All practice questions

Question 981Question →

A web hosting company charges each enterprise client a one-time fixed setup fee plus a constant monthly maintenance fee per server. A client operating $4$ servers pays a total of $\$1,100$ for the setup fee and the first $6$ months of server maintenance. A client operating $9$ servers pays a total of $\$1,850$ for the setup fee and the first $6$ months of server maintenance. What is the one-time fixed setup fee, in dollars?

Show answer & explanation

Answer: 500

Answer

500

The fixed setup fee is

\$500

. Letting

S

represent the fixed setup fee and

M

represent the 6-month maintenance fee per server, the two given scenarios yield

S + 4M = 1100

and

S + 9M = 1850

. Subtracting the first equation from the second gives

5M = 750

, which simplifies to

M = 150

. Substituting

M = 150

into

S + 4(150) = 1100

gives

S + 600 = 1100

, so

S = 500

.

Step-by-Step Solution

1

Define variables for the unknown fixed cost and per-server cost, and construct the system of linear equations.

Let

S

be the fixed setup fee in dollars and

M

be the 6-month maintenance fee per server in dollars. The equations are

S + 4M = 1100

and

S + 9M = 1850

.

Modeling the situational relationships as a linear system allows isolated solution of each unknown.

2

Subtract the two linear equations to eliminate the fixed fee

S

and solve for

M

.

5M = 750 \implies M = 150

.

Since the coefficient of

S

is 1 in both equations, elimination by subtraction directly isolates

M

.

3

Substitute the value of

M

back into the first equation to solve for

S

.

S + 4(150) = 1100 \implies S + 600 = 1100 \implies S = 500

.

Replacing

M

with 150 yields a linear equation in one variable for the setup fee.

Key Concept

Linear Equations in One and Two Variables

Estimated Time:1m 30s

Question 982Question →

A pharmacist has $80$ milliliters of a topical liquid formulation that is $5\%$ active ingredient by volume. How many milliliters of pure active ingredient must the pharmacist add to this formulation so that the resulting mixture is $24\%$ active ingredient by volume?

Show answer & explanation

Answer: 20

Answer

The pharmacist must add

20

milliliters of pure active ingredient.

Adding

20

mL of pure active ingredient increases the amount of solute from

4

mL to

24

mL and the total mixture volume from

80

mL to

100

mL, yielding a final concentration of

\frac{24}{100} = 24\%

.

Step-by-Step Solution

1

Determine initial amount of solute

Initial active ingredient volume =

80 \times 0.05 = 4

mL

Before adding pure active ingredient, the solution contains

5\%

active ingredient of the total

80

mL volume.

2

Formulate algebraic expressions for total active ingredient and total volume

Total active ingredient =

4 + x

mL; Total mixture volume =

80 + x

mL

Adding

x

mL of pure active ingredient increases both the solute volume and the total mixture volume by

x

.

3

Set up and solve the mixture concentration equation

\frac{4 + x}{80 + x} = \frac{6}{25} \implies 100 + 25x = 480 + 6x \implies 19x = 380 \implies x = 20

Setting the solute ratio equal to the target concentration of

24\%

yields a linear equation in

x

.

Key Concept

Mixture Concentration and Algebraic Dilution/Fortification

Question 983Question →

A corporation has three regional branches: Branch X, Branch Y, and Branch Z. The ratio of the number of employees in Branch X to Branch Y is $1 : 3$ . The average monthly salary of employees in Branch Y is $40\%$ higher than the average monthly salary of employees in Branch X. Branch Z has twice as many employees as Branch Y, and the average monthly salary of employees in Branch Z is $x\%$ lower than the combined average monthly salary of employees in Branches X and Y. If the overall average monthly salary across all three branches combined is $\$910$ , and the average monthly salary in Branch X is $\$1{}000$ , what is the value of $x$ ?

35

40

50

70

100

Show answer & explanation

Answer: 50

Answer

50

The combined average salary of Branches X and Y is calculated by weighting Branch X's salary (

\$1{}000

) with 1 unit of weight and Branch Y's salary (

\$1{}400

) with 3 units of weight, giving

\frac{1{}000 + 4{}200}{4} = \$1{}300

. For the entire company of

10

units of employees with an average salary of

\$910

, the total payroll is

10 \times 910 = \$9{}100

. Subtracting the combined payroll of Branches X and Y (

\$5{}200

) leaves

\$3{}900

for Branch Z's

6

units of employees. Thus, Branch Z's average salary is

\frac{3{}900}{6} = \$650

. Comparing

\$650

to

\$1{}300

gives a percentage decrease of

\frac{1{}300 - 650}{1{}300} \times 100\% = 50\%

.

Step-by-Step Solution

1

Determine the average monthly salary of Branch Y.

Branch Y average salary =

1{}000 \times 1.40 = \$1{}400

.

Branch Y's average salary is given as

40\%

higher than Branch X's average salary of

\$1{}000

.

2

Express employee counts in terms of a single variable

n

.

Branch X has

n

employees, Branch Y has

3n

employees, and Branch Z has

2 \times 3n = 6n

employees. Total employees =

n + 3n + 6n = 10n

.

The ratio of Branch X to Branch Y employees is

1:3

, and Branch Z has twice as many employees as Branch Y.

3

Calculate the combined average salary of Branches X and Y.

Combined average salary

S_{XY} = \frac{n(1{}000) + 3n(1{}400)}{n + 3n} = \frac{5{}200n}{4n} = \$1{}300

.

The combined average of two sets is total combined earnings divided by total combined size.

4

Calculate total company payroll and determine Branch Z's average salary.

Total payroll =

10n \times 910 = 9{}100n

. Branch Z payroll =

9{}100n - 5{}200n = 3{}900n

. Average salary for Branch Z

S_Z = \frac{3{}900n}{6n} = \$650

.

Total payroll is total employees times overall mean. Subtracting the combined payroll of X and Y yields Branch Z's payroll.

5

Compute the percentage by which Branch Z's average salary is lower than the combined average salary of Branches X and Y.

x = \frac{1{}300 - 650}{1{}300} \times 100\% = 50\%.

Percentage decrease is calculated as (Base Value - New Value) / Base Value.

Key Concept

Weighted Averages of Combined Sets

Estimated Time:2m 0s

Question 984Question →

How many integer values of $x$ satisfy the inequality $|x^2 - 5x| \le 6$ ?

Show answer & explanation

Answer: 8

Answer

The correct answer is 8.

To solve

|x^2 - 5x| \le 6

, express it as

-6 \le x^2 - 5x \le 6

. Solving

x^2 - 5x - 6 \le 0

gives

[-1, 6]

, while solving

x^2 - 5x + 6 \ge 0

gives

(-\infty, 2] \cup [3, \infty)

. Taking their intersection yields the solution set

[-1, 2] \cup [3, 6]

. The integers contained in this set are

-1, 0, 1, 2, 3, 4, 5, 6

, which gives 8 distinct integer values.

Step-by-Step Solution

1

Rewrite the absolute value inequality

-6 \le x^2 - 5x \le 6

For any real expression

A

and constant

k \ge 0

,

|A| \le k

is equivalent to

-k \le A \le k

.

2

Solve the upper bound condition

x^2 - 5x \le 6

-1 \le x \le 6

Subtract 6 from both sides to obtain

x^2 - 5x - 6 \le 0

. Factoring gives

(x - 6)(x + 1) \le 0

.

3

Solve the lower bound condition

x^2 - 5x \ge -6

x \le 2

or

x \ge 3

Add 6 to both sides to obtain

x^2 - 5x + 6 \ge 0

. Factoring gives

(x - 2)(x - 3) \ge 0

.

4

Combine the solution sets

[-1, 2] \cup [3, 6]

The intersection of

[-1, 6]

with

(-\infty, 2] \cup [3, \infty)

is the set of intervals

[-1, 2]

and

[3, 6]

.

5

Count all integer solutions in the combined set

8 integer values

The integer values in

[-1, 2]

are

-1, 0, 1, 2

(4 integers), and in

[3, 6]

are

3, 4, 5, 6

(4 integers), totaling

4 + 4 = 8

integers.

Key Concept

Solving quadratic absolute value inequalities using compound inequality decomposition.

Question 985Question →

A specialized coffee roastery produces three custom blends—Roast Alpha, Roast Beta, and Roast Gamma—using three varieties of single-origin beans: Grade A, Grade B, and Grade C.

- One batch of Roast Alpha requires 3 kg of Grade A, 1 kg of Grade B, and 2 kg of Grade C beans, and has a total raw material cost of $64. - One batch of Roast Beta requires 1 kg of Grade A, 4 kg of Grade B, and 2 kg of Grade C beans, and has a total raw material cost of$ 64.
- One batch of Roast Gamma requires 2 kg of Grade A, 2 kg of Grade B, and 5 kg of Grade C beans, and has a total raw material cost of $90.

What is the cost, in dollars, of 1 kg of Grade A beans?

Show answer & explanation

Answer: 12

Answer

The cost of 1 kg of Grade A beans is 12 dollars.

Setting up the 3-variable linear system

3x + y + 2z = 64

,

x + 4y + 2z = 64

, and

2x + 2y + 5z = 90

allows us to eliminate

z

by subtracting the second equation from the first, yielding

2x - 3y = 0

, or

x = 1.5y

. Substituting this relationship back into the system leads to

y = 8

and

x = 12

. Thus, 1 kg of Grade A beans costs 12 dollars.

Step-by-Step Solution

1

Formulate linear equations representing the total cost of each coffee blend batch.

Let

x

be the cost per kg of Grade A beans,

y

be the cost per kg of Grade B beans, and

z

be the cost per kg of Grade C beans:

\begin{aligned} (1)\quad 3x + y + 2z &= 64 \\ (2)\quad x + 4y + 2z &= 64 \\ (3)\quad 2x + 2y + 5z &= 90 \end{aligned}

Translating the word problem into a system of 3 linear equations with 3 variables.

2

Eliminate variable

z

by subtracting Equation (2) from Equation (1).

(3x + y + 2z) - (x + 4y + 2z) = 64 - 64 \implies 2x - 3y = 0 \implies x = 1.5y

Since both equations (1) and (2) contain the term

+2z

, subtracting them removes

z

directly and provides a simple relation between

x

and

y

.

3

Substitute

x = 1.5y

into Equation (1) and Equation (3) to obtain a system in terms of

y

and

z

.

From Equation (1):

3(1.5y) + y + 2z = 64 \implies 5.5y + 2z = 64 \implies 11y + 4z = 128 \quad (4)

From Equation (3):

2(1.5y) + 2y + 5z = 90 \implies 5y + 5z = 90 \implies y + z = 18 \implies z = 18 - y

Reducing the system from 3 variables down to 2 variables.

4

Substitute

z = 18 - y

into Equation (4) to solve for

y

, and subsequently calculate

x

.

11y + 4(18 - y) = 128 \implies 7y + 72 = 128 \implies 7y = 56 \implies y = 8

Using

x = 1.5y

:

x = 1.5 \times 8 = 12

Solving the single-variable linear equation for

y

, then substituting back to find the required cost

x

for Grade A beans.

Key Concept

Solving a 3-Variable System of Linear Equations via Variable Elimination

Question 986Question →

A specialty coffee roaster creates a signature espresso blend by combining Arabica coffee beans costing $18.00 per kilogram with Robusta coffee beans costing$ 12.00 per kilogram. The final blend weighs 50 kilograms and has an overall average cost of $15.60 per kilogram. How many kilograms of Arabica beans are included in the blend?

Show answer & explanation

Answer: 30

Answer

The blend contains 30 kilograms of Arabica coffee beans.

To determine the required quantity of Arabica beans, establish the total cost equation:

18A + 12(50 - A) = 50(15.60)

. Expanding the terms gives

18A + 600 - 12A = 780

, which simplifies to

6A = 180

, yielding

A = 30

kilograms.

Step-by-Step Solution

1

Define variables and relate component weights

Let

A

represent the mass of Arabica beans in kilograms. The mass of Robusta beans is

(50 - A)

kilograms.

The sum of the individual component weights must equal the total blend weight of 50 kilograms.

2

Formulate the weighted average total cost equation

18A + 12(50 - A) = 15.60 \times 50 = 780

The combined monetary cost of both bean types equals the total value of the 50 kg blend.

3

Solve for the unknown variable

A

6A + 600 = 780 \implies 6A = 180 \implies A = 30

Simplifying the algebraic linear equation determines the exact weight of Arabica beans.

Key Concept

Weighted Averages in Applied Contexts

Question 987Question →

In a plane, there are $n$ points such that no three points are collinear. If the number of distinct triangles that can be formed using these points as vertices is exactly $5$ times the number of distinct line segments that can be formed by joining pairs of these points, what is the value of $n$ ?

7

15

16

17

32

Show answer & explanation

Answer: $17$

Answer

The correct value of

n

is

17

.

The number of distinct triangles formed by

n

points (no three collinear) is

\binom{n}{3} = \frac{n(n-1)(n-2)}{6}

, and the number of line segments is

\binom{n}{2} = \frac{n(n-1)}{2}

. Setting

\binom{n}{3} = 5 \binom{n}{2}

gives

\frac{n(n-1)(n-2)}{6} = \frac{5n(n-1)}{2}

. Dividing both sides by

\frac{n(n-1)}{2}

simplifies the equation to

\frac{n-2}{3} = 5

, yielding

n - 2 = 15

, so

n = 17

.

Step-by-Step Solution

1

Express the number of triangles and line segments using combinations.

The number of triangles formed by selecting

3

non-collinear points from

n

points is

\binom{n}{3} = \frac{n(n-1)(n-2)}{6}

. The number of line segments formed by selecting

2

points from

n

points is

\binom{n}{2} = \frac{n(n-1)}{2}

.

Order does not matter when selecting vertices for triangles or endpoints for line segments.

2

Set up the given equation relating the two quantities.

\frac{n(n-1)(n-2)}{6} = 5 \times \frac{n(n-1)}{2}

The problem states that the number of triangles is

5

times the number of line segments.

3

Simplify the equation for

n \ge 3

.

Divide both sides by

\frac{n(n-1)}{2}

to get

\frac{n-2}{3} = 5

, which simplifies to

n - 2 = 15

.

Since

n \ge 3

,

n(n-1) \ne 0

, so dividing by common terms is valid.

4

Solve for

n

.

n = 15 + 2 = 17

Adding

2

to both sides isolates

n

.

Key Concept

Combinations in Geometry

Question 988Question →

A data set consists of $9$ positive integers. The set has a unique mode of $14$ , a median of $12$ , and an arithmetic mean of $11$ . If the range of the data set is $10$ , what is the maximum possible value of the largest integer in the set?

14

15

16

17

18

Show answer & explanation

Answer: $16$

Answer

The maximum possible value of the largest integer in the set is

16

.

The correct answer states that the maximum possible value is

16

. By ordering the

9

terms

x_1 \le x_2 \le \dots \le x_9

, the sum of all terms must equal

99

, with median

x_5 = 12

and

x_9 = x_1 + 10

. If

x_9 = 16

, then

x_1 = 6

, and we can construct a valid set

\{6, 6, 8, 9, 12, 14, 14, 14, 16\}

where

14

is the unique mode appearing

3

times. Trying a larger value such as

17

forces

x_1 = 7

, which makes it impossible to maintain

14

as the unique mode without violating the total sum of

99

.

Step-by-Step Solution

1

Express the total sum and set structure using ordered variables.

Let the

9

positive integers in non-decreasing order be

x_1 \le x_2 \le x_3 \le x_4 \le x_5 \le x_6 \le x_7 \le x_8 \le x_9

. The total sum is

9 \times 11 = 99

. The median is

x_5 = 12

. The range is

x_9 - x_1 = 10 \implies x_9 = x_1 + 10

.

Establishing the position of the median and the exact sum provides structural bounds for maximizing

x_9

.

2

Analyze the frequency requirements for the unique mode.

Since

14 > 12

(the median), the number

14

must lie in the upper half of the set (

x_6, x_7, x_8, x_9

). To be a unique mode,

14

must appear at least twice. If

x_9 = 16

, then

x_1 = 6

.

Maximizing

x_9

is equivalent to maximizing

x_1

, so we test the largest possible values for

x_1

and verify the uniqueness of the mode

14

.

3

Evaluate

x_1 = 7

(which would give

x_9 = 17

).

If

x_1 = 7

, then

x_9 = 17

. The sum equation becomes

7 + x_2 + x_3 + x_4 + 12 + x_6 + x_7 + x_8 + 17 = 99 \implies x_2 + x_3 + x_4 + x_6 + x_7 + x_8 = 63

. Since

x_2, x_3, x_4 \ge 7

and

x_6, x_7, x_8 \ge 12

, achieving this sum while keeping

14

as the unique mode is impossible (it would require

7

to appear

4

times, making

7

the mode instead of

14

). Thus,

x_9

cannot be

17

or greater.

Testing

x_9 = 17

shows a violation of the unique mode condition.

4

Construct a valid set for

x_1 = 6

(giving

x_9 = 16

).

If

x_1 = 6

and

x_9 = 16

, consider the set

\{6, 6, 8, 9, 12, 14, 14, 14, 16\}

. Sum =

6+6+8+9+12+14+14+14+16 = 99

. Median =

12

. Unique mode =

14

(frequency

3

). Range =

16 - 6 = 10

. All conditions are satisfied.

Constructing an explicit valid data set proves that

16

is attainable and is the maximum.

Key Concept

Maximizing elements in a constrained discrete data set using mean, median, mode, and range properties

Question 989Question →

A commercial real estate developer acquired a property for $\$P$ . Over the first year, the market value of the property increased by $25\%$ . During the second year, the property's value decreased by $x\%$ . In the third year, the value increased again by $20\%$ relative to its value at the end of the second year. If the final value of the property at the end of the third year was $14\%$ greater than the original acquisition price $\$P$ , what is the value of $x$ ?

21

24

30

31

76

Show answer & explanation

Answer: $24$

Answer

The value of

x

is 24.

To find the net effect of successive percentage changes, express each period's change as a multiplier of the value from the preceding period. A

25\%

increase corresponds to a multiplier of

1.25

, an

x\%

decrease corresponds to a multiplier of

\left(1 - \frac{x}{100}\right)

, and a

20\%

increase corresponds to a multiplier of

1.20

. Combined, these produce a final multiplier of

1.25 \times 1.20 \times \left(1 - \frac{x}{100}\right) = 1.50\left(1 - \frac{x}{100}\right)

. Setting this equal to the net

14\%

total increase (

1.14

) yields

1.50\left(1 - \frac{x}{100}\right) = 1.14

, which simplifies to

1 - \frac{x}{100} = 0.76

, giving

x = 24

.

Step-by-Step Solution

1

Express each sequential period's value using multiplier notation.

At Year 1 end:

V_1 = 1.25P

. At Year 2 end:

V_2 = 1.25P \left(1 - \frac{x}{100}\right)

. At Year 3 end:

V_3 = 1.25P \left(1 - \frac{x}{100}\right) \times 1.20

.

Successive percent changes compound on the intermediate values of each period, not on the original base price.

2

Equate the overall combined growth multiplier to the given total net change.

1.25 \times 1.20 \times \left(1 - \frac{x}{100}\right) = 1.14 \implies 1.50 \times \left(1 - \frac{x}{100}\right) = 1.14

The final value is given as

14\%

greater than

P

, which corresponds to a net multiplier of

1.14

.

3

Solve the algebraic equation for

x

.

1 - \frac{x}{100} = \frac{1.14}{1.50} = 0.76 \implies \frac{x}{100} = 0.24 \implies x = 24

Subtracting

0.76

from

1

yields the fractional decrease of

0.24

, which equals

24\%

.

Key Concept

Successive Percent Change and Multipliers

Estimated Time:2m 0s

Question 990Question →

If $a$ , $b$ , and $c$ are non-zero real numbers such that $a^2 b^3 c < 0$ , $\frac{a}{b^5 c^2} > 0$ , and $a c > b c$ , which of the following expressions MUST be true?

a > b

a c > 0

\frac{b - a}{c} < 0

b + c > 0

\frac{a}{c} > 0

Show answer & explanation

Answer: $\frac{b - a}{c} < 0$

Answer

The expression $\frac{b - a}{c} < 0$ MUST be true.

The condition

a^2 b^3 c < 0

requires

b c < 0

because

a^2 > 0

. The condition

\frac{a}{b^5 c^2} > 0

requires

a b > 0

because

c^2 > 0

. Combining these shows that

a

and

b

share the same sign, whereas

c

has the opposite sign. Consequently,

a c < 0

. From

a c > b c

, subtracting

b c

yields

(a - b) c > 0

. Multiplying by

-1

gives

(b - a) c < 0

, and dividing by

c^2 > 0

produces

\frac{b - a}{c} < 0

, which MUST be true in all cases.

Step-by-Step Solution

1

Determine the relative signs of

b

and

c

using

a^2 b^3 c < 0

.

b c < 0

, meaning

b

and

c

have opposite signs.

Since

a \neq 0

,

a^2 > 0

always. Dividing

a^2 b^3 c < 0

by

a^2

yields

b^3 c < 0

. Because

b^3

has the same sign as

b

,

b c < 0

.

2

Determine the relative signs of

a

and

b

using

\frac{a}{b^5 c^2} > 0

.

a b > 0

, meaning

a

and

b

have the same sign.

Since

c \neq 0

,

c^2 > 0

always. Multiplying by

c^2

gives

\frac{a}{b^5} > 0

, which implies

a

and

b^5

have the same sign. Thus

a b > 0

.

3

Determine the relationship between

a

and

c

.

a c < 0

, meaning

a

and

c

have opposite signs.

Since

a

and

b

have the same sign (

a b > 0

) while

b

and

c

have opposite signs (

b c < 0

),

a

and

c

must have opposite signs.

4

Analyze the inequality

a c > b c

.

\frac{b - a}{c} < 0

Rearranging

a c > b c

gives

a c - b c > 0 \implies (a - b) c > 0

. Multiplying both sides by

-1

flips the inequality:

(b - a) c < 0

. Dividing by the strictly positive quantity

c^2

yields

\frac{(b - a) c}{c^2} < 0 \implies \frac{b - a}{c} < 0

.

Key Concept

Deducing sign relationships of variables in inequalities and algebraic transformations without assuming positive signs.

Estimated Time:2m 0s

Question 991Question →

In an arithmetic sequence, the sum of the first $4$ terms is $28$ and the sum of the first $8$ terms is $88$ . What is the $10\text{th}$ term of this sequence?

18

20

22

24

26

Show answer & explanation

Answer: $22$

Answer

The 10th term of the sequence is 22.

Using the arithmetic series sum formula

S_n = \frac{n}{2}(2a_1 + (n-1)d)

, the conditions yield two linear equations:

2a_1 + 3d = 14

and

2a_1 + 7d = 22

. Subtracting these equations gives

4d = 8

, so

d = 2

. Substituting

d = 2

back into

2a_1 + 3(2) = 14

yields

a_1 = 4

. The 10th term is then calculated as

a_1 + 9d = 4 + 9(2) = 22

.

Step-by-Step Solution

1

Express the given sums using the arithmetic series sum formula

S_n = \frac{n}{2}(2a_1 + (n-1)d)

.

For

S_4 = 28

:

\frac{4}{2}(2a_1 + 3d) = 28 \implies 2a_1 + 3d = 14

.
For

S_8 = 88

:

\frac{8}{2}(2a_1 + 7d) = 88 \implies 2a_1 + 7d = 22

.

Setting up linear equations in terms of the first term

a_1

and common difference

d

allows us to solve for both sequence parameters.

2

Subtract the first equation from the second equation to solve for

d

.

(2a_1 + 7d) - (2a_1 + 3d) = 22 - 14 \implies 4d = 8 \implies d = 2

.

Eliminating

a_1

isolates the common difference

d

.

3

Substitute

d = 2

back into the first equation to solve for

a_1

.

2a_1 + 3(2) = 14 \implies 2a_1 + 6 = 14 \implies 2a_1 = 8 \implies a_1 = 4

.

Finding

a_1

completes the essential parameters of the sequence.

4

Calculate the 10th term using the formula

a_n = a_1 + (n-1)d

.

a_{10} = 4 + (10 - 1)(2) = 4 + 18 = 22

.

Evaluating the formula at

n = 10

provides the target term.

Key Concept

Arithmetic sequence term and series sum formulas.

Estimated Time:2m 0s

Question 992Question →

A software company audited $120$ applications for compliance with two protocols: Accessibility (Protocol A) and Security (Protocol B). The audit revealed that $75$ applications complied with Protocol A, $60$ applications complied with Protocol B, and $35$ applications complied with neither protocol. If an application selected at random from the audited group is known to comply with Protocol A, what is the probability that it also complies with Protocol B?

\frac{5}{12}

\frac{2}{3}

\frac{5}{6}

\frac{2}{15}

\frac{17}{24}

Show answer & explanation

Answer: $\frac{2}{3}$

Answer

\frac{2}{3}

Out of the 120 total applications, 35 comply with neither protocol, leaving 85 that comply with at least one. Using the inclusion-exclusion principle (

75 + 60 - 85 = 50

), exactly 50 applications comply with both protocols. Since the application is already known to comply with Protocol A, the denominator is restricted to the 75 applications in Protocol A. The conditional probability is therefore

50 / 75 = 2/3

.

Step-by-Step Solution

1

Determine the number of applications that comply with at least one protocol.

Total applications minus those complying with neither:

120 - 35 = 85

.

The total sample space is partitioned into applications complying with at least one protocol and those complying with neither.

2

Find the number of applications complying with both Protocol A and Protocol B using the Principle of Inclusion-Exclusion.

|A \cap B| = |A| + |B| - |A \cup B| = 75 + 60 - 85 = 50

.

Summing the counts of Protocol A and Protocol B counts applications in both protocols twice.

3

Calculate the conditional probability

P(B|A)

.

P(B|A) = \frac{|A \cap B|}{|A|} = \frac{50}{75} = \frac{2}{3}

.

Given that the selected application complies with Protocol A, the sample space is restricted to

|A| = 75

.

Key Concept

Conditional Probability with Overlapping Sets

Estimated Time:1m 30s

Question 993Question →

A sequence consists of $k$ consecutive positive odd integers. The arithmetic mean of the $3$ largest integers in the sequence is $29$ . If the sum of all $k$ integers in the sequence is $207$ , what is the value of $k$ ?

7

9

11

15

23

Show answer & explanation

Answer: 9

Answer

The total number of terms in the sequence,

k

, is 9.

The arithmetic mean of 3 consecutive odd integers is their middle term, so the three largest terms are 27, 29, and 31. The largest term is 31. Using the sum formula for an arithmetic progression,

S = k \cdot \frac{a + 31}{2} = 207

. Expressing the first term as

a = 31 - 2(k - 1) = 33 - 2k

yields

k(32 - k) = 207

, which simplifies to

k^2 - 32k + 207 = 0

. The roots are

k = 9

and

k = 23

. Because all integers in the sequence must be positive,

a = 33 - 2k > 0

, requiring

k \le 16

. Therefore,

k = 9

.

Step-by-Step Solution

1

Determine the largest integer in the sequence.

The largest integer is

31

.

For any 3 consecutive odd integers, the arithmetic mean is equal to the middle integer. Since the mean is

29

, the three largest integers are

27, 29, 31

, so the maximum term is

31

.

2

Express the smallest term

a

in terms of

k

.

a = 33 - 2k

.

The

k

-th term of a consecutive odd integer sequence starting at

a

is given by

31 = a + 2(k - 1) \implies a = 33 - 2k

.

3

Apply the positivity constraint.

k \le 16

.

Since all terms are positive integers, the smallest term must satisfy

a \ge 1 \implies 33 - 2k \ge 1 \implies k \le 16

.

4

Set up and solve the sum equation for

k

.

k = 9

.

The sum of an arithmetic sequence is

S = k \times \frac{a + L}{2}

. Substituting

S = 207

,

L = 31

, and

a = 33 - 2k

gives

207 = k \times \frac{(33 - 2k) + 31}{2} = k(32 - k) \implies k^2 - 32k + 207 = 0

. Factoring yields

(k - 9)(k - 23) = 0

. Since

k \le 16

,

k = 9

.

Key Concept

The sum of a sequence of consecutive evenly-spaced numbers equals the number of terms multiplied by the average of the first and last terms.

Estimated Time:1m 45s

Question 994Question →

A regional commercial airline compiled operational data for all scheduled flights across its Eastern Division during 2025. Analysis of the logs revealed that every flight experiencing a departure delay exceeding 45 minutes had undergone unscheduled engine maintenance earlier that same day. Additionally, none of the flights that underwent unscheduled engine maintenance earlier that day arrived at their destination on schedule.

Which of the following can be properly inferred from the passage above?

Every flight in the Eastern Division during 2025 that experienced a departure delay exceeding 45 minutes failed to arrive at its destination on schedule.

Unscheduled engine maintenance was the primary cause of departure delays exceeding 45 minutes in the Eastern Division in 2025.

Any flight in the Eastern Division that failed to arrive at its destination on schedule must have experienced a departure delay exceeding 45 minutes.

Flights in the Eastern Division that experienced departure delays of less than 45 minutes did not undergo unscheduled engine maintenance.

The Western Division of the airline experienced fewer departure delays exceeding 45 minutes than the Eastern Division did in 2025.

Show answer & explanation

Answer: Every flight in the Eastern Division during 2025 that experienced a departure delay exceeding 45 minutes failed to arrive at its destination on schedule.

Answer

Every flight in the Eastern Division during 2025 that experienced a departure delay exceeding 45 minutes failed to arrive at its destination on schedule.

The correct answer directly synthesizes the two stated factual premises using transitive deduction: every flight with a departure delay exceeding 45 minutes underwent unscheduled engine maintenance, and every flight that underwent unscheduled engine maintenance failed to arrive on schedule. Therefore, any flight with a departure delay exceeding 45 minutes necessarily failed to arrive on schedule.

Step-by-Step Solution

1

Identify the explicit premises given in the stimulus.

Premise 1: Departure delay > 45 min → Unscheduled engine maintenance. Premise 2: Unscheduled engine maintenance → NOT arrive on schedule.

Inference questions require strictly combining stated premises without introducing unstated assumptions.

2

Apply transitive conditional logic to connect the premises.

Departure delay > 45 min → Unscheduled engine maintenance → NOT arrive on schedule.

If A implies B, and B implies C, then A must imply C.

3

Evaluate the choices to find the one that MUST BE TRUE based solely on this deduction.

The statement that any flight delayed by over 45 minutes failed to arrive on schedule is a direct, inescapable logical deduction.

An inference must be guaranteed by the premises, whereas assumptions and speculations add unproven claims.

Key Concept

Distinguishing strict logical inferences (Must Be True) from unstated causal assumptions and out-of-scope speculations.

Question 995Question →

An autonomous electric shuttle completed a 240-mile test run from Point A to Point B. For the first 120 miles, the shuttle traveled at a constant speed of $v$ miles per hour. For the remaining 120 miles, due to battery efficiency controls, the shuttle traveled at a reduced speed equal to $\frac{2}{3}v$ miles per hour. On the return trip along the exact same 240-mile route from Point B to Point A, the shuttle maintained a uniform speed of $v + 10$ miles per hour. If the total time for the outbound trip from Point A to Point B was 2 hours longer than the total time for the return trip from Point B to Point A, what was the shuttle's average speed, in miles per hour, for the outbound trip from Point A to Point B?

32

40

41\frac{2}{3}

45

50

Show answer & explanation

Answer: 40

Answer

40 miles per hour

To find the average speed for the outbound trip, calculate the total time taken for both 120-mile legs. The first half takes

\frac{120}{v}

hours, and the second half at

\frac{2}{3}v

speed takes

\frac{120}{\frac{2}{3}v} = \frac{180}{v}

hours, giving a total outbound time of

\frac{300}{v}

hours. Setting up the time difference equation against the return trip gives

\frac{300}{v} - \frac{240}{v + 10} = 2

. Solving this quadratic equation yields

v = 50

mph. Substituting

v = 50

into the total outbound time formula gives

\frac{300}{50} = 6

hours. Finally, dividing the total distance of 240 miles by 6 hours gives an average speed of 40 miles per hour.

Step-by-Step Solution

1

Express the total outbound travel time in terms of

v

.

T_{\text{outbound}} = \frac{120}{v} + \frac{120}{\frac{2}{3}v} = \frac{120}{v} + \frac{180}{v} = \frac{300}{v}

hours.

Time equals distance divided by speed for each 120-mile segment.

2

Express the return travel time and set up the time-difference equation.

\frac{300}{v} - \frac{240}{v + 10} = 2

.

The return trip covers 240 miles at speed

v + 10

mph and takes 2 hours less than the outbound trip.

3

Solve the algebraic equation for

v

.

Simplifying

\frac{150}{v} - \frac{120}{v + 10} = 1

leads to

v^2 - 20v - 1500 = 0

, giving positive root

v = 50

.

Factoring

(v - 50)(v + 30) = 0

yields

v = 50

mph since speed must be positive.

4

Calculate total outbound time and outbound average speed.

Total outbound time =

\frac{300}{50} = 6

hours; Average speed =

\frac{240}{6} = 40

mph.

Average speed is defined as total distance divided by total elapsed time.

Key Concept

Average Speed across Multi-Leg Trips (Total Distance / Total Time)

Estimated Time:2m 30s

Question 996Question →

The table below shows the distribution of scores achieved by $50$ candidates on a professional certification assessment:

Score	Number of Candidates
$60$	$4$
$70$	$11$
$80$	$20$
$90$	$10$
$100$	$5$

What is the interquartile range (IQR) of the assessment scores?

10

20

30

35

40

Show answer & explanation

Answer: $20$

Answer

The interquartile range of the assessment scores is

20

.

To find the interquartile range (IQR), first find

Q_1

(

25\text{th}

percentile) and

Q_3

(

75\text{th}

percentile). With

50

total candidates,

Q_1

is situated around position

12.5

, which falls into the score of

70

.

Q_3

is situated around position

37.5

, which falls into the score of

90

. Subtracting

Q_1

from

Q_3

yields

90 - 70 = 20

.

Step-by-Step Solution

1

Calculate the cumulative frequency distribution to find score positions.

Score

60

: candidates

1

to

4

; Score

70

: candidates

5

to

15

; Score

80

: candidates

16

to

35

; Score

90

: candidates

36

to

45

; Score

100

: candidates

46

to

50

. Total

N = 50

.

Cumulative frequencies identify the precise position of ranked scores.

2

Determine the first quartile (

Q_1

), which represents the

25\text{th}

percentile.

The

25\text{th}

percentile corresponds to the

0.25 \times 50 = 12.5\text{th}

position. Looking at the cumulative frequencies, candidate positions

5

through

15

all scored

70

, so

Q_1 = 70

.

The first quartile marks the score boundary below which

25\%

of the dataset falls.

3

Determine the third quartile (

Q_3

), which represents the

75\text{th}

percentile.

The

75\text{th}

percentile corresponds to the

0.75 \times 50 = 37.5\text{th}

position. Candidate positions

36

through

45

all scored

90

, so

Q_3 = 90

.

The third quartile marks the score boundary below which

75\%

of the dataset falls.

4

Compute the interquartile range

(\text{IQR} = Q_3 - Q_1)

.

\text{IQR} = 90 - 70 = 20.

The interquartile range measures the spread of the middle

50\%

of the distribution.

Key Concept

Interquartile Range (IQR) and Quartile Positions in Frequency Distributions

Question 997Question →

At the beginning of 2021, a biotechnology firm allocated a fixed annual budget to its principal research laboratory. In 2022, the laboratory's budget was increased by $20\%$ relative to its 2021 budget. In 2023, the budget was reduced by $15\%$ from its 2022 level. In 2024, the budget was increased by $25\%$ over its 2023 level. If the budget in 2024 exceeded the budget in 2021 by $\$55,000$ , what was the laboratory's budget in 2021, in dollars?

Show answer & explanation

Answer: 200000

Answer

200,000 dollars

The initial 2021 budget

B

undergoes successive percentage changes over three years. A

20\%

increase in 2022 results in

1.20B

. A

15\%

decrease in 2023 yields

1.20B \times 0.85 = 1.02B

. A

25\%

increase in 2024 yields

1.02B \times 1.25 = 1.275B

. The net increase over the initial budget is

1.275B - B = 0.275B

. Setting

0.275B = 55,000

gives

B = \frac{55,000}{0.275} = 200,000

dollars.

Step-by-Step Solution

1

Represent the annual budgets sequentially in terms of the initial 2021 budget

B

2022 budget =

1.20B

, 2023 budget =

1.20B \times 0.85 = 1.02B

, 2024 budget =

1.02B \times 1.25 = 1.275B

Calculate successive percentage changes sequentially by multiplying the respective multipliers for each period

2

Calculate the net change in budget from 2021 to 2024

1.275B - B = 0.275B

Determine how much the final year's budget exceeds the initial base budget

3

Set up and solve the linear equation for

B

0.275B = 55,000 \implies B = \frac{55,000}{0.275} = 200,000

Equate the net algebraic difference to the given dollar amount to solve for the initial 2021 budget

Key Concept

Successive Percent Change and Base Value Tracking

Question 998Question →

Let $N = 504$ . If $M$ is the smallest positive integer such that $N \times M$ is a perfect cube, what is the value of $M$ ?

Show answer & explanation

Answer: 147

Answer

The smallest positive integer

M

such that

504 \times M

is a perfect cube is 147.

For an integer to be a perfect cube, the exponent of each prime factor in its prime factorization must be a multiple of 3. Prime factorizing 504 yields

2^3 \times 3^2 \times 7^1

. The exponent of 2 is 3 (already a multiple of 3). The exponent of 3 is 2, which requires 1 additional factor of 3 to reach 3. The exponent of 7 is 1, which requires 2 additional factors of 7 (

7^2

) to reach 3. Therefore, the minimum value for

M

is

3^1 \times 7^2 = 3 \times 49 = 147

.

Step-by-Step Solution

1

Express 504 as a product of its prime factors.

504 = 2^3 \times 3^2 \times 7^1

Decomposing

N

into prime factors allows analysis of the exponents required for perfect power conditions.

2

Apply the prime exponent rule for perfect cubes.

Every prime factor in

N \times M

must have an exponent that is a multiple of 3.

A number

K

is a perfect cube if and only if

K = p_1^{3a} \times p_2^{3b} \times \dots

3

Calculate the missing prime factors needed to complete the cube.

M = 3^{3-2} \times 7^{3-1} = 3^1 \times 7^2

To minimize

M

, we raise each prime to the smallest non-negative power that rounds the existing exponent up to the nearest multiple of 3.

4

Evaluate the value of

M

.

M = 3 \times 49 = 147

Direct arithmetic computation.

Key Concept

Prime Factorization and Exponent Requirements for Perfect Powers

Estimated Time:1m 30s

Question 999Question →

The positive integer $n$ has a prime factorization of the form $2^a \times 3^b \times 5^c$ , where $a$ , $b$ , and $c$ are positive integers. If $n$ is divisible by both 12 and 15, and $n$ has exactly 24 positive integer divisors, what is the least possible value of $n$ ?

180

360

480

540

600

Show answer & explanation

Answer: 360

Answer

The least possible value of

n

is 360.

To find the least value of

n = 2^a \times 3^b \times 5^c

divisible by

12 = 2^2 \times 3

and

15 = 3 \times 5

, we require

a \ge 2

,

b \ge 1

, and

c \ge 1

. The number of positive divisors is

(a+1)(b+1)(c+1) = 24

. Testing valid factor triples for 24 with

a+1 \ge 3

,

b+1 \ge 2

, and

c+1 \ge 2

gives possible values

n = 480

(from

a=5, b=1, c=1

),

n = 360

(from

a=3, b=2, c=1

),

n = 540

(from

a=2, b=3, c=1

), and

n = 600

(from

a=3, b=1, c=2

). The minimum among these valid integers is 360.

Step-by-Step Solution

1

Determine the minimum exponent constraints from divisibility conditions.

Since

n

is divisible by

12 = 2^2 \times 3^1

, we must have

a \ge 2

and

b \ge 1

. Since

n

is divisible by

15 = 3^1 \times 5^1

, we must have

b \ge 1

and

c \ge 1

. Combining these,

a \ge 2

,

b \ge 1

, and

c \ge 1

.

Divisibility requires that the prime factorization of

n

contains at least the prime powers present in the prime factorizations of 12 and 15.

2

Set up the total divisor count equation.

The total number of positive integer divisors of

n = 2^a \times 3^b \times 5^c

is given by

(a+1)(b+1)(c+1) = 24

.

The divisor counting formula states that for a number with prime factorization

p_1^{e_1} p_2^{e_2} \dots

, the total number of positive divisors is

(e_1+1)(e_2+1)\dots

3

Find all integer factor triples

(a+1, b+1, c+1)

that multiply to 24 subject to

a+1 \ge 3

,

b+1 \ge 2

, and

c+1 \ge 2

.

The valid factorizations of 24 into three factors meeting the bounds are:
-

(6, 2, 2) \implies (a, b, c) = (5, 1, 1)

, giving

n = 2^5 \times 3^1 \times 5^1 = 480

-

(4, 3, 2) \implies (a, b, c) = (3, 2, 1)

, giving

n = 2^3 \times 3^2 \times 5^1 = 360

-

(3, 4, 2) \implies (a, b, c) = (2, 3, 1)

, giving

n = 2^2 \times 3^3 \times 5^1 = 540

-

(4, 2, 3) \implies (a, b, c) = (3, 1, 2)

, giving

n = 2^3 \times 3^1 \times 5^2 = 600

To minimize

n = 2^a \times 3^b \times 5^c

, we must evaluate all valid permutations of exponents consistent with the divisor product constraint.

4

Compare the resulting values of

n

to find the minimum.

Comparing 480, 360, 540, and 600, the smallest value is 360.

Assigning the larger exponent 3 to the smallest prime base 2 and exponent 2 to prime base 3 minimizes the overall product.

Key Concept

Divisor Counting Formula and Prime Factorization Constraints

Estimated Time:2m 0s

Question 1000Question →

An executive board consisting of 8 distinct members needs to form a subcommittee of 4 members. However, board members Alex and Blair refuse to serve on the subcommittee together unless board member Morgan is also selected. How many different 4-member subcommittees can be formed under these conditions?

Show answer & explanation

Answer: 60

Answer

60

The solution uses complementary counting. First, calculate the total possible 4-member subcommittees from 8 members without restrictions, which is

\binom{8}{4} = 70

. Second, identify the restricted scenario that is not allowed: Alex and Blair are both selected, but Morgan is excluded. In this invalid scenario, 2 spots are taken by Alex and Blair, Morgan is excluded from consideration, leaving 2 spots to be filled from the remaining 5 board members, which equals

\binom{5}{2} = 10

invalid subcommittees. Subtracting the invalid subcommittees from the total gives

70 - 10 = 60

valid subcommittees.

Step-by-Step Solution

1

Calculate the total number of ways to choose a 4-member subcommittee from 8 members without restrictions.

The total unrestricted combinations is

\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70

.

Since the order of selecting members into a subcommittee does not matter, use the combination formula

\binom{n}{k}

.

2

Determine the condition under which a subcommittee selection is invalid.

A subcommittee is invalid if and only if both Alex and Blair are selected AND Morgan is excluded.

Alex and Blair agree to serve together only if Morgan is also present. Thus, having Alex and Blair together without Morgan violates the condition.

3

Calculate the number of invalid subcommittees.

The number of invalid subcommittees is

\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10

.

Alex and Blair take 2 of the 4 spots, and Morgan cannot take any spot. The remaining 2 spots must be filled from the remaining

8 - 3 = 5

members.

4

Subtract the invalid subcommittees from the total unrestricted subcommittees using complementary counting.

70 - 10 = 60

valid subcommittees.

Complementary counting yields the total number of subcommittees that satisfy the restriction.

Key Concept

Combinations with Restrictions and Complementary Counting

Estimated Time:2m 0s

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