ\). What is the area of the sector?
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\(\frac{60}{360}\pi(12)^2=24\pi\).
Answer: A
Question 30
A data set has mean 20. If 5 is added to every value, which statement is true?
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Adding the same constant to every value increases the mean by that constant but does not change spread.
The new mean is 25.
Answer: G
Question 31
Given \(i=\sqrt{-1}\), which is equal to \((3+2i)(1-4i)\)?
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Multiply: \(3-12i+2i-8i^2=3-10i+8=11-10i\).
Answer: C
Question 32
If \(5e^{2x}=80\), which expression equals \(x\)?
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\(e^{2x}=16\).
Taking natural logs gives \(2x=\ln16\): \(x=\frac{\ln16}{2}\).
Answer: J
Question 33
The graph of a piecewise function is shown. Which statement is true?
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At \(x=0\), the function value is determined by the closed point, not the open point.
Answer: A
Question 34
If \(A=\begin{bmatrix}2 & -1 \\ 3 & 4\end{bmatrix}\) and \(B=\begin{bmatrix}x & 5 \\ 1 & 2\end{bmatrix}\), and the top-left entry of \(A+B\) is 9, what is \(x\)?
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The top-left entry is \(2+x=9\), so \(x=7\).
Answer: G
Question 35
A sequence is defined by \(a_1=2\) and \(a_n=a_{n-1}+n^2\) for \(n\ge2\). What is \(a_4\)?
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Use the recurrence to build each term from the previous one.
Since \(n=2\) for the second term, \(a_2=a_1+2^2=2+4=6\).
Then \(a_3=6+3^2=15\) and \(a_4=15+4^2=31\).
Answer: C
Question 36
For \(x>0\), \(\log_2 x+\log_2(x-2)=3\). What is \(x\)?
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Combine logs: \(\log_2[x(x-2)]=3\).
\(x(x-2)=8\).
Thus \(x^2-2x-8=0\), giving \(x=4\) or \(-2\); the domain requires \(x=4\).
Answer: J
Question 37
The dashed lines shown are asymptotes of a hyperbola. Which feature can be concluded from the graph?
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A hyperbola approaches its asymptotes as its branches extend away from the center; the asymptotes are not branches of the graph.
Answer: A
Question 38
Suppose \(1\le m\le3\), \(4\le n\le8\), and \(2\le p\le5\). What is the greatest possible value of \(\frac{n}{mp}\)?
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To maximize \(\frac{n}{mp}\), choose the largest numerator and smallest denominator: \(n=8\), \(m=1\), \(p=2\).
The maximum is \(4\).
Answer: G
Question 39
In the right triangle shown, what is \(\cos\theta\)?
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The legs are 8 and 15.
The hypotenuse is 17.
Relative to \(\theta\), adjacent is 15, hence \(\cos\theta=\frac{15}{17}\).
Answer: C
Question 40
A ball’s height in meters is \(h(t)=-4.9t^2+19.6t+1.5\). At what time does it reach its maximum height?
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For \(at^2+bt+c\), the vertex occurs at \(t=-\frac{b}{2a}=-\frac{19.6}{2(-4.9)}=2\).
Answer: J
Question 41
Events \(A\) and \(B\) are mutually exclusive, with \(P(A)=0.35\) and \(P(B)=0.40\). Which must be true?
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Mutually exclusive events cannot occur together: \(P(A\cap B)=0\).
Answer: A
Question 42
A rectangle measures 20 by 12 and contains a circle of radius 5, as shown. What is the area inside the rectangle but outside the circle?
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Use the rectangle area formula \(A=lw\): \(A=20(12)=240\).
Use the circle area formula \(A=\pi r^2\): \(A=\pi(5)^2=25\pi\).
The requested area is \(240-25\pi\).
Answer: G
Question 43
The functions \(f\) and \(g\) are defined by \(f(x)=\frac{x-1}{2}\) and \(g(x)=3x+4\). For what value of \(x\) is \(f(g(x))=7\)?
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\(f(g(x))=\frac{3x+4-1}{2}=\frac{3x+3}{2}\).
Set equal to 7: \(3x+3=14\), so \(x=\frac{11}{3}\).
Answer: C
Question 44
The vector shown has terminal point \((5,4)\). What is its magnitude?
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The magnitude is \(\sqrt{5^2+4^2}=\sqrt{41}\).
Answer: J
Question 45
A population has mean 50 and standard deviation 8. A new data set is formed by multiplying every value by 3 and then subtracting 4. What are the new mean and standard deviation?
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The mean transforms as \(3(50)-4=146\).
Subtracting 4 does not affect spread, while multiplying by 3 makes the standard deviation \(3(8)=24\).
Answer: A