All practice questions

Question 961Question →

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

Show answer & explanation

Answer: 240

Answer

240

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

6! = 720

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

5! \times 2! = 240

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

\frac{240}{2} = 120

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

360 - 120 = 240

.

Step-by-Step Solution

1

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

360 arrangements

For 6 distinct letters, there are

6! = 720

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

\frac{720}{2} = 360

.

2

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

120 arrangements

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

5! = 120

ways. The block itself has

2! = 2

internal orderings, giving

120 \times 2 = 240

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

\frac{240}{2} = 120

.

3

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

240 arrangements

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

360 - 120 = 240

.

Key Concept

Permutations with Relative Position and Non-Adjacency Restrictions

Question 962Question →

For what value of the constant $k$ does the system of linear equations below have no solution?

\begin{aligned} (k - 2)x + 3y &= 6 \\ 4x + (k + 2)y &= 12 \end{aligned}

-2

4

-4

Both

-4

and

4

16

Show answer & explanation

Answer: $-4$

Answer

The constant value

k = -4

results in a system with no solution.

For a

2 \times 2

linear system

a_1 x + b_1 y = c_1

and

a_2 x + b_2 y = c_2

to have no solution, the equations must have proportional variable coefficients but non-proportional constant terms:

\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}

. Solving

\frac{k-2}{4} = \frac{3}{k+2}

gives

k^2 - 4 = 12

, so

k = \pm 4

. Substituting

k = -4

yields

-6x + 3y = 6

and

4x - 2y = 12

, which simplify to

-2x + y = 2

and

-2x + y = -6

. Since the constants

2

and

-6

differ, the lines are parallel and distinct, meaning the system has no solution.

Step-by-Step Solution

1

Set up the condition for parallel lines (equal slopes) by equating the ratio of the coefficients of

x

and

y

.

\frac{k - 2}{4} = \frac{3}{k + 2}

A system of two linear equations has either zero solutions (parallel, non-intersecting lines) or infinitely many solutions (coincident lines) when the slopes are equal.

2

Cross-multiply and solve the quadratic equation for

k

.

(k - 2)(k + 2) = 12 \implies k^2 - 4 = 12 \implies k^2 = 16 \implies k = 4 \text{ or } k = -4

Finding all values of

k

where the coefficient matrix determinant is zero.

3

Test

k = 4

in the original system.

(4 - 2)x + 3y = 6 \implies 2x + 3y = 6 \quad \text{and} \quad 4x + (4 + 2)y = 12 \implies 4x + 6y = 12

Dividing

4x + 6y = 12

by

2

yields

2x + 3y = 6

, which is identical to the first equation. Thus,

k = 4

yields infinitely many solutions.

4

Test

k = -4

in the original system.

(-4 - 2)x + 3y = 6 \implies -6x + 3y = 6 \quad \text{and} \quad 4x + (-4 + 2)y = 12 \implies 4x - 2y = 12

Simplifying both equations gives

-2x + y = 2

and

-2x + y = -6

. The lines have identical slopes but different constants, so they are parallel and distinct, producing no solution.

Key Concept

System Solvability and Linear Consistency

Estimated Time:2m 0s

Question 963Question →

A pharmaceutical research laboratory tested $120$ chemical compounds for the presence of three distinct bio-reactivity markers: Enzyme Inhibition ( $E$ ), Protein Binding ( $P$ ), and Receptor Affinity ( $R$ ). The test results revealed the following:

- $58$ compounds exhibited Enzyme Inhibition ( $E$ ).
- $52$ compounds exhibited Protein Binding ( $P$ ).
- $46$ compounds exhibited Receptor Affinity ( $R$ ).
- $8$ compounds exhibited all three markers.
- $15$ compounds exhibited none of the three markers.

How many of the tested compounds exhibited exactly two of the three markers?

27

35

43

51

66

Show answer & explanation

Answer: $35$

Answer

The number of compounds exhibiting exactly two markers is

35

.

The total number of compounds displaying at least one marker is

120 - 15 = 105

. The sum of the single set counts is

58 + 52 + 46 = 156

. Using the three-set relation

N(A) + N(B) + N(C) - N(\text{union}) = N(\text{exactly 2}) + 2N(\text{all 3})

, we get

156 - 105 = N(\text{exactly 2}) + 2(8)

. Simplifying gives

51 = N(\text{exactly 2}) + 16

, which yields

N(\text{exactly 2}) = 35

.

Step-by-Step Solution

1

Determine the total number of compounds exhibiting at least one marker.

\text{At least one marker} = 120 - 15 = 105

.

Subtracting the compounds with no markers from the total gives the union size

N(E \cup P \cup R)

.

2

Calculate the sum of the individual set counts.

N(E) + N(P) + N(R) = 58 + 52 + 46 = 156

.

This sum counts elements with exactly one marker once, elements with exactly two markers twice, and elements with all three markers three times.

3

Apply the 3-set region formula connecting single counts, region overlaps, and total union.

156 = (\text{Exactly 1}) + 2(\text{Exactly 2}) + 3(\text{All 3})

and

105 = (\text{Exactly 1}) + (\text{Exactly 2}) + (\text{All 3})

.

Subtracting the second equation from the first yields:

156 - 105 = (\text{Exactly 2}) + 2(\text{All 3})

.

4

Substitute the known value for all three markers to solve for compounds with exactly two markers.

51 = (\text{Exactly 2}) + 2(8) \implies 51 = (\text{Exactly 2}) + 16 \implies \text{Exactly 2} = 35

.

Solving the linear algebraic equation yields the exact number of compounds in the pairwise-only regions.

Key Concept

Three-Set Overlapping Venn Diagrams and Region Counting Equations

Estimated Time:2m 0s

Question 964Question →

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

Show answer & explanation

Answer: 9

Answer

The value of

a + b + c

is 9.

For

K = 2^a 5^b 11^c

, the number of divisors is given by

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

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

(a+1)(b+1)

additional divisors. Since this difference is 12,

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

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

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

divisors. Since this difference is 36,

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

. For positive integers

a

and

b

, the only factor pair of 12 that allows

36 / (a+b+3)

to be an integer is when

a+b = 6

. Consequently,

c+1 = 4

, so

c = 3

. Therefore,

a+b+c = 6+3 = 9

.

Step-by-Step Solution

1

Write the divisor count formulas for

K

,

10K

, and

11K

.

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

,

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

, and

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

.

The number of positive divisors of a number

2^x 5^y 11^z

is

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

.

2

Use

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

to solve for

(a+1)(b+1)

.

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

.

Factoring out

(a+1)(b+1)

simplifies the equation directly.

3

Use

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

to find the relationship between

a, b,

and

c

.

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

.

Expanding

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

.

4

Determine

a+b

and

c

using positive integer constraints.

a+b = 6

and

c = 3

.

Since

a, b \ge 1

, the pairs for

(a+1, b+1)

yielding 12 are

(2,6)

or

(3,4)

. If

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

,

a+b=6

, making

a+b+3=9

and

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

. If

(3,4)

,

a+b=5

, so

a+b+3=8

, but

36/8

is not an integer.

5

Sum

a

,

b

, and

c

.

a + b + c = 6 + 3 = 9

.

Adding the derived sum

a+b=6

and

c=3

gives 9.

Key Concept

Divisor Count Function from Prime Factorization

Estimated Time:2m 0s

Question 965Question →

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

Show answer & explanation

Answer: 50

Answer

The value of

x

is 50.

To find

x

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

x + y = 80

, which gives

y = 80 - x

. The concentrate equation is

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

. Substituting

y = 80 - x

yields

0.30x + 0.70(80 - x) = 36

. Expanding gives

0.30x + 56 - 0.70x = 36

, so

-0.40x = -20

, which results in

x = 50

.

Step-by-Step Solution

1

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

System equations:

x + y = 80

and

0.30x + 0.70y = 36

.

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

2

Substitute

y = 80 - x

into the concentrate equation to eliminate

y

.

0.30x + 0.70(80 - x) = 36

.

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

3

Simplify the single-variable linear equation and solve for

x

.

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

.

Combining like terms isolates the variable

x

.

Key Concept

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

Question 966Question →

For all positive real numbers $a$ and $b$ such that $b = 4a$ , if $a^b = b^a$ , then the value of $a$ is $\sqrt{2}$ .

True

False

Show answer & explanation

Answer: False

Answer

False. The statement is false because the exact value of

a

is

\sqrt[3]{4}

, which is not equal to

\sqrt{2}

.

The statement is false because simplifying

a^{4a} = (4a)^a

leads directly to

a^3 = 4

, giving

a = \sqrt[3]{4} = 2^{2/3}

. The claimed value

\sqrt{2} = 2^{1/2}

fails to satisfy the original equation, making the assertion mathematically false.

Step-by-Step Solution

1

Substitute

b = 4a

into the given exponent equation

a^b = b^a

.

a^{4a} = (4a)^a

Eliminate variable

b

to express the equation solely in terms of

a

.

2

Raise both sides of the equation to the power of

\frac{1}{a}

.

(a^{4a})^{1/a} = ((4a)^a)^{1/a} \implies a^4 = 4a

Apply power of a power exponent rule

(x^m)^n = x^{mn}

to simplify the exponents.

3

Divide both sides by

a

(since

a > 0

) and solve for

a

.

\frac{a^4}{a} = \frac{4a}{a} \implies a^3 = 4 \implies a = \sqrt[3]{4} = 2^{2/3}

Isolate

a

using standard division and radical extraction rules.

4

Compare the calculated value of

a

with

\sqrt{2}

.

a = 2^{2/3} \approx 1.587

, whereas

\sqrt{2} = 2^{1/2} \approx 1.414

. Thus

a \neq \sqrt{2}

.

Determine the truth value of the claimed conclusion.

Key Concept

Solving variable exponent equations of the form

a^b = b^a

using exponent power rules and root extraction.

Question 967Question →

For real numbers $x$ , $y$ , and $z$ , none of which is equal to zero, suppose that $x y^3 z > 0$ , $\frac{x^2 z}{y} < 0$ , and $x + y > 0$ . Which of the following inequalities MUST be true?

\frac{x + z}{y} < 0

\frac{z}{x} < 0

y + z > 0

x^2 > y^2

x y z^2 > 0

Show answer & explanation

Answer: $\frac{x + z}{y} < 0$

Answer

\frac{x + z}{y} < 0

The condition

x y^3 z > 0

reduces to

x y z > 0

because

y^2 > 0

. The condition

\frac{x^2 z}{y} < 0

reduces to

\frac{z}{y} < 0

because

x^2 > 0

, implying

y

and

z

have opposite signs (

y z < 0

). Substituting

y z < 0

into

x (y z) > 0

forces

x < 0

. Furthermore,

x + y > 0

with

x < 0

requires

y > 0

. Then

y > 0

and

y z < 0

force

z < 0

. Consequently,

x

and

z

are both negative numbers, so

x + z

is negative. Dividing this negative sum by the positive number

y

guarantees that

\frac{x + z}{y} < 0

MUST be true.

Step-by-Step Solution

1

Analyze the first given inequality to establish a sign relationship among the variables.

Since

y \neq 0

,

y^2 > 0

. Dividing

x y^3 z > 0

by

y^2

gives

x y z > 0

.

Dividing an inequality by a strictly positive quantity preserves the inequality sign.

2

Analyze the second given inequality to determine the relationship between

y

and

z

.

Since

x \neq 0

,

x^2 > 0

. Dividing

\frac{x^2 z}{y} < 0

by

x^2

gives

\frac{z}{y} < 0

, which means

z

and

y

must have opposite signs (i.e.,

y z < 0

).

A quotient of two non-zero numbers is negative if and only if the numerator and denominator have opposite signs.

3

Deduce the individual signs of

x

,

y

, and

z

using the results of Steps 1 and 2 and the condition

x + y > 0

.

From

x(yz) > 0

and

yz < 0

, we deduce

x < 0

. Since

x < 0

and

x + y > 0

,

y

must be positive (

y > 0

). Because

y > 0

and

yz < 0

,

z

must be negative (

z < 0

).

Product of two negative numbers is positive, and adding a positive number larger in magnitude than a negative number yields a positive sum.

4

Evaluate the expression

\frac{x + z}{y}

using the established signs.

Since

x < 0

and

z < 0

, their sum

x + z < 0

. Dividing the negative sum by positive

y

gives

\frac{x + z}{y} < 0

.

The sum of two negative numbers is negative, and dividing a negative number by a positive number gives a negative quotient.

Key Concept

Deduction of positive and negative variable signs from products, quotients, and sums in inequalities.

Estimated Time:2m 0s

Question 968Question →

An automated delivery van and an electric cargo scooter both travel along a straight route from Warehouse X to Warehouse Y, which are $D$ miles apart.

The van travels at a constant speed of $30$ miles per hour for the first half of the total distance, and at a constant speed of $60$ miles per hour for the second half of the total distance.

The scooter travels at a constant speed of $30$ miles per hour for the first half of its total travel time, and at a constant speed of $60$ miles per hour for the second half of its total travel time.

If the scooter completes the entire journey in $15$ minutes less time than the van, what is the value of $D$ , in miles?

30

45

60

90

180

Show answer & explanation

Answer: $90$

Answer

The distance

D

between Warehouse X and Warehouse Y is

90

miles.

The option stating

90

is correct because the van's total time is

\frac{D}{40}

hours (average speed

40

mph across equal distances) and the scooter's total time is

\frac{D}{45}

hours (average speed

45

mph across equal time intervals). Setting the difference

\frac{D}{40} - \frac{D}{45}

equal to

\frac{1}{4}

hour yields

D = 90

miles.

Step-by-Step Solution

1

Calculate the total travel time for the van in terms of

D

.

The van travels

\frac{D}{2}

miles at

30

mph and

\frac{D}{2}

miles at

60

mph. Time for first half:

\frac{D/2}{30} = \frac{D}{60}

hours. Time for second half:

\frac{D/2}{60} = \frac{D}{120}

hours. Total van time

T_{\text{van}} = \frac{D}{60} + \frac{D}{120} = \frac{3D}{120} = \frac{D}{40}

hours.

For equal distance legs, average speed is calculated via harmonic mean, giving total time

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

.

2

Calculate the total travel time for the scooter in terms of

D

.

Let

T_{\text{scooter}}

be the total travel time for the scooter. It spends

\frac{T_{\text{scooter}}}{2}

hours at

30

mph and

\frac{T_{\text{scooter}}}{2}

hours at

60

mph. Total distance

D = 30\left(\frac{T_{\text{scooter}}}{2}\right) + 60\left(\frac{T_{\text{scooter}}}{2}\right) = 15 T_{\text{scooter}} + 30 T_{\text{scooter}} = 45 T_{\text{scooter}}

. Thus,

T_{\text{scooter}} = \frac{D}{45}

hours.

For equal time legs, average speed is the arithmetic mean of the speeds (

45

mph).

3

Set up and solve the equation using the given time difference.

15

minutes is equal to

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

hours. The equation is

\frac{D}{40} - \frac{D}{45} = \frac{1}{4}

. Finding a common denominator of

360

:

\frac{9D - 8D}{360} = \frac{1}{4} \implies \frac{D}{360} = \frac{1}{4} \implies D = \frac{360}{4} = 90

miles.

Equating the difference between the van's travel time and the scooter's travel time to

15

minutes (

\frac{1}{4}

hour) allows solving for

D

.

Key Concept

Distinction between average speed over equal distances (harmonic mean) versus average speed over equal times (arithmetic mean).

Question 969Question →

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

Show answer & explanation

Answer: 35

Answer

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

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

10

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

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

.

Step-by-Step Solution

1

Identify the total count for each pairwise overlap of paradigms.

The pairwise counts are

|O \cap F| = 25

,

|F \cap R| = 18

, and

|O \cap R| = 22

, with all three paradigms

|O \cap F \cap R| = 10

.

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

2

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

Proficient in only

O

and

F

:

25 - 10 = 15

; proficient in only

F

and

R

:

18 - 10 = 8

; proficient in only

O

and

R

:

22 - 10 = 12

.

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

3

Sum the three distinct exclusive two-set counts.

15 + 8 + 12 = 35

.

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

Key Concept

Three-Set Venn Diagram Region Calculations

Alternative Method

Using a Venn diagram, enter

10

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

10

from each given pairwise total:

25 - 10 = 15

,

18 - 10 = 8

, and

22 - 10 = 12

. Adding these three region counts together gives

15 + 8 + 12 = 35

.

Estimated Time:1m 30s

Question 970Question →

For how many positive two-digit integers $n$ is the sum $n + (n + 2)$ equal to the product of exactly two distinct prime numbers?

20

21

22

24

25

Show answer & explanation

Answer: 21

Answer

There are 21 positive two-digit integers that satisfy the given condition.

Simplifying

n + (n + 2)

gives

2(n + 1)

. For

2(n + 1)

to equal the product of two distinct prime numbers,

n + 1

must be an odd prime number

p

(so that 2 and

p

are distinct). Because

n

is a two-digit positive integer (

10 \le n \le 99

),

p = n + 1

must satisfy

11 \le p \le 100

. There are 25 prime numbers under 100, and exactly 4 of them (2, 3, 5, 7) are less than 11. Therefore, there are

25 - 4 = 21

suitable prime numbers, yielding 21 valid values of

n

.

Step-by-Step Solution

1

Algebraically simplify the given sum.

n + (n + 2) = 2n + 2 = 2(n + 1)

Combining like terms isolates the common factor of 2.

2

Analyze the prime factorization structure.

The expression

2(n + 1)

has 2 as a prime factor. For it to be the product of exactly two distinct prime numbers,

n + 1

must be a prime number

p

distinct from 2 (i.e., an odd prime).

If

n + 1

were composite or equal to 2, the total number of distinct prime factors would not be exactly two distinct primes.

3

Determine the valid range for the prime

p = n + 1

.

Since

n

is a two-digit positive integer,

10 \le n \le 99

. Adding 1 to all parts gives

11 \le n + 1 \le 100

, so

11 \le p \le 100

.

The constraints on

n

dictate the bounds for the prime number

p

.

4

Count the number of prime numbers in the range

[11, 100]

.

There are 25 prime numbers less than 100. The primes less than 11 are 2, 3, 5, and 7 (4 primes). Thus, there are

25 - 4 = 21

primes in the range

[11, 100]

.

Subtracting primes outside the valid range yields the count of valid values for

n

.

Key Concept

Prime Factorization and Prime Number Properties

Estimated Time:1m 45s

Question 971Question →

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

Show answer & explanation

Answer: 7

Answer

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

To solve

|x^2 - 10| = 3x

, note that

3x \ge 0

(so

x \ge 0

). Splitting into two algebraic cases gives

x^2 - 3x - 10 = 0

(yielding

x = 5

and extraneous

x = -2

) and

x^2 + 3x - 10 = 0

(yielding

x = 2

and extraneous

x = -5

). The valid real solutions are

x = 5

and

x = 2

, whose sum is 7.

Step-by-Step Solution

1

Determine domain restriction based on the absolute value definition.

x \ge 0

Because an absolute value expression cannot be negative,

|x^2 - 10| = 3x

requires

3x \ge 0

, which means

x \ge 0

.

2

Solve Case 1:

x^2 - 10 = 3x

.

x = 5

Rearranging yields

x^2 - 3x - 10 = 0

, which factors as

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

. The roots are

x = 5

and

x = -2

. Reject

x = -2

because

x \ge 0

.

3

Solve Case 2:

-(x^2 - 10) = 3x

.

x = 2

Rearranging yields

x^2 + 3x - 10 = 0

, which factors as

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

. The roots are

x = 2

and

x = -5

. Reject

x = -5

because

x \ge 0

.

4

Sum all valid real solutions.

7

The valid real solutions are

x = 5

and

x = 2

. Their sum is

5 + 2 = 7

.

Key Concept

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

Estimated Time:1m 30s

Question 972Question →

Vessel A contains $60$ liters of a beverage mixture that is $75\%$ fruit concentrate by volume, and Vessel B contains $90$ liters of a beverage mixture that is $25\%$ fruit concentrate by volume. First, $10$ liters of pure fruit concentrate are added to Vessel B. Next, $x$ liters of the mixture in Vessel A are removed and replaced with $x$ liters of pure water. If the concentration of fruit concentrate in Vessel A is now equal to the concentration of fruit concentrate in Vessel B, what is the value of $x$ ?

10

20

25.5

34

40

Show answer & explanation

Answer: $34$

Answer

The value of

x

is

34

.

To find

x

, first determine the resulting concentration in Vessel B. Vessel B initially contains

0.25 \times 90 = 22.5

liters of concentrate. Adding

10

liters of pure concentrate gives

32.5

liters of concentrate in

100

liters of total mixture, which is a concentration of

32.5\%

. In Vessel A, there are initially

0.75 \times 60 = 45

liters of concentrate. Removing

x

liters of the mixture removes

0.75x

liters of concentrate. When replaced with

x

liters of pure water, the total volume remains

60

liters, so the new concentration is

\frac{45 - 0.75x}{60}

. Setting

\frac{45 - 0.75x}{60} = 0.325

gives

45 - 0.75x = 19.5

, which simplifies to

0.75x = 25.5

, yielding

x = 34

.

Step-by-Step Solution

1

Calculate the new concentration of fruit concentrate in Vessel B after adding

10

liters of pure concentrate.

Initial concentrate in B =

0.25 \times 90 = 22.5

liters. Adding

10

liters of pure concentrate yields

22.5 + 10 = 32.5

liters of concentrate in a total volume of

90 + 10 = 100

liters. Thus, the new concentration in B is

\frac{32.5}{100} = 32.5\%

(or

0.325

).

Adding pure concentrate increases both the solute amount and the total solution volume.

2

Express the concentration of fruit concentrate in Vessel A after removing

x

liters of mixture and replacing it with

x

liters of pure water.

Initial concentrate in A =

0.75 \times 60 = 45

liters. Removing

x

liters removes

0.75x

liters of concentrate. Replacing with

x

liters of pure water keeps the total volume at

60

liters. The remaining concentrate is

45 - 0.75x

, so the new concentration in A is

\frac{45 - 0.75x}{60}

.

The replacement maintains constant total volume while diluting the concentrate.

3

Equate the concentrations of Vessel A and Vessel B to solve for

x

.

\frac{45 - 0.75x}{60} = 0.325 \implies 45 - 0.75x = 19.5 \implies 0.75x = 25.5 \implies x = \frac{25.5}{0.75} = 34

.

Setting the two concentration expressions equal gives a single linear equation in terms of

x

.

Key Concept

Dilution and multi-vessel mixture balance

Question 973Question →

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

Show answer & explanation

Answer: 37

Answer

The maximum possible value of

M

is 37.

To maximize the largest element

M = x_8

, we express

M

in terms of the smallest element

x_1

using the range:

M = x_1 + 18

. Thus, maximizing

M

is equivalent to maximizing

x_1

. Testing

x_1 = 20

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

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

, which exceeds the required sum of 200. Testing

x_1 = 19

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

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

. Thus, the maximum possible value of

M

is

19 + 18 = 37

.

Step-by-Step Solution

1

Formulate the algebraic equations from the statistical properties given.

Sum of 8 terms =

8 \times 25 = 200

;

x_4 + x_5 = 48

;

x_8 = x_1 + 18 = M

.

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

2

Relate maximizing the maximum term

M

to maximizing the minimum term

x_1

.

Maximizing

M = x_1 + 18

requires making

x_1

as large as possible.

Since the range is fixed at 18,

M

increases directly as

x_1

increases.

3

Test

x_1 = 20

to determine feasibility.

Minimum possible sum for

x_1 = 20

is

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

, which is invalid.

Distinct integer constraints force

x_4 \ge 23

; since

x_4 + x_5 = 48

and

x_4 < x_5

,

x_4

must be 23 and

x_5

must be 25, forcing all lower bounds up.

4

Test

x_1 = 19

to confirm feasibility and construct a valid set.

The valid set

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

meets all criteria with a sum of 200.

The minimal sum for

x_1 = 19

is 198, leaving headroom to increase

x_7

to 29 to reach the sum of 200.

Key Concept

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

Question 974Question →

A biotechnology laboratory formulates three custom reagent mixtures—Solution $X$ , Solution $Y$ , and Solution $Z$ —using three chemical compounds: Alpha, Beta, and Gamma.

- Solution $X$ contains $2\text{ mL}$ of Alpha, $3\text{ mL}$ of Beta, and $1\text{ mL}$ of Gamma, and costs $\$13.00$ .
- Solution $Y$ contains $1\text{ mL}$ of Alpha, $2\text{ mL}$ of Beta, and $4\text{ mL}$ of Gamma, and costs $\$11.00$ .
- Solution $Z$ contains $3\text{ mL}$ of Alpha, $1\text{ mL}$ of Beta, and $2\text{ mL}$ of Gamma, and costs $\$13.00$ .

What is the cost of a mixture containing $5\text{ mL}$ of Alpha, $4\text{ mL}$ of Beta, and $3\text{ mL}$ of Gamma?

\$22.00

\$24.00

\$25.00

\$26.00

\$30.00

Show answer & explanation

Answer: $\$26.00$

Answer

\$26.00

The target quantity consists of

5\text{ mL}

of Alpha,

4\text{ mL}

of Beta, and

3\text{ mL}

of Gamma. By inspecting the given system, adding Solution

X

(

2\text{ mL}

Alpha,

3\text{ mL}

Beta,

1\text{ mL}

Gamma) and Solution

Z

(

3\text{ mL}

Alpha,

1\text{ mL}

Beta,

2\text{ mL}

Gamma) directly yields

5\text{ mL}

Alpha,

4\text{ mL}

Beta, and

3\text{ mL}

Gamma. Therefore, the required cost is simply the sum of the costs of Solution

X

and Solution

Z

:

\$13.00 + \$13.00 = \$26.00

.

Step-by-Step Solution

1

Set up the linear system of equations representing the cost of each solution.

Let

a

,

b

, and

g

be the cost per mL of Alpha, Beta, and Gamma, respectively.
Equation 1:

2a + 3b + g = 13

Equation 2:

a + 2b + 4g = 11

Equation 3:

3a + b + 2g = 13

Translate the given word problem into algebraic equations representing system relationships.

2

Identify the requested quantity and evaluate whether it can be formed as a linear combination of the given equations.

Target expression:

5a + 4b + 3g

Recognizing linear combinations avoids solving for individual variable values when not required.

3

Add Equation 1 and Equation 3.

(2a + 3b + g) + (3a + b + 2g) = 13 + 13 \implies 5a + 4b + 3g = 26

The sum of coefficients for Alpha (

2+3=5

), Beta (

3+1=4

), and Gamma (

1+2=3

) exactly matches the target mixture.

Key Concept

Solving systems of linear equations using linear combinations without full variable elimination

Estimated Time:2m 0s

Question 975Question →

If $x$ is a real number satisfying the equation $\sqrt{15 - 2x} = 2x - 3$ , what is the sum of all real values of $x$ that satisfy this equation?

3

\frac{5}{2}

-\frac{1}{2}

\frac{7}{2}

1

Show answer & explanation

Answer: $3$

Answer

The sum of all real values of

x

satisfying the equation is

3

.

Squaring both sides of

\sqrt{15 - 2x} = 2x - 3

gives

15 - 2x = 4x^2 - 12x + 9

, which simplifies to

2x^2 - 5x - 3 = 0

. Factoring yields candidate roots

x = 3

and

x = -1/2

. Substituting

x = 3

into the original equation gives

\sqrt{9} = 3

, which is true. Substituting

x = -1/2

yields

\sqrt{16} = -4

, which is false because principal radicals cannot evaluate to negative values. Therefore,

x = 3

is the sole valid solution, making the sum equal to

3

.

Step-by-Step Solution

1

Isolate the radical and state domain constraints.

The principal square root

\sqrt{15 - 2x}

must be non-negative, requiring

15 - 2x \ge 0 \implies x \le 7.5

, and

2x - 3 \ge 0 \implies x \ge 1.5

.

Radical expressions produce non-negative principal square roots.

2

Square both sides of the equation.

15 - 2x = (2x - 3)^2 \implies 15 - 2x = 4x^2 - 12x + 9

.

Eliminate the radical to form a polynomial equation.

3

Rearrange into standard quadratic form and solve for

x

.

4x^2 - 10x - 6 = 0 \implies 2x^2 - 5x - 3 = 0 \implies (2x + 1)(x - 3) = 0

, giving candidate solutions

x = 3

and

x = -1/2

.

Solve the quadratic equation using factoring.

4

Test candidate solutions in the original equation to eliminate extraneous roots.

For

x = 3

:

\sqrt{15 - 2(3)} = \sqrt{9} = 3

and

2(3) - 3 = 3

(Valid). For

x = -1/2

:

\sqrt{15 - 2(-1/2)} = \sqrt{16} = 4

, but

2(-1/2) - 3 = -4

(Extraneous).

Squaring an equation can introduce extraneous roots.

Key Concept

Solving Radical Equations and Identifying Extraneous Solutions

Question 976Question →

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

Show answer & explanation

Answer: 105

Answer

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

Out of the total

7! = 5,040

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

4! = 24

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

2! = 2

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

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

.

Step-by-Step Solution

1

Calculate slot selections for the odd digits

35 ways

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

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

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

2

Determine valid arrangements for the even digits

3 ways

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

3! = 6

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

\frac{6}{2} = 3

valid ways.

3

Multiply independent choices to find the total arrangements

105 codes

Applying the Fundamental Counting Principle,

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

.

Key Concept

Permutations with Relative Order Restrictions

Estimated Time:2m 0s

Question 977Question →

A management consulting firm charges clients based on three staff tiers: Junior Consultants, Senior Consultants, and Directors. Junior Consultants are billed at a rate of $\$120$ per hour, Senior Consultants at $\$210$ per hour, and Directors at $\$390$ per hour. On a strategic assignment, the ratio of hours worked by Junior Consultants to Senior Consultants to Directors was $5 : 3 : 2$ , respectively. What was the overall average hourly billing rate per consultant hour for this assignment?

\$201

\$240

\$255

\$282

\$670

Show answer & explanation

Answer: $\$201$

Answer

The overall average hourly billing rate per consultant hour for the assignment is

\$201

.

The weighted average is found by multiplying each tier's rate by its corresponding ratio weight, summing these products, and dividing by the sum of the ratio weights:

\frac{(5 \times 120) + (3 \times 210) + (2 \times 390)}{5 + 3 + 2} = \frac{600 + 630 + 780}{10} = \frac{2010}{10} = \$201

.

Step-by-Step Solution

1

Determine the total weight of the ratio elements.

Sum of ratio parts =

5 + 3 + 2 = 10

total parts of hours.

To calculate a weighted average, the sum of all weight parts must be calculated to serve as the denominator.

2

Calculate the total revenue generated for 10 representative hours based on the ratio.

Total revenue =

(5 \times \$120) + (3 \times \$210) + (2 \times \$390) = \$600 + \$630 + \$780 = \$2,010

.

Multiplying each tier's hourly billing rate by its respective weight yields the total weighted revenue.

3

Divide the total revenue by the total number of ratio hours.

Weighted average rate =

\frac{\$2,010}{10} = \$201

per hour.

Dividing total cost/revenue by total units yields the weighted average per unit.

Key Concept

Weighted Average formula:

\text{Weighted Average} = \frac{\sum (w_i \cdot x_i)}{\sum w_i}

, where

w_i

represents the weight of each component and

x_i

represents the value of each component.

Estimated Time:1m 30s

Question 978Question →

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

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

Show answer & explanation

Answer: 5

Answer

The maximum possible value of the expression

x + 2y - z

is 5.

By analyzing the given inequalities,

x

and

y

must have opposite signs, and

x

and

z

must have opposite signs, which means

y

and

z

must have the same sign. To maximize

x + 2y - z

, we examine the scenario where

x

is negative, while

y

and

z

are positive. Taking

z = 1

(the smallest positive integer),

y = 5

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

x = -4

satisfies

x + y \le 1

,

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

, and

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

. This yields

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

.

Step-by-Step Solution

1

Analyze product and quotient inequalities to deduce the relative signs of

x

,

y

, and

z

.

x

and

y

have opposite signs,

x

and

z

have opposite signs, and

|x| > |z| \ge 1

. Consequently,

y

and

z

share the same sign.

Because

z^2 > 0

for non-zero

z

,

x^3 y z^2 < 0

requires

x^3 y < 0

. Also

\frac{x}{z} < -1

implies

\frac{x}{z}

is negative and has magnitude greater than 1.

2

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

x + 2y - z

.

Case A (

x < 0, y > 0, z > 0

) allows positive contributions from

2y

and

-z

, whereas Case B (

x > 0, y < 0, z < 0

) bounds the expression below 2.

In Case B,

y

and

z

are negative, so

2y \le -2

suppresses the sum.

3

Apply integer domain bounds and inequality constraints to maximize

x + 2y - z

in Case A.

The maximum value is 5, achieved when

x = -4

,

y = 5

, and

z = 1

.

Setting

z = 1

(smallest positive integer) and

y = 5

(largest positive integer) with

x = -4

satisfies

x + y \le 1

and all problem conditions.

Key Concept

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

Estimated Time:2m 0s

Question 979Question →

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

Show answer & explanation

Answer: 3.5

Answer

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

In the first 2 hours, Cyclist A covers

18 \times 2 = 36

miles and Cyclist B covers

30 \times 2 = 60

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

30 \times 0.6

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

18 + 18 = 36

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

Step-by-Step Solution

1

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

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

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

2

Determine the distance separating the cyclists at t = 2 hours

150 - 96 = 54 miles remaining

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

3

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

30 * 0.60 = 18 mph

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

4

Calculate relative speed of approach after 2 hours

18 + 18 = 36 mph

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

5

Compute the time to cover the remaining distance

54 / 36 = 1.5 hours

Time equals distance divided by relative speed.

6

Sum the time segments to find the total elapsed time

2 + 1.5 = 3.5 hours

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

Key Concept

Relative Speed and Piecewise Motion in Converging Rate Problems

Question 980Question →

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

Show answer & explanation

Answer: 27

Answer

The value of

k + m

is 27.

For the arithmetic sequence, the

k

-th term is

a_k = a_1 + (k-1)d

. Setting

4 + 3(k-1) = 64

gives

3(k-1) = 60

, so

k-1 = 20

and

k = 21

. For the geometric sequence, the

m

-th term is

b_m = b_1 r^{m-1}

. Setting

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

gives

2^m = 64

, which implies

m = 6

. Adding the two values gives

k + m = 21 + 6 = 27

.

Step-by-Step Solution

1

Determine the term position

k

in the arithmetic sequence.

k = 21

Using

a_k = a_1 + (k-1)d

, set

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

.

2

Determine the term position

m

in the geometric sequence.

m = 6

Using

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

, set

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

.

3

Compute the sum of the two position indices

k

and

m

.

27

k + m = 21 + 6 = 27

.

Key Concept

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

Estimated Time:1m 30s

Answer

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