All practice questions

Question 221Question →

A local theater sold a total of $120$ tickets for a weekend performance, consisting only of adult tickets and student tickets. Adult tickets were sold for $\$15$ each, and student tickets were sold for $\$10$ each. If the total revenue from the ticket sales was $\$1{}500$ , how many adult tickets were sold?

Show answer & explanation

Answer: 60

Answer

The correct answer is 60 adult tickets.

By setting up the total ticket equation

a + s = 120

and total revenue equation

15a + 10s = 1500

, substituting

s = 120 - a

gives

15a + 10(120 - a) = 1500

. Simplifying yields

5a = 300

, so

a = 60

. Thus, 60 adult tickets were sold.

Step-by-Step Solution

1

Set up the linear equation in one variable.

15a + 10(120 - a) = 1500

Express the number of student tickets as

(120 - a)

so the revenue equation relies on a single variable.

2

Expand and combine like terms.

5a + 1200 = 1500

Distribute

10

across

(120 - a)

to simplify the equation.

3

Isolate the variable

a

.

a = 60

Subtract

1200

from both sides and divide by

5

to find the number of adult tickets.

Key Concept

Algebraic Modeling of Linear Systems in Two Variables

Estimated Time:1m 0s

Question 222Question →

How many integer values of $x$ satisfy the inequality $|x - 3| \leq 4$ ?

Show answer & explanation

Answer: 9

Answer

There are 9 integer values of

x

that satisfy the inequality.

Unfolding

|x - 3| \leq 4

yields

-4 \leq x - 3 \leq 4

. Adding 3 across all terms gives

-1 \leq x \leq 7

. The integer values satisfying this range are

-1, 0, 1, 2, 3, 4, 5, 6, 7

, which totals 9 values.

Step-by-Step Solution

1

Convert the absolute value inequality into a compound linear inequality.

-4 \leq x - 3 \leq 4

The absolute value inequality

|k| \leq c

(for

c \geq 0

) is equivalent to

-c \leq k \leq c

.

2

Isolate

x

by adding 3 across all parts of the inequality.

-1 \leq x \leq 7

Adding the same constant to all parts of an inequality maintains the inequality relationships.

3

Count the total number of integer values within the inclusive range

[-1, 7]

.

9

The number of integers in an inclusive range

[a, b]

is calculated as

b - a + 1

, which gives

7 - (-1) + 1 = 9

.

Key Concept

Solving absolute value inequalities and counting integer solutions within a bounded interval

Question 223Question →

A research group evaluated 150 renewable energy projects. Each project utilizes at least one of three primary energy technologies: solar, wind, or hydro. Exactly 85 projects utilize solar power, 60 projects utilize wind power, and 45 projects utilize hydro power. Furthermore, 20 projects utilize both solar and wind power, 15 projects utilize both solar and hydro power, and 10 projects utilize both wind and hydro power. How many of the surveyed projects utilize all three energy technologies?

Show answer & explanation

Answer: 5

Answer

5 projects utilize all three energy technologies.

Applying the three-set inclusion-exclusion principle Total = |A| + |B| + |C| - (|A ∩ B| + |A ∩ C| + |B ∩ C|) + |A ∩ B ∩ C| + Neither gives 150 = 85 + 60 + 45 - (20 + 15 + 10) + |A ∩ B ∩ C| + 0. Simplifying yields 150 = 145 + |A ∩ B ∩ C|, which gives |A ∩ B ∩ C| = 5.

Step-by-Step Solution

1

Identify known set sizes and recall the Principle of Inclusion-Exclusion for three sets.

Total = 150, |S| = 85, |W| = 60, |H| = 45, |S ∩ W| = 20, |S ∩ H| = 15, |W ∩ H| = 10, Neither = 0.

Setting up known quantities ensures proper application of set intersection rules.

2

Substitute the set sizes into the formula: Total = |S| + |W| + |H| - (|S ∩ W| + |S ∩ H| + |W ∩ H|) + |S ∩ W ∩ H| + Neither.

150 = 85 + 60 + 45 - (20 + 15 + 10) + |S ∩ W ∩ H| + 0.

Pairwise overlaps are double-counted when individual sets are summed and must be subtracted; the central triple overlap is over-subtracted and must be added back.

3

Combine known terms and isolate the unknown triple intersection.

150 = 190 - 45 + |S ∩ W ∩ H| => 150 = 145 + |S ∩ W ∩ H| => |S ∩ W ∩ H| = 5.

Subtracting 145 from 150 yields the exact count of projects belonging to all three sets.

Key Concept

Three-Set Inclusion-Exclusion Principle

Estimated Time:2m 0s

Question 224Question →

A boutique perfumery blends three essential oils—Jasmine, Sandalwood, and Bergamot—in an initial volume ratio of $4 : 3 : 2$ , respectively. To modify the fragrance profile, the perfumer adds $20\text{ milliliters}$ of Sandalwood and $40\text{ milliliters}$ of Bergamot to the mixture, leaving the amount of Jasmine unchanged. If the resulting volume ratio of Sandalwood to Bergamot is $5 : 6$ , what was the total initial volume, in milliliters, of the fragrance batch?

Show answer & explanation

Answer: 90

Answer

The total initial volume of the fragrance batch was 90 milliliters.

By representing the initial amounts of Jasmine, Sandalwood, and Bergamot as

4x

,

3x

, and

2x

respectively, the total initial volume is

9x

. Setting up the proportion

\frac{3x + 20}{2x + 40} = \frac{5}{6}

yields

x = 10

. Substituting

x = 10

into

9x

gives the initial volume of 90 milliliters.

Step-by-Step Solution

1

Define variables for the initial quantities using the given ratio.

Jasmine volume =

4x

, Sandalwood volume =

3x

, Bergamot volume =

2x

, where

x

is a positive multiplier. Total initial volume =

4x + 3x + 2x = 9x

.

Representing ratio parts algebraically allows setting up equations for the modified amounts.

2

Formulate an equation based on the new ratio after adding essential oils.

New Sandalwood volume =

3x + 20

, New Bergamot volume =

2x + 40

. Equating their ratio to

5 : 6

gives

\frac{3x + 20}{2x + 40} = \frac{5}{6}

.

The problem specifies the new relationship between Sandalwood and Bergamot.

3

Solve the proportion for

x

.

Cross-multiplying gives

6(3x + 20) = 5(2x + 40) \implies 18x + 120 = 10x + 200 \implies 8x = 80 \implies x = 10

.

Finding the multiplier

x

allows us to calculate the exact initial quantities.

4

Compute the total initial volume.

Total initial volume =

9x = 9 \times 10 = 90

milliliters.

The question asks for the total initial volume of the batch.

Key Concept

Solving multi-step ratio problems involving additions to individual components by establishing an algebraic multiplier.

Question 225Question →

For a certain constant $k$ , the quadratic equation $x^2 - 12x + k = 0$ has two real roots, $r_1$ and $r_2$ . If $r_1^2 + r_2^2 = 94$ , what is the value of $k$ ?

Show answer & explanation

Answer: 25

Answer

25

Applying Vieta's formulas gives

r_1 + r_2 = 12

and

r_1 r_2 = k

. Substituting these into the identity

(r_1 + r_2)^2 = r_1^2 + r_2^2 + 2r_1 r_2

yields

12^2 = 94 + 2k

. Solving

144 = 94 + 2k

gives

2k = 50

, so

k = 25

.

Step-by-Step Solution

1

Apply Vieta's formulas to the given quadratic equation

x^2 - 12x + k = 0

.

The sum of the roots is

r_1 + r_2 = 12

and the product of the roots is

r_1 r_2 = k

.

For a quadratic equation

ax^2 + bx + c = 0

, the sum of roots is

-b/a

and product is

c/a

.

2

Expand the square of the sum of the roots

(r_1 + r_2)^2

.

(r_1 + r_2)^2 = r_1^2 + r_2^2 + 2r_1 r_2

This algebraic identity connects the sum of roots, sum of squared roots, and product of roots.

3

Substitute

r_1 + r_2 = 12

,

r_1^2 + r_2^2 = 94

, and

r_1 r_2 = k

into the identity.

12^2 = 94 + 2k \implies 144 = 94 + 2k

Replacing terms with known numerical values creates a linear equation in

k

.

4

Isolate and solve for

k

.

2k = 144 - 94 = 50 \implies k = 25

Basic algebraic manipulation yields the exact value of

k

.

Key Concept

Vieta's Formulas and Symmetric Polynomial Identities

Estimated Time:1m 30s

Question 226Question →

A pharmaceutical laboratory prepares a liquid vaccine solution by blending three ingredients: Active Compound X, Active Compound Y, and Distilled Water in the volume ratio of $3 : 4 : 8$ , respectively. To meet updated formulation guidelines, a chemist adds $12$ liters of Active Compound Y and $36$ liters of Distilled Water to the solution, leaving the quantity of Active Compound X unchanged. If the new ratio of Active Compound X to Distilled Water in the resulting solution is $1 : 4$ , what was the total volume, in liters, of the solution before any ingredients were added?

Show answer & explanation

Answer: 135

Answer

The total volume of the solution before any ingredients were added was 135 liters.

Representing the initial volumes of Active Compound X, Active Compound Y, and Distilled Water as

3x

,

4x

, and

8x

, the initial total volume is

15x

liters. After adding

36

liters of Distilled Water while keeping Compound X at

3x

liters, the ratio of X to Water becomes

\frac{3x}{8x + 36} = \frac{1}{4}

. Solving

12x = 8x + 36

gives

4x = 36

, so

x = 9

. Multiplying

15

by

9

gives the initial total volume of

135

liters.

Step-by-Step Solution

1

Define the initial component quantities using a common ratio multiplier

The initial volumes of Compound X, Compound Y, and Distilled Water are

3x

,

4x

, and

8x

liters, giving an initial total volume of

15x

liters.

A ratio of

3 : 4 : 8

means the actual quantities are integer multiples of a common constant

x

.

2

Set up the ratio equation reflecting the additions

The equation comparing the unchanged Compound X to the updated volume of Distilled Water is

\frac{3x}{8x + 36} = \frac{1}{4}

.

No Compound X was added, so its volume stays

3x

, whereas

36

liters were added to the initial

8x

liters of Distilled Water.

3

Solve the algebraic equation for the multiplier

x

4(3x) = 1(8x + 36) \implies 12x = 8x + 36 \implies 4x = 36 \implies x = 9

.

Cross-multiplying isolates the terms with

x

and allows solving for the multiplier.

4

Calculate the target initial total volume

15 \times 9 = 135

liters.

Substituting

x = 9

into the initial total volume expression

15x

yields the final answer.

Key Concept

Solving multi-step ratio word problems by setting up unknown multiplier equations based on partial component alterations

Question 227Question →

A container initially holds $100$ liters of an acid solution that is $40\%$ acid by volume. First, $x$ liters of the solution are drained and replaced with an equal volume of pure water. After the mixture is thoroughly stirred, $x$ liters of the new solution are drained and replaced with an equal volume of pure acid. If the final solution is $45.6\%$ acid by volume, what is the value of $x$ ?

Show answer & explanation

Answer: 20

Answer

The value of

x

is

20

liters.

The solution requires tracking the amount of pure acid through two distinct replacement operations. Initially, there are

40

liters of acid in

100

liters of solution. Draining

x

liters removes

0.4x

liters of acid, leaving

(40 - 0.4x)

liters of acid after adding

x

liters of pure water. In the second step, draining

x

liters removes a fraction

\frac{x}{100}

of the remaining acid, leaving

(40 - 0.4x)(1 - \frac{x}{100})

liters of acid. Adding

x

liters of pure acid yields a total acid volume of

(40 - 0.4x)(1 - \frac{x}{100}) + x = 45.6

liters. Expanding and solving the resulting quadratic equation

x^2 + 50x - 1400 = 0

gives

x = 20

(since

x > 0

).

Step-by-Step Solution

1

Determine initial volume of solute (pure acid).

Initial acid volume =

40\% \times 100 = 40

liters.

Establishing the starting amount of pure acid in the

100

-liter container.

2

Model the acid amount after the first replacement (with water).

Acid volume after first replacement =

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

liters.

Draining

x

liters removes

\frac{x}{100}

of the total acid, and adding water adds zero acid.

3

Model the acid amount after the second replacement (with pure acid).

Final acid volume =

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

liters.

Draining

x

liters of the new solution removes

\frac{x}{100}

of its acid, and replacing with pure acid adds

x

liters of acid.

4

Set up and solve the quadratic equation given final acid volume of

45.6

liters.

x^2 + 50x - 1400 = 0 \implies (x+70)(x-20) = 0 \implies x = 20

.

Discarding the negative root (

x = -70

) because volume must be positive.

Key Concept

Multi-stage sequential mixture removal and replacement

Question 228Question →

A barista has $40$ ounces of a flavored drink mixture that is $30\%$ syrup by volume. How many ounces of pure water must be added to reduce the syrup concentration to $24\%$ by volume?

Show answer & explanation

Answer: 10

Answer

10 ounces of pure water must be added.

Adding 10 ounces of pure water increases the total volume of the mixture from 40 ounces to 50 ounces while keeping the volume of pure syrup constant at 12 ounces. The resulting concentration is 12 / 50 = 0.24, or 24% by volume.

Step-by-Step Solution

1

Calculate the volume of pure syrup present in the initial mixture

12

ounces of syrup

The initial 40-ounce mixture contains 30% syrup by volume (

40 \times 0.30 = 12

).

2

Set up the concentration equation after adding

x

ounces of pure water

\frac{12}{40 + x} = 0.24

Adding pure water increases the total volume to

40 + x

ounces without changing the amount of pure syrup.

3

Solve the linear equation for

x

x = 10

Multiplying both sides by

40 + x

yields

12 = 9.6 + 0.24x

, which simplifies to

2.4 = 0.24x

, giving

x = 10

.

Key Concept

Dilution of a mixture by adding pure solvent

Question 229Question →

The function

f

is defined for all real numbers by

f(x) = \begin{cases} 3x - 1 & \text{if } x \le 4 \\ x^2 - k & \text{if } x > 4 \end{cases}

where

k

is a constant. If

f(f(3)) = 14

, what is the value of

k

?

Show answer & explanation

Answer: 50

Answer

50

To solve for

k

, evaluate the composite function

f(f(3))

step-by-step. First, find

f(3)

. Since

3 \le 4

, we use the first branch of the definition:

f(3) = 3(3) - 1 = 8

. Next, evaluate

f(8)

. Since

8 > 4

, we use the second branch of the definition:

f(8) = 8^2 - k = 64 - k

. Given that

f(f(3)) = 14

, we set

64 - k = 14

, which yields

k = 50

.

Step-by-Step Solution

1

Evaluate the inner function value f(3)

f(3) = 8

Because 3 is less than or equal to 4, the definition f(x) = 3x - 1 applies, yielding f(3) = 3(3) - 1 = 8.

2

Express f(f(3)) in terms of k

f(8) = 64 - k

Since the inner value is 8 and 8 > 4, the piecewise condition specifies using f(x) = x^2 - k, giving f(8) = 8^2 - k = 64 - k.

3

Solve the linear equation for k

k = 50

Setting the calculated expression equal to the given value 14 gives 64 - k = 14, which simplifies to k = 50.

Key Concept

Piecewise Function Evaluation and Parameter Resolution

Estimated Time:1m 30s

Question 230Question →

For all non-zero real numbers $a$ and $b$ , the binary operator $\star$ is defined by $a \star b = \frac{a^2 + b}{a}$ . The function $f$ is defined by $f(x) = x \star 16$ for all $x \neq 0$ . If $k$ is a positive real number such that $f(f(k)) = 10$ , what is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is

4

.

By definition of the custom operator,

f(x) = x + \frac{16}{x}

. Evaluating

f(4) = 4 + \frac{16}{4} = 8

and then

f(f(4)) = f(8) = 8 + \frac{16}{8} = 10

confirms that

k = 4

satisfies the condition.

Step-by-Step Solution

1

Express

f(x)

by applying the custom operator definition

f(x) = x \star 16 = \frac{x^2 + 16}{x} = x + \frac{16}{x}

Using

a = x

and

b = 16

in

a \star b = \frac{a^2 + b}{a}

simplifies the function formula.

2

Set

m = f(k)

and solve

f(m) = 10

for

m

m + \frac{16}{m} = 10 \implies m^2 - 10m + 16 = 0 \implies (m - 2)(m - 8) = 0

, yielding

m = 2

or

m = 8

Replacing the inner function evaluation with an intermediate variable creates a manageable quadratic equation.

3

Solve

f(k) = 2

and

f(k) = 8

for positive real values of

k

For

f(k) = 2

:

k^2 - 2k + 16 = 0

has discriminant

(-2)^2 - 4(1)(16) = -60 < 0

, giving no real solutions. For

f(k) = 8

:

k^2 - 8k + 16 = 0 \implies (k - 4)^2 = 0 \implies k = 4

.

Only

m = 8

produces a real value for

k

because

k + \frac{16}{k} \geq 8

for all positive real numbers

k

.

Key Concept

Custom Binary Operators and Nested Function Evaluation

Question 231Question →

In a corporate workshop of $80$ executives, $52$ executives attended the Leadership module and $40$ executives attended the Negotiation module. If $18$ executives attended neither module, how many executives attended both modules?

Show answer & explanation

Answer: 30

Answer

30 executives attended both modules.

Applying the formula

\text{Total} = \text{Group A} + \text{Group B} - \text{Both} + \text{Neither}

yields

80 = 52 + 40 - \text{Both} + 18

. Simplifying gives

80 = 110 - \text{Both}

, so the number of executives who attended both modules is

30

.

Step-by-Step Solution

1

Determine the number of executives who attended at least one of the two modules.

Executives attending at least one module =

80 - 18 = 62

.

Subtracting those who attended neither module from the total gives the union of the two sets.

2

Apply the inclusion-exclusion principle to find the intersection of the two sets.

\text{Both} = \text{Leadership} + \text{Negotiation} - \text{At least one} = 52 + 40 - 62 = 30

.

Summing the participants of each module double-counts those who attended both, so subtracting the number attending at least one module isolates the overlap.

Key Concept

Two-Set Overlapping Sets Formula (Inclusion-Exclusion Principle)

Estimated Time:1m 0s

Question 232Question →

At a software firm, an audit of 200 developers evaluated proficiency in three programming languages: Python, Java, and C++.

- 120 developers are proficient in Python.
- 105 developers are proficient in Java.
- 95 developers are proficient in C++.
- 15 developers are proficient in all three languages.
- 10 developers are proficient in none of the three languages.
- The ratio of the number of developers proficient in BOTH Python and Java ONLY to the number of developers proficient in BOTH Python and C++ ONLY is $3 : 2$ .
- The number of developers proficient in ONLY Java is equal to the number of developers proficient in ONLY C++.

How many developers are proficient in ONLY Python?

Show answer & explanation

Answer: 55

Answer

55 developers are proficient in ONLY Python.

By decomposing the 3-set Venn diagram into 7 mutually exclusive regions, the given ratio

d : e = 3 : 2

gives

d = 3k

and

e = 2k

. Comparing the totals for Java (

105

) and C++ (

95

) under the condition that Java-only equals C++-only shows that

k = 10

. Thus,

30

developers are in Python and Java only,

20

are in Python and C++ only, and

15

are in all three. Subtracting these three regions from the total

120

Python developers yields

120 - 30 - 20 - 15 = 55

proficient in Python only.

Step-by-Step Solution

1

Assign variables to the Venn diagram regions and incorporate given ratios.

Represent

d = |P \cap J \text{ only}| = 3k

and

e = |P \cap C \text{ only}| = 2k

, with

g = 15

and

\text{None} = 10

.

Establishing explicit variables for non-overlapping regions simplifies system solving.

2

Set up set total equations for

|J|

and

|C|

to solve for

k

.

|J| \implies b + 3k + f = 90

and

|C| \implies b + 2k + f = 80

. Subtracting yields

k = 10

.

Since

b = c

(Java only = C++ only), subtracting the two set equations eliminates

b

and

f

, directly giving

k

.

3

Calculate the region values

d

and

e

, then solve for

a = |P \text{ only}|

.

d = 30

,

e = 20

. Then

a = 120 - (30 + 20 + 15) = 55

.

Subtracting all other regions of set

P

from the total proficient in Python isolates those proficient ONLY in Python.

Key Concept

Three-Set Overlapping Sets and Region Decomposition

Question 233Question →

At a research foundation, annual grant funding was initially allocated among three divisions—Artificial Intelligence, Biotechnology, and Renewable Energy—in the ratio $7 : 4 : 3$ , respectively. At mid-year, the foundation redistributed the funding by transferring $\$10$ million from the Artificial Intelligence division to the Renewable Energy division, while the Biotechnology division's funding remained unchanged. After this transfer, the ratio of Artificial Intelligence funding to Renewable Energy funding became $3 : 2$ . What was the total annual grant funding, in millions of dollars, allocated across all three divisions?

Show answer & explanation

Answer: 140

Answer

The total annual grant funding allocated across all three divisions was

140

million dollars.

By setting the initial funding amounts to

7x

,

4x

, and

3x

million dollars, the total funding is

14x

. The transfer alters Artificial Intelligence funding to

7x - 10

and Renewable Energy funding to

3x + 10

. Setting their ratio to

\frac{3}{2}

yields the equation

2(7x - 10) = 3(3x + 10)

, which simplifies to

5x = 50

, giving

x = 10

. Thus, total funding is

14 \times 10 = 140

million dollars.

Step-by-Step Solution

1

Define variables for the initial allocation based on the ratio

7 : 4 : 3

.

Artificial Intelligence funding

= 7x

, Biotechnology funding

= 4x

, Renewable Energy funding

= 3x

, and total funding

= 7x + 4x + 3x = 14x

.

Ratios represent relative parts of a whole multiplier

x

.

2

Model the redistribution of funds.

New Artificial Intelligence funding

= 7x - 10

; New Renewable Energy funding

= 3x + 10

.

Transferring

\$10

million decreases Artificial Intelligence funding by

10

and increases Renewable Energy funding by

10

.

3

Equate the new ratio to

3 : 2

and solve for

x

.

\frac{7x - 10}{3x + 10} = \frac{3}{2} \implies 14x - 20 = 9x + 30 \implies 5x = 50 \implies x = 10

.

Cross-multiplication converts ratio relationships into a solvable linear equation.

4

Calculate the total annual grant funding.

Total funding

= 14x = 14 \times 10 = 140

million dollars.

Substituting

x = 10

into the total funding expression

14x

gives the final answer.

Key Concept

Algebraic setup of multi-part ratio redistribution problems

Question 234Question →

In a group of $100$ students, $60$ students study Spanish and $40$ students study French. Among the $60$ students studying Spanish, $15$ also study French. If a student who studies Spanish is selected at random, what is the probability that the selected student also studies French?

Show answer & explanation

Answer: 0.25

Answer

The probability that a randomly selected Spanish-studying student also studies French is 0.25.

Since the student is chosen from the group of

60

Spanish-studying students, the sample space is restricted to

60

. Within this subset,

15

students study French. The conditional probability is therefore

\frac{15}{60} = 0.25

.

Step-by-Step Solution

1

Identify the total number of outcomes in the restricted sample space.

The sample space is restricted to students studying Spanish:

n(\text{Spanish}) = 60

.

The question specifies that the student is chosen from those who study Spanish.

2

Identify the number of favorable outcomes within this restricted sample space.

The number of students studying both Spanish and French is

n(\text{Spanish} \cap \text{French}) = 15

.

We need the count of students who satisfy both the given condition and the target event.

3

Calculate the conditional probability.

P(\text{French} | \text{Spanish}) = \frac{15}{60} = 0.25

.

Conditional probability is calculated by dividing the intersection count by the given condition's total count.

Key Concept

Conditional Probability:

P(A|B) = \frac{n(A \cap B)}{n(B)}

Question 235Question →

A quality control inspector tests items from a batch containing $n$ high-grade items and $4$ low-grade items, where $n > 1$ . Two items are selected sequentially. If the first item selected is high-grade, it is kept out of the batch for testing, and additionally $1$ low-grade item is also removed from the batch before the second selection. If the probability of selecting two high-grade items in succession is $\frac{3}{8}$ , what is the value of $n$ ?

Show answer & explanation

Answer: 6

Answer

The initial number of high-grade items

n

is

6

.

Because the outcome of the first draw alters both the count of high-grade items and the count of low-grade items, the two draws are dependent events. The probability of drawing a high-grade item on the first draw is

\frac{n}{n+4}

. Following this successful outcome, the remaining pool consists of

n-1

high-grade items and

3

low-grade items, yielding a total of

n+2

items. The probability of a second high-grade selection is therefore

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

. Multiplying these probabilities gives

\frac{n(n-1)}{(n+4)(n+2)} = \frac{3}{8}

. Solving the resulting quadratic equation

5n^2 - 26n - 24 = 0

gives

n = 6

(since

n

must be a positive integer).

Step-by-Step Solution

1

Calculate the probability of selecting a high-grade item on the first draw.

P(H_1) = \frac{n}{n+4}

The initial batch contains

n

high-grade items out of a total of

n+4

items.

2

Determine the conditional sample space for the second draw after event

H_1

occurs.

High-grade items remaining =

n-1

; low-grade items remaining =

4-1=3

; total remaining items =

(n-1)+3 = n+2

.

Drawing a high-grade item removes

1

high-grade item and

1

additional low-grade item from the batch.

3

Calculate the conditional probability of selecting a high-grade item on the second draw.

P(H_2 | H_1) = \frac{n-1}{n+2}

There are now

n-1

high-grade items out of

n+2

total remaining items.

4

Set up and solve the algebraic probability equation.

\frac{n(n-1)}{(n+4)(n+2)} = \frac{3}{8} \implies 8n^2 - 8n = 3(n^2 + 6n + 8) \implies 5n^2 - 26n - 24 = 0 \implies (5n+4)(n-6) = 0 \implies n = 6

.

Because

n > 1

is required, the negative fraction root

-\frac{4}{5}

is discarded.

Key Concept

Probability of dependent events with non-standard sample space modifications

Question 236Question →

A museum cataloged $250$ historical artifacts. Each artifact was subjected to at least one of three preservation procedures: micro-scanning, chemical analysis, or radiocarbon dating. Exactly $140$ artifacts underwent micro-scanning, $120$ underwent chemical analysis, and $100$ underwent radiocarbon dating. If $40$ artifacts underwent both micro-scanning and chemical analysis, $35$ underwent both chemical analysis and radiocarbon dating, and $45$ underwent both micro-scanning and radiocarbon dating, how many artifacts underwent all three procedures?

Show answer & explanation

Answer: 10

Answer

10 artifacts underwent all three preservation procedures.

Using the Inclusion-Exclusion Principle for three sets, the total union is equal to the sum of individual set sizes (

140 + 120 + 100 = 360

), minus the sum of two-set overlaps (

40 + 35 + 45 = 120

), plus the three-set overlap. Setting up

250 = 360 - 120 + x

yields

250 = 240 + x

, so

x = 10

.

Step-by-Step Solution

1

Set up the Inclusion-Exclusion formula for three overlapping sets.

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

The total size of the union equals the sum of the individual set sizes minus the pairwise intersections plus the triple intersection.

2

Substitute the given values into the formula.

250 = 140 + 120 + 100 - (40 + 35 + 45) + |A \cap B \cap C|

Since every artifact underwent at least one procedure, the total number of artifacts equals the union of all three sets.

3

Compute the sum of individual sets and pairwise intersections to isolate the unknown.

250 = 360 - 120 + |A \cap B \cap C| \Rightarrow 250 = 240 + |A \cap B \cap C|

Simplifying the arithmetic terms allows direct calculation of the triple intersection.

4

Subtract

240

from

250

to find the final value.

|A \cap B \cap C| = 10

Solving the linear equation yields the number of artifacts undergoing all three procedures.

Key Concept

Principle of Inclusion-Exclusion for Three Sets

Question 237Question →

A storage tank is completely filled with $150$ liters of a solution containing Compounds X, Y, and Water in a ratio of $2:3:5$ by volume. First, $30$ liters of the mixture are drained and replaced with $30$ liters of a liquid mixture that is $20\%$ Compound Y by volume. Next, $50$ liters of the resulting mixture are drained and replaced with $50$ liters of another liquid mixture that is $70\%$ Compound Y by volume. What is the percentage of Compound Y, by volume, in the final mixture?

Show answer & explanation

Answer: 42

Answer

The final mixture contains 42% of Compound Y by volume.

To find the final concentration, track the exact volume of Compound Y through both stages. Initially, the tank holds

150 \times \frac{3}{10} = 45

liters of Y. In the first replacement, draining

30

liters (

20\%

of the tank) leaves

80\%

of Y, which is

36

liters; adding

30

liters of

20\%

Y mixture adds

6

liters, bringing Y to

42

liters. In the second replacement, draining

50

liters (

\frac{1}{3}

of the tank) leaves

\frac{2}{3}

of Y, which is

28

liters; adding

50

liters of

70\%

Y mixture adds

35

liters, yielding

63

liters of Y. The final concentration is

\frac{63}{150} \times 100\% = 42\%

.

Step-by-Step Solution

1

Calculate initial volume of Compound Y in the tank

Initial Y volume = 45 liters

The ratio X:Y:Water is 2:3:5, giving a total of 2 + 3 + 5 = 10 parts. The fraction of Y is 3/10. For a 150-liter tank, Y = 150 * (3/10) = 45 liters.

2

Calculate Compound Y after the first draining and replacement cycle

Compound Y volume after first replacement = 42 liters

Draining 30 liters removes 30 / 150 = 1/5 of the mixture. Y remaining = 45 * (1 - 1/5) = 36 liters. Replacing with 30 liters of 20% Y adds 30 * 0.20 = 6 liters of Y. Total Y = 36 + 6 = 42 liters.

3

Calculate Compound Y after the second draining and replacement cycle

Compound Y volume after second replacement = 63 liters

Draining 50 liters removes 50 / 150 = 1/3 of the current mixture. Y remaining = 42 * (1 - 1/3) = 28 liters. Replacing with 50 liters of 70% Y adds 50 * 0.70 = 35 liters of Y. Total Y = 28 + 35 = 63 liters.

4

Compute final percentage of Compound Y

42%

The final volume of the tank remains 150 liters. Percentage of Y = (63 / 150) * 100% = 42%.

Key Concept

Multi-Stage Mixture Removal and Replacement

Question 238Question →

A delivery van travels from Warehouse A to Warehouse B at a constant speed of $40$ miles per hour and returns along the exact same route from Warehouse B to Warehouse A at a constant speed of $60$ miles per hour. If the distance between Warehouse A and Warehouse B is $120$ miles, what is the average speed of the delivery van for the entire round trip, in miles per hour?

Show answer & explanation

Answer: 48

Answer

The average speed for the entire round trip is

48

miles per hour.

To find the average speed for the round trip, divide the total distance (

240

miles) by the total time taken (

5

hours). The time taken for the first leg is

\frac{120}{40} = 3

hours, and for the return leg is

\frac{120}{60} = 2

hours. Thus, the average speed is

\frac{240}{5} = 48

miles per hour.

Step-by-Step Solution

1

Determine the total distance traveled during the round trip.

The distance from Warehouse A to B is

120

miles, making the total round-trip distance

120 + 120 = 240

miles.

Average speed requires the total distance for all legs of the journey.

2

Calculate time taken for each leg and find total time.

Time taken at

40

mph is

\frac{120}{40} = 3

hours. Time taken at

60

mph is

\frac{120}{60} = 2

hours. Total time =

3 + 2 = 5

hours.

Time equals distance divided by rate (

t = \frac{d}{r}

).

3

Divide total distance by total time to obtain average speed.

Average speed =

\frac{240 \text{ miles}}{5 \text{ hours}} = 48

miles per hour.

The defining formula for average speed is

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

.

Key Concept

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

Question 239Question →

If $x$ and $y$ are positive real numbers such that $x^2 - y^2 = 105$ and $x + y = 15$ , what is the value of $(x - 2y)^2$ ?

Show answer & explanation

Answer: 9

Answer

The value of

(x - 2y)^2

is 9.

Factoring

x^2 - y^2

into

(x - y)(x + y)

gives

(x - y)(15) = 105

, which simplifies to

x - y = 7

. Solving the system of equations

x + y = 15

and

x - y = 7

gives

x = 11

and

y = 4

. Substituting these values into the target expression

(x - 2y)^2

yields

(11 - 2(4))^2 = 3^2 = 9

.

Step-by-Step Solution

1

Apply the difference of squares factoring identity to

x^2 - y^2

(x - y)(x + y) = 105

The difference of two squares

x^2 - y^2

factors into

(x - y)(x + y)

.

2

Calculate the value of

x - y

x - y = 7

Since

x + y = 15

, dividing 105 by 15 gives

x - y = 7

.

3

Solve the system of equations for

x

and

y

x = 11

and

y = 4

Adding

(x + y) + (x - y) = 15 + 7

yields

2x = 22 \implies x = 11

. Subtracting

(x + y) - (x - y) = 15 - 7

yields

2y = 8 \implies y = 4

.

4

Evaluate the expression

(x - 2y)^2

9

Substitute

x = 11

and

y = 4

into

(x - 2y)^2

to obtain

(11 - 2(4))^2 = (11 - 8)^2 = 3^2 = 9

.

Key Concept

Difference of Squares Factoring and Systems of Equations

Question 240Question →

A private equity firm invested a total of $\$1,000,000$ divided between two portfolio companies, Company A and Company B. During the first year, the value of Company A increased by $x\%$ , while the value of Company B decreased by $x\%$ , where $x > 0$ . At the end of the first year, the value of Company A was $\$720,000$ , and the combined value of both companies was $\$1,040,000$ . If the value of Company B decreased by an additional $x\%$ during the second year relative to its value at the end of the first year, what was the value, in dollars, of Company B at the end of the second year?

Show answer & explanation

Answer: 256000

Answer

The value of Company B at the end of the second year was 256,000 dollars.

By representing the initial investments as algebraic expressions involving the multiplier

k = \frac{x}{100}

and setting up the total initial capital equation

A_0 + B_0 = 1,000,000

, we obtain a quadratic equation in

k

. Solving

(5k-1)^2 = 0

yields

k = 0.2

(

x = 20\%

). Decreasing Company B's Year 1 value of

\$320,000

by

20\%

gives

\$256,000

.

Step-by-Step Solution

1

Find the value of Company B at the end of Year 1

Year 1 value of Company B =

1,040,000 - 720,000 = 320,000

dollars

The total combined value of both companies at the end of Year 1 is given as $1,040,000.

2

Formulate algebraic expressions for initial values using rate

k = \frac{x}{100}

A_0 = \frac{720,000}{1+k}

and

B_0 = \frac{320,000}{1-k}

Company A increased by

x\%

so

A_1 = A_0(1+k)

; Company B decreased by

x\%

so

B_1 = B_0(1-k)

.

3

Set up and simplify the quadratic equation for the combined initial investment

\frac{720,000}{1+k} + \frac{320,000}{1-k} = 1,000,000 \implies 25k^2 - 10k + 1 = 0

Dividing by

80,000

yields

\frac{9}{1+k} + \frac{4}{1-k} = 12.5

. Multiplying through by

2(1-k^2)

leads to

2[9(1-k) + 4(1+k)] = 25(1-k^2)

.

4

Solve for

k

and determine percentage

x

(5k - 1)^2 = 0 \implies k = 0.2

, so

x = 20\%

Factoring the perfect square quadratic expression gives a unique solution for

k

.

5

Compute Company B's value at the end of Year 2

320,000 \times (1 - 0.2) = 256,000

dollars

Company B's value decreases by an additional

20\%

of its Year 1 value (

320,000

).

Key Concept

Algebraic Modeling of Rational/Quadratic Equations from Multi-Step Percent Change Scenarios

Estimated Time:2m 30s

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