All practice questions

Question 801Question →

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 802Question →

At a research institute, $120$ scientists are each assigned to at least one of three interdisciplinary projects: Alpha, Beta, and Gamma. The project assignments meet the following conditions:

- Exactly $70$ scientists work on Project Alpha.
- Exactly $65$ scientists work on Project Gamma.
- Exactly $25$ scientists work on both Project Alpha and Project Beta.
- Exactly $30$ scientists work on both Project Beta and Project Gamma.
- Exactly $20$ scientists work on both Project Alpha and Project Gamma.
- The number of scientists working on all three projects is equal to half the number of scientists who work on Project Gamma only.

How many scientists work on Project Beta only?

5

10

15

25

30

Show answer & explanation

Answer: $5$

Answer

Exactly

5

scientists work on Project Beta only.

The correct answer is

5

. By using the 3-set Venn diagram equation for Project Gamma, we find the number of scientists in all three projects is

x = 15

. Decomposing all disjoint regions gives Alpha only

= 40

, Gamma only

= 30

, and two-project overlaps of

10

,

15

, and

5

. Subtracting the sum of these six regions (

115

) from the total universe (

120

) leaves exactly

5

scientists working on Project Beta only.

Step-by-Step Solution

1

Define variables for the regions of the 3-set Venn diagram.

Let

x

be the number of scientists working on all three projects (Alpha

\cap

Beta

\cap

Gamma).

Setting the central intersection as

x

allows expressing all pairwise overlaps in terms of

x

.

2

Express the two-project-only regions and Project Gamma only in terms of

x

.

Alpha & Beta only

= 25 - x

; Beta & Gamma only

= 30 - x

; Alpha & Gamma only

= 20 - x

; Gamma only

= 2x

.

Each given two-set intersection includes the triple intersection

x

. Gamma only is twice

x

per the stem.

3

Set up an equation for the total number of scientists in Project Gamma (

65

) to solve for

x

.

\text{Gamma Total} = (2x) + (20 - x) + (30 - x) + x = x + 50 = 65 \implies x = 15

.

Summing all four regions comprising Project Gamma equates to the total given as

65

.

4

Calculate the numerical values of the intersection regions and Gamma only.

All three

= 15

; Gamma only

= 30

; Alpha & Beta only

= 10

; Beta & Gamma only

= 15

; Alpha & Gamma only

= 5

.

Substituting

x = 15

back into the expressions from Step 2 gives exact region values.

5

Calculate the number of scientists working on Project Alpha only.

\text{Alpha Total} = 70 = \text{Alpha only} + 10 + 5 + 15 \implies \text{Alpha only} = 40

.

Project Alpha is composed of Alpha only plus its three overlapping regions (

10

,

5

, and

15

).

6

Use the total universe of

120

scientists to solve for Project Beta only.

120 = \text{Alpha only} + \text{Beta only} + \text{Gamma only} + (A \cap B)_{\text{only}} + (B \cap G)_{\text{only}} + (A \cap G)_{\text{only}} + \text{All three} \implies 120 = 40 + \text{Beta only} + 30 + 10 + 15 + 5 + 15 = 115 + \text{Beta only} \implies \text{Beta only} = 5

.

Since every scientist is on at least one project, the sum of all 7 disjoint region counts equals the total population of

120

.

Key Concept

Three-Set Overlapping Sets and Venn Diagram Analysis

Question 803Question →

Three automated server clusters—Cluster $X$ , Cluster $Y$ , and Cluster $Z$ —process large-scale data analytics tasks. Working alone at their respective constant rates, Cluster $X$ can complete a standard workload in $12$ hours, Cluster $Y$ can complete it in $20$ hours, and Cluster $Z$ can complete it in $30$ hours. Cluster $X$ and Cluster $Y$ begin processing a standard workload together. After $4$ hours, Cluster $X$ goes offline due to maintenance while Cluster $Y$ continues working alone. Exactly $2$ hours after Cluster $X$ goes offline, Cluster $Z$ is brought online to assist Cluster $Y$ . How many additional hours will it take Cluster $Y$ and Cluster $Z$ working together to complete the remaining portion of the workload?

4.4

hours

5.6

hours

2.8

hours

18.33

hours

6.2

hours

Show answer & explanation

Answer: $4.4$ hours

Answer

The additional time required for Cluster Y and Cluster Z to finish the remaining workload is

4.4

hours.

The combined rate of Cluster X and Cluster Y is

\frac{1}{12} + \frac{1}{20} = \frac{2}{15}

job/hr. In

4

hours, they complete

4 \times \frac{2}{15} = \frac{8}{15}

of the job, leaving

\frac{7}{15}

. Cluster Y then works alone for

2

hours, completing

2 \times \frac{1}{20} = \frac{1}{10}

of the job. The remaining work is

\frac{7}{15} - \frac{1}{10} = \frac{11}{30}

. Finally, Cluster Y and Cluster Z work together at a combined rate of

\frac{1}{20} + \frac{1}{30} = \frac{1}{12}

job/hr. The time needed to finish is

\frac{11/30}{1/12} = 4.4

hours.

Step-by-Step Solution

1

Calculate individual work rates for each cluster per hour.

r_X = \frac{1}{12}

,

r_Y = \frac{1}{20}

, and

r_Z = \frac{1}{30}

of the total workload per hour.

Work rate is the reciprocal of the total time required to complete one full workload.

2

Calculate the work completed in Stage 1 when Cluster X and Cluster Y work together for 4 hours.

Combined rate

r_{X+Y} = \frac{1}{12} + \frac{1}{20} = \frac{5 + 3}{60} = \frac{8}{60} = \frac{2}{15}

. Work done in 4 hours =

4 \times \frac{2}{15} = \frac{8}{15}

. Remaining work =

1 - \frac{8}{15} = \frac{7}{15}

.

Multiply the combined rate of Clusters X and Y by 4 hours to find the fraction of work completed.

3

Calculate the work completed in Stage 2 when Cluster Y works alone for 2 hours.

Work done by Y in 2 hours =

2 \times \frac{1}{20} = \frac{1}{10} = \frac{3}{30}

. Remaining work =

\frac{7}{15} - \frac{1}{10} = \frac{14}{30} - \frac{3}{30} = \frac{11}{30}

.

Subtract the work done by Cluster Y during its solo 2-hour interval from the remaining work.

4

Calculate the time required for Cluster Y and Cluster Z working together to complete the remaining

11/30

of the work.

Combined rate

r_{Y+Z} = \frac{1}{20} + \frac{1}{30} = \frac{3 + 2}{60} = \frac{5}{60} = \frac{1}{12}

. Time

t = \frac{11/30}{1/12} = \frac{11}{30} \times 12 = \frac{132}{30} = 4.4

hours.

Divide the remaining workload fraction by the combined hourly work rate of Clusters Y and Z.

Key Concept

Combined Work Rate in Multi-Stage Problems

Estimated Time:2m 0s

Question 804Question →

Four sequence-derived values $S_1, S_2, S_3,$ and $S_4$ are defined below. Arrange these values in order from smallest to largest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from smallest to largest is

S_1 < S_3 < S_4 < S_2

(corresponding to

55 < 64 < 72 < 80

).

Computing each value yields

S_1 = 55

,

S_2 = 80

,

S_3 = 64

, and

S_4 = 72

. Arranging these values in ascending order results in

55 < 64 < 72 < 80

, corresponding to

S_1 < S_3 < S_4 < S_2

.

Step-by-Step Solution

1

Calculate the value of

S_1

S_1 = 55

Using the sum formula for an arithmetic sequence

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

, we have

S_5 = \frac{5}{2}[2(3) + (5-1)4] = \frac{5}{2}[6 + 16] = 55

.

2

Calculate the value of

S_2

S_2 = 80

Using the sum formula for a finite geometric sequence

S_n = \frac{a_1(r^n - 1)}{r - 1}

, we have

S_4 = \frac{2(3^4 - 1)}{3 - 1} = \frac{2(80)}{2} = 80

.

3

Calculate the value of

S_3

S_3 = 64

Using the formula for the

n

-th term of an arithmetic sequence

a_n = a_1 + (n-1)d

, we have

c_{10} = 10 + (10-1)6 = 10 + 54 = 64

.

4

Calculate the value of

S_4

S_4 = 72

Using the sum formula for an infinite geometric series

S_\infty = \frac{a_1}{1 - r}

, we have

S_4 = \frac{48}{1 - 1/3} = \frac{48}{2/3} = 72

.

5

Order the computed values from smallest to largest

S_1 (55) < S_3 (64) < S_4 (72) < S_2 (80)

Comparing the numerical values gives

55 < 64 < 72 < 80

.

Key Concept

Evaluation and comparison of finite and infinite arithmetic and geometric sequence terms and sums

Question 805Question →

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 806Question →

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 807Question →

A corporate venture capital firm evaluated $120$ technology startups for investment. Each startup met at least one of three key criteria: strong artificial intelligence capability, established revenue growth, or international market presence. Exactly $65$ startups met the artificial intelligence criteria, $55$ met the revenue growth criteria, and $50$ met the international presence criteria. Furthermore, $25$ startups met both the artificial intelligence and revenue growth criteria, $20$ met both the revenue growth and international presence criteria, and $15$ met both the artificial intelligence and international presence criteria. How many startups met all three investment criteria?

5

10

15

20

25

Show answer & explanation

Answer: $10$

Answer

The number of startups that met all three investment criteria is 10.

According to the Inclusion-Exclusion Principle for three sets, the total population is equal to the sum of the three individual sets minus the sum of the three pairwise intersections, plus the intersection of all three sets. Substituting the given values:

120 = 65 + 55 + 50 - (25 + 20 + 15) + x

, which simplifies to

120 = 110 + x

, giving

x = 10

. Thus,

10

startups met all three criteria.

Step-by-Step Solution

1

State the Principle of Inclusion-Exclusion for three sets

|A \cup B \cup C| = |A| + |B| + |C| - (|A \cap B| + |B \cap C| + |A \cap C|) + |A \cap B \cap C|

To account for startups counted in multiple overlapping categories without double-counting or over-subtracting.

2

Substitute the known values from the problem statement into the formula

120 = 65 + 55 + 50 - (25 + 20 + 15) + x

, where

x = |A \cap B \cap C|

Every startup meets at least one criterion, so the union size equals the total number of startups (

120

).

3

Simplify the sums and solve for

x

120 = 170 - 60 + x \implies 120 = 110 + x \implies x = 10

Subtracting

110

from

120

gives the exact number of startups meeting all three criteria.

Key Concept

Three-Set Inclusion-Exclusion Principle

Question 808Question →

A pharmaceutical facility uses two automated purification columns, Column $X$ and Column $Y$ , to process standard batches of a synthetic compound. Working alone at its constant rate, Column $X$ can purify a full batch in $15$ hours, while Column $Y$ can purify a full batch in $10$ hours working alone at its constant rate. Column $X$ begins purifying a batch alone. After $3$ hours, Column $Y$ is turned on to work alongside Column $X$ , and both columns work together at their respective constant rates until the batch is completely purified. What is the total time, in hours, from the moment Column $X$ started until the batch is completely purified?

4.8 hours

7.2 hours

7.8 hours

9.6 hours

11.0 hours

Show answer & explanation

Answer: 7.8 hours

Answer

7.8 hours

The correct answer of 7.8 hours is obtained by evaluating the work completed in stages. In the first 3 hours, Column X completes 3/15 or 1/5 of the batch, leaving 4/5 of the batch remaining. When Column Y joins Column X, their combined rate is 1/15 + 1/10 = 1/6 batch per hour. Completing the remaining 4/5 of the batch at a rate of 1/6 per hour requires (4/5) / (1/6) = 4.8 hours. Adding the initial 3 hours yields a total time of 7.8 hours.

Step-by-Step Solution

1

Determine the individual hourly work rates for each column

Rate of Column

X = \frac{1}{15}

batch per hour; Rate of Column

Y = \frac{1}{10}

batch per hour.

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

2

Calculate the fraction of work completed during the initial stage

Work done by

X = 3 \times \frac{1}{15} = \frac{3}{15} = \frac{1}{5}

of the batch.

Column

X

works alone for the first 3 hours.

3

Find the remaining fraction of work to be completed

Remaining work

= 1 - \frac{1}{5} = \frac{4}{5}

of the batch.

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

4

Determine the combined work rate of both columns working together

Combined rate

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

batch per hour.

Rates add when two entities work simultaneously.

5

Calculate the time required for both columns to finish the remaining work

Time together

= \frac{\text{Remaining Work}}{\text{Combined Rate}} = \frac{4/5}{1/6} = \frac{4}{5} \times 6 = \frac{24}{5} = 4.8

hours.

Time equals work divided by rate.

6

Calculate the total elapsed time from the start

Total time

= 3 \text{ hours (alone)} + 4.8 \text{ hours (together)} = 7.8

hours.

Add the initial duration to the combined working duration.

Key Concept

Work Rate and Combined Work

Question 809Question →

If $x$ is a real number satisfying the equation $2x^2 - 7x + 3 = 0$ and $y$ is a real number satisfying the equation $y^2 + 5y - 14 = 0$ , what is the maximum possible value of $\frac{x}{y}$ ?

\frac{3}{2}

\frac{3}{7}

\frac{1}{4}

-\frac{3}{7}

6

Show answer & explanation

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

Answer

The maximum possible value of

\frac{x}{y}

is

\frac{3}{2}

.

Factoring

2x^2 - 7x + 3 = 0

gives

(2x - 1)(x - 3) = 0

, so

x \in \left\{\frac{1}{2}, 3\right\}

. Factoring

y^2 + 5y - 14 = 0

gives

(y + 7)(y - 2) = 0

, so

y \in \{-7, 2\}

. The four possible values for

\frac{x}{y}

are

\frac{3}{2}

,

-\frac{3}{7}

,

\frac{1}{4}

, and

-\frac{1}{14}

. The greatest value among these is

\frac{3}{2}

.

Step-by-Step Solution

1

Solve the quadratic equation for

x

by factoring.

Factor

2x^2 - 7x + 3 = 0

as

(2x - 1)(x - 3) = 0

, yielding solutions

x = \frac{1}{2}

and

x = 3

.

Finding all valid real roots of the first equation determines possible values for the numerator.

2

Solve the quadratic equation for

y

by factoring.

Factor

y^2 + 5y - 14 = 0

as

(y + 7)(y - 2) = 0

, yielding solutions

y = -7

and

y = 2

.

Finding all valid real roots of the second equation determines possible values for the denominator.

3

Evaluate all possible combinations for the ratio

\frac{x}{y}

.

The possible ratios are

\frac{3}{2} = 1.5

,

\frac{3}{-7} = -\frac{3}{7}

,

\frac{1/2}{2} = \frac{1}{4} = 0.25

, and

\frac{1/2}{-7} = -\frac{1}{14}

.

Testing all root pairs ensures we identify the maximum overall value.

4

Select the maximum value among the evaluated ratios.

The largest value is

\frac{3}{2}

.

Comparing

1.5

,

-0.428

,

0.25

, and

-0.071

shows

1.5

is the greatest.

Key Concept

Solving quadratic equations via polynomial factoring and optimizing rational expressions over discrete solution sets.

Estimated Time:1m 30s

Question 810Question →

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 811Question →

A store owner purchases a bicycle for $150. To establish the list price, the store owner marks up the cost price by 40%. During a promotional sale, the bicycle is sold at a 20% discount off the list price. What is the store owner's net profit from the sale of the bicycle?

$18

$30

$42

$60

$168

Show answer & explanation

Answer: $18

Answer

$18

The correct answer of

18 is obtained by first calculating the list price (

150 + 40% of

150 =

210), then computing the selling price (

210 - 20% of

210 =

168), and finally finding the profit (

168 -

150 =

18).

Step-by-Step Solution

1

Calculate the list price after a 40% markup on the cost price of $150.

List Price =

150 × 1.40 =

210

Markup is applied directly to the cost price.

2

Calculate the selling price after a 20% discount on the list price of $210.

Selling Price =

210 × (1 - 0.20) =

210 × 0.80 = $168

Discounts are calculated based on the list price, not the cost price.

3

Subtract the original cost price from the final selling price to find the net profit.

Net Profit =

168 -

150 = $18

Profit is the difference between total revenue (selling price) and total cost.

Key Concept

Successive percentage changes with different bases (cost price vs. list price)

Question 812Question →

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 813Question →

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 814Question →

An international airline surveyed $360$ frequent flyers regarding three premium services they utilized during the past year: First-Class Lounge Access, Priority Baggage Handling, and In-Flight Wi-Fi. The survey revealed the following data:

- Every surveyed passenger utilized at least one of the three services.
- $220$ passengers utilized First-Class Lounge Access.
- $180$ passengers utilized Priority Baggage Handling.
- $140$ passengers utilized In-Flight Wi-Fi.
- $60$ passengers utilized all three services.

If the ratio of the number of passengers who utilized exactly one service to the number of passengers who utilized exactly two services is $4 : 1$ , how many passengers utilized exactly two services?

48

60

75

120

240

Show answer & explanation

Answer: $60$

Answer

The number of passengers who utilized exactly two services is 60.

By breaking the total set of passengers into disjoint groups—those using exactly one service (

E_1

), exactly two services (

E_2

), all three services (

E_3 = 60

), and none (

\text{None} = 0

)—we set up the equation

360 = E_1 + E_2 + 60 + 0

. Subtracting

60

from both sides gives

E_1 + E_2 = 300

. Given the ratio

E_1 : E_2 = 4 : 1

, we substitute

E_1 = 4E_2

, yielding

5E_2 = 300

, so

E_2 = 60

.

Step-by-Step Solution

1

Set up the fundamental 3-set total elements equation in terms of disjoint regions.

\text{Total} = E_1 + E_2 + E_3 + \text{None}

, where

E_1

is exactly 1 service,

E_2

is exactly 2 services, and

E_3

is all 3 services.

Categorizing members into mutually exclusive regions simplifies group accounting without double-counting.

2

Substitute the known values into the total elements formula.

360 = E_1 + E_2 + 60 + 0 \implies E_1 + E_2 = 300

.

Since every passenger used at least one service,

\text{None} = 0

, leaving

300

passengers who used either exactly 1 or exactly 2 services.

3

Apply the given ratio of

E_1 : E_2 = 4 : 1

to solve for

E_2

.

E_1 = 4E_2 \implies 4E_2 + E_2 = 300 \implies 5E_2 = 300 \implies E_2 = 60

.

The ratio implies that out of

5

total parts for

E_1 + E_2

,

1

part represents

E_2

.

Key Concept

3-Set Overlapping Sets Disjoint Regions Formula

Estimated Time:2m 0s

Question 815Question →

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 816Question →

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 817Question →

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 818Question →

A cyclist travels from Town A to Town B, a distance of $60$ miles, at a constant speed of $30$ miles per hour. On the return trip from Town B to Town A along the exact same route, she travels at a constant speed of $20$ miles per hour. What is the cyclist's average speed, in miles per hour, for the entire round trip?

25 miles per hour

24 miles per hour

50 miles per hour

12 miles per hour

10 miles per hour

Show answer & explanation

Answer: 24 miles per hour

Answer

24 miles per hour

To find the average speed for any multi-leg trip, divide total distance by total time. The first leg of

60

miles at

30

mph takes

2

hours. The return leg of

60

miles at

20

mph takes

3

hours. The total distance is

120

miles and total time is

5

hours. Therefore, the average speed is

\frac{120}{5} = 24

miles per hour.

Step-by-Step Solution

1

Calculate the time taken for the trip from Town A to Town B.

t_1 = \frac{60}{30} = 2

hours

Time equals distance divided by speed.

2

Calculate the time taken for the return trip from Town B to Town A.

t_2 = \frac{60}{20} = 3

hours

Time equals distance divided by speed.

3

Find the total distance and total time for the round trip.

Total Distance =

60 + 60 = 120

miles; Total Time =

2 + 3 = 5

hours

Average speed is based on the entire journey's distance and duration.

4

Compute the average speed.

\text{Average Speed} = \frac{120}{5} = 24\text{ miles per hour}

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}

Key Concept

Average Speed for a Multi-Leg Trip

Estimated Time:1m 0s

Question 819Question →

For all non-zero real numbers $x$ and $y$ , the binary operator $\diamondsuit$ is defined by $x \diamondsuit y = \frac{x}{y} - \frac{y}{x}$ . If the function $g$ is defined by $g(x) = x \diamondsuit 2$ for all non-zero real numbers $x$ , what is the value of $g(g(4))$ ?

-\frac{7}{12}

\frac{7}{12}

\frac{3}{2}

-\frac{15}{4}

0

Show answer & explanation

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

Answer

-\frac{7}{12}

Evaluating the inner expression

g(4)

yields

4 \diamondsuit 2 = \frac{4}{2} - \frac{2}{4} = \frac{3}{2}

. Evaluating

g\left(\frac{3}{2}\right)

yields

\frac{3/2}{2} - \frac{2}{3/2} = \frac{3}{4} - \frac{4}{3} = -\frac{7}{12}

, which is the correct final value.

Step-by-Step Solution

1

Evaluate the inner function expression

g(4)

using the given definition

g(x) = x \diamondsuit 2

.

g(4) = 4 \diamondsuit 2 = \frac{4}{2} - \frac{2}{4} = 2 - \frac{1}{2} = \frac{3}{2}

.

Nested function evaluations require evaluating the innermost expression first.

2

Substitute the result

g(4) = \frac{3}{2}

into the outer function to find

g\left(\frac{3}{2}\right)

.

g\left(\frac{3}{2}\right) = \frac{3}{2} \diamondsuit 2 = \frac{\frac{3}{2}}{2} - \frac{2}{\frac{3}{2}}

.

The output of the inner function becomes the input for the outer function.

3

Simplify the resulting fractions and subtract them.

\frac{3}{4} - \frac{4}{3} = \frac{9 - 16}{12} = -\frac{7}{12}

.

Finding a common denominator allows direct subtraction of fractions.

Key Concept

Nested Function Evaluation with Custom Operators

Estimated Time:1m 30s

Question 820Question →

A box contains $x$ blue spheres and $8$ yellow spheres, where $x$ is a positive integer. If two spheres are selected at random one after another without replacement, the probability that both selected spheres are blue is $\frac{5}{17}$ . What is the total number of spheres in the box initially?

10

15

18

20

25

Show answer & explanation

Answer: 18

Answer

The total number of spheres in the box initially is 18.

The probability of drawing two blue spheres sequentially without replacement is given by

\frac{x}{x+8} \times \frac{x-1}{x+7} = \frac{5}{17}

. Expanding and rearranging the equation gives

3x^2 - 23x - 70 = 0

, which factors into

(3x + 7)(x - 10) = 0

. Since

x

must be a positive integer,

x = 10

. The total number of spheres in the box initially is

x + 8 = 10 + 8 = 18

.

Step-by-Step Solution

1

Set up the probability expression for dependent sequential events.

P(\text{both blue}) = \frac{x}{x+8} \times \frac{x-1}{x+7}

Because the draws occur without replacement, the total count decreases from

x+8

to

x+7

and the number of blue spheres decreases from

x

to

x-1

for the second draw.

2

Equate to the given probability and clear denominators to form a quadratic equation.

\frac{x(x-1)}{(x+8)(x+7)} = \frac{5}{17} \implies 17(x^2 - x) = 5(x^2 + 15x + 56) \implies 3x^2 - 23x - 70 = 0

Cross-multiplying converts the rational probability equation into a standard quadratic equation.

3

Factor the quadratic equation to find the positive integer root.

(3x + 7)(x - 10) = 0 \implies x = 10

Since the count of spheres must be a positive integer,

x = 10

is the only valid solution for the number of blue spheres.

4

Calculate the total initial number of spheres.

\text{Total} = x + 8 = 10 + 8 = 18

The question asks for the total initial number of spheres, which includes both the

x

blue spheres and the 8 yellow spheres.

Key Concept

Probability of Dependent Events Without Replacement

Estimated Time:2m 0s

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Answer

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Answer

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Common Mistakes

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Common Mistakes

Alternative Method

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Common Mistakes

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