All practice questions

Question 921Question →

A corporate advisory firm executes two types of client projects: Type A and Type B. Each Type A project requires $x$ hours of financial modeling and $y$ hours of executive reporting. Each Type B project requires $x + 3$ hours of financial modeling and $2y - 2$ hours of executive reporting. During the first quarter, the firm completed 6 Type A projects and 4 Type B projects. If the total time spent on financial modeling across all 10 projects was 72 hours and the total time spent on executive reporting was 62 hours, what is the value of $3x + 4y$ ?

Show answer & explanation

Answer: 38

Answer

The value of

3x + 4y

is 38.

Setting up total financial modeling hours gives

6x + 4(x + 3) = 72

, which yields

x = 6

. Setting up total executive reporting hours gives

6y + 4(2y - 2) = 62

, which yields

y = 5

. Substituting these values into

3x + 4y

yields

3(6) + 4(5) = 38

.

Step-by-Step Solution

1

Formulate and solve the linear equation for financial modeling hours.

x = 6

6 Type A projects take

6x

hours and 4 Type B projects take

4(x + 3)

hours. Summing them yields

6x + 4x + 12 = 72

, which simplifies to

10x = 60

, giving

x = 6

.

2

Formulate and solve the linear equation for executive reporting hours.

y = 5

6 Type A projects take

6y

hours and 4 Type B projects take

4(2y - 2)

hours. Summing them yields

6y + 8y - 8 = 62

, which simplifies to

14y = 70

, giving

y = 5

.

3

Substitute

x = 6

and

y = 5

into the targeted expression

3x + 4y

.

38

3(6) + 4(5) = 18 + 20 = 38

.

Key Concept

Linear Equations in One and Two Variables

Question 922Question →

An express delivery truck travels from Warehouse A to Warehouse B along a $180$ -mile route. For the first $60$ miles, the truck maintains a constant speed of $v$ miles per hour. For the remaining $120$ miles, heavy traffic reduces its constant speed to $\frac{v}{2}$ miles per hour. If the average speed of the truck for the entire $180$ -mile trip was $36$ miles per hour, what was the truck's speed $v$ , in miles per hour, during the first $60$ miles?

36

48

54

60

75

Show answer & explanation

Answer: $60$

Answer

The speed

v

during the first leg of the trip was

60

miles per hour.

The correct answer is

60

miles per hour. Total trip time is the sum of time on the first leg (

\frac{60}{v}

hours) and time on the second leg (

\frac{120}{v/2} = \frac{240}{v}

hours), giving a total time of

\frac{300}{v}

hours. Since average speed equals total distance divided by total time,

\frac{180}{300/v} = \frac{3v}{5} = 36

, solving to

v = 60

.

Step-by-Step Solution

1

Express the time spent on each leg of the trip in terms of

v

.

Leg 1 time:

t_1 = \frac{60}{v}

hours. Leg 2 time:

t_2 = \frac{120}{v/2} = \frac{240}{v}

hours.

Time equals distance divided by rate. The rate for the second leg is half of

v

.

2

Calculate the total time for the entire

180

-mile trip in terms of

v

.

Total time

T = t_1 + t_2 = \frac{60}{v} + \frac{240}{v} = \frac{300}{v}

hours.

Summing the travel times of both legs gives the total trip duration.

3

Set up the average speed equation using total distance divided by total time.

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{180}{\frac{300}{v}} = \frac{180v}{300} = \frac{3v}{5}$.

Average speed for any multi-leg trip is defined as total distance divided by total time.

4

Equate the average speed expression to the given value of

36

mph and solve for

v

.

\frac{3v}{5} = 36 \implies 3v = 180 \implies v = 60$ miles per hour.

Multiplying both sides by

5

and dividing by

3

isolates

v

.

Key Concept

Average speed for a multi-leg journey is always calculated as total distance divided by total time, not by taking the arithmetic mean of the speeds.

Estimated Time:2m 0s

Question 923Question →

A courier service dispatches a delivery driver from Facility A to Facility B along a straight $120$ -mile route at a constant speed of $r$ miles per hour. On the return journey from Facility B to Facility A along the same route, traffic congestion reduces the driver's constant speed by $20$ miles per hour. If the total driving time for the entire round trip is $5$ hours, what was the driver's speed, in miles per hour, on the return trip?

38

40

48

60

8

Show answer & explanation

Answer: 40

Answer

The driver's speed on the return trip was 40 miles per hour.

Setting the total time equation to 120/r + 120/(r - 20) = 5 leads to the quadratic equation r^2 - 68r + 480 = 0. Factoring gives roots r = 60 and r = 8. Since r = 8 yields a negative return speed, r must be 60. Subtracting 20 gives the correct return speed of 40 mph.

Step-by-Step Solution

1

Set up expressions for outbound and return travel times in terms of r.

Outbound time = \frac{120}{r} hours; Return time = \frac{120}{r - 20} hours.

Time equals distance divided by speed.

2

Formulate the equation for total round-trip time.

\frac{120}{r} + \frac{120}{r - 20} = 5

The sum of the travel times for both legs equals 5 hours.

3

Simplify the equation and convert it into standard quadratic form.

r^2 - 68r + 480 = 0

Dividing by 5 gives \frac{24}{r} + \frac{24}{r - 20} = 1, and multiplying by r(r - 20) yields 24(r - 20) + 24r = r(r - 20).

4

Solve the quadratic equation for r and eliminate extraneous solutions.

r = 60 mph (since r = 8 gives a negative return speed).

Factoring gives (r - 60)(r - 8) = 0. The root r = 8 is physically invalid because r - 20 must be positive.

5

Calculate the return speed r - 20.

60 - 20 = 40 mph

The return speed is 20 mph slower than the outbound speed.

Key Concept

Algebraic Modeling of Motion & Distance-Rate-Time Relationships

Estimated Time:2m 0s

Question 924Question →

For any positive integer $n$ , what is the units digit of the expression $S = 7^{4n+1} + 9^{2n} + 3^{4n+3}$ ?

0

3

5

7

9

Show answer & explanation

Answer: 5

Answer

The units digit of the expression is 5.

Each base has a predictable units digit cyclicity. The pattern for powers of

7

is

7, 9, 3, 1

(period 4), giving a units digit of

7

for exponent

4n+1

. The pattern for powers of

9

is

9, 1

(period 2), giving a units digit of

1

for even exponent

2n

. The pattern for powers of

3

is

3, 9, 7, 1

(period 4), giving a units digit of

7

for exponent

4n+3

. Adding these units digits gives

7 + 1 + 7 = 15

, whose units digit is

5

.

Step-by-Step Solution

1

Determine the units digit of

7^{4n+1}

The units digit of powers of

7

follows a repeating pattern of length 4:

7, 9, 3, 1

. Since the exponent

4n+1

has a remainder of

1

when divided by

4

, the units digit is

7

.

Units digit cyclicity of base

7

repeats every 4 powers.

2

Determine the units digit of

9^{2n}

The units digit of powers of

9

follows a repeating pattern of length 2:

9, 1

. For any positive integer

n

, the exponent

2n

is even, so the units digit is

1

.

Even powers of

9

always end in

1

.

3

Determine the units digit of

3^{4n+3}

The units digit of powers of

3

follows a repeating pattern of length 4:

3, 9, 7, 1

. Since the exponent

4n+3

has a remainder of

3

when divided by

4

, the units digit is

7

.

Units digit cyclicity of base

3

repeats every 4 powers.

4

Sum the units digits and take the final units digit

7 + 1 + 7 = 15

, which has a units digit of

5

.

The units digit of a sum of numbers equals the units digit of the sum of their individual units digits.

Key Concept

Units digit cyclicity of integer powers

Estimated Time:1m 30s

Question 925Question →

Set $S$ consists of $n$ consecutive integers, where $n > 1$ . The sum of all the elements in Set $S$ except the greatest element is $360$ , and the sum of all the elements in Set $S$ except the least element is $440$ . What is the median of the elements in Set $S$ ?

0

4

5

8

10

Show answer & explanation

Answer: $5$

Answer

The median of the elements in Set

S

is

5

.

Subtracting the given partial sums gives the difference between the largest and smallest elements:

(T - a_1) - (T - a_n) = 440 - 360 = 80

. For a set of

n

consecutive integers,

a_n - a_1 = n - 1

, so

n = 81

. In any set of consecutive integers, the arithmetic mean equals the median,

m

. Therefore, the total sum of all

81

elements is

81m

. The middle term is the 41st element (

m

), which means the 81st element is

m + 40

. Substituting these into

T - a_n = 360

yields

81m - (m + 40) = 360

, simplifying to

80m = 400

, which gives

m = 5

.

Step-by-Step Solution

1

Set up equations for the total sum

T

of Set

S

.

Let

a_1

be the least element and

a_n

be the greatest element.

T - a_n = 360

and

T - a_1 = 440

.

Subtracting the greatest element leaves

360

, and subtracting the least element leaves

440

.

2

Find the difference between the greatest and least elements

a_n - a_1

.

(T - a_1) - (T - a_n) = 440 - 360 \implies a_n - a_1 = 80

.

Subtracting the two sum equations eliminates the total sum

T

.

3

Determine the number of elements

n

in Set

S

.

For consecutive integers,

a_n - a_1 = n - 1

. Thus,

n - 1 = 80 \implies n = 81

.

The difference between the

n

-th and 1st term of consecutive integers is

n - 1

.

4

Relate the total sum

T

and the greatest element

a_n

to the median

m

.

Since

n = 81

is odd, the mean equals the median

m

. Total sum

T = 81m

. The greatest element is

a_{81} = m + 40

.

In an evenly spaced set, total sum is

n \times m

, and the last term is

m + \frac{n-1}{2}

.

5

Solve for the median

m

.

T - a_n = 360 \implies 81m - (m + 40) = 360 \implies 80m - 40 = 360 \implies 80m = 400 \implies m = 5

.

Substituting

T = 81m

and

a_n = m + 40

into the first equation allows solving for

m

directly.

Key Concept

Mean-Median Equivalence and Counting Terms in Consecutive Integer Sets

Question 926Question →

What is the sum of all the distinct prime factors of the integer $6^8 - 6^6$ ?

Show answer & explanation

Answer: 17

Answer

The sum of the distinct prime factors of

6^8 - 6^6

is 17.

Factoring out

6^6

yields

6^6(6^2 - 1) = 6^6 \times 35

. The prime factors of

6^6

are 2 and 3, while the prime factors of 35 are 5 and 7. The distinct prime factors are 2, 3, 5, and 7. Summing these prime numbers gives

2 + 3 + 5 + 7 = 17

.

Step-by-Step Solution

1

Factor out the common term

6^6

6^8 - 6^6 = 6^6(6^2 - 1) = 6^6 \times 35

Factoring simplifies the large exponent expression into a product of smaller integers.

2

Break down each base into its prime factors

6^6 = (2 \times 3)^6 = 2^6 \times 3^6

and

35 = 5 \times 7

Prime factorization requires expressing all bases as prime numbers.

3

List the distinct prime bases and compute their sum

Distinct prime factors: 2, 3, 5, 7. Sum =

2 + 3 + 5 + 7 = 17

.

Exponents do not affect which prime numbers are factors, only how many times they divide the number.

Key Concept

Prime Factorization of Difference of Exponents

Question 927Question →

Biopharmaceuticals produced via cell culture require precise environmental stability inside bioreactors. Last year, several batches of a critical biologic drug at a manufacturing facility were lost to opportunistic bacterial contamination. To eliminate future batch rejections, plant engineers installed automated sensor arrays that monitor nutrient consumption rates in real time. The engineers claim that this system will prevent future bacterial contamination losses because early detection of abnormal nutrient consumption allows immediate, targeted anti-microbial treatment before bacteria proliferate. Which of the following, if true, most strongly supports the plant engineers' claim?

In pilot trials with identical bioreactors, bacterial strains consistently altered nutrient consumption rates several hours before reaching cell densities capable of causing batch loss.

The targeted anti-microbial agents currently used in the facility disrupt the metabolic pathways of host cells, significantly reducing overall yield.

Installing automated sensor arrays in bioreactors reduces overall facility electricity consumption by optimizing temperature regulation cycles.

Nutrient consumption rates in cell cultures frequently fluctuate due to harmless ambient temperature variations rather than bacterial growth.

Reducing batch rejections is essential for maintaining the manufacturing facility's long-term annual profit margins.

Show answer & explanation

Answer: In pilot trials with identical bioreactors, bacterial strains consistently altered nutrient consumption rates several hours before reaching cell densities capable of causing batch loss.

Answer

The argument is most strongly supported by evidence showing that in pilot trials, bacterial strains consistently altered nutrient consumption rates several hours before reaching cell densities capable of causing batch loss.

The correct answer provides critical empirical evidence establishing that the indicator (nutrient consumption changes) occurs significantly ahead of the harmful outcome (batch loss). This confirms the core assumption of the engineers' plan: that sensor detection provides a usable window of opportunity to intervene effectively before proliferation ruins the batch.

Step-by-Step Solution

1

Identify the conclusion and main premise of the argument.

Conclusion: Real-time nutrient sensor arrays will eliminate future batch contamination losses. Premise: Early detection of abnormal nutrient consumption allows immediate targeted treatment before bacteria proliferate.

Understanding the precise link between early detection and intervention timing is crucial for identifying assumptions.

2

Analyze the underlying gap in the reasoning.

The argument assumes that bacterial growth changes nutrient consumption detectably *before* contamination reaches a point where the batch is already ruined or beyond treatment.

If bacterial alteration of nutrient levels occurs too late, early intervention will fail.

3

Evaluate which option validates this necessary temporal link.

The statement verifying that bacterial strains alter nutrient consumption several hours before destructive cell densities are reached proves that the detection window provides sufficient time for successful anti-microbial intervention.

Directly confirming the feasibility and timing of the proposed causal mechanism provides the strongest support.

Key Concept

Strengthening Arguments via Confirming Necessary Assumptions and Causal Timelines

Estimated Time:2m 0s

Question 928Question →

If $n$ is an integer such that $|3 - 2n| \le 9$ and $|n - 1| \ge 2$ , what is the sum of all possible values of $n$ ?

Show answer & explanation

Answer: 12

Answer

The sum of all possible integer values of

n

is 12.

Solving

|3 - 2n| \le 9

gives

-3 \le n \le 6

. Solving

|n - 1| \ge 2

restricts

n

to

n \le -1

or

n \ge 3

. Taking the intersection yields the integer set

\{-3, -2, -1, 3, 4, 5, 6\}

, whose sum equals 12.

Step-by-Step Solution

1

Unpack and solve

|3 - 2n| \le 9

-3 \le n \le 6

Removing absolute value yields

-9 \le 3 - 2n \le 9

. Dividing by

-2

requires reversing the inequality direction.

2

Unpack and solve

|n - 1| \ge 2

n \le -1

or

n \ge 3

An absolute value greater than or equal to 2 implies distance from 1 is at least 2 units in either direction.

3

Determine the intersection set of integers

\{-3, -2, -1, 3, 4, 5, 6\}

Filters out integers 0, 1, and 2 from the continuous range

[-3, 6]

.

4

Calculate the sum of the valid integers

12

Summing

(-3) + (-2) + (-1) + 3 + 4 + 5 + 6 = 12

.

Key Concept

Combining system of absolute value inequalities and handling inequality sign flips when dividing by negative quantities.

Question 929Question →

What is the sum of all valid real solutions to the equation $|2x + 15| = 1 - 5x$ ?

-2

\frac{10}{3}

\frac{16}{3}

-\frac{30}{7}

2

Show answer & explanation

Answer: $-2$

Answer

The sum of all valid real solutions is

-2

.

Solving the absolute value equation

|2x + 15| = 1 - 5x

yields two candidate values:

x = -2

and

x = \frac{16}{3}

. Substituting

x = -2

into the right-hand side gives

1 - 5(-2) = 11

, which is non-negative and matches

|2(-2) + 15| = 11

. Substituting

x = \frac{16}{3}

gives

1 - 5\left(\frac{16}{3}\right) = -\frac{77}{3}

, which is negative and therefore invalid. Thus,

x = -2

is the unique valid solution, making the sum

-2

.

Step-by-Step Solution

1

Set up the two cases for the absolute value equation

|2x + 15| = 1 - 5x

.

Case 1:

2x + 15 = 1 - 5x

; Case 2:

2x + 15 = -(1 - 5x)

.

By definition,

|u| = v

implies

u = v

or

u = -v

, provided

v \geq 0

.

2

Solve Case 1 for

x

.

7x = -14 \implies x = -2

.

Adding

5x

and subtracting

15

from both sides isolates

x

.

3

Solve Case 2 for

x

.

2x + 15 = -1 + 5x \implies 3x = 16 \implies x = \frac{16}{3}

.

Distributing the negative sign and combining like terms yields

x = \frac{16}{3}

.

4

Check candidate solutions against the non-negativity constraint

1 - 5x \geq 0

.

For

x = -2

:

1 - 5(-2) = 11 \geq 0

(Valid). For

x = \frac{16}{3}

:

1 - 5\left(\frac{16}{3}\right) = -\frac{77}{3} < 0

(Extraneous).

An absolute value cannot equal a negative number, so candidate solutions that make the right-hand side negative must be discarded.

5

Sum all valid real solutions.

The only valid solution is

x = -2

, so the sum is

-2

.

Extraneous solutions are excluded from the final sum.

Key Concept

Solving absolute value linear equations requires checking candidate solutions against domain constraints to filter out extraneous roots.

Estimated Time:2m 0s

Question 930Question →

For all real numbers $x > 0$ such that $x \neq 1$ , the algebraic expression $\frac{\sqrt{x} - 1}{\sqrt[4]{x} + 1} + \frac{\sqrt{x} + 1}{\sqrt[4]{x} - 1}$ is equivalent to $\frac{2(\sqrt[4]{x}^3 + 1)}{\sqrt{x} - 1}$ .

True

False

Show answer & explanation

Answer: True

Answer

The statement is True.

The statement is true because substituting

u = \sqrt[4]{x}

allows both expressions to be simplified via algebraic factoring identities to the identical expression

\frac{2(u^2 - u + 1)}{u - 1}

.

Step-by-Step Solution

1

Perform a substitution to simplify the radical exponents.

Let

u = \sqrt[4]{x}

, so that

\sqrt{x} = u^2

. The left-hand side becomes

\frac{u^2 - 1}{u + 1} + \frac{u^2 + 1}{u - 1}

.

Converting fourth roots and square roots into polynomial terms makes factoring easier.

2

Simplify the left-hand side expression.

\frac{(u-1)(u+1)}{u+1} + \frac{u^2+1}{u-1} = (u-1) + \frac{u^2+1}{u-1} = \frac{(u-1)^2 + u^2 + 1}{u-1} = \frac{2(u^2 - u + 1)}{u-1}.

Factoring the numerator of the first fraction cancels out the

(u+1)

term prior to combining terms.

3

Factor the right-hand side expression using polynomial identities.

\frac{2(u^3+1)}{u^2-1} = \frac{2(u+1)(u^2-u+1)}{(u+1)(u-1)} = \frac{2(u^2-u+1)}{u-1}.

Applying the sum of cubes identity

u^3 + 1 = (u + 1)(u^2 - u + 1)

allows cancellation of

(u+1)

from the denominator.

4

Compare the simplified left-hand side and right-hand side expressions.

Both expressions reduce to

\frac{2(\sqrt[4]{x}^2 - \sqrt[4]{x} + 1)}{\sqrt[4]{x} - 1}

.

Because both sides reduce to the identical simplified form for all

x > 0, x \neq 1

, the equivalence holds true.

Key Concept

Algebraic manipulation of radicals using fractional exponent substitution, difference of squares, and sum of cubes factoring.

Question 931Question →

A commuter drives from home to work along a straight route at a constant speed of $v$ miles per hour. On the return trip along the exact same route, heavy traffic causes the drive to take $50\%$ longer than the outbound trip. If the commuter's average speed for the entire round trip was $48$ miles per hour, what was the value of $v$ , in miles per hour?

36

50

57.6

60

72

Show answer & explanation

Answer: 60

Answer

60 miles per hour

The total distance for the round trip is

2d

. If outbound time is

t_1 = \frac{d}{v}

, the return time taking

50\%

longer is

1.5t_1 = \frac{1.5d}{v}

. The total time is

2.5t_1 = \frac{5d}{2v}

. Dividing total distance

2d

by total time

\frac{5d}{2v}

gives average speed

\frac{4v}{5}

. Setting

\frac{4v}{5} = 48

gives

v = 60

miles per hour.

Step-by-Step Solution

1

Define variables for distance and outbound time.

Let the one-way distance be

d

miles. Outbound time is

t_1 = \frac{d}{v}

hours.

Relating time to distance and speed using the standard formula

t = \frac{d}{v}

.

2

Express return time and return speed in terms of outbound parameters.

Return time is

t_2 = 1.5 t_1 = \frac{1.5d}{v}

hours.

The return trip takes 50% longer, so

t_2 = t_1 + 0.5 t_1 = 1.5 t_1

.

3

Calculate total distance and total time for the round trip.

Total distance

= 2d

. Total time

= t_1 + t_2 = t_1 + 1.5 t_1 = 2.5 t_1 = \frac{2.5d}{v} = \frac{5d}{2v}

hours.

Average speed requires total distance divided by total time.

4

Set up the average speed equation and solve for

v

.

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{5d}{2v}} = \frac{4v}{5}

. Given average speed is 48 mph:

\frac{4v}{5} = 48 \implies 4v = 240 \implies v = 60$.

Solving the linear algebraic equation yields the exact outbound speed.

Key Concept

Average speed for multi-leg journeys must always be calculated as Total Distance divided by Total Time.

Estimated Time:2m 0s

Question 932Question →

A retail store sets the selling price, $P$ , of a custom item based on its wholesale cost, $C$ , according to the linear equation $P = 1.4C + 20$ . The store's profit on each item is defined as $P - C$ . If the profit on a certain item is also equal to $0.2P + 40$ , what is the wholesale cost, $C$ , of the item in dollars?

Show answer & explanation

Answer: 200

Answer

The wholesale cost

C

of the item is $200.

By writing profit both as

P - C = 0.4C + 20

and as

0.2P + 40 = 0.28C + 44

, we obtain a single linear equation in terms of

C

:

0.4C + 20 = 0.28C + 44

. Subtracting

0.28C

and

20

from both sides gives

0.12C = 24

, which simplifies to

C = 200

.

Step-by-Step Solution

1

Substitute

P = 1.4C + 20

into the standard profit expression

P - C

.

Profit =

0.4C + 20

This expresses the profit solely as a linear function of the wholesale cost

C

.

2

Substitute

P = 1.4C + 20

into the alternative profit expression

0.2P + 40

.

Profit =

0.28C + 44

This converts the given percentage-based profit condition into an expression dependent only on

C

.

3

Set the two profit expressions equal to each other and solve the resulting single-variable linear equation.

0.12C = 24 \implies C = 200

Equating two valid expressions for the same quantity allows solving for the unknown variable

C

.

Key Concept

Solving Systems of Linear Equations by Algebraic Substitution

Estimated Time:2m 0s

Question 933Question →

A security analyst is designing a 4-character access code consisting of a digit, followed by two letters, followed by another digit. The code must be constructed according to the following rules:

- The first character must be a prime digit chosen from the set of single-digit integers $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ .
- The second and third characters must be distinct uppercase letters selected from the set $\{A, B, C, D, E\}$ .
- The fourth character must be an odd digit chosen from the set of single-digit integers $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ , such that it is not equal to the first digit.

How many different 4-character access codes can be created following these rules?

300

340

400

420

500

Show answer & explanation

Answer: 340

Answer

340 access codes

The correct count of 340 is obtained by separating the problem into two distinct cases based on whether the prime first digit is even or odd. For the even prime digit (2), there are 5 possible odd digits for the fourth slot, yielding 100 codes. For the three odd prime digits (3, 5, 7), the fourth slot has 4 available choices to avoid duplicating the first digit, yielding 240 codes. Summing both cases gives 340.

Step-by-Step Solution

1

Identify the set of prime single-digit integers.

The prime digits among

0

through

9

are

2, 3, 5,

and

7

(a total of

4

options). Note that

2

is even, while

3, 5,

and

7

are odd.

1

is not prime by definition, and

0, 4, 6, 8, 9

are composite or neither.

2

Calculate the number of ways to choose the two distinct letter characters.

5 \times 4 = 20

possible two-letter arrangements.

There are

5

choices for the second character and

4

remaining choices for the third character since they must be distinct.

3

Evaluate Case 1: The first digit is the even prime digit (

2

).

1 \times 20 \times 5 = 100

codes.

There is

1

choice for the first digit (

2

). The odd digits available for the fourth character are

1, 3, 5, 7, 9

(

5

choices). Since

2

is even, it never matches any odd digit.

4

Evaluate Case 2: The first digit is an odd prime digit (

3, 5,

or

7

).

3 \times 20 \times 4 = 240

codes.

There are

3

choices for the first digit. The odd digits available for the fourth character are

1, 3, 5, 7, 9

(

5

total), but the fourth digit cannot equal the first digit, leaving

5 - 1 = 4

choices.

5

Sum the total codes across both mutually exclusive cases.

100 + 240 = 340

total codes.

By the Fundamental Counting Principle and Addition Principle for disjoint sets.

Key Concept

Fundamental Counting Principle with Conditional Restrictions and Case Analysis

Estimated Time:2m 0s

Question 934Question →

An artisanal coffee roasting company purchases unroasted green coffee beans at a cost of $\$12.00$ per kilogram. During the roasting process, moisture loss reduces the total weight of the coffee beans by $20\%$ . To set the regular retail price per kilogram of roasted beans, the company marks up the effective cost price per kilogram of roasted beans by $50\%$ . If the company sells the roasted beans at a promotional discount of $10\%$ off the regular retail price, what is the net profit percentage earned on the total cost of the unroasted green coffee beans?

Show answer & explanation

Answer: 35

Answer

The net profit percentage earned on the total cost of the unroasted green coffee beans is 35%.

The correct calculation yields a total revenue of

16.20 from 0.80 kg of roasted coffee produced from 1 kg of green coffee purchased for

12.00. This results in a net profit of

4.20, which is exactly 35% of the initial

12.00 cost.

Step-by-Step Solution

1

Calculate the effective cost per kilogram of roasted beans after weight loss.

Effective cost per kg of roasted beans =

12.00 / (1 - 0.20) =

15.00 per kg.

Because weight decreases by 20%, 1 kg of green beans yields 0.80 kg of roasted beans, raising the per-unit cost.

2

Calculate the regular retail price per kilogram of roasted beans.

Regular retail price =

15.00 * (1 + 0.50) =

22.50 per kg.

The 50% markup is applied to the effective cost price of the roasted beans.

3

Calculate the promotional selling price after a 10% discount.

Discounted selling price =

22.50 * (1 - 0.10) =

20.25 per kg.

The discount reduces the regular retail price by 10%.

4

Find total revenue generated from the yield of 1 kg of green coffee beans.

Total revenue = 0.80 kg *

20.25/kg =

16.20.

1 kg of initial green beans produces 0.80 kg of sellable roasted product.

5

Calculate the net profit percentage based on the initial cost.

Net profit percentage = ((

16.20 -

12.00) /

12.00) * 100% = (

4.20 / $12.00) * 100% = 35%.

Profit percentage is the net profit divided by the original total cost.

Key Concept

Multi-step profit calculation involving shrink/yield loss, markup, successive discounts, and base identification.

Question 935Question →

If $n = 18^4 \times 35^3 \times 22^2$ , what is the total number of prime factors of $n$ , counting multiplicities (the sum of the exponents in its prime factorization)?

5

9

18

22

23

Show answer & explanation

Answer: 22

Answer

The total number of prime factors of

n

, counting multiplicities, is 22.

The correct answer is obtained by rewriting each base into prime factors (

18 = 2 \times 3^2

,

35 = 5 \times 7

,

22 = 2 \times 11

), combining powers of equal bases to obtain

n = 2^6 \times 3^8 \times 5^3 \times 7^3 \times 11^2

, and summing the resulting exponents (

6 + 8 + 3 + 3 + 2 = 22

).

Step-by-Step Solution

1

Express each composite base in terms of its prime factors.

18 = 2 \times 3^2

,

35 = 5 \times 7

, and

22 = 2 \times 11

.

Prime factorization requires writing every base strictly as a product of prime numbers.

2

Substitute the prime factorizations into the expression for

n

and apply exponent rules.

n = (2 \times 3^2)^4 \times (5 \times 7)^3 \times (2 \times 11)^2 = (2^4 \times 3^8) \times (5^3 \times 7^3) \times (2^2 \times 11^2)

.

Distribute exponents over multiplication using

(a \times b)^k = a^k \times b^k

and

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

.

3

Combine like prime bases by adding their exponents.

n = 2^{4+2} \times 3^8 \times 5^3 \times 7^3 \times 11^2 = 2^6 \times 3^8 \times 5^3 \times 7^3 \times 11^2

.

Combine terms with identical bases according to exponent rules (

a^m \times a^n = a^{m+n}

).

4

Sum the exponents of all prime factors to find the total count including multiplicities.

6 + 8 + 3 + 3 + 2 = 22

.

The total number of prime factors counted with multiplicity is given by the sum of exponents in the canonical prime factorization.

Key Concept

Prime Factorization and Exponent Rules

Estimated Time:1m 30s

Question 936Question →

Set $S$ consists of $n$ consecutive odd integers. The sum of all elements in Set $S$ is $195$ . If the largest element in Set $S$ is $9$ times the smallest element in Set $S$ , what is the value of $n$ ?

3

9

12

13

15

Show answer & explanation

Answer: $13$

Answer

The total number of elements in Set

S

is

13

.

For any set of consecutive odd integers, the terms are spaced

2

units apart. Expressing the largest term as

a + 2(n-1) = 9a

yields

n = 4a + 1

. Using the average formula for evenly spaced sets,

\text{Sum} = n \times \frac{\text{first} + \text{last}}{2} = n(5a) = 195

, giving

an = 39

. Substituting

n = 4a + 1

gives

4a^2 + a - 39 = 0

, which factors as

(4a + 13)(a - 3) = 0

, so

a = 3

. Substituting

a = 3

into

n = 4a + 1

gives

n = 13

.

Step-by-Step Solution

1

Express the largest term in terms of the smallest term

a

and term count

n

.

The largest term is

a + 2(n - 1)

. Since the largest term is

9a

, we have

a + 2(n - 1) = 9a \implies 2(n - 1) = 8a \implies n - 1 = 4a \implies n = 4a + 1

.

Consecutive odd integers increase by increments of

2

.

2

Express the sum of the set using the arithmetic mean of an evenly spaced set.

\text{Average} = \frac{\text{Smallest} + \text{Largest}}{2} = \frac{a + 9a}{2} = 5a. \text{Sum} = n \times \text{Average} \implies 195 = n(5a) \implies an = 39.

For any set of consecutive odd integers, the average is the mean of the first and last terms.

3

Substitute

n = 4a + 1

into

an = 39

and solve for

a

.

a(4a + 1) = 39 \implies 4a^2 + a - 39 = 0 \implies (4a + 13)(a - 3) = 0. Since

a

must be a positive integer,

a = 3

.

The smallest element of a set of positive odd integers must be a positive odd integer.

4

Calculate

n

using

a = 3

.

n = 4(3) + 1 = 13.

Substitute the value of

a

back into the formula derived in Step 1.

Key Concept

Consecutive Integers and Number Sets

Question 937Question →

What is the value of $\sqrt{16^5 + 16^5 + 16^5 + 16^5}$ ?

2^{10}

2^{11}

2^{12}

2^{15}

2^{20}

Show answer & explanation

Answer: $2^{11}$

Answer

2^{11}

Combining the four identical terms inside the radical gives

4 \cdot 16^5

. Expressing both factors with base 2 yields

2^2 \cdot (2^4)^5 = 2^2 \cdot 2^{20} = 2^{22}

. Taking the square root gives

\sqrt{2^{22}} = 2^{11}

, which matches the value

2^{11}

.

Step-by-Step Solution

1

Factor out the common term inside the square root.

16^5 + 16^5 + 16^5 + 16^5 = 4 \cdot 16^5

Adding four identical terms is equivalent to multiplying the term by 4.

2

Convert both numbers to powers of 2.

4 = 2^2

and

16^5 = (2^4)^5 = 2^{20}

, so

4 \cdot 16^5 = 2^2 \cdot 2^{20} = 2^{22}

Expressing terms with a common prime base allows exponent rules to be applied.

3

Apply the square root to the simplified exponential expression.

\sqrt{2^{22}} = (2^{22})^{1/2} = 2^{11}

Taking the square root of a base raised to a power is equivalent to dividing the exponent by 2.

Key Concept

Combining like terms with exponents and applying prime factorization exponent laws under radical signs.

Estimated Time:1m 30s

Question 938Question →

An jeweler creates a custom silver alloy by melting together two available alloys. Alloy X is $40\%$ silver by weight, and Alloy Y is $70\%$ silver by weight. If the jeweler uses $15$ grams more of Alloy Y than Alloy X to produce a final alloy mixture that is $60\%$ silver by weight, what is the total weight, in grams, of the final alloy mixture?

15

grams

30

grams

45

grams

60

grams

75

grams

Show answer & explanation

Answer: $45$ grams

Answer

45

grams

The total weight of the mixture is

45

grams. Letting

x

represent the grams of Alloy X, the weight of Alloy Y is

x + 15

. Equating the pure silver content gives

0.40x + 0.70(x + 15) = 0.60(2x + 15)

. Solving yields

x = 15

grams for Alloy X and

30

grams for Alloy Y, making the total final mixture weight

15 + 30 = 45

grams.

Step-by-Step Solution

1

Define variables for the weights of the alloys.

Let the weight of Alloy X be

x

grams. Since Alloy Y has

15

grams more than Alloy X, the weight of Alloy Y is

x + 15

grams. The total weight of the mixture is

x + (x + 15) = 2x + 15

grams.

Establishing explicit algebraic terms for component and total weights is essential for setting up the mixture equation.

2

Set up the equation for total silver content.

0.40x + 0.70(x + 15) = 0.60(2x + 15)

The total amount of pure silver from Alloy X and Alloy Y combined must equal the amount of pure silver in the final mixture.

3

Solve the algebraic equation for

x

.

0.40x + 0.70x + 10.5 = 1.20x + 9 \Rightarrow 1.10x + 10.5 = 1.20x + 9 \Rightarrow 0.10x = 1.5 \Rightarrow x = 15

.

Isolating

x

determines the weight of Alloy X used.

4

Calculate the total weight of the final alloy mixture.

Total weight =

2x + 15 = 2(15) + 15 = 45

grams.

The question asks for the total weight of the mixture, requiring substitution of

x = 15

back into the total weight expression.

Key Concept

Weighted average and concentration conservation in mixtures

Question 939Question →

If $k$ is an integer such that $|2k - 7| \le 11$ and $|k + 2| > 4$ , how many possible values of $k$ exist?

Show answer & explanation

Answer: 7

Answer

The total number of possible integer values of

k

is 7.

Unfolding

|2k - 7| \le 11

gives

-2 \le k \le 9

, representing 12 possible integers. Unfolding

|k + 2| > 4

gives

k > 2

or

k < -6

. Intersecting these two regions for real integers eliminates

k < -6

(since

k \ge -2

) and restricts the set to

3 \le k \le 9

. The integers in this range are

3, 4, 5, 6, 7, 8,

and

9

, yielding a total of 7 valid values.

Step-by-Step Solution

1

Solve the first absolute value inequality

|2k - 7| \le 11

.

-2 \le k \le 9

Expanding the absolute value gives

-11 \le 2k - 7 \le 11

. Adding 7 to all parts yields

-4 \le 2k \le 18

, and dividing by 2 results in

-2 \le k \le 9

.

2

Solve the second absolute value inequality

|k + 2| > 4

.

k > 2

or

k < -6

Expanding the strict absolute value inequality gives two cases:

k + 2 > 4 \implies k > 2

, or

k + 2 < -4 \implies k < -6

.

3

Find the intersection of the two solution sets for integer values of

k

.

3 \le k \le 9

Since the first inequality requires

k \ge -2

, no integer can satisfy both

k \ge -2

and

k < -6

. Thus,

k

must satisfy

-2 \le k \le 9

and

k > 2

, which reduces to

3 \le k \le 9

.

4

Count the number of integers in the range

3 \le k \le 9

.

7

The valid integers in this range are

3, 4, 5, 6, 7, 8,

and

9

. The total count is

9 - 3 + 1 = 7

.

Key Concept

Solving compound absolute value inequalities for integer solution counts

Question 940Question →

For how many positive integer values of $n$ is $2^8 + 2^{11} + 2^n$ equal to the square of an integer?

Zero

One

Two

Three

Four

Show answer & explanation

Answer: One

Answer

There is exactly 1 positive integer value of

n

(specifically,

n = 12

).

The option stating 'One' is correct because factoring out

2^8

yields

2^8(9 + 2^{n-8})

. For this product to be a perfect square,

9 + 2^{n-8}

must equal

k^2

for some integer

k

. Rewriting this as

(k-3)(k+3) = 2^{n-8}

forces both factors to be powers of 2 whose difference is 6. The unique solution to

2^b - 2^a = 6

is

a = 1

and

b = 3

, which gives

n - 8 = 4

, so

n = 12

. No other positive integers

n \le 8

produce a square.

Step-by-Step Solution

1

Analyze the expression for

n > 8

by factoring out

2^8

.

2^8 + 2^{11} + 2^n = 2^8 (1 + 2^3 + 2^{n-8}) = 2^8 (9 + 2^{n-8})

.

Since

2^8 = (2^4)^2

is already a perfect square, the entire expression is a square if and only if

9 + 2^{n-8}

is a perfect square.

2

Set

9 + 2^{n-8} = k^2

for some integer

k > 3

and factor using prime power properties.

k^2 - 9 = 2^{n-8} \implies (k - 3)(k + 3) = 2^{n-8}

.

The difference of squares allows us to express the product of two integers as a power of 2.

3

Solve for the prime factors of the terms

(k-3)

and

(k+3)

.

Let

k - 3 = 2^a

and

k + 3 = 2^b

where

a + b = n - 8

and

b > a

. Subtracting the two equations gives

(k + 3) - (k - 3) = 2^b - 2^a = 6 \implies 2^a(2^{b-a} - 1) = 6 = 2^1 \cdot 3

.

The prime factorization of 6 uniquely dictates that

2^a = 2^1 \implies a = 1

, and

2^{b-a} - 1 = 3 \implies 2^{b-1} = 4 \implies b = 3

.

4

Determine

n

and test values of

n \le 8

.

Since

a = 1

and

b = 3

,

n - 8 = 1 + 3 = 4 \implies n = 12

. Testing

n \le 8

reveals no other squares (e.g., for

n=8

,

2^8(1+8+1)=10 \cdot 2^8

, not a square; for

n=3

,

2312 = 2^3 \cdot 17^2

, not a square).

This confirms that

n = 12

is the unique positive integer solution.

Key Concept

Prime factorization of differences of squares and prime power analysis

Estimated Time:2m 0s

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