All practice questions

The following passage is adapted from an essay on nineteenth-century optics and astronomy:

In the spring of 1800, the astronomer William Herschel conducted a series of experiments to measure the thermal energy associated with different colors of the visible solar spectrum. Passing sunlight through a glass prism, he projected the resulting rainbow spectrum onto a flat surface and placed sensitive glass-bulb thermometers within each colored band—violet, blue, green, yellow, orange, and red. To ensure that room temperature changes did not skew his measurements, Herschel positioned additional control thermometers in the shade near the spectrum table, away from the direct sunlight.

During his measurements, Herschel noticed a systematic increase in recorded temperature from the violet end toward the red end of the spectrum. Red light consistently registered a significantly higher temperature than green or violet light. Intrigued by this thermal progression, Herschel decided to measure the temperature just beyond the red edge of the visible spectrum, where no visible illumination was apparent. To his surprise, this dark region beyond the red boundary registered the highest temperature of all, proving the existence of invisible rays carrying heat energy—what Herschel initially termed "calorific rays" and what scientists later renamed infrared radiation.

In his papers presented to the Royal Society of London, Herschel carefully described his experimental apparatus. He mounted a glass prism on an adjustable stand positioned near a small opening in a window shutter, allowing a narrow beam of sunlight to enter the darkened room. The target surface holding the test thermometers was covered with thick white paper to render the boundaries of the spectral colors sharply visible. Herschel stressed that the shade-stationed control thermometers were crucial for verifying that ambient room air fluctuations were separate from the heat generated by the isolated light rays.

Based on the passage, where did Herschel place the control thermometers during his experiment?

In the shade near the table holding the spectrum

Directly inside the violet band of the visible spectrum

Immediately beyond the red edge of the visible light spectrum

Mounted directly onto the opening in the window shutter

Show answer & explanation

Answer: In the shade near the table holding the spectrum

Answer

In the shade near the table holding the spectrum

The correct answer accurately restates the explicit detail given in the passage, which mentions that Herschel placed the control thermometers in the shade near the spectrum table to monitor ambient room temperature.

Step-by-Step Solution

1

Locate the keywords 'control thermometers' in the passage text.

Found in Paragraph 1: 'Herschel positioned additional control thermometers in the shade near the spectrum table, away from the direct sunlight.'

Explicit detail questions require finding the exact sentence where the specific term is discussed.

2

Match the location described in the passage to the correct option.

The text explicitly specifies that the control thermometers were placed in the shade near the table.

Literal comprehension relies directly on stated facts rather than inferences.

Key Concept

Identifying Explicit Details in a Text

Estimated Time:1m 0s

Question 5542Question →

A pharmaceutical laboratory formulates a liquid medication by combining Active Ingredient X, Stabilizer Y, and Purified Water in a ratio of $3 : 4 : 8$ by volume. A technician needs to prepare a batch of this medication that contains exactly $360\text{ mL}$ of Stabilizer Y. How many total milliliters of liquid medication will this batch contain?

675\text{ mL}

720\text{ mL}

990\text{ mL}

1,350\text{ mL}

5,400\text{ mL}

Show answer & explanation

Answer: $1,350\text{ mL}$

Answer

1,350\text{ mL}

To find the total batch volume, determine the value of one part in the ratio. The three components (Active Ingredient X, Stabilizer Y, and Purified Water) are mixed in a ratio of

3 : 4 : 8

, giving a total of

3 + 4 + 8 = 15

parts. Stabilizer Y represents

4

of these parts and has a volume of

360\text{ mL}

. Dividing

360

by

4

gives

90\text{ mL}

per part. Multiplying

90\text{ mL}

by the total

15

parts yields

1,350\text{ mL}

.

Step-by-Step Solution

1

Find the total number of parts in the ratio.

Total parts =

3 + 4 + 8 = 15

parts.

The complete mixture is made up of the sum of the ratio components.

2

Determine the volume represented by a single ratio part.

Value of

1

part =

\frac{360\text{ mL}}{4} = 90\text{ mL}

.

Stabilizer Y accounts for

4

parts of the total ratio and has a volume of

360\text{ mL}

.

3

Calculate the total volume of the batch.

Total volume =

15 \times 90\text{ mL} = 1,350\text{ mL}

.

Multiply the single-part volume by the total number of parts.

Key Concept

Solving Part-to-Total Proportions with Multiple Components

Estimated Time:1m 15s

Question 5543Question →

An automated agricultural sprinkler sweeps across a section of a circular field, forming a circular sector. The area of the irrigated sector is $45\pi$ square meters, and the length of the outer boundary arc of the sector is $6\pi$ meters. What is the radius, in meters, of the circular field?

7.5

9

15

22.5

30

Show answer & explanation

Answer: 15

Answer

15 meters

Using the relationship between sector area, arc length, and radius,

A = \frac{1}{2} r s

, we substitute

A = 45\pi

and

s = 6\pi

to get

45\pi = \frac{1}{2} r (6\pi) = 3\pi r

. Solving for

r

gives

r = 15

meters.

Step-by-Step Solution

1

Recall the formulas for sector area (

A

) and arc length (

s

) in terms of radius (

r

) and central angle (

\theta

in radians).

s = r\theta

and

A = \frac{1}{2}r^2\theta

These are the fundamental geometric formulas for circular sectors.

2

Express the sector area formula in terms of arc length

s

.

A = \frac{1}{2} r (r\theta) = \frac{1}{2} r s

Substituting

s = r\theta

simplifies the calculation by eliminating the central angle

\theta

.

3

Substitute the given values

A = 45\pi

and

s = 6\pi

into the simplified formula and solve for

r

.

45\pi = \frac{1}{2} r (6\pi) \implies 45\pi = 3\pi r \implies r = 15

Dividing both sides by

3\pi

yields the radius

r = 15

meters.

Key Concept

Relationship between Sector Area, Arc Length, and Radius

Question 5544Question →

In the standard $(x, y)$ coordinate plane, a circle is defined by the equation $x^2 + y^2 + 14x - 6y + 9 = 0$ . What is the area, in square units, of the region enclosed by the circle?

7\pi

14\pi

40\pi

49\pi

58\pi

Show answer & explanation

Answer: $49\pi$

Answer

The area of the region enclosed by the circle is

49\pi

square units.

To find the area of the circle, we rewrite the equation

x^2 + y^2 + 14x - 6y + 9 = 0

in standard form

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

. Regrouping gives

(x^2 + 14x) + (y^2 - 6y) = -9

. Completing the square for

x

requires adding

(\frac{14}{2})^2 = 49

, and for

y

requires adding

(\frac{-6}{2})^2 = 9

. Adding both to the right side gives

-9 + 49 + 9 = 49

. Thus, the standard equation is

(x + 7)^2 + (y - 3)^2 = 49

, yielding

r^2 = 49

. The area enclosed by the circle is

\pi r^2 = 49\pi

.

Step-by-Step Solution

1

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

(x^2 + 14x) + (y^2 - 6y) = -9

Grouping

x

-terms and

y

-terms prepares the equation for completing the square.

2

Complete the square for both

x

and

y

expressions by adding

(\frac{14}{2})^2 = 49

and

(\frac{-6}{2})^2 = 9

to both sides.

(x^2 + 14x + 49) + (y^2 - 6y + 9) = -9 + 49 + 9

Adding these values turns the quadratic expressions into perfect square trinomials while keeping the equation balanced.

3

Write in standard circle equation form

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

to identify

r^2

.

(x + 7)^2 + (y - 3)^2 = 49

, so

r^2 = 49

The standard form directly gives the radius squared,

r^2 = 49

.

4

Calculate the area of the circle using the formula

A = \pi r^2

.

A = \pi (49) = 49\pi

Substituting

r^2 = 49

into the area formula gives the final enclosed area.

Key Concept

Converting general circle equation

x^2 + y^2 + Dx + Ey + F = 0

to standard form

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

by completing the square.

Question 5545Question →

A quality control laboratory tested four battery models to measure the proportion of initial charge retained after 24 hours of continuous operation. The recorded results for each model are listed below:

- Model Alpha: $\frac{13}{20}$
- Model Beta: $64.5\%$
- Model Gamma: $\frac{5}{8}$
- Model Delta: $0.638$

Which of the following represents the battery models arranged in order from least to greatest proportion of remaining charge?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest proportion of remaining charge is Model Gamma (

\frac{5}{8}

), Model Delta (

0.638

), Model Beta (

64.5\%

), and Model Alpha (

\frac{13}{20}

).

Converting each model's retained charge to decimal form gives Model Gamma

= \frac{5}{8} = 0.625

, Model Delta

= 0.638

, Model Beta

= 64.5\% = 0.645

, and Model Alpha

= \frac{13}{20} = 0.650

. Comparing these values yields

0.625 < 0.638 < 0.645 < 0.650

, which matches the order Model Gamma, Model Delta, Model Beta, Model Alpha.

Step-by-Step Solution

1

Convert each given fraction, percentage, and decimal into equivalent decimal values for direct comparison.

Model Alpha

= \frac{13}{20} = 0.650

; Model Beta

= 64.5\% = 0.645

; Model Gamma

= \frac{5}{8} = 0.625

; Model Delta

= 0.638

.

Converting all values to decimals standardizes the quantities so they can be compared easily by place value.

2

Compare the resulting decimal numbers from least to greatest.

0.625 < 0.638 < 0.645 < 0.650

Comparing digits from left to right: all have

0.6

in the tenths place; comparing the hundredths place shows

2 < 3 < 4 < 5

.

3

Map the sorted decimal values back to their original model designations.

Model Gamma (

\frac{5}{8}

), Model Delta (

0.638

), Model Beta (

64.5\%

), Model Alpha (

\frac{13}{20}

).

This establishes the required sequence from least to greatest.

Key Concept

Converting fractions, decimals, and percentages to a common format (decimals) to order rational numbers.

Question 5546Question →

A landscape architect is designing a public park with a total area of $14,400\text{ square feet}$ . The plan allocates space for a lawn, flowerbeds, and paved walkways in the ratio $7 : 3 : 2$ , respectively. How many more square feet of area are dedicated to the lawn than to the flowerbeds?

3,600

4,800

5,760

6,000

8,400

Show answer & explanation

Answer: 4,800

Answer

4,800 square feet

The sum of all ratio parts is

7 + 3 + 2 = 12

. Dividing the total park area of

14,400\text{ sq ft}

by

12

gives

1,200\text{ sq ft}

per ratio part. The lawn accounts for

7

parts (

8,400\text{ sq ft}

) and the flowerbeds account for

3

parts (

3,600\text{ sq ft}

). Subtracting the flowerbed area from the lawn area gives

8,400 - 3,600 = 4,800\text{ sq ft}

. Alternatively, taking the direct difference of

7 - 3 = 4

parts and multiplying by

1,200\text{ sq ft}

gives

4,800\text{ sq ft}

.

Step-by-Step Solution

1

Find the total number of equal ratio parts.

7 + 3 + 2 = 12\text{ parts}

The complete park area is divided into portions defined by the sum of all terms in the ratio.

2

Calculate the area corresponding to one ratio part.

\frac{14,400}{12} = 1,200\text{ square feet per part}

Dividing the total area by the total number of parts determines the size of a single unit part.

3

Determine the difference in ratio parts between the lawn and the flowerbeds.

7 - 3 = 4\text{ parts}

The lawn receives 7 parts and the flowerbeds receive 3 parts.

4

Multiply the difference in parts by the area per part.

4 \times 1,200 = 4,800\text{ square feet}

Multiplying the difference in ratio units by the unit area yields the total excess area of the lawn over the flowerbeds.

Key Concept

Solving multi-part ratio word problems using unit rate allocation per ratio part.

Estimated Time:1m 15s

Question 5547Question →

A woodworker starts with a wooden board of length $L$ inches. After cutting off a piece measuring $16.5$ inches to build a shelf bracket, the remaining portion of the board is $48.5$ inches long. Which of the following equations can be used to find $L$ , the original length of the wooden board in inches?

L - 16.5 = 48.5

L + 16.5 = 48.5

16.5L = 48.5

\frac{L}{16.5} = 48.5

48.5 - L = 16.5

Show answer & explanation

Answer: $L - 16.5 = 48.5$

Answer

L - 16.5 = 48.5

The original length of the board is

L

inches. Cutting off a piece measuring

16.5

inches decreases the total length by

16.5

, which is written algebraically as

L - 16.5

. Since the remaining length is given as

48.5

inches, setting these equal produces the equation

L - 16.5 = 48.5

.

Step-by-Step Solution

1

Identify the given variables and quantities in the scenario

Initial length =

L

, length cut off =

16.5

, remaining length =

48.5

Establishing clear variables allows systematic setup of the algebraic model.

2

Translate the physical action into a mathematical operation

Cutting off a piece represents subtraction from the starting quantity:

L - 16.5

Removal of a part from a whole corresponds to subtraction.

3

Equate the expression for remaining length to the given remaining value

L - 16.5 = 48.5

The remaining length after cutting is stated to be

48.5

inches.

Key Concept

Translating verbal removal scenarios into one-step subtraction equations

Estimated Time:1m 0s

Question 5548Question →

A venue manager allocates a total hospitality budget of $d$ dollars for an upcoming music festival. If the manager divides this budget equally among 14 performing artists, each artist receives exactly $\$185$ . Which of the following equations correctly models this situation to find the total budget $d$ ?

\frac{d}{14} = 185

14d = 185

d - 14 = 185

d + 185 = 14

\frac{185}{d} = 14

Show answer & explanation

Answer: $\frac{d}{14} = 185$

Answer

The correct equation is

\frac{d}{14} = 185

, which expresses that the total budget

d

divided equally among 14 artists yields

\$185

per artist.

Equal distribution of a total quantity

d

across 14 recipients is mathematically represented by dividing

d

by 14. Setting this quotient equal to the per-person amount of

\$185

yields the equation

\frac{d}{14} = 185

.

Step-by-Step Solution

1

Identify the given quantities and variable

Total budget =

d

, number of equal shares =

14

, amount per share =

\$185

Setting up an equation requires matching the verbal scenario to mathematical operations.

2

Translate equal sharing into an algebraic operation

The total budget

d

divided by

14

is written as

\frac{d}{14}

Dividing a total quantity into equal parts corresponds to division.

3

Equate the expression to the given share amount

\frac{d}{14} = 185

Each share is given as

\$185

.

Key Concept

Translating verbal division scenarios into one-step linear equations

Question 5549Question →

To decipher the damaged sections of the medieval ledger, paleographers analyzed the faded, intricate script, cataloged regional dialect variations, and translated forgotten trade terms.

Which of the following choices for the underlined portion best maintains standard English conventions?

NO CHANGE

faded, intricate script; cataloged regional dialect variations; and translated

faded intricate script, they cataloged regional dialect variations, and translated

faded, intricate script, cataloging regional dialect variations, and translated

Show answer & explanation

Answer: NO CHANGE

Answer

The original phrasing ('faded, intricate script, cataloged regional dialect variations, and translated') is correct as written.

The original text correctly places a comma between the coordinate adjectives 'faded' and 'intricate' because both independently describe the noun 'script'. Furthermore, it properly uses commas to separate all three parallel past-tense verb phrases ('analyzed...', 'cataloged...', and 'translated...').

Step-by-Step Solution

1

Evaluate the punctuation between the adjectives 'faded' and 'intricate'.

The adjectives are coordinate because you can insert 'and' between them ('faded and intricate script') or reverse their order ('intricate, faded script'). Therefore, a comma between them is required.

Coordinate adjectives modifying the same noun must be separated by a comma.

2

Check the parallel structure and series punctuation of the sentence predicate.

The sentence lists three parallel past-tense verb phrases: (1) analyzed the script, (2) cataloged regional dialect variations, and (3) translated forgotten trade terms.

Items in a simple series of three or more elements must be separated by commas, including a serial comma before the coordinating conjunction 'and'.

Key Concept

Commas with Coordinate Adjectives and Items in a Series

Estimated Time:1m 0s

Question 5550Question →

At an auto dealership, $\frac{2}{5}$ of the vehicles in the inventory are sedans, $35\%$ are SUVs, and the remaining 45 vehicles are trucks. What is the total number of vehicles in the dealership's inventory?

Show answer & explanation

Answer: 180

Answer

The total number of vehicles in the dealership's inventory is 180.

Converting

\frac{2}{5}

to a percentage yields

40\%

. Adding the

40\%

sedans and

35\%

SUVs shows that

75\%

of the inventory consists of sedans and SUVs. Consequently, the remaining

25\%

of the inventory consists of trucks. Since there are 45 trucks,

25\%

of the total inventory equals 45. Dividing 45 by

0.25

yields the total inventory of 180 vehicles.

Step-by-Step Solution

1

Convert fraction of sedans to a percentage

Sedans represent

40\%

of the total inventory.

To combine the amounts easily, convert

\frac{2}{5}

to percentage form:

\frac{2}{5} = 0.40 = 40\%

.

2

Calculate the combined percentage of sedans and SUVs

Sedans and SUVs together make up

75\%

of the inventory.

Add the two percentages:

40\% + 35\% = 75\%

.

3

Determine the remaining percentage representing trucks

Trucks make up

25\%

of the inventory.

The entire inventory equals

100\%

, so

100\% - 75\% = 25\%

.

4

Solve for the total number of vehicles

The total inventory is 180 vehicles.

Set up the equation

25\% \text{ of Total} = 45

, which gives

0.25 \times \text{Total} = 45

, so

\text{Total} = \frac{45}{0.25} = 180

.

Key Concept

Converting fractions to percentages to find an unknown total quantity from a remaining part

Question 5551Question →

At a community theater, ticket sales for a weekend musical were divided among three ticket categories: advance online purchases, box office walk-ups, and discounted student passes. Advance online purchases accounted for $\frac{2}{5}$ of total ticket sales, and box office walk-up sales accounted for $35\%$ of total ticket sales. If the remaining 125 tickets sold were discounted student passes, what was the total number of tickets sold for the musical?

375

500

167

313

94

Show answer & explanation

Answer: 500

Answer

The total number of tickets sold for the musical was 500.

Advance online tickets represent

\frac{2}{5} = 40\%

of the total. Adding the

35\%

from box office walk-ups gives

75\%

of total sales accounted for. The remaining

25\%

of sales corresponds to the 125 student passes. Setting

25\%

of the total equal to 125 and solving gives a total of 500 tickets.

Step-by-Step Solution

1

Convert the fraction of advance online tickets to a percentage.

\frac{2}{5} = \frac{40}{100} = 40\%

Converting all proportions to percentages allows for direct comparison and addition.

2

Calculate the combined percentage of advance online and box office ticket sales.

40\% + 35\% = 75\%

Adding the two known categories determines the total portion of non-student ticket sales.

3

Find the percentage corresponding to the discounted student passes.

100\% - 75\% = 25\%

The sum of all categories must equal

100\%

of total sales.

4

Set up an equation relating the remaining percentage to the total ticket count

T

.

0.25 \times T = 125 \implies T = \frac{125}{0.25} = 500

Dividing the part by its corresponding percentage rate yields the whole amount.

Key Concept

Solving multi-step word problems involving conversions between fractions, percentages, and whole numbers.

Estimated Time:1m 30s

Question 5552Question →

Although subterranean mycorrhizal networks transport vital nutrients between canopy trees, forest ecologists still struggle to map these fragile complex fungal threads, for their delicate structures break down rapidly when exposed to air.

Which of the following choices produces a grammatically correct sentence that properly uses commas?

NO CHANGE

fragile, complex, fungal threads for their delicate structures break down rapidly

fragile, complex fungal threads, for their delicate structures break down rapidly

fragile, complex fungal threads; for, their delicate structures break down rapidly

Show answer & explanation

Answer: fragile, complex fungal threads, for their delicate structures break down rapidly

Answer

The correct choice is the option that places a comma between 'fragile' and 'complex', leaves 'fungal' unseparated, and includes a comma before the coordinating conjunction 'for'.

The correct answer accurately separates the coordinate adjectives 'fragile' and 'complex' with a single comma, refrains from putting a comma before the cumulative adjective 'fungal', and correctly places a comma prior to the coordinating conjunction 'for', which joins two complete independent clauses.

Step-by-Step Solution

1

Evaluate the adjectives preceding 'threads' to determine coordinate versus cumulative relationship.

'Fragile' and 'complex' are coordinate adjectives modifying 'fungal threads' (testable by reversing them: 'complex, fragile fungal threads' or inserting 'and': 'fragile and complex'), requiring a separating comma. 'Fungal' is a cumulative adjective directly classifying 'threads', so no comma belongs between 'complex' and 'fungal'.

Coordinate adjectives independently modify the noun and require a comma, whereas cumulative adjectives build upon each other without punctuation.

2

Analyze the clause structure surrounding the conjunction 'for'.

'forest ecologists still struggle to map these fragile, complex fungal threads' and 'their delicate structures break down rapidly when exposed to air' are both independent clauses.

When two independent clauses are connected by a coordinating conjunction (FANBOYS: for, and, nor, but, or, yet, so), a comma must precede the conjunction.

Key Concept

Commas with Coordinate Adjectives and Coordinating Conjunctions

Estimated Time:1m 0s

Question 5553Question →

A chemist uses the algebraic formula $E = \frac{2x^2 - yz}{x + y^2}$ to determine the stability index of a synthesized compound. If $x = -4$ , $y = -3$ , and $z = \frac{2}{3}$ , what is the value of $E$ ?

-6

6

\frac{34}{49}

\frac{34}{5}

-\frac{34}{13}

Show answer & explanation

Answer: $\frac{34}{5}$

Answer

\frac{34}{5}

Substituting

x = -4

,

y = -3

, and

z = \frac{2}{3}

into the numerator yields

2(-4)^2 - (-3)\left(\frac{2}{3}\right) = 2(16) - (-2) = 32 + 2 = 34

. Substituting into the denominator yields

-4 + (-3)^2 = -4 + 9 = 5

. Thus, the value of the expression is

\frac{34}{5}

.

Step-by-Step Solution

1

Substitute the given values into the numerator

2x^2 - yz

.

2(-4)^2 - (-3)\left(\frac{2}{3}\right) = 2(16) - (-2) = 32 + 2 = 34

Squaring a negative number yields a positive result

(-4)^2 = 16

, and multiplying

-3

by

\frac{2}{3}

gives

-2

, which is then subtracted.

2

Substitute the given values into the denominator

x + y^2

.

-4 + (-3)^2 = -4 + 9 = 5

Evaluating the exponent first gives

(-3)^2 = 9

, then adding

-4

gives

5

.

3

Divide the numerator by the denominator.

\frac{34}{5}

Combine the evaluated numerator and denominator to get the final value of the expression.

Key Concept

Evaluating algebraic expressions involving multiple variables with negative bases and fractional terms using standard order of operations.

Estimated Time:1m 15s

Question 5554Question →

If $a = -2$ , $b = 3$ , and $c = -\frac{1}{2}$ , what is the value of the algebraic expression $\frac{a^3 b + 4c}{a^2 - 2b}$ ?

Fill in the blanks below

If

a = -2

,

b = 3

, and

c = -\frac{1}{2}

, the value of the algebraic expression

\frac{a^3 b + 4c}{a^2 - 2b}

is .

Show answer & explanation

Answer

13

Substituting

a = -2

,

b = 3

, and

c = -\frac{1}{2}

into the given expression yields a numerator of

(-2)^3(3) + 4(-\frac{1}{2}) = -24 - 2 = -26

and a denominator of

(-2)^2 - 2(3) = 4 - 6 = -2

. Dividing the numerator by the denominator gives

\frac{-26}{-2} = 13

.

Step-by-Step Solution

1

Evaluate the terms in the numerator individually.

a^3 b = (-2)^3 \cdot 3 = -8 \cdot 3 = -24

and

4c = 4\left(-\frac{1}{2}\right) = -2

.

Exponents take precedence before multiplication, and multiplying a positive by a negative yields a negative number.

2

Combine the terms to calculate the numerator.

Numerator

= -24 + (-2) = -26

.

Adding two negative numbers sums their magnitudes with a negative sign.

3

Evaluate the terms in the denominator.

a^2 = (-2)^2 = 4

and

2b = 2(3) = 6

.

Squaring a negative base results in a positive value.

4

Calculate the denominator.

Denominator

= 4 - 6 = -2

.

Subtracting a larger number from a smaller number produces a negative result.

5

Divide the numerator by the denominator to find the final value.

\frac{-26}{-2} = 13

.

Dividing a negative number by a negative number yields a positive quotient.

Key Concept

Evaluating Algebraic Expressions

Estimated Time:1m 30s

Question 5555Question →

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

-\frac{17}{2}

-\frac{13}{2}

\frac{7}{10}

\frac{7}{2}

\frac{43}{2}

Show answer & explanation

Answer: $\frac{7}{2}$

Answer

\frac{7}{2}

Substituting

x = -3

and

y = 2

gives a numerator of

(-3)^2(2) - (-3 - 2)^2 = 18 - 25 = -7

and a denominator of

2(-3) + 2^2 = -6 + 4 = -2

. Dividing

-7

by

-2

gives

\frac{7}{2}

.

Step-by-Step Solution

1

Evaluate the terms in the numerator

x^2 y = (-3)^2(2) = 9 \cdot 2 = 18

and

(x - y)^2 = (-3 - 2)^2 = (-5)^2 = 25

Apply order of operations by performing parenthetical subtraction before squaring negative numbers.

2

Subtract the terms to find the total numerator value

18 - 25 = -7

Subtract the squared binomial result from the first product.

3

Evaluate the denominator

2x + y^2 = 2(-3) + 2^2 = -6 + 4 = -2

Multiply and square the variable values before adding.

4

Divide the numerator by the denominator and simplify

\frac{-7}{-2} = \frac{7}{2}

Dividing a negative number by a negative number yields a positive result.

Key Concept

Evaluating Algebraic Expressions with Negative Bases and Parentheses

Estimated Time:1m 0s

Question 5556Question →

If $x = -2$ , $y = \frac{1}{3}$ , and $z = -4$ , what is the value of the algebraic expression $\frac{x^3 y^{-1} - (x - z)^2}{x + y z}$ ?

\frac{42}{5}

\frac{18}{5}

\frac{33}{5}

\frac{2}{5}

14

Show answer & explanation

Answer: $\frac{42}{5}$

Answer

\frac{42}{5}

Substituting

x = -2

,

y = \frac{1}{3}

, and

z = -4

into the expression yields

(-2)^3 = -8

,

(\frac{1}{3})^{-1} = 3

, and

(-2 - (-4))^2 = 2^2 = 4

. Thus, the numerator equals

-8 \times 3 - 4 = -28

. The denominator equals

-2 + (\frac{1}{3})(-4) = -2 - \frac{4}{3} = -\frac{10}{3}

. Dividing the numerator by the denominator gives

\frac{-28}{-\frac{10}{3}} = \frac{84}{10} = \frac{42}{5}

.

Step-by-Step Solution

1

Substitute the given values into the numerator of the expression:

x^3 y^{-1} - (x - z)^2

.

Numerator =

(-2)^3 \left(\frac{1}{3}\right)^{-1} - (-2 - (-4))^2 = (-8)(3) - (2)^2 = -24 - 4 = -28

.

First apply exponent rules and basic arithmetic inside the parentheses following standard order of operations.

2

Substitute the given values into the denominator of the expression:

x + y z

.

Denominator =

-2 + \left(\frac{1}{3}\right)(-4) = -2 - \frac{4}{3} = -\frac{6}{3} - \frac{4}{3} = -\frac{10}{3}

.

Multiply

y

and

z

first, then find a common denominator to add the fraction to the integer.

3

Divide the simplified numerator by the simplified denominator.

\frac{-28}{-\frac{10}{3}} = -28 \times \left(-\frac{3}{10}\right) = \frac{84}{10} = \frac{42}{5}.

Dividing by a fraction is equivalent to multiplying by its reciprocal.

Key Concept

Evaluating Algebraic Expressions with Negative Numbers and Negative Exponents

Estimated Time:1m 15s

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