All practice questions

Question 1781Question →

For all positive real numbers $a$ and $b$ , the custom operation $\diamondsuit$ is defined by $a \diamondsuit b = \frac{a^2 + b^2}{ab}$ . The function $f$ is defined for all $x > 0$ by $f(x) = x \diamondsuit 4$ . If $f(x) = 2.5$ , what is the value of $x$ that is greater than $4$ ?

Show answer & explanation

Answer: 8

Answer

The value of

x

greater than

4

is

8

.

Applying the custom operator gives

f(x) = \frac{x^2 + 16}{4x}

. Setting this equal to

2.5

yields

\frac{x^2 + 16}{4x} = \frac{5}{2}

, which simplifies to

x^2 - 10x + 16 = 0

. The roots are

x = 2

and

x = 8

. Since

x

must be greater than

4

, the only valid answer is

8

.

Step-by-Step Solution

1

Substitute

a = x

and

b = 4

into the custom operation definition

a \diamondsuit b = \frac{a^2 + b^2}{ab}

.

f(x) = \frac{x^2 + 16}{4x}

This establishes the explicit algebraic rule for the function

f(x)

.

2

Set

f(x)

equal to

2.5

and clear the fraction.

\frac{x^2 + 16}{4x} = 2.5 \implies x^2 + 16 = 10x

Multiplying both sides by

4x

converts the rational equation into a standard polynomial equation.

3

Rearrange into standard quadratic form and solve by factoring.

x^2 - 10x + 16 = 0 \implies (x - 2)(x - 8) = 0 \implies x = 2

or

x = 8

Factoring determines all potential positive real solutions for

x

.

4

Select the solution satisfying the constraint

x > 4

.

x = 8

The question explicitly specifies that

x

must be greater than

4

, eliminating

x = 2

.

Key Concept

Evaluating custom binary operations and solving algebraic function equations involving quadratic constraints.

Question 1782Question →

A container holds $n$ spheres, exactly 5 of which are blue and the remaining $n - 5$ are green. Two spheres are drawn at random from the container one after another without replacement. If the probability that at least one of the selected spheres is green is $\frac{14}{15}$ , what is the total number of spheres $n$ in the container?

15

20

25

30

75

Show answer & explanation

Answer: 25

Answer

The total number of spheres

n

in the container is 25.

The probability of at least one green sphere is complementary to drawing zero green spheres (meaning both spheres drawn are blue). Subtracting

\frac{14}{15}

from 1 yields

P(\text{both blue}) = \frac{1}{15}

. Since the selection is without replacement,

P(\text{both blue}) = \frac{5}{n} \times \frac{4}{n-1} = \frac{20}{n(n-1)}

. Equating this to

\frac{1}{15}

gives

n(n-1) = 300

. Solving the quadratic equation

n^2 - n - 300 = 0

gives

n = 25

, which correctly represents the total number of spheres.

Step-by-Step Solution

1

Use the complement rule to determine the probability that both selected spheres are blue.

P(\text{both blue}) = 1 - P(\text{at least one green}) = 1 - \frac{14}{15} = \frac{1}{15}

.

The event that at least one sphere is green is the complement of the event that both drawn spheres are blue.

2

Set up the joint probability equation for drawing two blue spheres sequentially without replacement.

P(\text{both blue}) = \frac{5}{n} \times \frac{4}{n - 1} = \frac{20}{n(n - 1)}

.

There are 5 blue spheres initially out of

n

. After drawing one blue sphere, 4 blue spheres remain out of

n - 1

total spheres.

3

Equate the expressions and solve for

n

.

\begin{aligned} \frac{20}{n(n - 1)} &= \frac{1}{15} \\ n(n - 1) &= 300 \\ n^2 - n - 300 &= 0 \\ (n - 25)(n + 12) &= 0 \end{aligned}

Cross-multiplying gives a quadratic equation in terms of

n

.

4

Select the valid positive integer solution for

n

.

n = 25

(since

n > 0

).

The total number of spheres must be a positive integer.

Key Concept

Probability of Complementary Events and Dependent Sequential Events

Estimated Time:2m 0s

Question 1783Question →

A courier service calculates the shipping cost for a package based on its weight $w$ , in pounds. For packages weighing up to $20$ pounds, the cost is a flat fee of $\$12.00$ plus $\$2.50$ per pound. For packages weighing more than $20$ pounds, the cost is a flat fee of $\$20.00$ plus $\$2.00$ per pound. If the total shipping cost for a package was strictly greater than $\$45.00$ and at most $\$70.00$ , which of the following could be the weight of the package, in pounds? Select all such weights.

Select all that apply

12

pounds

15

pounds

20

pounds

24

pounds

26

pounds

Show answer & explanation

Answer: $15$ pounds; $20$ pounds; $24$ pounds

Answer

The valid weights of the package are 15 pounds, 20 pounds, and 24 pounds.

The weight of the package must result in a total cost

C(w)

such that

\$45.00 < C(w) \le \$70.00

. Evaluating the options:
- For 15 pounds:

12 + 2.50(15) = \$49.50

, which falls within the range.
- For 20 pounds:

12 + 2.50(20) = \$62.00

, which falls within the range.
- For 24 pounds:

20 + 2.00(24) = \$68.00

, which falls within the range.

Step-by-Step Solution

1

Formulate the cost function

C(w)

as a piecewise model.

C(w) = 12 + 2.50w

for

0 < w \le 20

, and

C(w) = 20 + 2.00w

for

w > 20

.

The cost structure changes depending on whether the weight exceeds

20

pounds.

2

Set up and solve the inequality

45 < C(w) \le 70

for packages up to

20

pounds.

45 < 12 + 2.50w \le 70 \implies 33 < 2.50w \le 58 \implies 13.2 < w \le 23.2

. Restricting to

w \le 20

gives

13.2 < w \le 20

.

This determines the valid weight interval for packages billed under the first tier.

3

Set up and solve the inequality

45 < C(w) \le 70

for packages over

20

pounds.

45 < 20 + 2.00w \le 70 \implies 25 < 2.00w \le 50 \implies 12.5 < w \le 25

. Restricting to

w > 20

gives

20 < w \le 25

.

This determines the valid weight interval for packages billed under the second tier.

4

Combine the valid intervals and evaluate each option.

The overall valid weight interval is

13.2 < w \le 25

. Testing the values: 12 pounds is outside the range; 15 pounds, 20 pounds, and 24 pounds are within the range; 26 pounds is outside the range.

Any weight strictly greater than

13.2

pounds and up to

25

pounds produces a cost between

\$45.00

and

\$70.00

.

Key Concept

Algebraic Modeling of Piecewise Functions and Compound Inequalities

Question 1784Question →

A short-pulse laser emits a single pulse lasting $8.4 \times 10^{-9}$ seconds. A high-speed optical sensor completes one measurement cycle every $1.4 \times 10^{-11}$ seconds. How many measurement cycles does the sensor complete during the duration of a single laser pulse?

Show answer & explanation

Answer: 600

Answer

600

Dividing the pulse duration (

8.4 \times 10^{-9}

seconds) by the sensor cycle time (

1.4 \times 10^{-11}

seconds) yields

\frac{8.4}{1.4} \times 10^{-9 - (-11)} = 6 \times 10^2 = 600

complete cycles.

Step-by-Step Solution

1

Set up the division expression for the total number of cycles.

\frac{8.4 \times 10^{-9}\text{ seconds}}{1.4 \times 10^{-11}\text{ seconds}}

To find how many cycle intervals fit within the total pulse duration.

2

Divide the decimal coefficients.

\frac{8.4}{1.4} = 6

Separating the numerical coefficients from the powers of ten.

3

Apply exponent rules to divide powers of 10.

10^{-9 - (-11)} = 10^{-9 + 11} = 10^2 = 100

Dividing powers with the same base requires subtracting the denominator exponent from the numerator exponent.

4

Combine results to find total cycles.

6 \times 100 = 600

Multiplying coefficient quotient by the simplified power of ten.

Key Concept

Division of Numbers in Scientific Notation and Exponent Rules

Question 1785Question →

A dataset $S$ consists of 7 distinct positive integers. The arithmetic mean of the numbers in $S$ is 20, and the median is 18. If the largest integer in $S$ is 35, what is the maximum possible value for the second-largest integer in $S$ ?

34

33

35

62

19

Show answer & explanation

Answer: 34

Answer

34

To find the maximum possible value of the second-largest integer, we arrange the 7 distinct positive integers in ascending order:

x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7

. The median

x_4 = 18

and the largest element

x_7 = 35

. The total sum of all 7 elements is

7 \times 20 = 140

. Since

x_6

must be strictly less than

x_7 = 35

, the maximum potential integer value for

x_6

is 34. If

x_6 = 34

, the remaining four elements (

x_1, x_2, x_3, x_5

) must sum to

87 - 34 = 53

. Choosing

x_5 = 19

(the smallest integer greater than 18) leaves a sum of 34 for

x_1 + x_2 + x_3

, which can be satisfied by distinct positive integers such as 1, 16, and 17. Thus, 34 is achievable.

Step-by-Step Solution

1

Calculate the total sum of the 7 integers in the dataset.

Total sum =

7 \times 20 = 140

.

The sum of a dataset equals the number of elements multiplied by its arithmetic mean.

2

Identify the positions of known elements when the dataset is ordered in ascending order

x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7

.

Median

x_4 = 18

and largest element

x_7 = 35

.

For a 7-element dataset, the median is the 4th element.

3

Calculate the combined sum of the remaining five unknown elements.

x_1 + x_2 + x_3 + x_5 + x_6 = 140 - (18 + 35) = 87

.

Subtracting the median and the largest element from the total sum gives the sum of the remaining five elements.

4

Determine the theoretical upper bound for the second-largest integer

x_6

.

Since

x_6 < x_7 = 35

and all integers are distinct,

x_6 \le 34

.

The second-largest integer must be strictly less than the largest integer.

5

Verify if

x_6 = 34

can produce a valid dataset of distinct positive integers.

If

x_6 = 34

, then

x_1 + x_2 + x_3 + x_5 = 87 - 34 = 53

. Setting

x_5 = 19

,

x_3 = 17

,

x_2 = 16

, and

x_1 = 1

gives

1 + 16 + 17 + 19 = 53

, forming the valid set

\{1, 16, 17, 18, 19, 34, 35\}

.

Since a valid set of distinct positive integers exists satisfying all constraints, 34 is the maximum possible value.

Key Concept

Measures of Central Tendency and Data Constraints

Question 1786Question →

An automated risk-management system uses three independent algorithms—Algorithm X, Algorithm Y, and Algorithm Z—to detect fraudulent transactions. The probability that Algorithm X detects a given fraudulent transaction is $\frac{3}{5}$ , the probability that Algorithm Y detects it is $\frac{2}{3}$ , and the probability that Algorithm Z detects it is $\frac{3}{4}$ . If a fraudulent transaction occurs, what is the probability that it will be detected by at least two of these three algorithms?

Show answer & explanation

Answer: 0.75

Answer

The probability that the transaction is detected by at least two of the three algorithms is 0.75 (or 3/4).

Because the algorithms operate independently, the event 'at least two algorithms detect the transaction' consists of four mutually exclusive outcomes: exactly X and Y detect (probability 6/60 = 0.10), exactly X and Z detect (probability 9/60 = 0.15), exactly Y and Z detect (probability 12/60 = 0.20), and all three detect (probability 18/60 = 0.30). Summing these four probabilities gives 0.10 + 0.15 + 0.20 + 0.30 = 0.75.

Step-by-Step Solution

1

Determine the complementary probabilities of non-detection for each algorithm.

P(X does not detect) = 2/5, P(Y does not detect) = 1/3, and P(Z does not detect) = 1/4.

The probability of an event's complement is 1 minus the probability of the event.

2

Calculate the probability for each scenario where exactly two algorithms detect the transaction.

P(X and Y only) = 6/60, P(X and Z only) = 9/60, P(Y and Z only) = 12/60.

Since the algorithms operate independently, joint probabilities are calculated by multiplying individual probabilities.

3

Calculate the probability that all three algorithms detect the transaction.

P(X and Y and Z) = 18/60.

Multiplying the individual detection probabilities of all three independent algorithms.

4

Sum the probabilities of all qualifying mutually exclusive outcomes.

(6/60) + (9/60) + (12/60) + (18/60) = 45/60 = 0.75.

The events representing different combinations of detections are mutually exclusive, so their probabilities add directly.

Key Concept

Independent Events and Addition Rule for Mutually Exclusive Outcomes

Question 1787Question →

If $p$ , $p + 2$ , and $p + 4$ are all prime numbers, what is the value of $p^2 + 5$ ?

6

9

14

19

30

Show answer & explanation

Answer: 14

Answer

The value of

p^2 + 5

is 14.

For any three consecutive odd integers

p

,

p + 2

, and

p + 4

, exactly one of them must be a multiple of 3. Because all three expressions represent prime numbers, the term divisible by 3 must be equal to 3 (since 3 is the only prime divisible by 3). Setting

p = 3

gives

p + 2 = 5

and

p + 4 = 7

, both of which are prime. Any choice of

p > 3

forces either

p + 2

or

p + 4

to be a multiple of 3 greater than 3, making it composite. Thus

p = 3

is uniquely determined, and evaluating

p^2 + 5

gives

3^2 + 5 = 14

.

Step-by-Step Solution

1

Analyze the possible remainders when the prime

p

is divided by 3.

Any positive integer

p

can be written in one of three forms:

3k

,

3k + 1

, or

3k + 2

for some integer

k

.

Dividing any integer by 3 leaves a remainder of 0, 1, or 2.

2

Test each remainder case for the expressions

p

,

p + 2

, and

p + 4

.

If

p = 3k + 1

, then

p + 2 = 3k + 3 = 3(k + 1)

, which is divisible by 3. If

p = 3k + 2

, then

p + 4 = 3k + 6 = 3(k + 2)

, which is divisible by 3.

Among any three consecutive odd numbers of the form

p, p+2, p+4

, exactly one of them must be a multiple of 3.

3

Deduce the unique value of

p

.

The only way all three numbers

p

,

p + 2

, and

p + 4

can be prime is if the one divisible by 3 is equal to 3 itself, which forces

p = 3

.

The only prime number divisible by 3 is 3 itself; any larger multiple of 3 is composite.

4

Evaluate the target expression

p^2 + 5

using

p = 3

.

3^2 + 5 = 9 + 5 = 14

.

Substitute the uniquely determined value

p = 3

into the given algebraic expression.

Key Concept

Divisibility properties of consecutive odd integers and prime number definitions

Estimated Time:1m 30s

Question 1788Question →

A logistics company operates two delivery vans, Van A and Van B. Van A consumes fuel at a rate of $1$ gallon for every $15$ miles driven, and Van B consumes fuel at a rate of $1$ gallon for every $25$ miles driven. On a specific trip, the two vans were driven a total combined distance of $450$ miles and together consumed exactly $22$ gallons of fuel. What was the total distance, in miles, driven by Van A?

100 miles

150 miles

225 miles

250 miles

300 miles

Show answer & explanation

Answer: 150 miles

Answer

The total distance driven by Van A was 150 miles.

The correct answer is 150 miles. By setting up the total fuel consumption equation as

\frac{d_A}{15} + \frac{450 - d_A}{25} = 22

, multiplying by the least common multiple

75

gives

5d_A + 1350 - 3d_A = 1650

, which simplifies to

2d_A = 300

and

d_A = 150

miles.

Step-by-Step Solution

1

Define variables for the distance traveled by each van.

Let

d_A

be the distance driven by Van A. Then the distance driven by Van B is

450 - d_A

.

The total combined distance for both vans is

450

miles.

2

Express the total fuel consumed in terms of

d_A

.

\frac{d_A}{15} + \frac{450 - d_A}{25} = 22

Fuel used equals distance divided by miles per gallon for each vehicle.

3

Clear denominators by multiplying the entire equation by the common denominator 75.

5d_A + 3(450 - d_A) = 1650

Simplifies fractions to solve the linear algebraic equation easily.

4

Solve for

d_A

.

5d_A + 1350 - 3d_A = 1650 \implies 2d_A = 300 \implies d_A = 150

Isolates the target variable

d_A

to find the distance driven by Van A.

Key Concept

Linear Equations and Rate-Distance Modeling

Question 1789Question →

In the $xy$ -plane, line $k$ has an $x$ -intercept of $(8, 0)$ and a $y$ -intercept of $(0, 6)$ . Line $p$ is perpendicular to line $k$ and intersects line $k$ at its $y$ -intercept. What is the $x$ -intercept of line $p$ ?

-8

-\frac{9}{2}

\frac{9}{2}

8

-\frac{32}{3}

Show answer & explanation

Answer: $-\frac{9}{2}$

Answer

-\frac{9}{2}

The line

k

passes through

(8, 0)

and

(0, 6)

, giving a slope of

m_k = \frac{6 - 0}{0 - 8} = -\frac{3}{4}

. A line perpendicular to

k

must have a slope equal to the negative reciprocal of

-\frac{3}{4}

, which is

\frac{4}{3}

. Because line

p

intersects line

k

at

(0, 6)

, its

y

-intercept is also

6

, making its equation

y = \frac{4}{3}x + 6

. Setting

y = 0

gives

0 = \frac{4}{3}x + 6

, which solves to

x = -\frac{9}{2}

.

Step-by-Step Solution

1

Calculate the slope of line

k

Slope of line

k

is

m_k = \frac{6 - 0}{0 - 8} = -\frac{6}{8} = -\frac{3}{4}

The slope formula is

m = \frac{y_2 - y_1}{x_2 - x_1}

using given intercepts

(8, 0)

and

(0, 6)

.

2

Determine the slope of line

p

Slope of line

p

is

m_p = -\frac{1}{m_k} = \frac{4}{3}

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.

3

Write the equation of line

p

Equation of line

p

is

y = \frac{4}{3}x + 6

Line

p

intersects line

k

at its

y

-intercept

(0, 6)

, so line

p

has a

y

-intercept of

6

.

4

Find the

x

-intercept of line

p

x = -\frac{9}{2}

(or

-4.5

)

Set

y = 0

in the line equation:

0 = \frac{4}{3}x + 6 \implies \frac{4}{3}x = -6 \implies x = -6 \cdot \frac{3}{4} = -\frac{9}{2}

.

Key Concept

Perpendicular Slopes and Line Intercepts

Estimated Time:1m 30s

Question 1790Question →

In triangle $ABC$ , the measure of angle $A$ is $45^\circ$ and the measure of angle $B$ is $105^\circ$ . The length of side $AB$ is $6$ units. Which of the following statements must be true? Select all that apply.

Select all that apply

The length of the altitude from vertex

B

to side

AC

is

3\sqrt{2}

units.

The length of side

BC

is

6\sqrt{2}

units.

The area of triangle

ABC

is

9 + 9\sqrt{3}

square units.

The length of side

AC

is

6\sqrt{2} + 6\sqrt{6}

units.

The area of triangle

ABC

is

18 + 18\sqrt{3}

square units.

Show answer & explanation

Answer: The length of the altitude from vertex $B$ to side $AC$ is $3\sqrt{2}$ units.; The length of side $BC$ is $6\sqrt{2}$ units.; The area of triangle $ABC$ is $9 + 9\sqrt{3}$ square units.

Answer

The correct statements are: the length of the altitude from vertex

B

to side

AC

is

3\sqrt{2}

units, the length of side

BC

is

6\sqrt{2}

units, and the area of triangle

ABC

is

9 + 9\sqrt{3}

square units.

By drawing altitude

BD

perpendicular to

AC

, triangle

ABC

decomposes into two special right triangles. In the

45^\circ-45^\circ-90^\circ

triangle

ABD

, the hypotenuse is

6

, yielding leg lengths

BD = AD = 3\sqrt{2}

. In the

30^\circ-60^\circ-90^\circ

triangle

BCD

, side

BD = 3\sqrt{2}

is opposite the

30^\circ

angle, making hypotenuse

BC = 2 \times 3\sqrt{2} = 6\sqrt{2}

and long leg

CD = 3\sqrt{6}

. Combining

AD

and

CD

gives base

AC = 3\sqrt{2} + 3\sqrt{6}

, leading to an area of

\frac{1}{2}(3\sqrt{2} + 3\sqrt{6})(3\sqrt{2}) = 9 + 9\sqrt{3}

.

Step-by-Step Solution

1

Determine the third angle of triangle

ABC

\angle C = 180^\circ - (45^\circ + 105^\circ) = 30^\circ

.

The sum of interior angles in any triangle is

180^\circ

.

2

Drop an altitude

BD

perpendicular to side

AC

Altitude

BD

splits triangle

ABC

into two right triangles:

\triangle ABD

(

45^\circ-45^\circ-90^\circ

) and

\triangle BCD

(

30^\circ-60^\circ-90^\circ

).

In

\triangle ABD

,

\angle A = 45^\circ

and

\angle ADB = 90^\circ

, leaving

\angle ABD = 45^\circ

. In

\triangle BCD

,

\angle CBD = 105^\circ - 45^\circ = 60^\circ

and

\angle C = 30^\circ

.

3

Calculate side lengths in

45^\circ-45^\circ-90^\circ

triangle

ABD

AD = BD = \frac{AB}{\sqrt{2}} = \frac{6}{\sqrt{2}} = 3\sqrt{2}

units.

The ratio of sides in a

45^\circ-45^\circ-90^\circ

triangle is

1:1:\sqrt{2}

.

4

Calculate side lengths in

30^\circ-60^\circ-90^\circ

triangle

BCD

Hypotenuse

BC = 2 \times BD = 6\sqrt{2}

units, and long leg

CD = BD \times \sqrt{3} = 3\sqrt{2} \times \sqrt{3} = 3\sqrt{6}

units.

The ratio of sides opposite

30^\circ:60^\circ:90^\circ

is

1:\sqrt{3}:2

.

5

Calculate total base

AC

and area of triangle

ABC

AC = AD + CD = 3\sqrt{2} + 3\sqrt{6}

units. Area =

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} (3\sqrt{2} + 3\sqrt{6})(3\sqrt{2}) = \frac{1}{2} (18 + 18\sqrt{3}) = 9 + 9\sqrt{3}

square units.

Standard formula for triangle area is

\frac{1}{2} b h

.

Key Concept

Decomposing an oblique triangle with

45^\circ

and

30^\circ

angles into

45^\circ-45^\circ-90^\circ

and

30^\circ-60^\circ-90^\circ

special right triangles.

Estimated Time:2m 0s

Question 1791Question →

A university department tracked the number of research articles published by its 8 faculty members over a five-year period. The numbers of publications for 7 of the faculty members were $4, 7, 9, 12, 15, 18,$ and $23$ . If the arithmetic mean of the number of publications for all 8 faculty members is equal to $1.25$ times their median, what is the number of publications for the 8th faculty member?

Show answer & explanation

Answer: 47

Answer

47

The sum of the 7 known values is

88

, making the total sum

88 + x

and the arithmetic mean

\frac{88 + x}{8}

. Assuming

x \ge 15

, the 4th and 5th numbers when sorted are

12

and

15

, yielding a median of

13.5

. Setting the mean equal to

1.25 \times 13.5 = 16.875

gives

\frac{88 + x}{8} = 16.875

, which simplifies to

88 + x = 135

and yields

x = 47

.

Step-by-Step Solution

1

Calculate the sum of the 7 known data points and express the mean in terms of the unknown 8th value

x

.

Sum of 7 known values =

4 + 7 + 9 + 12 + 15 + 18 + 23 = 88

. Total mean =

\frac{88 + x}{8}

.

The arithmetic mean of

n

values is the sum of all values divided by

n

.

2

Analyze the position of

x

in sorted order to determine the median.

Assuming

x \ge 15

, the 4th and 5th values in ascending order are

12

and

15

, giving a median of

\frac{12 + 15}{2} = 13.5

.

For an even number of data points (

8

), the median is the average of the 4th and 5th terms in sorted order.

3

Formulate and solve the equation linking the mean and median.

\frac{88 + x}{8} = 1.25 \times 13.5 = 16.875 \implies 88 + x = 135 \implies x = 47

.

The problem specifies that the mean is equal to

1.25

times the median.

Key Concept

Calculating mean and median of a dataset containing an unknown value.

Estimated Time:1m 30s

Question 1792Question →

A laboratory technician has $12$ liters of a solution containing $15\%$ salt by weight. The technician wants to increase the salt concentration to $25\%$ by adding a second solution that contains $40\%$ salt by weight. How many liters of the $40\%$ solution must be added?

Show answer & explanation

Answer: 8

Answer

8 liters of the 40% solution must be added.

The total amount of salt contributed by the two solutions is

0.15(12) + 0.40x = 1.8 + 0.40x

. The final mixture volume is

(12 + x)

liters, and its concentration must be

25\%

. Equating the total salt to

25\%

of total volume yields

1.8 + 0.40x = 0.25(12 + x)

. Expanding gives

1.8 + 0.40x = 3.0 + 0.25x

, which simplifies to

0.15x = 1.2

, giving

x = 8

liters.

Step-by-Step Solution

1

Define the unknown variable and calculate the initial solute quantity.

Let

x

be the volume of the

40\%

solution added (in liters). Salt in initial solution =

0.15 \times 12 = 1.8

liters.

Establishing the mass balance of the salt solute is necessary to construct the algebraic equation.

2

Set up the algebraic concentration equation.

1.8 + 0.40x = 0.25(12 + x)

The combined salt from both solutions must equal

25\%

of the total combined liquid volume

(12 + x)

liters.

3

Solve the linear equation for

x

.

1.8 + 0.40x = 3.0 + 0.25x \implies 0.15x = 1.2 \implies x = 8

Isolating

x

yields the exact number of liters required.

Key Concept

Algebraic mixture problems using mass balance equations.

Estimated Time:1m 30s

Question 1793Question →

A committee of $3$ members is to be selected at random without replacement from a group of $5$ data scientists and $5$ software engineers. What is the conditional probability that at least two data scientists are selected in total, given that the first person selected is a data scientist?

\frac{5}{18}

\frac{1}{2}

\frac{5}{9}

\frac{13}{18}

\frac{3}{4}

Show answer & explanation

Answer: $\frac{13}{18}$

Answer

The conditional probability is

\frac{13}{18}

.

The conditional probability is calculated by adjusting the sample space after the first draw. With 1 data scientist already selected, there are 4 data scientists and 5 software engineers left (9 total). Selecting at least 1 more data scientist in the next 2 draws has a probability complementary to selecting 0 more data scientists. The probability of selecting 0 additional data scientists is

\frac{\binom{5}{2}}{\binom{9}{2}} = \frac{10}{36} = \frac{5}{18}

. Therefore, the required conditional probability is

1 - \frac{5}{18} = \frac{13}{18}

.

Step-by-Step Solution

1

Determine the remaining pool of candidates after the first selection.

Since 1 data scientist is selected first, 4 data scientists and 5 software engineers remain (total of 9 candidates). Two more candidates must be selected.

The conditional statement fixes the first selection, changing the sample space for the remaining 2 selections.

2

Identify the condition required for the target event to occur.

Since 1 data scientist is already selected, having at least two data scientists in total means selecting at least 1 additional data scientist in the remaining 2 draws.

Total data scientists = 1 (first draw) + (number of data scientists in next 2 draws).

3

Calculate the complementary probability (selecting 0 additional data scientists).

The number of ways to pick 2 software engineers from 5 is

\binom{5}{2} = 10

. The total ways to pick 2 people from 9 is

\binom{9}{2} = 36

. The probability of 0 additional data scientists is

\frac{10}{36} = \frac{5}{18}

.

Selecting 0 additional data scientists is equivalent to selecting 2 software engineers from the remaining group.

4

Subtract the complementary probability from 1.

1 - \frac{5}{18} = \frac{13}{18}

.

The sum of the probability of an event and its complement equals 1.

Key Concept

Conditional Probability without Replacement

Question 1794Question →

A museum display curator is arranging $6$ distinct historical coins— $4$ silver coins and $2$ gold coins—in a single row inside a display case. If the $2$ gold coins cannot be placed adjacent to each other, how many different linear arrangements of the $6$ coins are possible?

240

360

480

576

720

Show answer & explanation

Answer: $480$

Answer

The total number of valid linear arrangements is

480

.

Arranging the

4

distinct silver coins yields

4! = 24

orderings. These

4

coins create

5

available gaps (including both ends). To guarantee the

2

distinct gold coins are not adjacent, each gold coin must occupy a separate gap. The number of ways to assign

2

distinct gold coins to

5

gaps is

P(5, 2) = 5 \times 4 = 20

. By the Fundamental Counting Principle, the total number of valid linear arrangements is

24 \times 20 = 480

.

Step-by-Step Solution

1

Calculate the total number of ways to arrange the unrestricted silver coins.

The

4

distinct silver coins can be arranged in

4! = 24

ways.

The relative positions of the silver coins matter because each coin is distinct.

2

Determine the number of available gap positions created by the silver coins to separate the gold coins.

Arranging

4

silver coins creates

5

possible gaps (one before the first coin, three between adjacent silver coins, and one after the last coin: \_ S \_ S \_ S \_ S \_).

Placing at most one gold coin per gap guarantees that the two gold coins will not be adjacent.

3

Calculate the number of ways to place the

2

distinct gold coins into the

5

available gaps.

The number of ways to place the

2

distinct gold coins into

5

distinct slots is

P(5, 2) = 5 \times 4 = 20

.

Order matters because the gold coins are distinct objects.

4

Apply the Fundamental Counting Principle to find the total arrangements.

Total arrangements =

24 \times 20 = 480

.

The arrangement of the silver coins and the placement of the gold coins are independent sequential choices.

Key Concept

Slot method for non-adjacent arrangements using permutations and the Fundamental Counting Principle

Question 1795Question →

In the $xy$ -plane, line $L$ has a slope of $-\frac{3}{4}$ and passes through the point $(4, 1)$ . Which of the following statements about line $L$ must be true? Select all such statements.

Select all that apply

Line

L

passes through the point

(-4, 7)

.

Line

L

passes through Quadrant III.

Line

L

is perpendicular to the line defined by

4x - 3y = 12

.

The

x

-intercept of line

L

is

\left(\frac{4}{3}, 0\right)

.

Line

L

is parallel to the line defined by

3x + 4y = 25

.

Show answer & explanation

Answer: Line $L$ passes through the point $(-4, 7)$ .; Line $L$ is perpendicular to the line defined by $4x - 3y = 12$ .; Line $L$ is parallel to the line defined by $3x + 4y = 25$ .

Answer

The true statements are that line L passes through (-4, 7), is perpendicular to 4x - 3y = 12, and is parallel to 3x + 4y = 25.

The correct statements are those identifying that

(-4, 7)

satisfies

y = -\frac{3}{4}x + 4

, that

4x - 3y = 12

has a perpendicular slope of

\frac{4}{3}

, and that

3x + 4y = 25

has an identical parallel slope of

-\frac{3}{4}

.

Step-by-Step Solution

1

Find the equation of line L using point-slope form.

y - 1 = -\frac{3}{4}(x - 4) \implies y = -\frac{3}{4}x + 4

, or in standard form

3x + 4y = 16

.

Establishing the linear equation allows systematic verification of points, intercepts, and slope properties.

2

Verify point

(-4, 7)

on line L.

3(-4) + 4(7) = -12 + 28 = 16

, which satisfies the equation.

Checking if the point coordinates satisfy

3x + 4y = 16

confirms it lies on the line.

3

Determine quadrant coverage.

The

y

-intercept is

(0, 4)

and the

x

-intercept is

\left(\frac{16}{3}, 0\right)

. For

x < 0

,

y > 4

(Quadrant II); for

0 < x < \frac{16}{3}

,

y > 0

(Quadrant I); for

x > \frac{16}{3}

,

y < 0

(Quadrant IV).

Line L passes through Quadrants I, II, and IV only, so it does not enter Quadrant III.

4

Analyze slope relationships for perpendicularity and parallelism.

Line

4x - 3y = 12

has slope

\frac{4}{3}

(negative reciprocal of

-\frac{3}{4}

). Line

3x + 4y = 25

has slope

-\frac{3}{4}

(identical slope).

Perpendicular lines have slopes that multiply to

-1

, while parallel lines have equal slopes and different intercepts.

Key Concept

Linear line equations, parallel/perpendicular slopes, and point/intercept evaluations in coordinate geometry.

Estimated Time:1m 30s

Question 1796Question →

A robot starting at point $P$ travels due east for $4$ meters to point $Q$ . At point $Q$ , the robot turns $60^\circ$ counterclockwise from its original direction and travels in a straight line for $8$ meters, stopping at point $R$ . What is the straight-line distance, in meters, between point $P$ and point $R$ ?

4\sqrt{3}

4\sqrt{5}

4\sqrt{7}

8 + 4\sqrt{3}

12

Show answer & explanation

Answer: $4\sqrt{7}$

Answer

The straight-line distance between point

P

and point

R

is

4\sqrt{7}

meters.

Dropping a vertical perpendicular from

R

to the extended line

PQ

forms a

30^\circ

-

60^\circ

-

90^\circ

right triangle

QSR

with hypotenuse

8

. The base extension

QS

equals

4

and the altitude

RS

equals

4\sqrt{3}

. Applying the Pythagorean theorem to the right triangle

PSR

with legs

PS = 4 + 4 = 8

and

RS = 4\sqrt{3}

gives

PR = \sqrt{8^2 + (4\sqrt{3})^2} = \sqrt{64 + 48} = \sqrt{112} = 4\sqrt{7}

meters.

Step-by-Step Solution

1

Construct a right triangle by extending segment

PQ

past

Q

to a point

S

directly below

R

, dropping altitude

RS \perp PS

.

Triangle

QSR

is formed with

\angle RQS = 60^\circ

and

\angle QSR = 90^\circ

.

Decomposing the angled path into perpendicular horizontal and vertical components allows the use of special right triangles and the Pythagorean theorem.

2

Use

30^\circ

-

60^\circ

-

90^\circ

side ratios (

1 : \sqrt{3} : 2

) in triangle

QSR

where hypotenuse

QR = 8

.

Horizontal segment

QS = 8 \times \frac{1}{2} = 4

meters, and vertical altitude

RS = 8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3}

meters.

The leg opposite the

30^\circ

angle is half the hypotenuse, and the leg opposite the

60^\circ

angle is

\sqrt{3}

times the short leg.

3

Calculate the total horizontal distance

PS

and vertical distance

RS

for the large right triangle

PSR

.

Total base

PS = PQ + QS = 4 + 4 = 8

meters, and height

RS = 4\sqrt{3}

meters.

Point

P

, point

Q

, and point

S

are collinear on the horizontal line.

4

Apply the Pythagorean theorem to right triangle

PSR

to find distance

PR

.

PR = \sqrt{PS^2 + RS^2} = \sqrt{8^2 + (4\sqrt{3})^2} = \sqrt{64 + 48} = \sqrt{112} = 4\sqrt{7}

meters.

The straight-line distance

PR

is the hypotenuse of right triangle

PSR

.

Key Concept

Combining

30^\circ

-

60^\circ

-

90^\circ

special right triangles with the Pythagorean theorem to solve multi-step 2D path/distance problems.

Estimated Time:1m 45s

Question 1797Question →

A healthcare clinic logged the waiting times, in minutes, for 7 patients on Monday: $12, 41, 24, 19, 33, 28,$ and $49$ . On Tuesday, 3 additional patients were logged. The arithmetic mean waiting time for all 10 patients combined was $30$ minutes, and the median waiting time of the 3 patients logged on Tuesday was $34$ minutes. If one of the patients logged on Tuesday had a waiting time of $22$ minutes and another had a waiting time of $34$ minutes, what was the waiting time, in minutes, of the third patient logged on Tuesday?

28 minutes

32 minutes

34 minutes

38 minutes

48 minutes

Show answer & explanation

Answer: 38 minutes

Answer

38 minutes

The option stating 38 minutes is correct because the total sum for all 10 patients combined is

10 \times 30 = 300

minutes. The sum of Monday's 7 patient waiting times is

12 + 41 + 24 + 19 + 33 + 28 + 49 = 206

minutes. Therefore, Tuesday's 3 patients must sum to

300 - 206 = 94

minutes. Since two of Tuesday's patients waited 22 minutes and 34 minutes, the third patient's waiting time is

94 - (22 + 34) = 38

minutes. Ordering the three Tuesday times

(22, 34, 38)

confirms the median is indeed 34 minutes.

Step-by-Step Solution

1

Calculate the total sum of waiting times for all 10 patients.

Total sum =

10 \times 30 = 300

minutes.

The arithmetic mean of

n

values is

\text{Sum} / n

, so

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

.

2

Calculate the sum of waiting times for the 7 patients logged on Monday.

Monday sum =

12 + 41 + 24 + 19 + 33 + 28 + 49 = 206

minutes.

Adding the individual values gives the total Monday waiting time.

3

Find the total sum of waiting times for the 3 patients logged on Tuesday.

Tuesday sum =

300 - 206 = 94

minutes.

Subtracting Monday's total from the combined total yields Tuesday's total.

4

Determine the third patient's waiting time on Tuesday using the given values.

Third patient's waiting time =

94 - (22 + 34) = 94 - 56 = 38

minutes.

The sum of the 3 Tuesday patients is 94 minutes, and two known patients waited 22 and 34 minutes. Note that the sorted set

(22, 34, 38)

has a median of 34 minutes, consistent with the problem statement.

Key Concept

Combined Mean and Median Properties

Question 1798Question →

A municipal water treatment facility uses two intake pipes, Pipe $X$ and Pipe $Y$ , to fill a main reservoir. Operating alone at its constant rate, Pipe $X$ can fill the empty reservoir in $10$ hours. Operating alone at its constant rate, Pipe $Y$ can fill the empty reservoir in $15$ hours. Pipe $X$ is turned on first and operates alone for $3$ hours. Then, Pipe $Y$ is also turned on, and both pipes operate together until the reservoir is completely full. What is the total number of hours Pipe $X$ operates from the moment it is turned on until the reservoir is completely filled?

4.2

4.8

6.0

7.2

7.5

Show answer & explanation

Answer: $7.2$

Answer

The total number of hours Pipe X operates is

7.2

hours.

The correct answer is

7.2

hours. Pipe X completes

3/10

of the reservoir in 3 hours, leaving

7/10

of the reservoir to be filled. The combined rate of Pipe X and Pipe Y is

1/10 + 1/15 = 1/6

per hour. The joint time required to fill the remaining

7/10

is

(7/10) / (1/6) = 4.2

hours. Adding the initial 3 hours Pipe X worked alone yields a total of

3 + 4.2 = 7.2

hours.

Step-by-Step Solution

1

Determine the individual hourly work rates of Pipe X and Pipe Y.

Rate of Pipe X =

\frac{1}{10}

reservoir per hour; Rate of Pipe Y =

\frac{1}{15}

reservoir per hour.

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

2

Calculate the fraction of the reservoir filled by Pipe X during its initial 3-hour solo operation.

Work done in first 3 hours =

3 \times \frac{1}{10} = \frac{3}{10}

of the reservoir.

Multiplying Pipe X's rate by its solo operating time gives the completed portion of the work.

3

Find the remaining fraction of the reservoir that needs to be filled.

Remaining work =

1 - \frac{3}{10} = \frac{7}{10}

of the reservoir.

Subtracting the completed portion from the whole (1) leaves the uncompleted portion.

4

Determine the combined hourly rate when both pipes operate simultaneously.

Combined rate =

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

of the reservoir per hour.

Simultaneous operation rates are additive.

5

Calculate the time

t

during which both pipes operate together to finish the remaining work.

t = \frac{\frac{7}{10}}{\frac{1}{6}} = \frac{7}{10} \times 6 = \frac{42}{10} = 4.2

hours.

Dividing the remaining work by the combined rate yields the joint operation time.

6

Add Pipe X's solo time to the joint operation time to get Pipe X's total operating time.

Total time =

3 + 4.2 = 7.2

hours.

Pipe X was active during both the initial solo period and the joint operating period.

Key Concept

Combined Rate and Staggered Work Modeling

Estimated Time:2m 0s

Question 1799Question →

A technology committee has a pool of $8$ available guest speakers consisting of $5$ computer scientists and $3$ data privacy experts. A $4$ -person panel is to be formed from this pool. Which of the following statements regarding the possible panel selections must be true? Select all such statements.

Select all that apply

The total number of different

4

-person panels that can be formed without any restriction on specialty is equal to

70

.

The number of different

4

-person panels that consist of exactly

2

computer scientists and

2

data privacy experts is equal to

30

.

The number of different

4

-person panels containing at least

1

data privacy expert is equal to

65

.

The number of ways to select and arrange

4

computer scientists in a specific speaking order on stage is equal to

20

.

The number of different

4

-person panels containing no computer scientists is equal to

1

.

Show answer & explanation

Answer: The total number of different $4$ -person panels that can be formed without any restriction on specialty is equal to $70$ .; The number of different $4$ -person panels that consist of exactly $2$ computer scientists and $2$ data privacy experts is equal to $30$ .; The number of different $4$ -person panels containing at least $1$ data privacy expert is equal to $65$ .

Answer

The correct statements are those asserting that total unrestricted panels equal 70, panels with exactly 2 computer scientists and 2 privacy experts equal 30, and panels with at least 1 privacy expert equal 65.

The total unrestricted 4-person panels from 8 speakers is 8C4 = 70. Selecting 2 computer scientists (5C2 = 10) and 2 privacy experts (3C2 = 3) yields 10 × 3 = 30 panels. Using complementary counting, panels with at least 1 privacy expert equal total panels (70) minus panels composed entirely of computer scientists (5C4 = 5), giving 70 - 5 = 65.

Step-by-Step Solution

1

Calculate unrestricted combinations of 4 speakers out of 8

\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70

Selection order does not matter for committee membership, so we use combinations.

2

Calculate combinations with restricted counts (2 CS and 2 PE)

\binom{5}{2} \times \binom{3}{2} = 10 \times 3 = 30

By the Fundamental Counting Principle, independent choices for each subgroup are multiplied.

3

Calculate combinations with 'at least 1' condition using the complementary counting method

Total panels minus panels with 0 PE:

70 - \binom{5}{4} = 70 - 5 = 65

Subtracting outcomes that violate the constraint from total possible outcomes simplifies 'at least 1' calculations.

4

Evaluate the permutation and impossible constraint statements

Arranging 4 CS out of 5 gives

P(5,4) = 120 \neq 20

. A 4-person panel with 0 CS requires 4 PE out of 3, which gives

0 \neq 1

.

Arrangement requires permutations, and panel requirements exceeding available pool members yield zero valid groups.

Key Concept

Combinations, Permutations, and Complementary Counting Principle

Question 1800Question →

A quality control analyst evaluates a manufacturing process in which two specific types of flaws, Flaw $X$ and Flaw $Y$ , can occur on produced glass panels. The probability that a randomly selected panel has Flaw $X$ is $P(X) = 0.20$ , and the probability that it has Flaw $Y$ is $P(Y) = 0.30$ . The analyst confirms that Flaw $X$ and Flaw $Y$ are mutually exclusive events.

Which of the following statements MUST be true regarding these two flaw types? Select all such statements.

Select all that apply

The probability that a randomly selected panel has at least one of the two flaws is

0.50

.

The conditional probability of a panel having Flaw

X

given that it has Flaw

Y

,

P(X \mid Y)

, is equal to

0

.

The probability that a randomly selected panel has both Flaw

X

and Flaw

Y

is

0.06

.

Flaw

X

and Flaw

Y

are independent events.

The probability that a randomly selected panel has neither Flaw

X

nor Flaw

Y

is

0.94

.

Show answer & explanation

Answer: The probability that a randomly selected panel has at least one of the two flaws is $0.50$ .; The conditional probability of a panel having Flaw $X$ given that it has Flaw $Y$ , $P(X \mid Y)$ , is equal to $0$ .

Answer

The statement that the probability of having at least one flaw is 0.50 and the statement that the conditional probability P(X | Y) is 0 are both true.

Because Flaw

X

and Flaw

Y

are mutually exclusive, their intersection

P(X \cap Y) = 0

. By the addition rule, the probability of at least one flaw is

P(X \cup Y) = P(X) + P(Y) = 0.20 + 0.30 = 0.50

. Furthermore, the conditional probability

P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)} = \frac{0}{0.30} = 0

. Thus, both the statement claiming the union probability is

0.50

and the statement claiming the conditional probability is

0

are correct.

Step-by-Step Solution

1

Analyze the definition of mutually exclusive events

Since Flaw

X

and Flaw

Y

are mutually exclusive, they cannot occur simultaneously on the same panel. Therefore,

P(X \cap Y) = 0

.

By definition, mutually exclusive events have an intersection probability of zero.

2

Calculate the union probability P(X or Y)

Using the addition rule for mutually exclusive events:

P(X \cup Y) = P(X) + P(Y) - P(X \cap Y) = 0.20 + 0.30 - 0 = 0.50

.

The probability of at least one event occurring is the sum of their individual probabilities when the intersection is zero.

3

Calculate the conditional probability P(X | Y)

P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)} = \frac{0}{0.30} = 0

.

If Flaw

Y

is known to occur, Flaw

X

cannot occur due to mutual exclusivity.

4

Evaluate independence between the two events

For independence,

P(X \cap Y)

must equal

P(X) \times P(Y) = 0.20 \times 0.30 = 0.06

. Since

0 \neq 0.06

, the events are dependent.

Two events with non-zero probabilities that are mutually exclusive are always dependent because the occurrence of one guarantees the non-occurrence of the other.

Key Concept

Mutually Exclusive vs. Independent Events

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept