Pre-Algebra Practice Questions — ACT — Page 6

What is the value of the expression $\frac{\sqrt{144} \times 2^{-3}}{3^2}$ ?

-8

\frac{1}{4}

\frac{1}{6}

\frac{2}{9}

\frac{32}{3}

Show answer & explanation

Answer: $\frac{1}{6}$

Answer

The correct answer is the option representing

\frac{1}{6}

.

The correct answer is obtained by evaluating each part of the expression: the square root

\sqrt{144}

is

12

, the negative exponent

2^{-3}

is

\frac{1}{8}

, and the denominator

3^2

is

9

. Multiplying the numerator terms yields

12 \times \frac{1}{8} = 1.5

, and dividing by the denominator

9

gives

\frac{1.5}{9} = \frac{1}{6}

.

Step-by-Step Solution

1

Simplify the square root term in the numerator,

\sqrt{144}

.

\sqrt{144} = 12

Since

12^2 = 144

, the principal square root of

144

is

12

.

2

Simplify the negative exponent term in the numerator,

2^{-3}

.

2^{-3} = \frac{1}{2^3} = \frac{1}{8}

A negative exponent indicates the reciprocal of the base raised to the positive power,

a^{-n} = \frac{1}{a^n}

.

3

Calculate the value of the numerator by multiplying the simplified terms.

Numerator

= 12 \times \frac{1}{8} = \frac{12}{8} = \frac{3}{2}

Multiply the two simplified parts of the numerator.

4

Simplify the denominator,

3^2

.

3^2 = 9

3^2

means

3 \times 3

, which equals

9

.

5

Divide the simplified numerator by the simplified denominator to find the final value.

\frac{3/2}{9} = \frac{3}{2 \times 9} = \frac{3}{18} = \frac{1}{6}

Divide fractions by multiplying by the reciprocal of the denominator.

Key Concept

Evaluating expressions involving square roots, negative integer exponents, and positive integer exponents.

Question 102Question →

A machine at a factory seals $15$ boxes every $4$ minutes. At this rate, how many minutes will it take the machine to seal $90$ boxes?

6

10

24

60

360

Show answer & explanation

Answer: 24

Answer

The correct answer is

24

minutes. Since the machine seals

15

boxes every

4

minutes, sealing

90

boxes requires

6

times the number of boxes, which will take

6

times as long:

6 \times 4 = 24

minutes.

The correct answer is

24

. At a constant rate, the ratio of boxes sealed to time remains equivalent. Setting up the proportion

\frac{15}{4} = \frac{90}{x}

and cross-multiplying gives

15x = 360

, which simplifies to

x = 24

. Alternatively, since

90

boxes is

6

times the initial amount of

15

boxes (

90 \div 15 = 6

), the time required must also be

6

times the initial time:

6 \times 4 = 24

minutes.

Step-by-Step Solution

1

Determine the scaling factor for the number of boxes.

Scaling factor is

6

.

Divide the target number of boxes by the initial rate's number of boxes:

90 \div 15 = 6

.

2

Multiply the time interval by the scaling factor to find the total time required.

24

minutes.

Multiply the

4

minutes per interval by the scaling factor of

6

:

6 \times 4 = 24

.

Key Concept

Solving direct proportion problems by finding equivalent rates.

Alternative Method

Find the unit rate first: the machine seals

\frac{15}{4} = 3.75

boxes per minute. To find the time to seal

90

boxes, divide the total boxes by the unit rate:

\frac{90}{3.75} = 24

minutes.

Estimated Time:45s

Question 103Question →

Three lighthouse beacons flash at regular intervals. Beacon A flashes every $24$ seconds, Beacon B flashes every $36$ seconds, and Beacon C flashes every $s$ seconds, where $s$ is a positive integer. If all three beacons flash at the same instant, and the next time they all flash at the same instant is exactly $6$ minutes later, what is the number of possible values for $s$ ?

Show answer & explanation

Answer: 12

Answer

12

To find the number of possible values for

s

, we convert the joint flashing time to seconds (

6 \text{ minutes} = 360 \text{ seconds}

). The least common multiple (LCM) of the three intervals must equal this joint interval:

\text{LCM}(24, 36, s) = 360

. Writing the prime factorizations gives

24 = 2^3 \cdot 3^1

,

36 = 2^2 \cdot 3^2

, and

360 = 2^3 \cdot 3^2 \cdot 5^1

. For any positive integer

s = 2^a \cdot 3^b \cdot 5^c

, the power of each prime in the LCM is the maximum of the powers in the individual prime factorizations. This gives the constraints:

a \in \{0, 1, 2, 3\}

(4 options),

b \in \{0, 1, 2\}

(3 options), and

c = 1

(1 option). Multiplying these possibilities gives

4 \times 3 \times 1 = 12

possible values for

s

.

Step-by-Step Solution

1

Convert the joint interval to seconds.

360

seconds

The individual intervals are given in seconds, so the joint interval must be converted to the same unit to perform calculations.

2

Set up the LCM equation.

\text{LCM}(24, 36, s) = 360

The beacons will flash together at intervals that are multiples of all three individual intervals. The first time they flash together again represents the least common multiple.

3

Find the prime factorizations of the known numbers.

24 = 2^3 \cdot 3^1

,

36 = 2^2 \cdot 3^2

, and

360 = 2^3 \cdot 3^2 \cdot 5^1

Prime factorization allows us to analyze the relationship between the individual numbers and their least common multiple.

4

Analyze the exponents of the prime factors of

s

.

For

s = 2^a \cdot 3^b \cdot 5^c

, we must have

a \in \{0, 1, 2, 3\}

,

b \in \{0, 1, 2\}

, and

c = 1

.

The exponent of each prime factor in the LCM is the maximum of the exponents of that prime factor in the numbers being combined. Since the LCM has

2^3

, the maximum exponent of 2 must be 3, which is already satisfied by

24 = 2^3 \cdot 3^1

, so

a

can be any integer from 0 to 3. Since the LCM has

3^2

, the maximum exponent of 3 must be 2, which is already satisfied by

36 = 2^2 \cdot 3^2

, so

b

can be any integer from 0 to 2. Since the LCM has

5^1

and neither 24 nor 36 has a factor of 5,

s

must provide exactly one factor of 5 (

c=1

).

5

Calculate the total number of combinations for

s

.

4 \times 3 \times 1 = 12

Since the choice of each exponent is independent, we multiply the number of choices for each prime factor's exponent.

Key Concept

Using prime factorizations to determine the relationship between numbers and their least common multiple (LCM).

Question 104Question →

Simplify each of the following expressions and arrange them in order of their value from least to greatest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order of the expressions from least to greatest value is: the expression with square root of four times ten to the fourth power (200), the product expression (250), the division expression (300), and finally the cube root expression (400).

Evaluating all four expressions yields the values 200, 400, 300, and 250 respectively. Arranging these values from least to greatest yields 200, 250, 300, and 400.

Step-by-Step Solution

1

Evaluate the first expression:

\sqrt{4.0 \times 10^4}

.

200

Using the properties of square roots and exponents, rewrite as

\sqrt{4.0} \times \sqrt{10^4} = 2 \times 10^2 = 200

.

2

Evaluate the second expression:

\sqrt[3]{6.4 \times 10^7}

.

400

Adjust the scientific notation to make the exponent divisible by 3:

6.4 \times 10^7 = 64 \times 10^6

. Then evaluate the cube root:

\sqrt[3]{64} \times \sqrt[3]{10^6} = 4 \times 10^2 = 400

.

3

Evaluate the third expression:

\frac{4.8 \times 10^5}{1.6 \times 10^3}

.

300

Divide the coefficients

\frac{4.8}{1.6} = 3

and subtract the exponents in the denominator from the numerator

10^{5-3} = 10^2

, yielding

3 \times 10^2 = 300

.

4

Evaluate the fourth expression:

(5.0 \times 10^3) \times (5.0 \times 10^{-2})

.

250

Multiply the coefficients

5.0 \times 5.0 = 25.0

and add the exponents

3 + (-2) = 1

. The result is

25.0 \times 10^1 = 250

.

5

Compare the evaluated values to arrange them from least to greatest.

200 < 250 < 300 < 400

Comparing the values gives the final order:

\sqrt{4.0 \times 10^4}

(200), followed by

(5.0 \times 10^3) \times (5.0 \times 10^{-2})

(250), then

\frac{4.8 \times 10^5}{1.6 \times 10^3}

(300), and finally

\sqrt[3]{6.4 \times 10^7}

(400).

Key Concept

Applying exponent rules, square and cube root properties, and arithmetic operations on numbers in scientific notation.

Question 105Question →

A chef uses $\frac{1}{3}$ cup of milk and $\frac{1}{4}$ cup of water for a recipe. What fraction of a cup is the total liquid used in the recipe?

\frac{2}{7}

\frac{1}{12}

\frac{7}{12}

\frac{1}{7}

\frac{1}{6}

Show answer & explanation

Answer: $\frac{7}{12}$

Answer

The correct fraction of a cup of liquid is

\frac{7}{12}

.

The correct answer is found by finding a common denominator for the two fractions. The least common multiple of 3 and 4 is 12. Converting the fractions gives

\frac{4}{12}

and

\frac{3}{12}

. Adding these fractions yields

\frac{7}{12}

.

Step-by-Step Solution

1

Identify the mathematical operation needed to find the total liquid.

The total amount of liquid is the sum of the two fractions:

\frac{1}{3} + \frac{1}{4}

.

To find the combined total of two separate quantities, addition is required.

2

Find a common denominator for the two fractions.

The least common multiple of 3 and 4 is 12. Convert the fractions:

\frac{1}{3} = \frac{4}{12}

and

\frac{1}{4} = \frac{3}{12}

.

Fractions must have the same denominator before they can be added.

3

Add the converted fractions.

\frac{4}{12} + \frac{3}{12} = \frac{7}{12}

.

Add the numerators together and keep the common denominator.

Key Concept

Adding fractions with unlike denominators.

Estimated Time:45s

Question 106Question →

A school supply drive has $72$ notebooks, $96$ pens, and $120$ pencils. Volunteers want to pack these supplies into identical boxes so that each box contains the same number of notebooks, the same number of pens, and the same number of pencils, with no supplies left over. If they pack the maximum possible number of boxes, how many pens will be in each box?

24

4

8

12

1440

Show answer & explanation

Answer: 4

Answer

4

To find the number of pens in each box, we must first find the greatest common divisor (GCD) of the three quantities (

72

,

96

, and

120

) to determine the maximum number of identical boxes that can be created. The prime factorizations are

72 = 2^3 \times 3^2

,

96 = 2^5 \times 3

, and

120 = 2^3 \times 3 \times 5

. The GCD is the product of the lowest powers of the common prime factors:

2^3 \times 3 = 24

. This means a maximum of

24

boxes can be packed. Dividing the

96

pens by the

24

boxes gives

4

pens per box.

Step-by-Step Solution

1

Find the prime factorization of each of the three quantities: 72, 96, and 120.

72 = 2^3 \times 3^2

,

96 = 2^5 \times 3

, and

120 = 2^3 \times 3 \times 5

.

To find the greatest common divisor, we first need to break down each number into its prime factors.

2

Determine the Greatest Common Divisor (GCD) of 72, 96, and 120 by taking the lowest power of each common prime factor.

The common prime factors are 2 and 3. The lowest power of 2 is

2^3

, and the lowest power of 3 is

3^1

. Thus,

\text{GCD}(72, 96, 120) = 2^3 \times 3 = 8 \times 3 = 24

.

The GCD represents the maximum number of identical boxes that can be packed with no leftover items.

3

Divide the total number of pens by the number of boxes to find the number of pens per box.

96 \div 24 = 4

.

Since there are 96 pens distributed equally among 24 boxes, dividing the total pens by the number of boxes gives the amount in each box.

Key Concept

Using the Greatest Common Divisor (GCD) to solve distribution word problems.

Estimated Time:1m 15s

Question 107Question →

The rates at which three pipes, $A$ , $B$ , and $C$ , can fill a pool are in the ratio $3:4:6$ , respectively. To fill an empty pool, pipes $A$ and $B$ are turned on. After $2\text{ hours}$ , pipe $A$ is turned off and pipe $C$ is turned on. It takes another $3\text{ hours}$ for pipes $B$ and $C$ to finish filling the pool. What fraction of the pool's total volume was filled by pipe $B$ ?

\frac{4}{13}

\frac{3}{11}

\frac{5}{11}

\frac{8}{17}

\frac{20}{41}

Show answer & explanation

Answer: $\frac{5}{11}$

Answer

The fraction of the pool's total volume filled by Pipe B is

\frac{5}{11}

.

To find the fraction of the pool's volume filled by Pipe B, we determine the total volume of the pool and the volume contributed by Pipe B. If the rates of pipes

A

,

B

, and

C

are

3r

,

4r

, and

6r

per hour, the first

2\text{ hours}

fill

2(3r+4r) = 14r

units, and the next

3\text{ hours}

fill

3(4r+6r) = 30r

units, for a total pool volume of

44r

units. Since Pipe B runs for the entire

5\text{ hours}

, it fills

5 \times 4r = 20r

units. The fraction is

\frac{20r}{44r} = \frac{5}{11}

.

Step-by-Step Solution

1

Define the rates of the three pipes using a constant multiplier

r

.

Let the filling rates of pipes

A

,

B

, and

C

be

3r

,

4r

, and

6r

units of volume per hour, respectively.

This translates the given ratio

3:4:6

into algebraic expressions for their individual rates.

2

Calculate the volume of the pool filled during the first

2\text{ hours}

when pipes

A

and

B

are active.

Volume 1 =

2\text{ hours} \times (3r + 4r) = 2 \times 7r = 14r

units.

The rate of pipes

A

and

B

combined is the sum of their individual rates, and volume is rate multiplied by time.

3

Calculate the volume of the pool filled during the next

3\text{ hours}

when pipes

B

and

C

are active.

Volume 2 =

3\text{ hours} \times (4r + 6r) = 3 \times 10r = 30r

units.

The rate of pipes

B

and

C

combined is the sum of their individual rates, and volume is rate multiplied by time.

4

Find the total volume of the pool.

Total Volume =

14r + 30r = 44r

units.

The total volume is the sum of the volumes filled in both time intervals.

5

Calculate the total volume filled specifically by Pipe

B

across both periods.

Pipe

B

was active for the entire duration of

2 + 3 = 5\text{ hours}

. Volume filled by Pipe

B

=

5\text{ hours} \times 4r = 20r

units.

To find Pipe B's contribution, multiply its rate by the total time it operated.

6

Divide the volume filled by Pipe

B

by the total volume of the pool to find the fraction.

Fraction =

\frac{20r}{44r} = \frac{20}{44} = \frac{5}{11}

.

The fraction is the ratio of the part filled by Pipe B to the whole volume of the pool.

Key Concept

Using rates and ratios to find the fraction of work completed by a specific component in a multi-stage work problem.

Alternative Method

Instead of using a variable

r

, you can assume a concrete rate for the pipes that simplifies the arithmetic. For example, let the rates of pipes

A

,

B

, and

C

be

3

,

4

, and

6

gallons per hour, respectively. In the first

2\text{ hours}

, pipes

A

and

B

fill

2 \times (3 + 4) = 14\text{ gallons}

. In the next

3\text{ hours}

, pipes

B

and

C

fill

3 \times (4 + 6) = 30\text{ gallons}

. The total pool capacity is

14 + 30 = 44\text{ gallons}

. Pipe

B

works for the entire

5\text{ hours}

, filling

5 \times 4 = 20\text{ gallons}

. The fraction of the pool filled by Pipe

B

is

\frac{20}{44} = \frac{5}{11}

.

Estimated Time:3m 0s

Question 108Question →

An event coordinator is preparing identical welcome packets for a conference. She has $72$ promotional pens, $108$ notebooks, and $130$ keychains. She plans to distribute all of the pens and notebooks into the maximum number of identical packets such that no pens or notebooks are left over. She also wants to distribute the keychains equally among these packets. Finally, the coordinator requires that the total number of items (pens, notebooks, and keychains combined) in each packet is a multiple of $6$ . To meet these conditions, she must purchase additional keychains. What is the minimum number of additional keychains she needs to buy?

14

50

86

122

230

Show answer & explanation

Answer: 122

Answer

122

To maximize the number of identical packets using all 72 pens and 108 notebooks without leftovers, we find the greatest common factor (GCF) of 72 and 108, which is 36. This means 36 packets are created, with each packet containing 2 pens and 3 notebooks. If each packet contains q keychains, the total number of keychains is 36q. Since she starts with 130 keychains, we must have

36q \\geq 130

, which means

q \\geq 4

. The total number of items in each packet is

2 + 3 + q = 5 + q

. For this total to be a multiple of 6, and given

q \\geq 4

, the smallest possible integer value for q is 7 (since

5 + 7 = 12

). With 7 keychains per packet, the total number of keychains needed is

36 \\times 7 = 252

. Subtracting the 130 keychains she already has, she must buy a minimum of 122 keychains.

Step-by-Step Solution

1

Find the maximum number of packets that can be created using all 72 pens and 108 notebooks.

The number of packets must be the greatest common factor (GCF) of 72 and 108, which is 36.

Since all pens and notebooks must be distributed equally with none left over, the number of packets must divide both 72 and 108. To maximize the packets, we find their GCF.

2

Calculate the number of pens and notebooks in each of the 36 packets.

Each packet contains 2 pens (

72 \\div 36

) and 3 notebooks (

108 \\div 36

).

Dividing the total quantities of each item by the number of packets gives the quantity of that item per packet.

3

Set up the inequality for the number of keychains per packet, q.

Since she already has 130 keychains, the total keychains needed is

36q \\geq 130

, which simplifies to

q \\geq 3.61

. Since q must be an integer,

q \\geq 4

.

The keychains must be distributed equally, meaning each packet gets an integer number of keychains q. She must buy additional keychains, so the total keychains must be at least 130.

4

Apply the constraint that the total items per packet must be a multiple of 6 to find the minimum value of q.

The total items per packet is

2 + 3 + q = 5 + q

. The smallest integer

q \\geq 4

that makes

5 + q

a multiple of 6 is

q = 7

(since

5 + 7 = 12

, which is a multiple of 6).

This satisfies the divisibility condition for the total packet size while respecting the minimum quantity of keychains.

5

Calculate the number of additional keychains to buy.

Total keychains needed is

36 \\times 7 = 252

. The number of additional keychains to buy is

252 - 130 = 122

.

Subtracting the starting quantity of keychains from the total required quantity gives the number of additional keychains to purchase.

Key Concept

Using the Greatest Common Factor (GCF) to solve division and optimization problems with multiple constraints.

Alternative Method

Alternatively, a student can test the answer choices by working backwards. Add each choice to 130 to find the total keychains, check if the result is divisible by 36 (the GCF of 72 and 108), and verify if the resulting number of keychains per packet makes the total number of items per packet a multiple of 6.

Estimated Time:2m 0s

Question 109Question →

A chemist has three different acid solutions: Solution $X$ , Solution $Y$ , and Solution $Z$ . In Solution $X$ , the ratio of acid to water is $1:3$ by volume. In Solution $Y$ , the ratio of acid to water is $3:2$ by volume. In Solution $Z$ , the ratio of acid to water is $5:1$ by volume. If the chemist mixes Solution $X$ , Solution $Y$ , and Solution $Z$ in a volume ratio of $4:5:3$ , respectively, what is the ratio of acid to water in the final mixture?

17:7

34:25

13:11

3:2

101:79

Show answer & explanation

Answer: $13:11$

Answer

The ratio of acid to water in the final mixture is

13:11

.

The correct answer is the ratio

13:11

. To find the ratio of acid to water in the final mixture, first convert the individual ratios to part-to-whole fractions: Solution

X

is

\frac{1}{4}

acid and

\frac{3}{4}

water; Solution

Y

is

\frac{3}{5}

acid and

\frac{2}{5}

water; and Solution

Z

is

\frac{5}{6}

acid and

\frac{1}{6}

water. Next, scale these fractions by the volume mixing ratio of

4:5:3

. Assuming

4\text{ units}

of

X

,

5\text{ units}

of

Y

, and

3\text{ units}

of

Z

, the total acid is

4(\frac{1}{4}) + 5(\frac{3}{5}) + 3(\frac{5}{6}) = 1 + 3 + 2.5 = 6.5\text{ units}

, and the total water is

4(\frac{3}{4}) + 5(\frac{2}{5}) + 3(\frac{1}{6}) = 3 + 2 + 0.5 = 5.5\text{ units}

. The ratio of acid to water is

6.5:5.5

, which simplifies to

13:11

.

Step-by-Step Solution

1

Determine the part-to-whole fractions of acid and water for each solution.

For Solution

X

, the ratio of acid to water is

1:3

, which means the acid fraction is

\frac{1}{4}

and the water fraction is

\frac{3}{4}

. For Solution

Y

, the ratio is

3:2

, meaning the acid fraction is

\frac{3}{5}

and the water fraction is

\frac{2}{5}

. For Solution

Z

, the ratio is

5:1

, meaning the acid fraction is

\frac{5}{6}

and the water fraction is

\frac{1}{6}

.

Converting part-to-part ratios into fractions of the total volume is necessary to scale each solution correctly.

2

Choose convenient volumes representing the

4:5:3

mixing ratio, and calculate the volume of acid and water contributed by each solution.

Assume we mix

4\text{ liters}

of Solution

X

,

5\text{ liters}

of Solution

Y

, and

3\text{ liters}

of Solution

Z

. Acid from

X

is

4 \times \frac{1}{4} = 1\text{ liter}

; water is

3\text{ liters}

. Acid from

Y

is

5 \times \frac{3}{5} = 3\text{ liters}

; water is

2\text{ liters}

. Acid from

Z

is

3 \times \frac{5}{6} = 2.5\text{ liters}

; water is

3 \times \frac{1}{6} = 0.5\text{ liters}

.

Multiplying the part-to-whole fractions by the respective mixing volumes yields the absolute amounts of acid and water contributed.

3

Sum the total volumes of acid and water in the final mixture.

Total acid =

1 + 3 + 2.5 = 6.5\text{ liters}

. Total water =

3 + 2 + 0.5 = 5.5\text{ liters}

.

Finding the total amounts of each component allows us to determine the final composition of the mixture.

4

Find the simplified integer ratio of total acid to total water.

The ratio of acid to water is

6.5 : 5.5

. Multiplying both terms by

2

to convert to integers gives

13:11

.

Ratios are standardly expressed as simplified integers.

Key Concept

Calculating mixture compositions by converting part-to-part ratios to part-to-whole fractions and applying weighted average proportions.

Alternative Method

Instead of choosing arbitrary volumes like 4, 5, and 3, you can write the total volume as

V

and use algebraic fractions:

\text{Acid} = 4k(\frac{1}{4}) + 5k(\frac{3}{5}) + 3k(\frac{5}{6}) = 6.5k

, and

\text{Water} = 4k(\frac{3}{4}) + 5k(\frac{2}{5}) + 3k(\frac{1}{6}) = 5.5k

. The ratio remains

6.5k : 5.5k = 13:11

.

Estimated Time:3m 0s

Question 110Question →

On a standard number line, the coordinates of three distinct points $A$ , $B$ , and $C$ are represented by the integers $a$ , $b$ , and $c$ , respectively, such that $a < b < c$ . The distance between each point and the origin is represented by its absolute value, and these distances satisfy the inequality $|a| < |b| < |c|$ . If the product of the three coordinates is negative ( $abc < 0$ ) and the sum of their absolute values is $20$ , what is the maximum possible value of the coordinate $b$ ?

6

8

9

10

18

Show answer & explanation

Answer: 9

Answer

9

The correct answer is 9. By analyzing the signs of the coordinates, we determine that only

a

is negative, so

a < 0 < b < c

. Since

a

is a negative integer, the minimum value of

|a|

is 1. Under the constraint

|a| < |b| < |c|

and

|a| + |b| + |c| = 20

, we let

x = -a

,

y = b

, and

z = c

. We have

x + y + z = 20

with

1 \le x < y < z

. Since

z > y

, we get

20 = x + y + z > 1 + 2y

, which implies

2y < 19

, or

y < 9.5

. The maximum integer value for

y

(which is

b

) is 9, achieved when the coordinates are -1, 9, and 10.

Step-by-Step Solution

1

Determine the signs of the coordinates based on the given constraints.

The sign configuration must be

a < 0 < b < c

.

Since the product

abc < 0

, either all three coordinates are negative or exactly one is negative. If all three were negative (

a < b < c < 0

), their absolute values would satisfy

|a| > |b| > |c|

, which contradicts

|a| < |b| < |c|

. Thus, exactly one coordinate (

a

) is negative, and the other two (

b

and

c

) are positive.

2

Translate the absolute values and sum constraint into algebraic terms.

-a + b + c = 20

, where

a < 0

and

b, c > 0

.

For a negative number

a

, the absolute value

|a| = -a

. For positive numbers

b

and

c

,

|b| = b

and

|c| = c

. Therefore, the sum

|a| + |b| + |c| = 20

simplifies to

-a + b + c = 20

.

3

Set up inequalities to bound the coordinate of the middle point.

1 \le -a < b < c

.

Since

a

is a non-zero negative integer, the smallest possible value for its absolute value

|a| = -a

is

1

. The condition

|a| < |b| < |c|

then becomes

1 \le -a < b < c

.

4

Use the constraints to find the maximum value of

b

.

The maximum possible value of

b

is

9

.

Let

x = -a

,

y = b

, and

z = c

. We have

x + y + z = 20

with

1 \le x < y < z

. Since

z > y

, we have

20 = x + y + z > x + 2y \ge 1 + 2y

. This simplifies to

19 > 2y

, or

y < 9.5

. Since

y

must be an integer, the maximum possible value for

y

(which is

b

) is

9

. We can verify this maximum by setting

a = -1

,

b = 9

, and

c = 10

, which perfectly satisfies all conditions.

Key Concept

Analyzing coordinate signs and absolute value inequalities on a number line to perform optimization under integer constraints.

Question 111Question →

Let $S$ be the set of all integers $n$ that satisfy the inequality $|n - 3| + |n + 5| \leq 12$ . What is the sum of the absolute values of all integers in set $S$ ?

43

-13

12

28

37

Show answer & explanation

Answer: 43

Answer

43

The correct answer is 43. Solving the inequality

|n - 3| + |n + 5| \leq 12

by casework on the number line yields the solution set of integers

S = \{-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}

. Summing the absolute values of these elements gives

7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 + 1 + 2 + 3 + 4 + 5 = 43

.

Step-by-Step Solution

1

Divide the number line into intervals to remove the absolute value bars based on critical points

n = 3

and

n = -5

.

Three cases to analyze: Case 1 (

n \geq 3

), Case 2 (

-5 \leq n < 3

), and Case 3 (

n < -5

).

Evaluating absolute values requires knowing the signs of the expressions inside them.

2

Solve the inequality

|n - 3| + |n + 5| \leq 12

for each case.

For Case 1 (

n \geq 3

):

(n - 3) + (n + 5) \leq 12 \Rightarrow 2n + 2 \leq 12 \Rightarrow n \leq 5

, yielding integers

\{3, 4, 5\}

. For Case 2 (

-5 \leq n < 3

):

-(n - 3) + (n + 5) \leq 12 \Rightarrow 8 \leq 12

(always true), yielding integers

\{-5, -4, -3, -2, -1, 0, 1, 2\}

. For Case 3 (

n < -5

):

-(n - 3) - (n + 5) \leq 12 \Rightarrow -2n - 2 \leq 12 \Rightarrow -2n \leq 14 \Rightarrow n \geq -7

, yielding integers

\{-7, -6\}

.

Determining the integer values of

n

that satisfy the inequality in each segment of the number line.

3

Combine the intervals to construct the complete set

S

.

S = \{-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}

.

The union of all valid cases gives the entire set of solutions.

4

Compute the sum of the absolute values of the integers in

S

.

|-7| + |-6| + |-5| + |-4| + |-3| + |-2| + |-1| + |0| + |1| + |2| + |3| + |4| + |5| = 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 + 1 + 2 + 3 + 4 + 5 = 43

.

Adding the absolute values of each element in the solution set.

Key Concept

Solving multi-interval absolute value inequalities with integers and finding absolute values

Estimated Time:2m 0s

Question 112Question →

A certain cloud database stores a total of $3.6 \times 10^{15}$ bytes of data. If this data is distributed equally among $8.0 \times 10^4$ servers, how many megabytes of data are stored on each server? (Note: $1\text{ megabyte} = 10^6\text{ bytes}$ )

4.5 \times 10^3

4.5 \times 10^4

4.5 \times 10^5

4.5 \times 10^{10}

2.88 \times 10^{20}

Show answer & explanation

Answer: $4.5 \times 10^4$

Answer

4.5 \times 10^4

megabytes

The correct answer is the value that correctly represents the data per server in megabytes. Dividing the total data of

3.6 \times 10^{15}

bytes by the

8.0 \times 10^4

servers gives

4.5 \times 10^{10}

bytes per server. Converting this to megabytes by dividing by

10^6

gives

4.5 \times 10^4

megabytes.

Step-by-Step Solution

1

Calculate the bytes of data stored on each server by dividing the total data by the number of servers.

0.45 \times 10^{11}\text{ bytes} = 4.5 \times 10^{10}\text{ bytes}

Dividing the total bytes (

3.6 \times 10^{15}

) by the total number of servers (

8.0 \times 10^4

) gives the share of data per server. Mathematically,

\frac{3.6}{8.0} = 0.45

, and

\frac{10^{15}}{10^4} = 10^{15-4} = 10^{11}

.

2

Convert the result from bytes to megabytes using the conversion factor

1\text{ megabyte} = 10^6\text{ bytes}

.

4.5 \times 10^4\text{ megabytes}

To convert bytes to megabytes, divide the number of bytes by

10^6

. Since

\frac{4.5 \times 10^{10}}{10^6} = 4.5 \times 10^{10-6} = 4.5 \times 10^4

.

Key Concept

Division of numbers in scientific notation and unit conversion.

Question 113Question →

A store sells a winter coat that is originally priced at $120. During a weekend clearance sale, the coat's price is reduced by 25%. What is the sale price of the coat, in dollars?

Show answer & explanation

Answer: 90

Answer

The sale price of the coat is $90.

The correct answer is

90. To find the sale price, we compute the discount by taking 25% of

120, which is

30, and then subtract that discount from the original price of

120.

Step-by-Step Solution

1

Find 25% of $120 to determine the discount amount.

$30

A 25% reduction means we calculate

120 multiplied by 0.25, which equals

30.

2

Subtract the discount from the original price.

$90

The sale price is the original price minus the discount:

120 -

30 = $90.

Key Concept

Calculating a percentage markdown to determine a final price

Estimated Time:45s

Question 114Question →

Let $x = 2^{100}$ , $y = 3^{75}$ , and $z = 5^{50}$ . Which of the following inequalities correctly represents the relationship among the values of $x$ , $y$ , and $z$ ?

x < y < z

x < z < y

z < y < x

y < z < x

z < x < y

Show answer & explanation

Answer: $x < z < y$

Answer

x < z < y

To find the correct relationship, rewrite

x = 2^{100}

,

y = 3^{75}

, and

z = 5^{50}

with a common exponent. The greatest common divisor of

100

,

75

, and

50

is

25

. Using the rule

(a^m)^n = a^{m \cdot n}

, rewrite the terms:

x = (2^4)^{25} = 16^{25}

,

y = (3^3)^{25} = 27^{25}

, and

z = (5^2)^{25} = 25^{25}

. Comparing the bases shows

16 < 25 < 27

, which means

16^{25} < 25^{25} < 27^{25}

, so

x < z < y

.

Step-by-Step Solution

1

Find the greatest common divisor (GCD) of the exponents of the three expressions.

The exponents are

100

,

75

, and

50

. The greatest common divisor of these numbers is

25

.

Finding a common exponent allows us to rewrite each expression with the same power so we can compare their bases directly.

2

Rewrite each expression using the power of a power rule:

(a^m)^n = a^{m \cdot n}

.

x = 2^{100} = (2^4)^{25}

,

y = 3^{75} = (3^3)^{25}

, and

z = 5^{50} = (5^2)^{25}

.

This expresses all three values in the form

b^{25}

, where

b

is the base to be evaluated.

3

Evaluate the bases inside the parentheses.

2^4 = 16

, so

x = 16^{25}

;

3^3 = 27

, so

y = 27^{25}

;

5^2 = 25

, so

z = 25^{25}

.

Evaluating the base values simplifies each expression to a single number raised to the power of

25

.

4

Compare the evaluated bases and write the resulting inequality.

Since

16 < 25 < 27

, it follows that

16^{25} < 25^{25} < 27^{25}

. This corresponds to

x < z < y

.

Since the exponent

25

is positive, raising larger positive bases to this exponent results in larger values.

Key Concept

Comparing exponential expressions by rewriting them with a common exponent using exponent rules.

Question 115Question →

The table below shows the number of books read by 7 students during their summer vacation:

Student	Books Read
Amy	16
Ben	8
Chris	3
Diana	12
Ethan	14
Fiona	6
Gabe	11

What is the median number of books read by these students?

8

10

11

12

13

Show answer & explanation

Answer: 11

Answer

11

To find the median of a data set, the values must first be arranged in order from least to greatest: 3, 6, 8, 11, 12, 14, 16. Since there are 7 values, the median is the single middle value, which is 11.

Step-by-Step Solution

1

Sort the data set from least to greatest.

3, 6, 8, 11, 12, 14, 16

The median of a data set is the middle value when the numbers are ordered.

2

Identify the middle value of the sorted list.

11

Since there are 7 data points (an odd number), the median is the 4th value, which has exactly 3 values below it and 3 values above it.

Key Concept

Finding the median of a set of data

Question 116Question →

A positive integer $N$ has a prime factorization of the form $p^2 \times q$ , where $p$ and $q$ are distinct prime numbers. If the sum of all the positive factors of $N$ (including $1$ and $N$ ) is $78$ , what is the value of $N$ ?

Show answer & explanation

Answer: 45

Answer

The value of

N

is 45.

By applying the sum of divisors formula, the sum of factors of

N = p^2 \times q

is represented as

(1 + p + p^2)(1 + q) = 78

. Factoring 78 into two integers that satisfy the prime constraints of

p \ge 2

and

q \ge 2

yields

13 \times 6 = 78

. Solving

1 + p + p^2 = 13

gives

p = 3

, and

1 + q = 6

gives

q = 5

. Since 3 and 5 are distinct primes,

N = 3^2 \times 5 = 45

.

Step-by-Step Solution

1

Express the sum of factors of

N

algebraically

(1 + p + p^2)(1 + q) = 78

The sum of all positive factors of a number with prime factorization

p^a q^b

is given by the product of the sums of the powers of each prime factor.

2

Determine the constraints on

p

and

q

based on them being prime numbers

1 + p + p^2 \ge 7

and

1 + q \ge 3

The smallest prime number is 2, so

p \ge 2

and

q \ge 2

.

3

Find the factor pairs of 78 that satisfy the constraints

(1 + p + p^2, 1 + q) \in \{(13, 6), (26, 3)\}

The factors of 78 are 1, 2, 3, 6, 13, 26, 39, 78. We pair them such that one factor is at least 7 and the other is at least 3.

4

Solve for

p

and

q

for each possible factor pair

p = 3

and

q = 5

If

1 + p + p^2 = 13

, then

p^2 + p - 12 = 0

, which solves to

p = 3

(since

p > 0

). This leaves

1 + q = 6 \implies q = 5

. Both 3 and 5 are distinct primes. The other case,

1 + p + p^2 = 26

, has no integer solution for

p

.

5

Calculate the value of

N

N = 3^2 \times 5 = 45

Substitute the prime values back into the expression for

N

.

Key Concept

Sum of Divisors Formula and Prime Factorization

Estimated Time:1m 30s

Question 117Question →

If $x = -3$ and $y = 2$ , what is the value of the expression $x^2 - y(x - y^3)$ ?

-77

-1

13

27

31

Show answer & explanation

Answer: 31

Answer

31

Substituting

x = -3

and

y = 2

into the expression gives

x^2 - y(x - y^3) = (-3)^2 - 2(-3 - 2^3)

. Evaluating exponents first, we get

9 - 2(-3 - 8)

. Simplifying inside the parentheses gives

9 - 2(-11)

. Multiplying before subtracting yields

9 - (-22) = 9 + 22 = 31

.

Step-by-Step Solution

1

Substitute the given values into the expression

(-3)^2 - 2((-3) - 2^3)

To evaluate the expression for the specific numbers

x = -3

and

y = 2

.

2

Evaluate the exponents outside and inside the parentheses

9 - 2(-3 - 8)

Exponents must be calculated before multiplication and subtraction, according to PEMDAS rules. Note that

(-3)^2 = 9

and

2^3 = 8

.

3

Perform the operations inside the parentheses

9 - 2(-11)

Parentheses have the highest priority in the order of operations.

4

Multiply the terms

9 - (-22) = 9 + 22

Multiplication must be performed before subtraction.

5

Perform final addition

31

Adding 22 to 9 yields the final simplified value.

Key Concept

Order of operations (PEMDAS) and substitution of negative numbers in exponents.

Question 118Question →

A supercomputer simulation requires a total of $1.62 \times 10^{14}$ operations. The simulation is divided equally among 12 identical processors running in parallel. If each processor can perform $4.5 \times 10^9$ operations per second, how many seconds will it take to complete the simulation?

6.075 \times 10^{22}

3.6 \times 10^4

3.0 \times 10^3

4.32 \times 10^5

3.0 \times 10^4

Show answer & explanation

Answer: $3.0 \times 10^3$

Answer

3.0 \times 10^3

The correct answer is

3.0 \times 10^3

. To find the total time to complete the simulation, first compute the total operations the 12 parallel processors can perform per second:

12 \times (4.5 \times 10^9) = 5.4 \times 10^{10}

operations/second. Next, divide the total number of required operations by this combined rate:

\frac{1.62 \times 10^{14}}{5.4 \times 10^{10}} = 0.3 \times 10^{14-10} = 0.3 \times 10^4 = 3.0 \times 10^3

seconds.

Step-by-Step Solution

1

Determine the combined processing rate of the 12 processors working in parallel.

Combined Rate =

12 \times (4.5 \times 10^9) = 5.4 \times 10^{10}

operations per second.

Since the 12 processors run in parallel, their individual speeds are added together to find the total rate of operations performed per second.

2

Divide the total number of operations required by the combined rate of all processors to find the time in seconds.

Time =

\frac{1.62 \times 10^{14}}{5.4 \times 10^{10}}

Time equals the total workload divided by the combined rate of work.

3

Perform the division and convert the result to standard scientific notation.

Time =

0.3 \times 10^{14 - 10} = 0.3 \times 10^4 = 3.0 \times 10^3

seconds.

Subtracting the exponents during division gives

10^4

. The coefficient

0.3

is rewritten in standard scientific notation as

3.0 \times 10^{-1}

, so

0.3 \times 10^4 = 3.0 \times 10^3

.

Key Concept

Scientific notation operations and rate division

Question 119Question →

On a standard number line, the coordinate of point $P$ is an integer $x$ . If the sum of the distances from $P$ to $-2$ and from $P$ to $4$ is equal to the distance from $P$ to $10$ , what is the sum of all possible values of $x$ ?

Show answer & explanation

Answer: -4

Answer

The sum of all possible values of the integer

x

is

-4

.

The correct answer is

-4

. The distance between any two coordinates

a

and

b

on a number line is defined as

|a - b|

. Translating the problem gives the equation

|x + 2| + |x - 4| = |x - 10|

. Breaking the number line into intervals around the critical points

x = -2

,

x = 4

, and

x = 10

yields two valid integer solutions:

x = -8

and

x = 4

. The sum of these values is

-8 + 4 = -4

.

Step-by-Step Solution

1

Express the distances on the number line using absolute values.

The distance from

P(x)

to

-2

is

|x - (-2)| = |x + 2|

. The distance from

P(x)

to

4

is

|x - 4|

. The distance from

P(x)

to

10

is

|x - 10|

. The equation is

|x + 2| + |x - 4| = |x - 10|

.

The absolute value

|a - b|

represents the distance between points

a

and

b

on a standard number line.

2

Solve the equation by testing the intervals defined by the critical points

x = -2

,

x = 4

, and

x = 10

.

We analyze the four intervals:
- For

x < -2

:

-(x + 2) - (x - 4) = -(x - 10) \implies -2x + 2 = -x + 10 \implies x = -8

. Since

-8 < -2

, this is a valid solution.
- For

-2 \leq x < 4

:

(x + 2) - (x - 4) = -(x - 10) \implies 6 = -x + 10 \implies x = 4

. Since

4

is not in

[-2, 4)

, there is no solution in this interval.
- For

4 \leq x < 10

:

(x + 2) + (x - 4) = -(x - 10) \implies 2x - 2 = -x + 10 \implies 3x = 12 \implies x = 4

. Since

4

is in

[4, 10)

, this is a valid solution.
- For

x \geq 10

:

(x + 2) + (x - 4) = x - 10 \implies 2x - 2 = x - 10 \implies x = -8

. Since

-8 < 10

, there is no solution in this interval.

Absolute value terms change sign at their critical points, requiring case-by-case evaluation.

3

Sum all valid integer solutions.

The valid values for

x

are

-8

and

4

. Their sum is

-8 + 4 = -4

.

The problem asks for the sum of all possible values of

x

.

Key Concept

Representing geometric distances on a number line using absolute value equations and solving them using interval analysis.

Question 120Question →

A red blood cell has a diameter of approximately $0.000008$ meters. When this number is written in scientific notation as $a \times 10^n$ , where $1 \leq a < 10$ , what is the value of $n$ ?

Show answer & explanation

Answer: -6

Answer

The value of the exponent is

-6

.

To write the number

0.000008

in scientific notation, we shift the decimal point 6 places to the right to get

8

. Since the original value is less than 1, the exponent is negative, giving

8 \times 10^{-6}

. The value of

n

is therefore

-6

.

Step-by-Step Solution

1

Locate the decimal point and determine how many places it must be shifted to obtain a coefficient between 1 and 10.

The decimal point must be shifted 6 places to the right to get the number

8

.

Scientific notation requires the lead coefficient

a

to satisfy

1 \leq a < 10

.

2

Determine the sign and value of the exponent based on the decimal shift.

The exponent

n

is

-6

.

Moving the decimal point to the right to write a decimal value less than 1 results in a negative exponent equal to the number of shifts.

Key Concept

Scientific Notation for Decimals Less Than One

Pre-Algebra

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept