All practice questions

Question 241Question →

For a constant $a > 0$ , the quadratic equation $x^2 - ax + (2a + 1) = 0$ has two real roots, $r$ and $s$ . If $r^2 + s^2 = 43$ , what is the value of $a$ ?

Show answer & explanation

Answer: 9

Answer

The value of

a

is 9.

According to Vieta's formulas, the sum of the roots of

x^2 - ax + (2a + 1) = 0

is

r + s = a

and the product of the roots is

rs = 2a + 1

. Using the identity

r^2 + s^2 = (r + s)^2 - 2rs

, substitute the Vieta expressions to obtain

r^2 + s^2 = a^2 - 2(2a + 1) = a^2 - 4a - 2

. Setting this equal to 43 yields

a^2 - 4a - 2 = 43

, which simplifies to

a^2 - 4a - 45 = 0

. Factoring the quadratic gives

(a - 9)(a + 5) = 0

, giving solutions

a = 9

or

a = -5

. Because the problem specifies that

a > 0

,

a

must be 9. Checking the discriminant

\Delta = (-9)^2 - 4(1)(19) = 5 > 0

confirms that two real roots exist.

Step-by-Step Solution

1

Express the sum and product of the roots in terms of

a

using Vieta's formulas.

r + s = a

and

rs = 2a + 1

For a standard quadratic equation

x^2 + bx + c = 0

, the sum of the roots is

-b

and the product of the roots is

c

.

2

Relate

r^2 + s^2

to

(r + s)

and

rs

.

r^2 + s^2 = (r + s)^2 - 2rs = a^2 - 2(2a + 1) = a^2 - 4a - 2

Expanding

(r + s)^2 = r^2 + 2rs + s^2

allows expressing

r^2 + s^2

in terms of known quantities.

3

Substitute

r^2 + s^2 = 43

into the equation and solve for

a

.

a^2 - 4a - 2 = 43 \implies a^2 - 4a - 45 = 0 \implies (a - 9)(a + 5) = 0

Rearranging terms forms a new quadratic equation in terms of

a

.

4

Apply the positivity constraint

a > 0

and verify that the roots are real.

Since

a > 0

,

a = 9

. The discriminant of the original equation is

\Delta = (-9)^2 - 4(1)(19) = 5 > 0

, confirming real roots exist.

The question specifies that

a

is positive and that

r

and

s

are real numbers.

Key Concept

Vieta's Formulas and Quadratic Modeling

Question 242Question →

At an intellectual property law firm, a senior partner reviewed $180$ patent applications across three technical domains: Artificial Intelligence, Biotechnology, and Clean Energy. The audit revealed that $25$ applications belonged to none of these three domains. Furthermore, $90$ applications involved Artificial Intelligence, $85$ involved Biotechnology, and $80$ involved Clean Energy. If the number of applications that belonged to exactly two of these domains was three times the number of applications that belonged to all three domains, how many patent applications belonged to exactly one domain?

Show answer & explanation

Answer: 75

Answer

The total number of patent applications that belonged to exactly one domain is

75

.

The total number of applications in at least one domain is

180 - 25 = 155

. Let

E_1

be the number of applications in exactly one domain,

E_2

in exactly two domains, and

x

in all three domains. The total union is

E_1 + E_2 + x = 155

, while the sum of individual set sizes is

|A| + |B| + |C| = E_1 + 2E_2 + 3x = 90 + 85 + 80 = 255

. Subtracting the union equation from the sum equation gives

E_2 + 2x = 100

. Given

E_2 = 3x

, substituting gives

5x = 100 \implies x = 20

, which implies

E_2 = 60

. Finally, subtracting

E_2

and

x

from the total union yields

E_1 = 155 - 60 - 20 = 75

.

Step-by-Step Solution

1

Calculate the total number of applications in at least one domain

|A \cup B \cup C| = 180 - 25 = 155

Subtracting the applications belonging to none of the domains from the overall total yields the total number of unique applications covered by the three domains combined.

2

Formulate regional Venn diagram equations

E_1 + E_2 + x = 155

and

E_1 + 2E_2 + 3x = 255

Summing individual set counts counts elements in exactly two domains twice and elements in all three domains three times.

3

Deduce the relationship between

E_2

and

x

E_2 + 2x = 100

Subtracting the equation for total union from the sum of individual sets isolates the overcounted regions.

4

Use the given proportion

E_2 = 3x

to solve for

x

and

E_2

x = 20

and

E_2 = 60

Substituting

E_2 = 3x

into

E_2 + 2x = 100

gives

5x = 100

, so

x = 20

and

E_2 = 60

.

5

Find the number of applications belonging to exactly one domain (

E_1

)

E_1 = 75

Subtracting the count of applications in exactly two domains (

60

) and all three domains (

20

) from the total union (

155

) gives

155 - 60 - 20 = 75

.

Key Concept

Three-Set Venn Diagram Region Partitioning

Question 243Question →

A wholesaler purchases a desk for $\$250$ and marks up the cost price by $20\%$ to establish the retail price. If the desk is sold at the retail price, what is the profit, in dollars, earned on the sale?

Show answer & explanation

Answer: 50

Answer

The profit earned on the sale of the desk is

\$50

.

The profit is calculated by finding

20\%

of the cost price of

\$250

, which is

0.20 \times 250 = 50

.

Step-by-Step Solution

1

Identify the cost price and markup rate from the problem statement.

Cost price

= \$250

; Markup rate

= 20\%

.

Markup is calculated directly as a percentage of the original cost price.

2

Multiply the cost price by the markup percentage to find the profit.

\text{Profit} = 0.20 \times 250 = 50

.

Taking

20\%

of the

\$250

purchase price gives the exact dollar markup/profit.

Key Concept

Dollar profit is calculated as the markup percentage multiplied by the cost price.

Question 244Question →

A chemical storage vessel contains three industrial solvents—Solvent A, Solvent B, and Solvent C—blended in the initial ratio of $2 : 5 : 7$ , respectively. After $24$ liters of Solvent A and $12$ liters of Solvent B are added to the vessel, the ratio of Solvent A to Solvent B becomes $4 : 7$ . What was the initial volume, in liters, of Solvent C in the vessel?

Show answer & explanation

Answer: 140

Answer

The initial volume of Solvent C in the vessel was 140 liters.

Defining the initial amounts of Solvents A, B, and C as

2x

,

5x

, and

7x

, respectively, the addition of

24

liters of Solvent A and

12

liters of Solvent B yields the ratio equation

\frac{2x + 24}{5x + 12} = \frac{4}{7}

. Cross-multiplying gives

14x + 168 = 20x + 48

, which simplifies to

6x = 120

and

x = 20

. The initial volume of Solvent C is therefore

7 \times 20 = 140

liters.

Step-by-Step Solution

1

Express the initial quantities of the three solvents in terms of a common ratio multiplier

x

.

Solvent A =

2x

, Solvent B =

5x

, Solvent C =

7x

.

A ratio of

2 : 5 : 7

implies the actual quantities are proportional to 2, 5, and 7 by a factor

x

.

2

Set up an equation representing the modified ratio of Solvent A to Solvent B.

\frac{2x + 24}{5x + 12} = \frac{4}{7}

Adding 24 liters to Solvent A yields

2x + 24

liters, and adding 12 liters to Solvent B yields

5x + 12

liters.

3

Solve the algebraic equation for

x

.

7(2x + 24) = 4(5x + 12) \implies 14x + 168 = 20x + 48 \implies 6x = 120 \implies x = 20

.

Cross-multiplication converts the ratio comparison into a linear equation.

4

Compute the initial volume of Solvent C.

Initial volume of Solvent C =

7x = 7(20) = 140

liters.

Substitute

x = 20

into the expression

7x

for the original amount of Solvent C.

Key Concept

Ratio scaling and algebraic modeling of partial additions

Question 245Question →

A chemist prepares a mixture by combining $20$ liters of a $10\%$ saline solution with $30$ liters of a $20\%$ saline solution. What is the saline concentration, as a percentage, of the resulting mixture?

Show answer & explanation

Answer: 16

Answer

The concentration of the resulting mixture is

16\%

.

The total amount of pure saline in the mixture is

8

liters out of a total solution volume of

50

liters. The overall concentration is

\frac{8}{50} \times 100\% = 16\%

.

Step-by-Step Solution

1

Calculate the total amount of pure saline solute contributed by both solutions.

20 \text{ L} \times 0.10 + 30 \text{ L} \times 0.20 = 2 + 6 = 8 \text{ liters}

The total solute is the sum of the solute amounts in each component solution.

2

Calculate the total volume of the combined mixture.

20 + 30 = 50 \text{ liters}

The combined volume is the sum of the individual volumes.

3

Compute the weighted average concentration of the mixture.

\frac{8}{50} = 0.16 = 16\%

The weighted average concentration equals total solute divided by total solution volume.

Key Concept

Weighted Average in Mixture Problems

Question 246Question →

For all real numbers $x$ and $y$ , the custom binary operator $\Delta$ is defined by $x \Delta y = x^2 - 5xy + 4y^2$ . The function $f$ is defined by $f(x) = x \Delta 1$ . What is the positive integer value of $k$ such that $f(f(k)) = 0$ ?

Show answer & explanation

Answer: 5

Answer

The positive integer value of

k

is

5

.

Substituting

y = 1

into

x \Delta y = x^2 - 5xy + 4y^2

gives

f(x) = x^2 - 5x + 4

. For

f(f(k)) = 0

, the outer function evaluation requires

f(k)

to be a root of

f(x) = 0

. Solving

x^2 - 5x + 4 = 0

yields roots

1

and

4

. Setting

f(k) = 4

yields

k^2 - 5k = 0

, which has roots

k = 0

and

k = 5

. Since

k

must be a positive integer,

k = 5

. Setting

f(k) = 1

yields irrational values, so

5

is the unique solution.

Step-by-Step Solution

1

Substitute

y = 1

into the operator definition to find

f(x)

.

f(x) = x^2 - 5x + 4

f(x) = x \Delta 1 = x^2 - 5x(1) + 4(1)^2

.

2

Set

f(u) = 0

for

u = f(k)

and solve for

u

.

u = 1

or

u = 4

Factoring

u^2 - 5u + 4 = 0

yields

(u - 1)(u - 4) = 0

.

3

Solve

f(k) = 4

for

k

.

k = 0

or

k = 5

k^2 - 5k + 4 = 4 \implies k^2 - 5k = 0 \implies k(k - 5) = 0

.

4

Solve

f(k) = 1

for

k

and check for integer solutions.

k = \frac{5 \pm \sqrt{13}}{2}

(irrational roots)

k^2 - 5k + 4 = 1 \implies k^2 - 5k + 3 = 0

.

5

Select the positive integer solution.

k = 5

The value

k = 0

is not positive, and the roots from

f(k) = 1

are not integers.

Key Concept

Custom operator evaluation, composite/nested functions, and quadratic root analysis.

Question 247Question →

A set of $5$ integers has an arithmetic mean of $12$ . If four of the integers are $8$ , $10$ , $14$ , and $15$ , what is the value of the fifth integer?

Show answer & explanation

Answer: 13

Answer

The value of the fifth integer is

13

.

The total sum of a set of numbers is given by the formula

\text{Sum} = \text{Mean} \times n

. For

5

numbers with a mean of

12

, the total sum is

5 \times 12 = 60

. The sum of the four provided numbers is

8 + 10 + 14 + 15 = 47

. Subtracting

47

from

60

yields

13

, which is the value of the fifth integer.

Step-by-Step Solution

1

Find the total sum of the 5 integers

The total sum is

60

The sum of a set of numbers equals the arithmetic mean multiplied by the total count of numbers (

12 \times 5 = 60

).

2

Sum the four given integers

The sum of the four integers is

47

Adding the given numbers:

8 + 10 + 14 + 15 = 47

.

3

Subtract the sum of the known integers from the total sum

The fifth integer is

13

Subtracting

47

from

60

gives

60 - 47 = 13

.

Key Concept

Arithmetic Mean

Estimated Time:45s

Question 248Question →

Let $S$ be the set of all real numbers $x$ that satisfy the inequality $x^2 - 6x \leq 0$ . How many integer values of $k$ are there such that the equation $||x - 4| - |x + 2|| = k$ has at least one solution $x \in S$ ?

Show answer & explanation

Answer: 7

Answer

The correct answer is 7.

Solving the quadratic inequality

x^2 - 6x \leq 0

gives the domain

S = [0, 6]

. Over this closed interval, the function

g(x) = ||x - 4| - |x + 2||

is continuous and attains its minimum value of 0 at

x = 1

and its maximum value of 6 at

x = 4

(and throughout

[4, 6]

). By the Intermediate Value Theorem,

g(x)

takes on all real values in the interval

[0, 6]

. The integer values of

k

for which

g(x) = k

has a solution in

S

are

0, 1, 2, 3, 4, 5,

and

6

, making a total of 7 integers.

Step-by-Step Solution

1

Solve the quadratic inequality to define the set

S

.

x^2 - 6x \leq 0 \iff x(x - 6) \leq 0 \iff 0 \leq x \leq 6

. Thus,

S = [0, 6]

.

The solution set of

x(x-6) \leq 0

lies between the roots

x = 0

and

x = 6

inclusive.

2

Analyze the inner function

f(x) = |x - 4| - |x + 2|

for

x \in [0, 6]

.

Critical points of absolute values occur at

x = -2

and

x = 4

. Within

[0, 6]

, we split at

x = 4

.

The signs of

(x - 4)

and

(x + 2)

determine how the absolute value bars simplify.

3

Evaluate

g(x) = |f(x)|

on the sub-interval

[0, 4]

.

For

0 \leq x \leq 4

:

|x - 4| = 4 - x

and

|x + 2| = x + 2

. Thus,

f(x) = 2 - 2x

and

g(x) = |2 - 2x|

.

Evaluating

g(x)

at key points gives

g(0) = 2

,

g(1) = 0

, and

g(4) = 6

. By continuity,

g(x)

covers all values in

[0, 6]

on this interval.

4

Evaluate

g(x)

on the sub-interval

[4, 6]

.

For

4 \leq x \leq 6

:

|x - 4| = x - 4

and

|x + 2| = x + 2

. Thus,

f(x) = -6

and

g(x) = |-6| = 6

.

g(x)

remains constant at 6 for

x \in [4, 6]

.

5

Determine the range of

g(x)

on

S

and count the integer values of

k

.

The range of

g(x)

for

x \in [0, 6]

is

[0, 6]

. The integer values in this interval are

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

.

The equation

g(x) = k

has a solution in

S

if and only if

k

lies in the range of

g(x)

over

S

. There are

6 - 0 + 1 = 7

such integers.

Key Concept

Absolute value functions case evaluation and finding the range over a restricted domain.

Question 249Question →

An integer $n$ is chosen at random from the set of all positive integers less than or equal to $120$ . What is the probability that $n$ is a multiple of either $3$ or $5$ , but not a multiple of $15$ ?

Show answer & explanation

Answer: 0.4

Answer

The probability that the selected integer is a multiple of either 3 or 5, but not 15, is 0.4 (or 2/5).

The total number of integers from 1 to 120 is 120. Multiples of 3 up to 120 total 40, multiples of 5 total 24, and multiples of 15 total 8. Integers that are multiples of 3 but not 15 number 40 - 8 = 32. Integers that are multiples of 5 but not 15 number 24 - 8 = 16. The total number of favorable outcomes is 32 + 16 = 48. Thus, the probability is 48/120 = 2/5 = 0.4.

Step-by-Step Solution

1

Count total possible outcomes in the sample space.

The total number of integers from 1 to 120 is 120.

Each integer in the set {1, 2, ..., 120} is equally likely to be selected.

2

Count the number of multiples of 3, 5, and 15 within the range.

Multiples of 3: 40; Multiples of 5: 24; Multiples of 15: 8.

Since 120 is divisible by 3, 5, and 15, the count of multiples of k up to 120 is 120/k.

3

Calculate the number of integers that are multiples of 3 or 5, but not 15.

Number of favorable outcomes = (Multiples of 3 only) + (Multiples of 5 only) = (40 - 8) + (24 - 8) = 32 + 16 = 48.

Multiples of 15 are common multiples of both 3 and 5 and must be excluded completely according to the condition 'not a multiple of 15'.

4

Compute the single-event probability.

Probability = 48 / 120 = 2 / 5 = 0.4.

Probability of a single event is defined as the ratio of favorable outcomes to total possible outcomes.

Key Concept

Basic Single-Event Probability with Set Restrictions

Question 250Question →

A quality control inspector reviews a shipment containing a total of $N$ customized components, of which exactly 3 are defective. The inspector randomly selects 3 components from the shipment one by one without replacement. If the probability that at least one of the selected components is defective is equal to $\frac{31}{35}$ , what is the value of $N$ ?

Show answer & explanation

Answer: 7

Answer

The total number of components in the shipment,

N

, is 7.

To find

N

, apply the complementary probability formula

P(\text{at least 1 defective}) = 1 - P(\text{0 defective})

. Given that

P(\text{at least 1 defective}) = \frac{31}{35}

, the probability of drawing zero defective components is

1 - \frac{31}{35} = \frac{4}{35}

. Out of

N

total components,

N-3

are non-defective. Drawing 3 non-defective components without replacement gives

P(\text{0 defective}) = \frac{(N-3)(N-4)(N-5)}{N(N-1)(N-2)}

. Setting this equal to

\frac{4}{35}

and testing integer values starting at

N=6

yields

N=7

, since

\frac{4 \times 3 \times 2}{7 \times 6 \times 5} = \frac{24}{210} = \frac{4}{35}

.

Step-by-Step Solution

1

Calculate the probability of the complementary event (selecting no defective components).

P(\text{0 defective}) = 1 - \frac{31}{35} = \frac{4}{35}

.

Calculating 'at least one' directly requires summing three separate cases (1 defective, 2 defective, 3 defective), whereas using the complement

P(\text{at least 1}) = 1 - P(\text{none})

requires evaluating only one scenario.

2

Formulate the algebraic expression for picking 3 non-defective components without replacement.

P(\text{0 defective}) = \frac{\binom{N-3}{3}}{\binom{N}{3}} = \frac{(N-3)(N-4)(N-5)}{N(N-1)(N-2)}

.

There are

N-3

non-defective components out of

N

total components, and 3 are selected without replacement.

3

Equate the algebraic probability to the known complementary probability and solve for

N

.

\frac{(N-3)(N-4)(N-5)}{N(N-1)(N-2)} = \frac{4}{35} \implies N = 7.

Testing

N = 7

yields

\frac{4 \times 3 \times 2}{7 \times 6 \times 5} = \frac{24}{210} = \frac{4}{35}

. The function is strictly increasing for

N \ge 6

, making

N = 7

the unique integer solution.

Key Concept

Complementary Probability and Dependent Sampling (Without Replacement)

Question 251Question →

If $|3x + 4| = 19$ , what is the positive value of $x$ ?

Show answer & explanation

Answer: 5

Answer

The positive value of

x

is 5.

Solving the equation

|3x + 4| = 19

yields two cases:

3x + 4 = 19

, giving

x = 5

, and

3x + 4 = -19

, giving

x = -\frac{23}{3}

. The positive value among these solutions is 5.

Step-by-Step Solution

1

Split the absolute value equation into two linear equations.

3x + 4 = 19

or

3x + 4 = -19

The equation

|a| = b

for

b \ge 0

implies

a = b

or

a = -b

.

2

Solve for

x

in both cases.

x = 5

or

x = -\frac{23}{3}

Subtract 4 from both sides and divide by 3.

3

Select the value satisfying the problem constraint.

5

The problem asks specifically for the positive value of

x

.

Key Concept

Solving Linear Absolute Value Equations

Question 252Question →

A municipal water treatment facility uses three pumps—Pump $P$ , Pump $Q$ , and Pump $R$ —to fill a main reservoir. Working alone at their respective constant rates, Pump $P$ can fill the reservoir in 20 hours, Pump $Q$ in 30 hours, and Pump $R$ in 40 hours.

All three pumps begin filling the empty reservoir simultaneously. After 4 hours, Pump $P$ shuts off. Pumps $Q$ and $R$ continue operating together for another 8 hours before Pump $Q$ is also shut off. Pump $R$ continues to run alone until the reservoir is completely filled.

How many total hours does it take to fill the reservoir from the start of the process until it is completely filled?

Show answer & explanation

Answer: 16

Answer

The total time required to fill the reservoir from start to finish is 16 hours.

Converting the individual completion times into work rates per hour (1/20, 1/30, and 1/40 of the reservoir per hour) allows us to determine the combined output per phase. In the first 4 hours, all three pumps fill 52/120 of the reservoir. In the next 8 hours, Pumps Q and R fill 56/120 of the reservoir. This leaves 12/120 (or 1/10) of the job remaining. Pump R, working at a rate of 1/40 per hour, takes 4 hours to complete the final 1/10. Adding all three durations (4 + 8 + 4) gives a total of 16 hours.

Step-by-Step Solution

1

Determine the individual hourly rates of work for each pump.

Pump P completes

\frac{1}{20}

of the job per hour, Pump Q completes

\frac{1}{30}

of the job per hour, and Pump R completes

\frac{1}{40}

of the job per hour.

Work rate is the reciprocal of the total time required to complete one whole task.

2

Calculate the work completed during Stage 1 (4 hours with all 3 pumps working).

Combined rate =

\frac{1}{20} + \frac{1}{30} + \frac{1}{40} = \frac{6 + 4 + 3}{120} = \frac{13}{120}

. Work done =

4 \times \frac{13}{120} = \frac{52}{120}

.

Work done equals combined rate multiplied by time spent working together.

3

Calculate the work completed during Stage 2 (8 hours with Pumps Q and R working).

Combined rate of Q and R =

\frac{1}{30} + \frac{1}{40} = \frac{4 + 3}{120} = \frac{7}{120}

. Work done =

8 \times \frac{7}{120} = \frac{56}{120}

.

Only Pumps Q and R are active during this second period.

4

Calculate the remaining fraction of work after Stage 1 and Stage 2.

Total work done so far =

\frac{52}{120} + \frac{56}{120} = \frac{108}{120} = \frac{9}{10}

. Remaining work =

1 - \frac{9}{10} = \frac{1}{10}

.

Subtracting completed work from the whole (1) leaves the remaining work to be completed.

5

Calculate the time required for Pump R to finish the remaining work alone.

Time =

\frac{1/10}{1/40} = \frac{40}{10} = 4

hours.

Time equals remaining work divided by the individual rate of Pump R.

6

Sum the time spent across all stages.

Total time =

4 \text{ hours (Stage 1)} + 8 \text{ hours (Stage 2)} + 4 \text{ hours (Stage 3)} = 16

hours.

The total elapsed time is the sum of the durations of each distinct phase.

Key Concept

Work Rate and Combined Work in Multi-Stage Processes

Alternative Method

Assume a convenient total reservoir volume equal to the least common multiple of the times: 120 units. Pump P produces 6 units/hr, Pump Q produces 4 units/hr, and Pump R produces 3 units/hr. Stage 1 (4 hrs): 4 × (6 + 4 + 3) = 52 units. Stage 2 (8 hrs): 8 × (4 + 3) = 56 units. Total filled = 108 units. Remaining = 12 units. Stage 3 (Pump R alone): 12 / 3 = 4 hrs. Total time = 4 + 8 + 4 = 16 hours.

Estimated Time:2m 0s

Question 253Question →

Train X and Train Y depart simultaneously from Station A and Station B, respectively, heading toward each other along a straight, 360-mile track. Train X travels at a constant speed of 40 miles per hour for the first 2 hours, after which its speed increases by 50 percent. Train Y travels at a constant speed of 50 miles per hour for the first hour, stops for 30 minutes due to a signal delay, and then resumes its journey at a constant speed of 70 miles per hour. How many hours after their simultaneous departure will the two trains meet?

Show answer & explanation

Answer: 3.5

Answer

3.5 hours

Evaluating distance piecewise up to

t = 2

hours shows Train X covers 80 miles and Train Y covers 85 miles, leaving 195 miles between them. After

t = 2

hours, their relative rate of convergence is

60 + 70 = 130

mph. Dividing 195 miles by 130 mph yields 1.5 additional hours, bringing the total time since departure to 3.5 hours.

Step-by-Step Solution

1

Determine distance traveled by Train X in the first 2 hours and its new speed.

Train X distance =

40 \text{ mph} \times 2 \text{ hours} = 80 \text{ miles}

. New speed =

40 \times (1 + 0.50) = 60 \text{ mph}

.

Train X increases its speed by 50% after the 2-hour mark.

2

Determine distance traveled by Train Y during the first 2 hours.

In hour 1 (

t=0

to

t=1

):

50 \text{ mph} \times 1 \text{ h} = 50 \text{ miles}

. During delay (

t=1

to

t=1.5

):

0 \text{ miles}

. From

t=1.5

to

t=2.0

:

70 \text{ mph} \times 0.5 \text{ h} = 35 \text{ miles}

. Total distance =

50 + 0 + 35 = 85 \text{ miles}

.

Accounting for Train Y's initial leg, 30-minute delay, and new speed up to

t = 2

hours.

3

Find the remaining distance separating the two trains at

t = 2

hours.

Total distance covered by both trains =

80 + 85 = 165 \text{ miles}

. Remaining distance =

360 - 165 = 195 \text{ miles}

.

Subtracting total distance covered by both trains from the track length of 360 miles.

4

Calculate the time required to cover the remaining distance using relative speed.

Relative closing speed for

t > 2

is

60 + 70 = 130 \text{ mph}

. Additional time =

\frac{195}{130} = 1.5 \text{ hours}

.

When two objects move toward each other, their rates add together to find the closing rate.

5

Calculate total elapsed time from departure.

Total time =

2.0 + 1.5 = 3.5 \text{ hours}

.

Combining the initial 2-hour evaluation window with the additional 1.5 hours required to meet.

Key Concept

Relative speed and piecewise distance calculations for converging vehicles with variable speeds and delays.

Estimated Time:3m 0s

Question 254Question →

The table below categorizes 100 analysts at a consulting firm by their department and experience level:

Department	Junior (1–3 yrs)	Senior (4–7 yrs)	Lead (8+ yrs)	Total
Technology	14	16	10	40
Analytics	12	18	5	35
Operations	9	11	5	25
Total	35	45	20	100

If one analyst is selected at random from this group, what is the probability that the selected analyst works in the Analytics department or has Lead experience, but NOT both?

Show answer & explanation

Answer: 0.45

Answer

0.45

To find the probability of selecting an analyst who is either in the Analytics department or at the Lead level, but not both, we count the analysts in Analytics who are not Leads (

12 + 18 = 30

) and the analysts at the Lead level who are not in Analytics (

10 + 5 = 15

). The total number of favorable outcomes is

30 + 15 = 45

. Dividing by the total pool of

100

analysts yields a probability of

\frac{45}{100} = 0.45

.

Step-by-Step Solution

1

Determine the total size of the sample space.

Total analysts

N = 100

.

Basic probability requires dividing favorable outcomes by total possible outcomes.

2

Calculate the number of analysts satisfying 'Analytics, but NOT Lead'.

Junior Analytics (

12

) + Senior Analytics (

18

) =

30

.

Excludes the

5

Lead analysts in the Analytics department.

3

Calculate the number of analysts satisfying 'Lead, but NOT Analytics'.

Technology Lead (

10

) + Operations Lead (

5

) =

15

.

Excludes the

5

Lead analysts in the Analytics department.

4

Sum the non-overlapping favorable counts and compute probability.

Favorable outcomes

= 30 + 15 = 45

; Probability

= \frac{45}{100} = 0.45

.

Probability of a single event is defined as

\frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}

.

Key Concept

Basic Single-Event Probability from Two-Way Tabular Data with Mutually Exclusive Set Conditions

Question 255Question →

The roots of the quadratic polynomial $P(x) = x^2 + bx + c$ are $r_1$ and $r_2$ , where $b$ and $c$ are constants. If $r_1 + 2$ and $r_2 + 2$ are the roots of the quadratic equation $x^2 - 10x + 21 = 0$ , what is the value of $c$ ?

Show answer & explanation

Answer: 5

Answer

The value of

c

is 5.

Factoring

x^2 - 10x + 21 = 0

yields roots

3

and

7

. Because these roots represent

r_1 + 2

and

r_2 + 2

, subtracting

2

from each root gives the original roots

r_1 = 1

and

r_2 = 5

. In the monic polynomial

P(x) = x^2 + bx + c

, the constant coefficient

c

equals the product of the roots

r_1 \times r_2 = 1 \times 5 = 5

.

Step-by-Step Solution

1

Solve for the roots of the given equation

x^2 - 10x + 21 = 0

The roots are

3

and

7

.

Factoring the quadratic expression yields

(x - 3)(x - 7) = 0

, so

x = 3

or

x = 7

.

2

Determine the roots

r_1

and

r_2

of

P(x)

r_1 = 1

and

r_2 = 5

.

Since the roots of the second equation are shifted by

+2

, we set

r_1 + 2 = 3 \implies r_1 = 1

and

r_2 + 2 = 7 \implies r_2 = 5

.

3

Calculate the constant term

c

c = 5

.

By Vieta's formulas, for any monic quadratic polynomial

x^2 + bx + c

, the constant term

c

is the product of the roots

r_1 r_2 = 1 \times 5 = 5

.

Key Concept

Root transformation of quadratic equations and relationship between roots and coefficients via Vieta's formulas.

Question 256Question →

What is the numerical value of $\frac{2^8 \cdot 3^4}{6^4}$ ?

Show answer & explanation

Answer: 16

Answer

16

Rewriting the base

6

as

2 \cdot 3

gives

6^4 = 2^4 \cdot 3^4

. Substituting this into the original expression yields

\frac{2^8 \cdot 3^4}{2^4 \cdot 3^4}

. Cancelling

3^4

leaves

\frac{2^8}{2^4} = 2^{8-4} = 2^4 = 16

.

Step-by-Step Solution

1

Rewrite composite bases into prime factors

6^4 = (2 \cdot 3)^4 = 2^4 \cdot 3^4

Applying the power of a product rule

(ab)^n = a^n b^n

allows base matching.

2

Simplify the quotient using exponent subtraction

\frac{2^8 \cdot 3^4}{2^4 \cdot 3^4} = 2^{8-4} \cdot 3^{4-4} = 2^4 \cdot 3^0 = 16 \cdot 1 = 16

Applying the quotient rule

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

for powers with equal bases.

Key Concept

Exponent rules: Power of a Product

(ab)^n = a^n b^n

and Quotient of Powers

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

Question 257Question →

A commuter drives from home to work at a constant speed of $45$ miles per hour and arrives $10$ minutes late. On the following day, driving along the exact same route at a constant speed of $60$ miles per hour, the commuter arrives $5$ minutes early. What is the distance, in miles, from the commuter's home to work?

Show answer & explanation

Answer: 45

Answer

The distance from the commuter's home to work is 45 miles.

Let

d

be the distance in miles between home and work. At

45

miles per hour, the time taken is

\frac{d}{45}

hours. At

60

miles per hour, the time taken is

\frac{d}{60}

hours. The difference between being

10

minutes late and

5

minutes early is

15

minutes, or

\frac{15}{60} = \frac{1}{4}

hour. Setting up the equation

\frac{d}{45} - \frac{d}{60} = \frac{1}{4}

and finding a common denominator of

180

gives

\frac{4d - 3d}{180} = \frac{1}{4}

, which simplifies to

\frac{d}{180} = \frac{1}{4}

. Multiplying both sides by

180

yields

d = 45

miles.

Step-by-Step Solution

1

Calculate the difference in travel time between the two trips in hours.

The time difference is 10 minutes late - (-5 minutes early) = 15 minutes = 1/4 hour.

Since speeds are given in miles per hour, time must be expressed in hours.

2

Set up an equation relating the two travel times using the formula time = distance / speed.

d/45 - d/60 = 1/4, where d represents the distance in miles.

The slower speed (45 mph) takes 1/4 hour longer than the faster speed (60 mph).

3

Solve the equation for the distance variable d.

(4d - 3d)/180 = 1/4 => d/180 = 1/4 => d = 45.

Find a common denominator for 45 and 60, which is 180, and solve for d.

Key Concept

Rate, Time, and Distance Problems

Question 258Question →

At a commercial bank, an audit of $300$ business loan applications revealed that each application underwent at least one of three specialized risk evaluations: Credit Risk, Market Risk, and Operational Risk. Exactly $180$ applications underwent Credit Risk evaluation, $150$ underwent Market Risk evaluation, and $135$ underwent Operational Risk evaluation. If exactly $95$ applications underwent exactly two of the three risk evaluations, how many loan applications underwent all three risk evaluations?

Show answer & explanation

Answer: 35

Answer

35 loan applications underwent all three risk evaluations.

The total number of applications is

300

. The sum of individual evaluations is

180 + 150 + 135 = 465

. Using the Venn diagram region formula,

\text{Sum of individual sets} - \text{Total} = \text{Exactly } 2 + 2 \times (\text{Exactly } 3)

. Substituting the known values gives

465 - 300 = 95 + 2x

, which simplifies to

165 = 95 + 2x

. Solving for

x

gives

2x = 70

, so

x = 35

.

Step-by-Step Solution

1

State the fundamental overlapping set region equation for 3 sets.

\text{Total} = \text{Exactly } 1 + \text{Exactly } 2 + \text{Exactly } 3 + \text{Neither}

Every application falls into exactly one of these non-overlapping region categories.

2

Calculate the sum of the individual set counts.

180 + 150 + 135 = 465

Summing individual set counts counts items in 1 set once, 2 sets twice, and 3 sets three times.

3

Formulate the algebraic identity linking individual sums to region totals.

\text{Sum of Individual Sets} = \text{Exactly } 1 + 2(\text{Exactly } 2) + 3(\text{Exactly } 3)

This accounts for the multiple counting of overlapping regions.

4

Subtract the Total equation from the Sum equation to eliminate the 'Exactly 1' term.

465 - 300 = \text{Exactly } 2 + 2(\text{Exactly } 3) \Rightarrow 165 = 95 + 2(\text{Exactly } 3)

Subtracting

(\text{Exactly } 1 + \text{Exactly } 2 + \text{Exactly } 3)

from

(\text{Exactly } 1 + 2\cdot\text{Exactly } 2 + 3\cdot\text{Exactly } 3)

leaves

1\cdot\text{Exactly } 2 + 2\cdot\text{Exactly } 3

.

5

Solve the linear equation for the 'Exactly 3' intersection.

2(\text{Exactly } 3) = 165 - 95 = 70 \Rightarrow \text{Exactly } 3 = 35

Dividing the remaining difference of 70 by 2 gives the number of applications in all three sets.

Key Concept

Three-Set Venn Diagram Region Decomposition

Question 259Question →

For all real numbers $x$ and $y$ , the custom binary operator $\odot$ is defined by $x \odot y = x^2 y - x y^2$ . The function $f$ is defined by $f(x) = x \odot 3$ . What is the value of $f(f(2))$ ?

Show answer & explanation

Answer: 162

Answer

The value of

f(f(2))

is 162.

To evaluate the nested function

f(f(2))

, first determine

f(2)

using the definition

f(x) = x \odot 3

. Applying the definition of the custom operator

x \odot y = x^2 y - x y^2

with

x = 2

and

y = 3

yields

2 \odot 3 = (2)^2(3) - (2)(3)^2 = 12 - 18 = -6

. Then, evaluate

f(-6) = -6 \odot 3 = (-6)^2(3) - (-6)(3)^2 = 108 - (-54) = 162

.

Step-by-Step Solution

1

Evaluate the inner function expression f(2)

f(2) = 2 \odot 3 = (2)^2(3) - (2)(3)^2 = 12 - 18 = -6

By definition, f(x) = x \odot 3. Substituting x = 2 gives 2 \odot 3, which applies the custom operator definition x^2 y - x y^2.

2

Evaluate the outer function expression f(f(2)) = f(-6)

f(-6) = -6 \odot 3 = (-6)^2(3) - (-6)(3)^2 = 36(3) - (-54) = 108 + 54 = 162

Substitute the inner result -6 into the function f(x).

Key Concept

Custom Binary Operators and Nested Function Evaluation

Question 260Question →

An agricultural facility uses two automated loading conveyors, Conveyor $A$ and Conveyor $B$ , to fill a grain storage silo. Working alone at its constant rate, Conveyor $A$ can fill the silo in 10 hours. Working alone at its constant rate, Conveyor $B$ can fill the silo in 15 hours. Conveyor $A$ begins filling the empty silo alone. After 4 hours, Conveyor $B$ is turned on, and both conveyors work together at their respective constant rates until the silo is completely full. How many total hours does it take, from the moment Conveyor $A$ starts, to completely fill the silo?

Show answer & explanation

Answer: 7.6

Answer

The total time required to fill the silo is 7.6 hours.

Conveyor A fills 1/10 of the silo per hour and works alone for 4 hours, completing 2/5 of the total job. The remaining 3/5 of the job is completed by both conveyors working together at a combined rate of 1/10 + 1/15 = 1/6 silo per hour. Dividing 3/5 by 1/6 gives 3.6 hours for the second stage. Adding the initial 4 hours yields a total time of 7.6 hours.

Step-by-Step Solution

1

Determine the individual hourly rates of Conveyor A and Conveyor B.

Conveyor A rate =

1/10

silo per hour; Conveyor B rate =

1/15

silo per hour.

Work rate is the reciprocal of the time required to complete the total job.

2

Calculate the fraction of the silo filled by Conveyor A working alone for 4 hours.

Work completed =

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

of the silo.

Work done equals rate multiplied by time.

3

Determine the remaining fraction of work to be completed.

Remaining work =

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

of the silo.

Subtract the completed fraction from the total job (1).

4

Calculate the combined work rate when both conveyors operate together.

Combined rate =

\frac{1}{10} + \frac{1}{15} = \frac{3 + 2}{30} = \frac{5}{30} = \frac{1}{6}

silo per hour.

When working together, individual rates add up.

5

Determine the time required for both conveyors to finish the remaining work.

Time =

\frac{3/5}{1/6} = \frac{18}{5} = 3.6

hours.

Divide remaining work by the combined work rate.

6

Sum the time spent in both stages to find the total time.

Total time =

4 + 3.6 = 7.6

hours.

The total duration includes 4 hours of Conveyor A operating alone plus 3.6 hours of combined operation.

Key Concept

Work Rate and Combined Work

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Answer

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Common Mistakes

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