All practice questions

Question 301Question →

Car 1 departs from City A at 8:00 AM and travels along a straight highway toward City B at a constant speed of $50$ miles per hour. At 8:30 AM, Car 2 departs from City B and travels along the same highway toward City A at a constant speed of $60$ miles per hour. At 9:15 AM, Car 1 encounters heavy traffic and reduces its speed by $20$ percent, maintaining this reduced speed for the rest of its journey, while Car 2 continues at its original speed. If the two cars pass each other at exactly 10:00 AM, what is the distance, in miles, between City A and City B?

Show answer & explanation

Answer: 182.5

Answer

The distance between City A and City B is 182.5 miles.

The total distance between the two cities is the sum of the distances traveled by both cars prior to their meeting point at 10:00 AM. Car 1 travels for 1.25 hours at 50 mph (62.5 miles) and 0.75 hours at 40 mph (30 miles), totaling 92.5 miles. Car 2 travels for 1.5 hours at 60 mph (90 miles). Summing these gives 92.5 + 90 = 182.5 miles.

Step-by-Step Solution

1

Determine the distance traveled by Car 1 prior to its speed reduction

Car 1 traveled 62.5 miles between 8:00 AM and 9:15 AM

Car 1 drove for 1 hour and 15 minutes (1.25 hours) at 50 mph, so distance = 50 × 1.25 = 62.5 miles.

2

Determine the distance traveled by Car 1 after its speed reduction

Car 1 traveled 30 miles between 9:15 AM and 10:00 AM

Car 1 reduced its speed by 20%, bringing its rate to 50 × 0.80 = 40 mph. It traveled at 40 mph for 45 minutes (0.75 hours), so distance = 40 × 0.75 = 30 miles.

3

Determine the distance traveled by Car 2

Car 2 traveled 90 miles between 8:30 AM and 10:00 AM

Car 2 drove from 8:30 AM to 10:00 AM, which is 1 hour and 30 minutes (1.5 hours), at a constant speed of 60 mph, so distance = 60 × 1.5 = 90 miles.

4

Calculate the total distance between City A and City B

The total distance is 182.5 miles

Because the two cars traveled toward each other from opposite cities until they met, the total distance between the cities equals the sum of the individual distances: 62.5 + 30 + 90 = 182.5 miles.

Key Concept

Converging motion with variable rates and time offsets

Question 302Question →

An event catering service charges a fixed setup fee plus a constant hourly rate for event staff. For a 4-hour event, the total charge is $680. For a 7-hour event with the same staff requirements, the total charge is$ 1,070. What is the fixed setup fee, in dollars, charged by the catering service?

Show answer & explanation

Answer: 160

Answer

The fixed setup fee charged by the catering service is 160 dollars.

Modeling the total cost as

C = S + rt

, where

S

is the fixed setup fee and

r

is the hourly rate, yields the system of equations

S + 4r = 680

and

S + 7r = 1070

. Subtracting the first equation from the second yields

3r = 390

, so

r = 130

. Substituting

r = 130

back into the first equation yields

S + 520 = 680

, which gives

S = 160

.

Step-by-Step Solution

1

Define the algebraic model for total cost.

Let

S

be the fixed setup fee in dollars and

r

be the hourly rate in dollars per hour. The total cost

C

for

t

hours is modeled by

C = S + rt

.

The problem presents a linear relationship between time and total cost, consisting of a fixed component and a variable component.

2

Construct a system of linear equations.

S + 4r = 680

and

S + 7r = 1070

.

Substitute the two given combinations of time (

t = 4

and

t = 7

) and total cost (

C = 680

and

C = 1070

) into the algebraic model.

3

Solve for the variable hourly rate

r

.

(S + 7r) - (S + 4r) = 1070 - 680 \implies 3r = 390 \implies r = 130

.

Subtracting the first equation from the second eliminates the fixed setup fee variable

S

.

4

Calculate the fixed setup fee

S

.

S + 4(130) = 680 \implies S + 520 = 680 \implies S = 160

.

Substitute

r = 130

back into the first equation to find the value of

S

.

Key Concept

Linear Equation Modeling and Systems of Equations

Question 303Question →

An airline creates a $5$ -character identification code for each of its cargo routes. The characters in the code must satisfy the following sequential rules from left to right:

1. The first character must be a non-zero even digit.
2. The second and third characters must be distinct uppercase vowels chosen from the set $\{A, E, I, O, U\}$ .
3. The fourth character must be a single-digit prime number.
4. The fifth character must be a single digit ( $0$ through $9$ ) such that the sum of the first digit and the fifth digit is odd.

How many such unique identification codes can the airline create?

Show answer & explanation

Answer: 1600

Answer

1,600 unique identification codes

To find the total number of unique identification codes, we evaluate the number of available options for each character position and multiply them together using the Fundamental Counting Principle:

1. First position (non-zero even digit): The even digits from 0 to 9 are

0, 2, 4, 6, 8

. Excluding 0 leaves

4

options (

2, 4, 6, 8

).
2. Second and third positions (distinct vowels): There are

5

vowels (

A, E, I, O, U

). The second position has

5

options, and the third position has

4

remaining options, yielding

5 \times 4 = 20

combinations.
3. Fourth position (prime digit): The single-digit primes are

2, 3, 5, 7

, giving

4

options.
4. Fifth position (parity condition): The sum of the first digit and fifth digit must be odd. Since the first digit is an even number, the fifth digit must be an odd number to make the sum odd. The single-digit odd numbers are

1, 3, 5, 7, 9

, giving

5

options.

Multiplying the choices for all positions gives:

4 \times 20 \times 4 \times 5 = 1,600

Step-by-Step Solution

1

Determine the number of valid choices for the 1st position (non-zero even digit).

4 choices (

2, 4, 6, 8

).

The digit 0 is excluded because the first character must be non-zero.

2

Determine the number of valid choices for the 2nd and 3rd positions (distinct vowels).

5 \times 4 = 20

choices.

There are 5 vowels available for the 2nd position, and 4 remaining vowels for the 3rd position since they must be distinct.

3

Determine the number of valid choices for the 4th position (prime digit).

4 choices (

2, 3, 5, 7

).

The single-digit prime numbers are 2, 3, 5, and 7.

4

Determine the number of valid choices for the 5th position (parity constraint).

5 choices (

1, 3, 5, 7, 9

).

The sum of an even integer and another integer is odd if and only if the second integer is odd. Since the 1st digit is guaranteed to be even, the 5th digit must be odd.

5

Apply the Fundamental Counting Principle across all 5 sequential stages.

4 \times 20 \times 4 \times 5 = 1,600

unique codes.

Each stage represents an independent decision stage with a fixed number of valid choices.

Key Concept

Fundamental Counting Principle with Restricted Positions and Parity Rules

Estimated Time:2m 0s

Question 304Question →

A positive integer $n$ can be expressed in the form $p^3 q^2$ , where $p$ and $q$ are distinct prime numbers. If $n$ is divisible by 45 and is a factor of 4,050, what is the value of $n$ ?

Show answer & explanation

Answer: 675

Answer

675

Prime factorizing

45 = 3^2 \cdot 5

and

4,050 = 2 \cdot 3^4 \cdot 5^2

shows that

n

must be composed of the prime factors 3 and 5. The two candidate values for

p^3 q^2

are

3^3 \cdot 5^2 = 675

and

5^3 \cdot 3^2 = 1,125

. Both are multiples of 45, but only 675 is a factor of 4,050 because its power of 5 does not exceed

5^2

.

Step-by-Step Solution

1

Determine the prime factorizations of 45 and 4,050

45 = 3^2 \cdot 5^1

and

4,050 = 2^1 \cdot 3^4 \cdot 5^2

Decomposing the given numbers into prime factorizations determines the prime building blocks for

n

.

2

Identify the distinct prime factors

p

and

q

The primes

p

and

q

must be 3 and 5

Since

n

is divisible by 45, its prime factorization must contain at least

3^2

and

5^1

. Since

n = p^3 q^2

has exactly two distinct prime factors, those factors must be 3 and 5.

3

Evaluate both possible assignments for

p

and

q

n = 3^3 \cdot 5^2 = 675

or

n = 5^3 \cdot 3^2 = 1,125

Assigning 3 to

p

and 5 to

q

yields 675, while assigning 5 to

p

and 3 to

q

yields 1,125.

4

Filter using the constraint that

n

is a factor of 4,050

675 is a factor of 4,050; 1,125 is not a factor of 4,050

For a number to be a factor of

4,050 = 2^1 \cdot 3^4 \cdot 5^2

, the exponent of prime 5 cannot exceed 2. In 1,125, the exponent of 5 is 3, which makes 1,125 invalid.

Key Concept

Prime Factorization, Divisibility Rules, and Factors of Integers

Question 305Question →

A publishing firm evaluated $160$ manuscript submissions based on three quality criteria: Plot Originality ( $P$ ), Character Development ( $C$ ), and Dialogue Quality ( $D$ ). Each manuscript was evaluated against all three criteria.

- $75$ manuscripts met the criteria for Plot Originality ( $P$ ).
- $80$ manuscripts met the criteria for Character Development ( $C$ ).
- $70$ manuscripts met the criteria for Dialogue Quality ( $D$ ).
- $30$ manuscripts met the criteria for both Plot Originality and Character Development ( $P \cap C$ ).
- $25$ manuscripts met the criteria for both Character Development and Dialogue Quality ( $C \cap D$ ).
- $20$ manuscripts met the criteria for both Plot Originality and Dialogue Quality ( $P \cap D$ ).
- $10$ manuscripts met all three criteria ( $P \cap C \cap D$ ).

How many manuscripts met exactly one of the three criteria?

Show answer & explanation

Answer: 105

Answer

The number of manuscripts that met exactly one of the three criteria is 105.

To find the number of manuscripts meeting exactly one criterion, we isolate the unique regions for each set. Subtracting the overlapping intersections from the total of each set yields 35 manuscripts for Plot Originality only, 35 for Character Development only, and 35 for Dialogue Quality only. Combining these gives 105.

Step-by-Step Solution

1

Find the number of elements in the double-intersection regions only.

Exactly (P and C) = 20; Exactly (C and D) = 15; Exactly (P and D) = 10.

The given two-set intersections include manuscripts that satisfied all three criteria, so the triple intersection of 10 must be subtracted from each.

2

Calculate the count of manuscripts meeting only P, only C, and only D.

Only P = 35, Only C = 35, Only D = 35.

Subtract all shared regions (two-set-only overlaps plus the three-set overlap) from each individual total.

3

Sum the three single-criterion regions.

35 + 35 + 35 = 105.

The question asks specifically for the total number of manuscripts that met exactly one criterion.

Key Concept

3-Set Overlapping Venn Diagrams and Region Separation

Question 306Question →

If $x$ is an integer such that $|x - 3| + |x + 5| \le 12$ , how many possible values of $x$ exist?

Show answer & explanation

Answer: 13

Answer

13

Evaluating the inequality

|x - 3| + |x + 5| \le 12

across the three intervals defined by critical points

x = -5

and

x = 3

yields the continuous real solution set

[-7, 5]

. Counting all integers in this inclusive range gives

5 - (-7) + 1 = 13

possible values.

Step-by-Step Solution

1

Identify the critical points of the absolute value expressions.

The critical points where the expressions inside the absolute values change sign are

x = 3

and

x = -5

.

Setting

x - 3 = 0

gives

x = 3

, and setting

x + 5 = 0

gives

x = -5

.

2

Solve the inequality for the region

x < -5

.

-(x - 3) - (x + 5) \le 12 \implies -2x - 2 \le 12 \implies -2x \le 14 \implies x \ge -7

. Thus,

-7 \le x < -5

.

When

x < -5

, both

x - 3 < 0

and

x + 5 < 0

, so

|x - 3| = -(x - 3)

and

|x + 5| = -(x + 5)

.

3

Solve the inequality for the region

-5 \le x \le 3

.

-(x - 3) + (x + 5) \le 12 \implies 8 \le 12

, which is universally true for all

x

in this interval.

When

-5 \le x \le 3

,

x - 3 \le 0

and

x + 5 \ge 0

, so

|x - 3| = -(x - 3)

and

|x + 5| = x + 5

.

4

Solve the inequality for the region

x > 3

.

(x - 3) + (x + 5) \le 12 \implies 2x + 2 \le 12 \implies 2x \le 10 \implies x \le 5

. Thus,

3 < x \le 5

.

When

x > 3

, both

x - 3 > 0

and

x + 5 > 0

.

5

Combine the valid intervals and count the integer solutions.

The full solution range is

[-7, 5]

. The number of integer values is

5 - (-7) + 1 = 13

.

The number of integers in an inclusive range

[a, b]

is given by

b - a + 1

.

Key Concept

Solving multi-term absolute value inequalities by dividing the domain at critical points into distinct cases.

Question 307Question →

An athlete completes a three-leg training course consisting of cycling, running, and swimming, covering a total distance of $60$ miles in a total time of $4$ hours. The athlete cycles at a constant speed of $24$ miles per hour, runs at a constant speed of $10$ miles per hour, and swims at a constant speed of $2$ miles per hour. If the time spent cycling is equal to the combined time spent running and swimming, how many miles did the athlete run?

Show answer & explanation

Answer: 10

Answer

The athlete ran 10 miles.

Because cycling time equals combined running and swimming time, the 4-hour total splits evenly into 2 hours for cycling and 2 hours for the remaining legs. Cycling at 24 mph covers 48 miles, leaving 12 miles and 2 hours for running and swimming. Writing the total time equation as (d_running / 10) + ((12 - d_running) / 2) = 2 and solving yields d_running = 10 miles.

Step-by-Step Solution

1

Determine the time spent cycling

2 hours

Since the total time is 4 hours and cycling time equals the sum of running and swimming times, 2 * t_cycling = 4.

2

Calculate the distance covered while cycling

48 miles

Distance = Speed * Time = 24 mph * 2 hours = 48 miles.

3

Determine the combined distance and time for running and swimming

Combined distance = 12 miles, Combined time = 2 hours

Total distance minus cycling distance is 60 - 48 = 12 miles; total time minus cycling time is 4 - 2 = 2 hours.

4

Set up and solve the system of equations for running distance

Running distance = 10 miles

Using time = distance / speed gives (d_running / 10) + ((12 - d_running) / 2) = 2. Multiplying by 10 yields d_running + 60 - 5 * d_running = 20, leading to d_running = 10 miles.

Key Concept

Multi-leg rate, time, and distance problems using systems of linear equations.

Question 308Question →

Six distinct letters— $A, B, C, D, E,$ and $F$ —are to be arranged in a single line. How many different linear arrangements are possible such that letter $A$ appears somewhere to the left of letter $B$ , and letters $C$ and $D$ are not adjacent to each other?

Show answer & explanation

Answer: 240

Answer

240

To find the number of valid linear arrangements, apply symmetry and the complement rule. First, in half of all

6! = 720

arrangements (360 arrangements), letter A appears to the left of letter B. Next, find the number of arrangements where A is to the left of B AND letters C and D are adjacent. Treating C and D as one combined block gives

5! \times 2! = 240

total arrangements where C and D are adjacent. By symmetry, letter A appears to the left of letter B in half of these cases (

\frac{240}{2} = 120

). Subtracting these 120 restricted arrangements from the 360 total arrangements where A precedes B gives

360 - 120 = 240

.

Step-by-Step Solution

1

Determine the number of linear arrangements in which letter A appears somewhere to the left of letter B.

360 arrangements

For 6 distinct letters, there are

6! = 720

total linear arrangements. By symmetry, letter A appears before letter B in exactly half of all arrangements:

\frac{720}{2} = 360

.

2

Determine the number of arrangements where letter A is to the left of letter B AND letters C and D are adjacent.

120 arrangements

Treating C and D as a single block yields 5 items to arrange, which can be done in

5! = 120

ways. The block itself has

2! = 2

internal orderings, giving

120 \times 2 = 240

arrangements where C and D are adjacent. By symmetry, letter A appears before letter B in half of these arrangements:

\frac{240}{2} = 120

.

3

Subtract the arrangements where C and D are adjacent from the total arrangements where A is to the left of B.

240 arrangements

The number of arrangements where letter A is to the left of letter B and letters C and D are not adjacent is

360 - 120 = 240

.

Key Concept

Permutations with Relative Position and Non-Adjacency Restrictions

Question 309Question →

Let $K$ be a positive integer with prime factorization $K = 2^a \times 5^b \times 11^c$ , where $a$ , $b$ , and $c$ are positive integers. If $10K$ has 36 more positive divisors than $K$ , and $11K$ has 12 more positive divisors than $K$ , what is the value of $a + b + c$ ?

Show answer & explanation

Answer: 9

Answer

The value of

a + b + c

is 9.

For

K = 2^a 5^b 11^c

, the number of divisors is given by

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

. Multiplying by 11 increases the exponent of 11 by 1, yielding

(a+1)(b+1)

additional divisors. Since this difference is 12,

(a+1)(b+1) = 12

. Multiplying by 10 increases the exponents of both 2 and 5 by 1, yielding an additional

(c+1)(a+b+3)

divisors. Since this difference is 36,

(c+1)(a+b+3) = 36

. For positive integers

a

and

b

, the only factor pair of 12 that allows

36 / (a+b+3)

to be an integer is when

a+b = 6

. Consequently,

c+1 = 4

, so

c = 3

. Therefore,

a+b+c = 6+3 = 9

.

Step-by-Step Solution

1

Write the divisor count formulas for

K

,

10K

, and

11K

.

d(K) = (a+1)(b+1)(c+1)

,

d(11K) = (a+1)(b+1)(c+2)

, and

d(10K) = (a+2)(b+2)(c+1)

.

The number of positive divisors of a number

2^x 5^y 11^z

is

(x+1)(y+1)(z+1)

.

2

Use

d(11K) - d(K) = 12

to solve for

(a+1)(b+1)

.

(a+1)(b+1)(c+2) - (a+1)(b+1)(c+1) = 12 \implies (a+1)(b+1) = 12

.

Factoring out

(a+1)(b+1)

simplifies the equation directly.

3

Use

d(10K) - d(K) = 36

to find the relationship between

a, b,

and

c

.

(c+1)[(a+2)(b+2) - (a+1)(b+1)] = 36 \implies (c+1)(a+b+3) = 36

.

Expanding

(a+2)(b+2) - (a+1)(b+1) = ab + 2a + 2b + 4 - (ab + a + b + 1) = a + b + 3

.

4

Determine

a+b

and

c

using positive integer constraints.

a+b = 6

and

c = 3

.

Since

a, b \ge 1

, the pairs for

(a+1, b+1)

yielding 12 are

(2,6)

or

(3,4)

. If

(a+1,b+1)=(2,6)

,

a+b=6

, making

a+b+3=9

and

c+1=36/9=4 \implies c=3

. If

(3,4)

,

a+b=5

, so

a+b+3=8

, but

36/8

is not an integer.

5

Sum

a

,

b

, and

c

.

a + b + c = 6 + 3 = 9

.

Adding the derived sum

a+b=6

and

c=3

gives 9.

Key Concept

Divisor Count Function from Prime Factorization

Estimated Time:2m 0s

Question 310Question →

A beverage producer creates two liquid mixtures, Mixture $A$ and Mixture $B$ . Mixture $A$ consists of $30\%$ fruit concentrate by volume, and Mixture $B$ consists of $70\%$ fruit concentrate by volume. A lab technician combines $x$ liters of Mixture $A$ with $y$ liters of Mixture $B$ to prepare an $80$ -liter batch that contains $45\%$ fruit concentrate by volume. What is the value of $x$ ?

Show answer & explanation

Answer: 50

Answer

The value of

x

is 50.

To find

x

, we construct two linear equations based on total liquid volume and total fruit concentrate. The total volume equation is

x + y = 80

, which gives

y = 80 - x

. The concentrate equation is

0.30x + 0.70y = 0.45(80) = 36

. Substituting

y = 80 - x

yields

0.30x + 0.70(80 - x) = 36

. Expanding gives

0.30x + 56 - 0.70x = 36

, so

-0.40x = -20

, which results in

x = 50

.

Step-by-Step Solution

1

Formulate a system of two linear equations representing total volume and total concentrate volume.

System equations:

x + y = 80

and

0.30x + 0.70y = 36

.

The sum of the component volumes equals the total mixture volume, and the sum of the pure concentrate from each component equals the total concentrate in the final mixture.

2

Substitute

y = 80 - x

into the concentrate equation to eliminate

y

.

0.30x + 0.70(80 - x) = 36

.

Substituting one variable reduces the system to a single linear equation in one variable.

3

Simplify the single-variable linear equation and solve for

x

.

0.30x + 56 - 0.70x = 36 \implies -0.40x = -20 \implies x = 50

.

Combining like terms isolates the variable

x

.

Key Concept

Solving systems of two linear equations formed by weighted mixture word problems.

Question 311Question →

At a technology firm, an evaluation was conducted for 150 software developers to test their proficiency in three programming paradigms: Object-Oriented ( $O$ ), Functional ( $F$ ), and Reactive ( $R$ ). The evaluation revealed that 85 developers are proficient in $O$ , 60 are proficient in $F$ , and 45 are proficient in $R$ . Furthermore, 25 developers are proficient in both $O$ and $F$ , 18 are proficient in both $F$ and $R$ , 22 are proficient in both $O$ and $R$ , and 10 developers are proficient in all three paradigms. How many of the software developers surveyed are proficient in exactly two of these three programming paradigms?

Show answer & explanation

Answer: 35

Answer

The number of software developers proficient in exactly two of the three programming paradigms is 35.

To find the number of developers proficient in exactly two paradigms, subtract the number of developers proficient in all three paradigms (

10

) from each of the pairwise totals and sum the results:

(25 - 10) + (18 - 10) + (22 - 10) = 15 + 8 + 12 = 35

.

Step-by-Step Solution

1

Identify the total count for each pairwise overlap of paradigms.

The pairwise counts are

|O \cap F| = 25

,

|F \cap R| = 18

, and

|O \cap R| = 22

, with all three paradigms

|O \cap F \cap R| = 10

.

Each given two-set intersection includes individuals who are also proficient in all three paradigms.

2

Subtract the triple intersection from each pairwise total to isolate those proficient in exactly two paradigms.

Proficient in only

O

and

F

:

25 - 10 = 15

; proficient in only

F

and

R

:

18 - 10 = 8

; proficient in only

O

and

R

:

22 - 10 = 12

.

The region representing 'exactly two' sets excludes the central region where all three sets overlap.

3

Sum the three distinct exclusive two-set counts.

15 + 8 + 12 = 35

.

Adding these mutually exclusive categories provides the total count of developers in exactly two sets.

Key Concept

Three-Set Venn Diagram Region Calculations

Alternative Method

Using a Venn diagram, enter

10

in the central triple-intersection region. Next, fill in the three surrounding two-set-only regions by subtracting

10

from each given pairwise total:

25 - 10 = 15

,

18 - 10 = 8

, and

22 - 10 = 12

. Adding these three region counts together gives

15 + 8 + 12 = 35

.

Estimated Time:1m 30s

Question 312Question →

What is the sum of all real solutions to the equation $|x^2 - 10| = 3x$ ?

Show answer & explanation

Answer: 7

Answer

The sum of all real solutions to the equation is 7.

To solve

|x^2 - 10| = 3x

, note that

3x \ge 0

(so

x \ge 0

). Splitting into two algebraic cases gives

x^2 - 3x - 10 = 0

(yielding

x = 5

and extraneous

x = -2

) and

x^2 + 3x - 10 = 0

(yielding

x = 2

and extraneous

x = -5

). The valid real solutions are

x = 5

and

x = 2

, whose sum is 7.

Step-by-Step Solution

1

Determine domain restriction based on the absolute value definition.

x \ge 0

Because an absolute value expression cannot be negative,

|x^2 - 10| = 3x

requires

3x \ge 0

, which means

x \ge 0

.

2

Solve Case 1:

x^2 - 10 = 3x

.

x = 5

Rearranging yields

x^2 - 3x - 10 = 0

, which factors as

(x - 5)(x + 2) = 0

. The roots are

x = 5

and

x = -2

. Reject

x = -2

because

x \ge 0

.

3

Solve Case 2:

-(x^2 - 10) = 3x

.

x = 2

Rearranging yields

x^2 + 3x - 10 = 0

, which factors as

(x + 5)(x - 2) = 0

. The roots are

x = 2

and

x = -5

. Reject

x = -5

because

x \ge 0

.

4

Sum all valid real solutions.

7

The valid real solutions are

x = 5

and

x = 2

. Their sum is

5 + 2 = 7

.

Key Concept

Solving absolute value equations with a variable expression on one side requires verifying non-negativity constraints to eliminate extraneous roots.

Estimated Time:1m 30s

Question 313Question →

A logistics company recorded the daily number of deliveries made by each of its 8 delivery vans on a given day. Each van completed a distinct positive integer number of deliveries. The arithmetic mean of the number of deliveries made by the 8 vans was 25, the median was 24, and the range was 18. If $M$ represents the maximum number of deliveries completed by any single van that day, what is the maximum possible value of $M$ ?

Show answer & explanation

Answer: 37

Answer

The maximum possible value of

M

is 37.

To maximize the largest element

M = x_8

, we express

M

in terms of the smallest element

x_1

using the range:

M = x_1 + 18

. Thus, maximizing

M

is equivalent to maximizing

x_1

. Testing

x_1 = 20

forces the minimal possible sum of the 8 distinct terms to be

20 + 21 + 22 + 23 + 25 + 26 + 27 + 38 = 202

, which exceeds the required sum of 200. Testing

x_1 = 19

allows a minimal sum of 198, which can be adjusted to 200 by setting the set to

\{19, 20, 21, 23, 25, 26, 29, 37\}

. Thus, the maximum possible value of

M

is

19 + 18 = 37

.

Step-by-Step Solution

1

Formulate the algebraic equations from the statistical properties given.

Sum of 8 terms =

8 \times 25 = 200

;

x_4 + x_5 = 48

;

x_8 = x_1 + 18 = M

.

Mean gives total sum, even number of items gives median as average of 4th and 5th terms, and range links the maximum and minimum values.

2

Relate maximizing the maximum term

M

to maximizing the minimum term

x_1

.

Maximizing

M = x_1 + 18

requires making

x_1

as large as possible.

Since the range is fixed at 18,

M

increases directly as

x_1

increases.

3

Test

x_1 = 20

to determine feasibility.

Minimum possible sum for

x_1 = 20

is

20 + 21 + 22 + 23 + 25 + 26 + 27 + 38 = 202 > 200

, which is invalid.

Distinct integer constraints force

x_4 \ge 23

; since

x_4 + x_5 = 48

and

x_4 < x_5

,

x_4

must be 23 and

x_5

must be 25, forcing all lower bounds up.

4

Test

x_1 = 19

to confirm feasibility and construct a valid set.

The valid set

\{19, 20, 21, 23, 25, 26, 29, 37\}

meets all criteria with a sum of 200.

The minimal sum for

x_1 = 19

is 198, leaving headroom to increase

x_7

to 29 to reach the sum of 200.

Key Concept

Optimization of Extreme Values in Finite Ordered Sets using Mean, Median, and Range

Question 314Question →

A security system generates unique 7-digit access codes using each of the digits $1, 2, 3, 4, 5, 6,$ and $7$ exactly once. How many such codes can be formed in which all odd digits appear in strictly ascending order from left to right and the digit $2$ appears somewhere to the left of the digit $4$ ?

Show answer & explanation

Answer: 105

Answer

The total number of valid 7-digit security access codes is 105.

Out of the total

7! = 5,040

unrestricted linear arrangements of the 7 distinct digits, the 4 odd digits can be ordered in

4! = 24

ways, but only 1 of these orderings satisfies the strictly ascending condition. Furthermore, the digits 2 and 4 can be ordered in

2! = 2

ways, with digit 2 appearing to the left of digit 4 in exactly 1 of those orderings. Therefore, the number of valid arrangements is given by

\frac{7!}{4! \times 2!} = \frac{5,040}{24 \times 2} = \frac{5,040}{48} = 105

.

Step-by-Step Solution

1

Calculate slot selections for the odd digits

35 ways

From 7 positions, selecting 4 positions for the odd digits {1, 3, 5, 7} can be done in

\binom{7}{4} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35

ways. Since the odd digits must be ordered in strictly ascending order (1, then 3, then 5, then 7), there is only 1 valid relative order for any selection of 4 slots.

2

Determine valid arrangements for the even digits

3 ways

The remaining 3 slots are occupied by the even digits {2, 4, 6}. The total number of linear arrangements of 3 distinct digits is

3! = 6

. By symmetry, in exactly half of these arrangements, digit 2 appears to the left of digit 4, yielding

\frac{6}{2} = 3

valid ways.

3

Multiply independent choices to find the total arrangements

105 codes

Applying the Fundamental Counting Principle,

35 \text{ (slot choices for odd digits)} \times 1 \text{ (ordering of odd digits)} \times 3 \text{ (arrangements of even digits)} = 105

.

Key Concept

Permutations with Relative Order Restrictions

Estimated Time:2m 0s

Question 315Question →

Let $x$ , $y$ , and $z$ be integers such that $-5 \le x \le 5$ , $-5 \le y \le 5$ , and $-5 \le z \le 5$ . If these integers satisfy all of the following conditions:
1. $x^3 y z^2 < 0$
2. $\frac{x}{z} < -1$
3. $x + y \le 1$

What is the maximum possible value of the expression $x + 2y - z$ ?

Show answer & explanation

Answer: 5

Answer

The maximum possible value of the expression

x + 2y - z

is 5.

By analyzing the given inequalities,

x

and

y

must have opposite signs, and

x

and

z

must have opposite signs, which means

y

and

z

must have the same sign. To maximize

x + 2y - z

, we examine the scenario where

x

is negative, while

y

and

z

are positive. Taking

z = 1

(the smallest positive integer),

y = 5

(the largest positive integer within the given range), and

x = -4

satisfies

x + y \le 1

,

\frac{x}{z} = -4 < -1

, and

x^3 y z^2 = -320 < 0

. This yields

x + 2y - z = -4 + 10 - 1 = 5

.

Step-by-Step Solution

1

Analyze product and quotient inequalities to deduce the relative signs of

x

,

y

, and

z

.

x

and

y

have opposite signs,

x

and

z

have opposite signs, and

|x| > |z| \ge 1

. Consequently,

y

and

z

share the same sign.

Because

z^2 > 0

for non-zero

z

,

x^3 y z^2 < 0

requires

x^3 y < 0

. Also

\frac{x}{z} < -1

implies

\frac{x}{z}

is negative and has magnitude greater than 1.

2

Evaluate the sign cases to determine which case yields a larger value for

x + 2y - z

.

Case A (

x < 0, y > 0, z > 0

) allows positive contributions from

2y

and

-z

, whereas Case B (

x > 0, y < 0, z < 0

) bounds the expression below 2.

In Case B,

y

and

z

are negative, so

2y \le -2

suppresses the sum.

3

Apply integer domain bounds and inequality constraints to maximize

x + 2y - z

in Case A.

The maximum value is 5, achieved when

x = -4

,

y = 5

, and

z = 1

.

Setting

z = 1

(smallest positive integer) and

y = 5

(largest positive integer) with

x = -4

satisfies

x + y \le 1

and all problem conditions.

Key Concept

Deduction of variable signs from inequality products and quotients, combined with integer range optimization.

Estimated Time:2m 0s

Question 316Question →

Two cyclists, Cyclist A and Cyclist B, depart simultaneously from Town X and Town Y, respectively, traveling toward each other along a straight 150-mile path. Cyclist A travels at a constant speed of 18 miles per hour throughout the journey. Cyclist B initially travels at a constant speed of 30 miles per hour. After traveling for 2 hours, Cyclist B encounters a steep incline and decreases speed by 40 percent, maintaining this reduced speed for the remainder of the trip. How many hours after their departure will Cyclist A and Cyclist B meet?

Show answer & explanation

Answer: 3.5

Answer

The total time elapsed from departure until Cyclist A and Cyclist B meet is 3.5 hours.

In the first 2 hours, Cyclist A covers

18 \times 2 = 36

miles and Cyclist B covers

30 \times 2 = 60

miles, bringing their combined distance to 96 miles and leaving 54 miles remaining. Cyclist B's speed then decreases by 40% to 18 mph (

30 \times 0.6

). Moving toward each other, their combined relative rate becomes

18 + 18 = 36

mph. Dividing the remaining 54 miles by 36 mph yields 1.5 hours for the second phase. Adding the initial 2 hours gives a total time of 3.5 hours.

Step-by-Step Solution

1

Calculate cumulative distance traveled by both cyclists during the first 2 hours

Cyclist A travels 36 miles; Cyclist B travels 60 miles; Total = 96 miles

Both cyclists travel for 2 full hours at their initial constant speeds.

2

Determine the distance separating the cyclists at t = 2 hours

150 - 96 = 54 miles remaining

Subtract the combined distance covered from the total initial separation of 150 miles.

3

Calculate Cyclist B's new speed after the 40% decrease

30 * 0.60 = 18 mph

A 40% reduction means retaining 60% of the original speed of 30 mph.

4

Calculate relative speed of approach after 2 hours

18 + 18 = 36 mph

When two objects move toward each other, their relative speed is the sum of their individual speeds.

5

Compute the time to cover the remaining distance

54 / 36 = 1.5 hours

Time equals distance divided by relative speed.

6

Sum the time segments to find the total elapsed time

2 + 1.5 = 3.5 hours

The trip consists of an initial 2-hour phase plus an additional 1.5-hour phase.

Key Concept

Relative Speed and Piecewise Motion in Converging Rate Problems

Question 317Question →

An arithmetic sequence $a_1, a_2, a_3, \dots$ has a first term $a_1 = 4$ and a common difference $d = 3$ . A geometric sequence $b_1, b_2, b_3, \dots$ has a first term $b_1 = 2$ and a common ratio $r = 2$ . If the $k$ -th term of the arithmetic sequence and the $m$ -th term of the geometric sequence are both equal to $64$ , what is the value of $k + m$ ?

Show answer & explanation

Answer: 27

Answer

The value of

k + m

is 27.

For the arithmetic sequence, the

k

-th term is

a_k = a_1 + (k-1)d

. Setting

4 + 3(k-1) = 64

gives

3(k-1) = 60

, so

k-1 = 20

and

k = 21

. For the geometric sequence, the

m

-th term is

b_m = b_1 r^{m-1}

. Setting

2 \cdot 2^{m-1} = 64

gives

2^m = 64

, which implies

m = 6

. Adding the two values gives

k + m = 21 + 6 = 27

.

Step-by-Step Solution

1

Determine the term position

k

in the arithmetic sequence.

k = 21

Using

a_k = a_1 + (k-1)d

, set

4 + 3(k-1) = 64 \implies 3(k-1) = 60 \implies k - 1 = 20 \implies k = 21

.

2

Determine the term position

m

in the geometric sequence.

m = 6

Using

b_m = b_1 \cdot r^{m-1}

, set

2 \cdot 2^{m-1} = 64 \implies 2^m = 64 \implies m = 6

.

3

Compute the sum of the two position indices

k

and

m

.

27

k + m = 21 + 6 = 27

.

Key Concept

Calculating term indices in arithmetic and geometric sequences using general term formulas

Estimated Time:1m 30s

Question 318Question →

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 319Question →

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 320Question →

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.

Answer

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