ACT Math — Number & Quantity Domain Test

Calculator is allowed on all questions. Figures are not necessarily drawn to scale.

Question 1

What is \(\frac{3}{4}+\frac{5}{8}\)?

A) \(1\frac{3}{8}\)
B) \(1\frac{1}{2}\)
C) \(1\frac{5}{8}\)
D) \(1\frac{7}{8}\)

Show solution

Using eighths, \(\frac{3}{4}=\frac{6}{8}\).

Thus \(\frac{6}{8}+\frac{5}{8}=\frac{11}{8}=1\frac{3}{8}\).

Answer: A

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Question 2

Which expression is equal to \(0.000072\) in scientific notation?

F) \(7.2\times10^{-6}\)
G) \(7.2\times10^{-5}\)
H) \(7.2\times10^{5}\)
J) \(72\times10^{-5}\)

Show solution

Moving the decimal 5 places to the right gives \(7.2\): \(0.000072=7.2\times10^{-5}\).

Answer: G

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Question 3

A quantity increases from 80 to 100. What is the percent increase?

A) \(20\%\)
B) \(25\%\)
C) \(40\%\)
D) \(80\%\)

Show solution

The increase is 20.

Relative to the original 80, \(\frac{20}{80}=0.25=25\%\).

Answer: B

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Question 4

The first term of an arithmetic sequence is 7, and each term after the first is 4 greater than the preceding term. What is the fifth term?

F) 11
G) 15
H) 19
J) 23

Show solution

For an arithmetic sequence, add the common difference each time.

Starting at 7 and adding 4 gives \(7,11,15,19,23\).

The fifth term is 23.

Answer: J

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Question 5

Which expression is equivalent to \(x^3\cdot x^5\)?

A) \(x^8\)
B) \(x^{15}\)
C) \(2x^8\)
D) \(x^2\)

Show solution

When multiplying powers with the same base, add exponents: \(x^{3+5}=x^8\).

Answer: A

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Question 6

What is \(\sqrt{72}\) in simplest radical form?

F) \(3\sqrt{8}\)
G) \(6\sqrt{2}\)
H) \(8\sqrt{2}\)
J) \(12\sqrt{2}\)

Show solution

\(72=36\cdot2\), so \(\sqrt{72}=\sqrt{36}\sqrt2=6\sqrt2\).

Answer: G

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Question 7

A recipe uses flour and sugar in the ratio \(5:2\). If 15 cups of flour are used, how many cups of sugar are needed?

A) 4
B) 5
C) 6
D) 7.5

Show solution

The scale factor from 5 cups of flour to 15 is 3.

The sugar amount is \(2(3)=6\) cups.

Answer: C

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Question 8

An account contains $2,000 and earns \(5\%\) interest compounded annually. If no money is added or withdrawn, how much is in the account after 2 years?

F) $2,100
G) $2,150
H) $2,200
J) $2,205

Show solution

After 2 years, the balance is \(2000(1.05)^2=2205\).

Answer: J

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Question 9

A geometric sequence begins \(3,12,48,\ldots\). What is the fifth term?

A) 768
B) 1,536
C) 3,072
D) 12,288

Show solution

For a geometric sequence, multiply by the common ratio each time.

The common ratio is \(12/3=4\).

Continue the sequence: \(3,12,48,192,768\), so the fifth term is 768.

Answer: A

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Question 10

If \(\sqrt{x+5}=7\), what is \(x\)?

F) 42
G) 44
H) 49
J) 54

Show solution

Square both sides: \(x+5=49\), so \(x=44\).

Answer: G

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Question 11

What is the value of \(i^{23}\)?

A) \(-1\)
B) \(1\)
C) \(-i\)
D) \(i\)

Show solution

Powers of \(i\) repeat every 4.

Since \(23\equiv3\pmod4\), \(i^{23}=i^3=-i\).

Answer: C

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Question 12

What is \((4-3i)+(2+5i)\)?

F) \(2-8i\)
G) \(2+2i\)
H) \(6-8i\)
J) \(6+2i\)

Show solution

Add real and imaginary parts separately: \((4+2)+(-3+5)i=6+2i\).

Answer: J

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Question 13

If \(A=\begin{bmatrix}2&-1\\3&4\end{bmatrix}\), what is \(3A\)?

A) \(\begin{bmatrix}6&-3\\9&12\end{bmatrix}\)
B) \(\begin{bmatrix}5&2\\6&7\end{bmatrix}\)
C) \(\begin{bmatrix}6&-1\\3&12\end{bmatrix}\)
D) \(\begin{bmatrix}8&2\\12&16\end{bmatrix}\)

Show solution

Multiply every entry by 3: \(3A=\begin{bmatrix}6&-3\\9&12\end{bmatrix}\).

Answer: A

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Question 14

The vector shown has horizontal component 6 and vertical component 8. What is its magnitude?

F) 8
G) 10
H) 12
J) 14

Show solution

The magnitude is \(\sqrt{6^2+8^2}=\sqrt{100}=10\).

Answer: G

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Question 15

What is \((3+2i)(4-i)\)?

A) \(10+5i\)
B) \(12+5i\)
C) \(14+5i\)
D) \(14+11i\)

Show solution

\((3+2i)(4-i)=12-3i+8i-2i^2=14+5i\).

Answer: C

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Question 16

If \(\begin{bmatrix}x&2\\-1&5\end{bmatrix}+\begin{bmatrix}4&-2\\3&1\end{bmatrix}=\begin{bmatrix}11&0\\2&6\end{bmatrix}\), what is \(x\)?

F) 4
G) 5
H) 6
J) 7

Show solution

Compare the top-left entries: \(x+4=11\), so \(x=7\).

Answer: J

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Question 17

A sequence is defined by \(a_1=2\) and \(a_n=2a_{n-1}+1\) for \(n\ge2\). What is \(a_5\)?

A) 47
B) 63
C) 95
D) 127

Show solution

Use the recurrence to build each term from the previous one.

\(a_2=2(2)+1=5\), \(a_3=2(5)+1=11\), \(a_4=2(11)+1=23\), and \(a_5=2(23)+1=47\).

Therefore, \(a_5=47\).

Answer: A

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Question 18

Point \(P\) is shown in the coordinate plane. If \(P\) is represented in polar coordinates as \((r,\theta)\) with \(0^\circ<\theta<90^\circ\), what is \(r\)?

F) 4
G) 5
H) 6
J) 7

Show solution

The point has Cartesian coordinates \((3,4)\): \(r=\sqrt{3^2+4^2}=5\).

Answer: G

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Question 19

A machine produces 18 parts every 12 minutes. At this constant rate, how many parts will it produce in 3 hours?

A) 180
B) 240
C) 270
D) 324

Show solution

Three hours is 180 minutes.

There are \(180/12=15\) production intervals.

The machine produces \(15(18)=270\) parts.

Answer: C

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Question 20

The equation \(x^2-6x+13=0\) has two complex solutions. Which of the following is one of them?

F) \(-3-2i\)
G) \(-3+2i\)
H) \(3-i\)
J) \(3-2i\)

Show solution

Using the quadratic formula, \(x=\frac{6\pm\sqrt{36-52}}{2}=\frac{6\pm4i}{2}=3\pm2i\).

One solution is \(3-2i\).

Answer: J

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