All practice questions

Question 501Question →

An artisanal bakery prepares a specialty flour blend consisting of rye, oat, and wheat flour. By weight, $\frac{3}{10}$ of the blend is rye flour and $0.45$ of the blend is oat flour. The remaining portion of the blend is wheat flour. If a single batch of the blend contains $4.5$ pounds of wheat flour, what is the total weight, in pounds, of the flour blend prepared for the batch?

Show answer & explanation

Answer: 18

Answer

The total weight of the flour blend prepared for the batch is 18 pounds.

To find the total weight of the mixture, first determine the decimal fraction representing wheat flour. Converting

\frac{3}{10}

to

0.30

, the combined proportion of rye and oat flour is

0.30 + 0.45 = 0.75

. Wheat flour makes up the remaining

1 - 0.75 = 0.25

(

25\%

) of the mixture. Setting up the equation

0.25 \times \text{Total Weight} = 4.5

gives

\text{Total Weight} = \frac{4.5}{0.25} = 18

pounds.

Step-by-Step Solution

1

Convert the fraction of rye flour to a decimal and sum it with the oat flour portion.

Rye portion =

\frac{3}{10} = 0.30

; Combined Rye and Oat portion =

0.30 + 0.45 = 0.75

.

Converting all given values to a consistent decimal format allows for straightforward addition.

2

Calculate the remaining portion representing wheat flour.

Wheat portion =

1 - 0.75 = 0.25

.

The sum of all proportional components in a whole mixture equals 1.

3

Divide the weight of the wheat flour by its decimal proportion to solve for the total batch weight.

Total weight =

\frac{4.5}{0.25} = 18

pounds.

Since

25\%

(

0.25

) of the total weight is

4.5

pounds, dividing the part by its decimal rate yields the total whole.

Key Concept

Solving multi-step word problems involving conversions between fractions, decimals, and percentages to find an unknown total.

Question 502Question →

A circle graphed on the coordinate plane is represented by the equation $x^2 + y^2 - 12x + 16y + 19 = 0$ . What is the length of the radius of the circle?

Show answer & explanation

Answer: 9

Answer

The radius of the circle is 9.

Converting the given general equation

x^2 + y^2 - 12x + 16y + 19 = 0

into standard form

(x - h)^2 + (y - k)^2 = r^2

by completing the square yields

(x - 6)^2 + (y + 8)^2 = 81

. Taking the square root of 81 gives a radius of 9.

Step-by-Step Solution

1

Rearrange the equation to group the x-terms and y-terms together and move the constant term to the right side.

(x^2 - 12x) + (y^2 + 16y) = -19

This sets up the expression for completing the square.

2

Complete the square for both the

x

and

y

variable groups by adding

(12/2)^2 = 36

and

(16/2)^2 = 64

to both sides of the equation.

(x - 6)^2 + (y + 8)^2 = -19 + 36 + 64 = 81

Adding these values balances the equation and converts the quadratic expressions into perfect square binomials.

3

Compare to the standard equation of a circle

(x - h)^2 + (y - k)^2 = r^2

to determine the radius

r

.

r^2 = 81 \implies r = \sqrt{81} = 9

The constant term on the right side of the standard form equation equals the square of the radius.

Key Concept

Standard form equation of a circle and completing the square

Question 503Question →

If $x = -2$ and $y = 5$ , what is the value of the algebraic expression $\frac{x^3 + 4y}{x + y}$ ?

Show answer & explanation

Answer: 4

Answer

The value of the expression is 4.

Substituting

x = -2

and

y = 5

yields

(-2)^3 + 4(5) = -8 + 20 = 12

in the numerator, and

-2 + 5 = 3

in the denominator. Dividing

12

by

3

gives

4

.

Step-by-Step Solution

1

Substitute the given numerical values into the algebraic expression.

\frac{(-2)^3 + 4(5)}{(-2) + 5}

Replace each occurrence of

x

with

-2

and

y

with

5

.

2

Evaluate the numerator using order of operations.

-8 + 20 = 12

Calculate

(-2)^3 = -8

and

4(5) = 20

, then add the terms together.

3

Evaluate the denominator.

-2 + 5 = 3

Add

-2

and

5

.

4

Divide the numerator by the denominator.

\frac{12}{3} = 4

Simplify the fraction to get the final integer answer.

Key Concept

Evaluating algebraic expressions requires substituting specific numerical values into the expression and carefully applying the order of operations, especially when handling negative numbers raised to powers.

Question 504Question →

A software development team consists of $4$ front-end developers, $5$ back-end developers, and $3$ quality assurance (QA) engineers. The project manager needs to select a $3$ -member subcommittee to present a new product feature. If the subcommittee must contain at least $1$ front-end developer and at least $1$ back-end developer, how many different $3$ -member subcommittees can be formed?

Show answer & explanation

Answer: 130

Answer

The total number of valid 3-member subcommittees that can be formed is 130.

To form a 3-member committee with at least 1 front-end developer and at least 1 back-end developer from 4 front-end, 5 back-end, and 3 QA engineers, we break the problem into three mutually exclusive valid scenarios: selecting 1 member from each role gives

4 \times 5 \times 3 = 60

ways; selecting 2 front-end and 1 back-end developer gives

C(4,2) \times C(5,1) = 6 \times 5 = 30

ways; and selecting 1 front-end and 2 back-end developers gives

C(4,1) \times C(5,2) = 4 \times 10 = 40

ways. Summing these possibilities gives

60 + 30 + 40 = 130

. Alternatively, subtracting invalid committees (those with no front-end developers:

C(8,3) = 56

, no back-end developers:

C(7,3) = 35

, minus double-counted 3 QA engineers:

1

) from total 3-member committees (

C(12,3) = 220

) yields

220 - (56 + 35 - 1) = 130

.

Step-by-Step Solution

1

Determine the required subcommittee size and role counts

The team has 4 front-end developers, 5 back-end developers, and 3 QA engineers (12 members total). A subcommittee of size 3 is required.

Establishing the total population and category counts is necessary before computing combination constraints.

2

Enumerate the mutually exclusive cases that satisfy all constraints

Case 1: (1 front-end, 1 back-end, 1 QA)
Case 2: (2 front-end, 1 back-end, 0 QA)
Case 3: (1 front-end, 2 back-end, 0 QA)

The committee requires at least 1 front-end and at least 1 back-end member in a 3-person group.

3

Compute combinations for each valid case using the combination formula

C(n, k) = \frac{n!}{k!(n-k)!}

Case 1:

C(4,1) \times C(5,1) \times C(3,1) = 4 \times 5 \times 3 = 60

Case 2:

C(4,2) \times C(5,1) \times C(3,0) = 6 \times 5 \times 1 = 30

Case 3:

C(4,1) \times C(5,2) \times C(3,0) = 4 \times 10 \times 1 = 40

Order of selection does not matter when forming a committee, so combinations are used.

4

Add the counts from all mutually exclusive cases

Total valid subcommittees =

60 + 30 + 40 = 130

According to the addition rule of counting, the total number of outcomes across disjoint cases is the sum of their individual counts.

Key Concept

Counting combinations across multiple groups with specific distribution constraints

Question 505Question →

In the standard $(x, y)$ coordinate plane, a circle is defined by the equation $x^2 + y^2 - 12x + 6y + 20 = 0$ . What is the distance from the center of this circle to the point $(9, 1)$ ?

Show answer & explanation

Answer: 5

Answer

The distance from the center of the circle to the point

(9, 1)

is 5.

Completing the square transforms

x^2 + y^2 - 12x + 6y + 20 = 0

into standard form

(x - 6)^2 + (y + 3)^2 = 25

, establishing the center of the circle at

(6, -3)

. Applying the coordinate distance formula between

(6, -3)

and

(9, 1)

yields

\sqrt{(9 - 6)^2 + (1 - (-3))^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5

.

Step-by-Step Solution

1

Group the

x

and

y

terms and move the constant term to the right side of the equation.

(x^2 - 12x) + (y^2 + 6y) = -20

Grouping terms isolates the variables to prepare for completing the square.

2

Complete the square for both variable expressions.

(x - 6)^2 + (y + 3)^2 = -20 + 36 + 9 = 25

Adding

( -12 / 2 )^2 = 36

and

( 6 / 2 )^2 = 9

to both sides converts the equation to standard form

(x - h)^2 + (y - k)^2 = r^2

.

3

Determine the coordinates of the center

(h, k)

.

Center is

(6, -3)

In the standard circle equation

(x - h)^2 + (y - k)^2 = r^2

, the center is given by

(h, k)

.

4

Calculate the distance between the center

(6, -3)

and the point

(9, 1)

using the distance formula.

d = \sqrt{(9 - 6)^2 + (1 - (-3))^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5

The distance formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

measures the straight-line distance between two points.

Key Concept

Equations of Circles and Coordinate Distance

Question 506Question →

A solar power system generated a total of $450$ kilowatt-hours (kWh) of electricity during a 3-day period. On Day 1, the system generated $\frac{2}{9}$ of the total 3-day electricity. On Day 2, it generated $40\%$ of the remaining electricity after Day 1. How many kilowatt-hours of electricity were generated on Day 3?

Show answer & explanation

Answer: 210

Answer

210 kWh

To find the electricity generated on Day 3, first determine the Day 1 portion by multiplying

\frac{2}{9}

by

450

, which gives

100

kWh. The amount remaining for Days 2 and 3 is

450 - 100 = 350

kWh. Day 2 accounts for

40\%

of this remaining amount, which is

0.40 \times 350 = 140

kWh. Finally, subtract Day 2's portion from the remaining amount:

350 - 140 = 210

kWh.

Step-by-Step Solution

1

Calculate the amount of electricity generated on Day 1.

100

kWh

Multiply the fraction

\frac{2}{9}

by the total electricity (

450

kWh).

2

Determine the remaining electricity after Day 1.

350

kWh

Subtract Day 1's generation (

100

kWh) from the total amount (

450

kWh).

3

Calculate the amount of electricity generated on Day 2.

140

kWh

Convert

40\%

to a decimal (

0.40

) and multiply by the remaining

350

kWh.

4

Calculate the amount of electricity generated on Day 3.

210

kWh

Subtract Day 2's generation (

140

kWh) from the

350

kWh remaining after Day 1.

Key Concept

Multi-step word problems involving fractions, percentage calculations, and remaining quantities.

Estimated Time:1m 15s

Question 507Question →

A community library's summer program requires participants to create a 3-book reading list consisting of exactly $1$ biography, $1$ science fiction novel, and $1$ history book selected from a featured list. The featured list contains $5$ biographies (2 of which have over $400$ pages), $6$ science fiction novels (3 of which have over $400$ pages), and $4$ history books (1 of which has over $400$ pages). If a participant selects $1$ book of each genre at random from the featured list, what is the probability that at least $1$ of the selected books has over $400$ pages? Express your answer as a decimal.

Show answer & explanation

Answer: 0.775

Answer

0.775

To find the probability of selecting at least one book over

400

pages, it is most efficient to use the complement rule:

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

. There are

5 \times 6 \times 4 = 120

total 3-book combinations. The number of books with

400

pages or fewer in each category are

3

biographies,

3

sci-fi novels, and

3

history books. Thus, there are

3 \times 3 \times 3 = 27

combinations with no books over

400

pages. The probability of choosing no books over

400

pages is

\frac{27}{120} = 0.225

. Subtracting this from

1

gives

1 - 0.225 = 0.775

.

Step-by-Step Solution

1

Calculate the total number of possible combinations of selecting 1 book from each genre

Total combinations =

5 \times 6 \times 4 = 120

By the Fundamental Counting Principle, multiplying the number of choices in each independent category yields the total outcomes.

2

Determine the complement event: selecting a reading list where NO book has over 400 pages

Available books of 400 pages or fewer: 3 biographies, 3 sci-fi novels, and 3 history books

Subtracting the number of books over 400 pages from the total in each category gives the count of books with 400 pages or fewer.

3

Calculate the number of combinations consisting entirely of books with 400 pages or fewer

Complement combinations =

3 \times 3 \times 3 = 27

Applying the Fundamental Counting Principle to the non-qualifying choices gives the total outcomes for the complement event.

4

Find the probability of the complement event and subtract from 1 to find the target probability

P(\text{at least one over } 400) = 1 - \frac{27}{120} = 1 - 0.225 = 0.775

The complement rule states that

P(A) = 1 - P(A')

, which is much more efficient than calculating probabilities for 1, 2, or 3 long books separately.

Key Concept

Complementary Probability and Fundamental Counting Principle

Estimated Time:1m 30s

Question 508Question →

A circle in the standard $(x, y)$ coordinate plane is represented by the equation $x^2 + y^2 + 4x - 6y - 12 = 0$ . What is the distance between the two points where this circle intersects the $x$ -axis?

Show answer & explanation

Answer: 8

Answer

The distance between the two points where the circle intersects the

x

-axis is 8 units.

Setting

y = 0

gives the quadratic equation

x^2 + 4x - 12 = 0

. Factoring yields

(x + 6)(x - 2) = 0

, giving solutions

x = -6

and

x = 2

. The distance between these two

x

-intercepts is

2 - (-6) = 8

units.

Step-by-Step Solution

1

Substitute

y = 0

into the circle equation.

x^2 + 4x - 12 = 0

Points on the

x

-axis have a

y

-coordinate of 0.

2

Solve the quadratic equation for

x

by factoring.

x = -6

and

x = 2

Factoring

(x + 6)(x - 2) = 0

yields the

x

-coordinates of the intersection points.

3

Find the distance between

(-6, 0)

and

(2, 0)

.

2 - (-6) = 8

The horizontal distance between two points

(x_1, 0)

and

(x_2, 0)

is

|x_2 - x_1|

.

Key Concept

Finding

x

-intercepts of a circle given in general form

Question 509Question →

A security code consists of four characters generated in order: a letter, a digit, a letter, and a digit.

- The 1st character is a letter chosen from the 5-element set $\{A, B, C, D, E\}$ .
- The 2nd character is a digit chosen from the 9 non-zero digits $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ .
- The 3rd character is a letter chosen from $\{A, B, C, D, E\}$ , but it cannot be the same as the 1st character.
- The 4th character is an even digit chosen from $\{2, 4, 6, 8\}$ , but it cannot be the same as the 2nd character.

How many different security codes can be formed following these rules?

Show answer & explanation

Answer: 640

Answer

640

The problem requires counting combinations with dependent restrictions, which can be solved by splitting into two mutually exclusive cases. If the 2nd character is an odd digit (5 possibilities), the 4th character has all 4 even digits available, giving 5 × 5 × 4 × 4 = 400 valid codes. If the 2nd character is an even digit (4 possibilities), the 4th character cannot match it, leaving 3 options, giving 5 × 4 × 4 × 3 = 240 valid codes. Summing the two distinct cases gives 400 + 240 = 640.

Step-by-Step Solution

1

Analyze the conditions for the digit positions (2nd and 4th characters).

The restrictions require dividing the counting into two cases based on the parity of the 2nd character.

The 4th character must be chosen from the 4 even digits {2, 4, 6, 8}. If the 2nd character is an odd digit, it does not reduce the choices for the 4th character. If the 2nd character is an even digit, it reduces the remaining choices for the 4th character from 4 to 3.

2

Calculate total possibilities for Case 1 (2nd character is odd).

5 × 5 × 4 × 4 = 400 codes.

There are 5 options for the 1st letter, 5 odd digits {1, 3, 5, 7, 9} for the 2nd character, 4 remaining letters for the 3rd character, and 4 even digits for the 4th character.

3

Calculate total possibilities for Case 2 (2nd character is even).

5 × 4 × 4 × 3 = 240 codes.

There are 5 options for the 1st letter, 4 even digits {2, 4, 6, 8} for the 2nd character, 4 remaining letters for the 3rd character, and 3 remaining even digits for the 4th character.

4

Sum the results of the two cases.

400 + 240 = 640 security codes.

According to the Addition Principle of counting, the total number of outcomes for mutually exclusive events is the sum of their individual outcomes.

Key Concept

Fundamental Counting Principle with Case Analysis

Question 510Question →

In the standard $(x, y)$ coordinate plane, a circle is tangent to the horizontal line $y = 7$ . The equation of the circle is $x^2 + y^2 - 10x + 4y + c = 0$ , where $c$ is a constant. What is the value of $c$ ?

Show answer & explanation

Answer: -52

Answer

The value of

c

is

-52

.

Completing the square gives

(x - 5)^2 + (y + 2)^2 = 29 - c

. The center is

(5, -2)

. The distance from

(5, -2)

to the horizontal line

y = 7

is

|7 - (-2)| = 9

, which is the radius

r

. Squaring the radius gives

r^2 = 81

. Setting

29 - c = 81

gives

c = -52

.

Step-by-Step Solution

1

Complete the square for

x

and

y

terms.

(x - 5)^2 + (y + 2)^2 = 29 - c

Writing the equation in standard form

(x - h)^2 + (y - k)^2 = r^2

reveals the center

(h, k)

and the expression for

r^2

.

2

Calculate the radius using the distance from the center to the tangent line.

Radius

r = 9

The center is

(5, -2)

and the tangent line is

y = 7

. The distance between

y = -2

and

y = 7

is

|7 - (-2)| = 9

.

3

Set

r^2 = 81

and solve for

c

.

c = -52

Equating the radius squared expression

29 - c

to

81

yields

29 - c = 81 \implies c = -52

.

Key Concept

Converting circle equations to standard form and using distance to tangent lines to determine radius.

Question 511Question →

A high school cafeteria offers a lunch meal combination consisting of 1 main dish, 1 side dish, and 1 drink. There are 4 main dish options, 5 side dish options, and 3 drink options available. However, due to dietary restrictions, 1 specific main dish option cannot be paired with 2 specific side dish options. How many different valid lunch meal combinations of 1 main dish, 1 side dish, and 1 drink can a student choose?

Show answer & explanation

Answer: 54

Answer

The total number of valid lunch meal combinations is 54.

The total number of unrestricted meal options is

4 \times 5 \times 3 = 60

. The restricted main dish cannot be paired with 2 specific side dishes across any of the 3 drink options, creating

1 \times 2 \times 3 = 6

invalid combinations. Subtracting these from the total gives

60 - 6 = 54

valid combinations.

Step-by-Step Solution

1

Calculate the total number of unrestricted meal choices

4 \times 5 \times 3 = 60

total combinations

By the Fundamental Counting Principle, multiplying the available choices for each dish component yields the total possible combinations without restrictions.

2

Determine the number of invalid combinations caused by dietary restrictions

1 \times 2 \times 3 = 6

invalid combinations

The 1 restricted main dish cannot be paired with 2 specific side dishes, and each of those 2 pairs can still be matched with any of the 3 drinks.

3

Subtract invalid combinations from the unrestricted total

60 - 6 = 54

valid combinations

Subtracting the disallowed combinations leaves only the combinations that satisfy all dietary restrictions.

Key Concept

Fundamental Counting Principle with Restrictions

Question 512Question →

A coffee shop blend initially consists of a $20$ -pound mixture of Arabica and Robusta beans, of which $45\%$ by weight is Arabica. The shop manager adds pure Arabica beans to the mixture to increase the proportion of Arabica beans to $56\%$ by weight. How many pounds of pure Arabica beans must be added to the mixture?

Show answer & explanation

Answer: 5

Answer

5 pounds of pure Arabica beans must be added to the mixture.

To solve for the required added weight, first compute the original mass of Arabica beans:

45\%

of

20

pounds is

0.45 \times 20 = 9

pounds. Letting

x

represent the pounds of pure Arabica added, the updated mass of Arabica is

9 + x

pounds and the updated total mass of the coffee batch is

20 + x

pounds. Setting the ratio

\frac{9 + x}{20 + x} = 0.56

gives

9 + x = 0.56(20 + x)

. Expanding the right side results in

9 + x = 11.2 + 0.56x

. Subtracting

0.56x

and

9

from both sides gives

0.44x = 2.2

, which simplifies to

x = 5

pounds.

Step-by-Step Solution

1

Determine the initial weight of Arabica beans.

Arabica weight =

0.45 \times 20 = 9

pounds.

The starting

20

-pound batch contains

45\%

Arabica beans.

2

Formulate the mixture fraction with the unknown added weight

x

.

\frac{9 + x}{20 + x} = 0.56

Adding

x

pounds of pure Arabica increases both the total Arabica weight (numerator) and the total mixture weight (denominator).

3

Solve the algebraic equation for

x

.

9 + x = 11.2 + 0.56x \implies 0.44x = 2.2 \implies x = 5

Isolating

x

yields the exact amount of pure Arabica required.

Key Concept

Solving percentage mixture problems using algebraic proportions

Estimated Time:1m 30s

Question 513Question →

In the standard $(x, y)$ coordinate plane, the equation of a circle is given by $3x^2 + 3y^2 - 12x + 24y - 15 = 0$ . What is the radius of this circle?

Show answer & explanation

Answer: 5

Answer

The radius of the circle is 5 units.

Dividing the equation

3x^2 + 3y^2 - 12x + 24y - 15 = 0

by 3 gives

x^2 + y^2 - 4x + 8y - 5 = 0

. Completing the square for both variables results in

(x - 2)^2 + (y + 4)^2 = 25

. Because

r^2 = 25

, taking the square root gives the radius

r = 5

.

Step-by-Step Solution

1

Divide the entire equation by 3.

x^2 + y^2 - 4x + 8y - 5 = 0

The standard form of a circle equation

(x-h)^2 + (y-k)^2 = r^2

requires the coefficients of

x^2

and

y^2

to be 1.

2

Group terms and move the constant to the right side of the equation.

(x^2 - 4x) + (y^2 + 8y) = 5

Isolating quadratic terms makes completing the square straightforward.

3

Complete the square for both variables by adding

(\frac{-4}{2})^2 = 4

and

(\frac{8}{2})^2 = 16

to both sides.

(x^2 - 4x + 4) + (y^2 + 8y + 16) = 5 + 4 + 16

Adding these values turns the expressions into perfect square trinomials while preserving equality.

4

Express in standard form and solve for the radius

r

.

(x - 2)^2 + (y + 4)^2 = 25 \implies r^2 = 25 \implies r = 5

The radius is the principal square root of

r^2

.

Key Concept

Converting general form of a circle equation to standard form using completing the square.

Question 514Question →

A technology firm allocated its annual equipment budget among three divisions. The engineering division received $\frac{3}{8}$ of the total budget, and the operations division received $30\%$ of the total budget. The remaining $\$52,000$ was allocated to the design division. What was the total equipment budget, in dollars, for the technology firm?

Show answer & explanation

Answer: 160000

Answer

The total equipment budget for the technology firm was $160,000 dollars.

Converting both allocated parts into decimals gives

\frac{3}{8} = 0.375

for engineering and

30\% = 0.30

for operations. Combined, these two divisions account for

0.375 + 0.30 = 0.675

(or

67.5\%

) of the total budget. The remaining fraction for the design division is

1 - 0.675 = 0.325

(or

32.5\%

). Dividing the design budget of

\$52,000

by

0.325

yields the total budget of

\$160,000

.

Step-by-Step Solution

1

Convert the percentage and fraction to equivalent decimal values.

Engineering receives

\frac{3}{8} = 0.375

, and operations receives

30\% = 0.30

.

Expressing all portions in decimal form allows for straightforward addition and subtraction.

2

Sum the portions allocated to the engineering and operations divisions.

0.375 + 0.30 = 0.675

.

This finds the total proportion of the budget that has already been accounted for.

3

Determine the remaining decimal portion allocated to the design division.

1 - 0.675 = 0.325

.

The complete budget represents 1 whole (100%). Subtracting the allocated portions yields the design division's share.

4

Divide the dollar amount of the design division by its decimal portion to calculate the total budget.

\frac{52,000}{0.325} = 160,000

.

Since

\text{Design Portion} \times \text{Total Budget} = \text{Design Amount}

, dividing the amount by the decimal portion gives the total budget.

Key Concept

Solving multi-step applied word problems by converting between fractions, decimals, and percentages to determine an unknown total value.

Question 515Question →

A security system requires a 4-digit passcode created using the digits $1, 2, 3, 4, 5, 6,$ and $7$ . No digit may be repeated in a passcode. If the first digit and the last digit must both be odd numbers, how many different passcodes can be formed?

Show answer & explanation

Answer: 240

Answer

240 passcodes can be formed under the given conditions.

To form a 4-digit passcode with distinct digits where the first and last digits are odd: first choose the first digit from the 4 available odd digits (

1, 3, 5, 7

). Next, choose the fourth digit from the remaining 3 odd digits. Then, choose the second digit from the remaining 5 available digits in the full set, and the third digit from the remaining 4 available digits. Multiplying these choices gives

4 \times 5 \times 4 \times 3 = 240

.

Step-by-Step Solution

1

Categorize available digits into odd and even sets

4 odd digits ({1, 3, 5, 7}) and 3 even digits ({2, 4, 6}) out of 7 total digits

The first and last positions require odd digits, so we need to track odd digits separately.

2

Determine options for the first and last positions

4 options for position 1; 3 options for position 4

Position 1 must be odd (4 choices). Because digits cannot repeat, position 4 has 3 remaining odd choices.

3

Determine options for the second and third positions

5 options for position 2; 4 options for position 3

Two digits have already been assigned. Out of 7 total digits, 5 remain for position 2, leaving 4 for position 3.

4

Multiply choices across all positions

4 * 5 * 4 * 3 = 240

By the Fundamental Counting Principle, the total number of combinations is the product of the number of choices at each step.

Key Concept

Fundamental Counting Principle with Position Restrictions and Non-Replacement

Question 516Question →

If $x = -3$ , $y = -\frac{1}{3}$ , and $z = 16$ , what is the value of the algebraic expression $\frac{x^3 y^{-2} + 3 z^{1/2}}{(xy + 2)^3 - \frac{3}{2} z^{1/2}}$ ?

Show answer & explanation

Answer: -11

Answer

The evaluated value of the expression is -11.

Substituting the given values into the numerator yields

(-3)^3 \left(-\frac{1}{3}\right)^{-2} + 3(16)^{1/2} = (-27)(9) + 3(4) = -243 + 12 = -231

. Substituting into the denominator yields

\left((-3)\left(-\frac{1}{3}\right) + 2\right)^3 - \frac{3}{2}(16)^{1/2} = (1 + 2)^3 - \frac{3}{2}(4) = 27 - 6 = 21

. Dividing

-231

by

21

produces

-11

.

Step-by-Step Solution

1

Evaluate the terms in the numerator

Numerator = -231

Calculate

x^3 = (-3)^3 = -27

,

y^{-2} = \left(-\frac{1}{3}\right)^{-2} = 9

, giving

x^3 y^{-2} = -243

. Then calculate

3 z^{1/2} = 3(4) = 12

. Adding these gives

-243 + 12 = -231

.

2

Evaluate the terms in the denominator

Denominator = 21

Calculate

xy = (-3)\left(-\frac{1}{3}\right) = 1

, so

(xy + 2)^3 = (1 + 2)^3 = 27

. Then calculate

\frac{3}{2} z^{1/2} = \frac{3}{2}(4) = 6

. Subtracting gives

27 - 6 = 21

.

3

Divide the evaluated numerator by the evaluated denominator

-11

\frac{-231}{21} = -11

.

Key Concept

Evaluating Algebraic Expressions with Negative and Fractional Exponents

Question 517Question →

If

p = -3

,

q = \frac{1}{2}

, and

r = -27

, what is the value of the algebraic expression below?

\frac{p^2 q^{-3} + r^{1/3}}{(2pq + 5)^2 + q^{-1}}

Show answer & explanation

Answer: 11.5

Answer

11.5

Evaluating each component of the expression step-by-step using order of operations:
1. Numerator:

p^2 q^{-3} + r^{1/3} = (-3)^2 \left(\frac{1}{2}\right)^{-3} + (-27)^{1/3} = 9 \cdot 8 + (-3) = 72 - 3 = 69

.
2. Denominator:

(2pq + 5)^2 + q^{-1} = \left(2(-3)\left(\frac{1}{2}\right) + 5\right)^2 + \left(\frac{1}{2}\right)^{-1} = (-3 + 5)^2 + 2 = 2^2 + 2 = 4 + 2 = 6

.
3. Final Quotient:

\frac{69}{6} = 11.5

.

Step-by-Step Solution

1

Evaluate the terms in the numerator

Numerator = 69

Since

p = -3

,

p^2 = 9

. With

q = \frac{1}{2}

,

q^{-3} = 2^3 = 8

. Thus,

p^2 q^{-3} = 9 \cdot 8 = 72

. Also,

r^{1/3} = (-27)^{1/3} = -3

. Adding these values gives

72 + (-3) = 69

.

2

Evaluate the terms in the denominator

Denominator = 6

First,

2pq = 2(-3)\left(\frac{1}{2}\right) = -3

. Then

(2pq + 5)^2 = (-3 + 5)^2 = 2^2 = 4

. Next,

q^{-1} = \left(\frac{1}{2}\right)^{-1} = 2

. Adding these components gives

4 + 2 = 6

.

3

Divide the numerator by the denominator

11.5

Dividing the numerator (69) by the denominator (6) yields

\frac{69}{6} = 11.5

(or

\frac{23}{2}

).

Key Concept

Evaluating algebraic expressions with negative bases, negative exponents, fractional exponents, and order of operations.

Question 518Question →

A high school planning committee needs to select a subcommittee of $3$ people from a pool of $5$ teachers and $4$ students. How many different subcommittees can be formed consisting of exactly $2$ teachers and $1$ student?

Show answer & explanation

Answer: 40

Answer

The total number of different subcommittees that can be formed is 40.

To find the number of unique subcommittees with 2 teachers and 1 student, calculate the combination for choosing 2 teachers from 5, which equals 10, and 1 student from 4, which equals 4. Multiplying these independent choices together results in 40 possible subcommittees.

Step-by-Step Solution

1

Calculate the number of ways to select 2 teachers from the pool of 5 teachers.

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

ways

Selection order does not matter for committee members, so combinations are used.

2

Calculate the number of ways to select 1 student from the pool of 4 students.

\binom{4}{1} = 4

ways

Choosing 1 item from a set of 4 offers 4 distinct possibilities.

3

Apply the Fundamental Counting Principle to determine total combinations.

10 \times 4 = 40

subcommittees

Multiply the number of choices for teachers by the number of choices for students.

Key Concept

Combinations and Fundamental Counting Principle

Estimated Time:1m 0s

Question 519Question →

Bioacousticians measured the echolocation click rate (in clicks per minute, $\text{clicks/min}$ ) of a harbor porpoise at four different water depths across three 15-minute observation trials. The results are summarized in the table below:

Water Depth (m)	Trial 1 (clicks/min)	Trial 2 (clicks/min)	Trial 3 (clicks/min)
10	120	115	125
20	140	148	138
30	185	170	185
40	210	205	215

Based on the table, what is the average echolocation click rate, in $\text{clicks/min}$ , across all three trials at a depth of 30 m?

Show answer & explanation

Answer: 180

Answer

The average echolocation click rate at a depth of 30 m across all three trials is 180 clicks/min.

To calculate the average echolocation click rate at 30 m depth, locate the row for 30 m, add the click rates from the three trials (

185 + 170 + 185 = 540

), and divide by the number of trials (

540 \div 3 = 180\text{ clicks/min}

).

Step-by-Step Solution

1

Locate the depth row for 30 m in the provided data table.

Retrieved values for Trial 1 (185), Trial 2 (170), and Trial 3 (185).

The question specifically requests calculations for the 30 m depth.

2

Sum the recorded click rates across the three trials.

185 + 170 + 185 = 540\text{ clicks/min}

.

Obtaining the sum is required before dividing by the total count of trials.

3

Divide the calculated sum by 3.

\frac{540}{3} = 180\text{ clicks/min}

.

Calculating the arithmetic mean yields the average click rate.

Key Concept

Calculating the arithmetic mean from tabular data

Estimated Time:1m 0s

Question 520Question →

A high school photography club has $12$ active members, consisting of $7$ seniors and $5$ juniors. The club needs to elect an executive board composed of a President, a Vice President, and a Secretary, where no member can hold more than one position. If the President must be a senior, how many different executive board arrangements are possible?

Show answer & explanation

Answer: 770

Answer

The total number of different executive board arrangements possible is

770

.

To calculate the total number of distinct executive board arrangements, use the Fundamental Counting Principle by calculating the number of options for each position sequentially. The President must be a senior, so there are 7 choices for President. Once the President is chosen, any of the remaining 11 members can serve as Vice President. After filling the Vice President role, 10 members remain for Secretary. Multiplying these independent choices gives

7 \times 11 \times 10 = 770

.

Step-by-Step Solution

1

Determine the available choices for the President position

7 possibilities

The President position is restricted to seniors only, and there are 7 seniors in the club.

2

Determine the available choices for the Vice President position

11 possibilities

After 1 member is chosen as President, 11 of the original 12 members remain available for the Vice President role.

3

Determine the available choices for the Secretary position

10 possibilities

After 2 members are chosen for President and Vice President, 10 members remain available for Secretary.

4

Calculate the total number of distinct outcomes using the Fundamental Counting Principle

770 total arrangements

Multiplying the choices for each position gives

7 \times 11 \times 10 = 770

.

Key Concept

Fundamental Counting Principle with Position Restrictions

Estimated Time:1m 15s

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