Elementary Algebra Alıştırma Soruları — ACT — Sayfa 12

Soru 221Soru →

Two volunteer teams, Team A and Team B, are packing food boxes for a local shelter. Team A packs at a constant rate of $15$ boxes per hour and begins packing at 8:00 a.m. Team B packs at a constant rate of $20$ boxes per hour and begins packing at 9:30 a.m. If both teams pack continuously at their respective rates, how many hours after Team A begins will both teams have packed the same total number of boxes?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

Both teams will have packed the same total number of boxes

6

hours after Team A begins.

The correct answer is

6

. Let

x

be the number of hours Team A packs. Team B begins 1.5 hours later, so Team B packs for

x - 1.5

hours. Setting their total boxes packed equal gives

15x = 20(x - 1.5)

. Distributing yields

15x = 20x - 30

. Subtracting

20x

from both sides gives

-5x = -30

, which simplifies to

x = 6

.

Adım Adım Çözüm

1

Define the variable for time and identify the time difference.

Let

x

be the number of hours Team A packs. Since Team B starts 1 hour and 30 minutes (which is

1.5

hours) later, Team B's time is represented as

x - 1.5

hours.

Establishing correct algebraic representations for time is necessary to set up the equation.

2

Set up an equation equating the total boxes packed by both teams.

The equation is

15x = 20(x - 1.5)

.

Since both teams pack a constant number of boxes per hour, multiplying their rate by their active time gives the total boxes packed.

3

Solve the linear equation for

x

.

Distribute the 20:

15x = 20x - 30

. Subtract

20x

from both sides:

-5x = -30

. Divide by

-5

:

x = 6

.

Solving the equation gives the number of hours after Team A starts when their packed boxes are equal.

Anahtar Kavram

Translating real-world rates and time shifts into linear equations and solving them.

Soru 222Soru →

A local community garden allocates a plot of land for different types of vegetables. Tomatoes occupy $\frac{2}{5}$ of the total area, peppers occupy $\frac{1}{3}$ of the total area, and the remaining 80 square feet are planted with herbs. What is the total area, in square feet, of the community garden plot?

100

120

128

200

300

Cevabı ve açıklamayı göster

Cevap: 300

Cevap

300 square feet

Let

x

be the total area of the garden plot. Since the garden is divided into tomatoes (

\frac{2}{5}

of the total area), peppers (

\frac{1}{3}

of the total area), and herbs (

80

square feet), the sum of these parts equals the total area:

\frac{2}{5}x + \frac{1}{3}x + 80 = x

. Finding a common denominator of 15 for the fractions gives

\frac{6}{15}x + \frac{5}{15}x + 80 = x

, which simplifies to

\frac{11}{15}x + 80 = x

. Subtracting

\frac{11}{15}x

from both sides yields

80 = \frac{4}{15}x

. Multiplying both sides by the reciprocal

\frac{15}{4}

gives

x = 80 \times \frac{15}{4} = 300

square feet.

Adım Adım Çözüm

1

Define the variable and translate the given fractions and numbers into an equation. Let

x

represent the total area of the garden plot.

\frac{2}{5}x + \frac{1}{3}x + 80 = x

The sum of the areas of the tomato plot, the pepper plot, and the herb plot must equal the total area of the garden.

2

Find a common denominator to combine the fractional coefficients of

x

. The least common multiple of 5 and 3 is 15. Convert

\frac{2}{5}

and

\frac{1}{3}

to fractions with a denominator of 15.

\frac{6}{15}x + \frac{5}{15}x + 80 = x \implies \frac{11}{15}x + 80 = x

Combining like terms simplifies the expression to help isolate the variable.

3

Subtract

\frac{11}{15}x

from both sides of the equation to group all

x

terms on one side.

80 = x - \frac{11}{15}x \implies 80 = \frac{4}{15}x

This isolates the constant term on one side of the equation.

4

Multiply both sides of the equation by the reciprocal of

\frac{4}{15}

, which is

\frac{15}{4}

, to solve for

x

.

x = 80 \times \frac{15}{4} = 20 \times 15 = 300

This yields the value of the total area of the garden plot.

Anahtar Kavram

Solving linear equations in one variable with fractional coefficients

Soru 223Soru →

A pet care service, Canine Care, charges a flat monthly registration fee of $25 plus$ 15 per dog walk. A competing service, Paws & Claws, charges a flat monthly registration fee of $45 plus$ 10 per dog walk. If a client's total monthly cost with Canine Care is $30 more than their total monthly cost with Paws & Claws for the same number of walks, how many dog walks did the client's dog receive during that month?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The client's dog received 10 dog walks during the month.

The correct answer is 10. By translating the verbal descriptions into algebraic expressions, we find that Canine Care's monthly cost is

15w + 25

and Paws & Claws' monthly cost is

10w + 45

. Setting up the equation where Canine Care is

30

more than Paws & Claws gives

15w + 25 = 10w + 45 + 30

. Simplifying the right side results in

15w + 25 = 10w + 75

. Subtracting

10w

from both sides gives

5w + 25 = 75

. Subtracting

25

from both sides gives

5w = 50

, and dividing by

5

yields

w = 10

.

Adım Adım Çözüm

1

Define the variable and write the cost expression for Canine Care.

15w + 25

, where

w

is the number of walks.

To represent the total monthly cost of Canine Care algebraically based on the flat fee and per-walk rate.

2

Write the cost expression for Paws & Claws.

10w + 45

, where

w

is the number of walks.

To represent the total monthly cost of Paws & Claws algebraically based on the flat fee and per-walk rate.

3

Set up the equation relating the two costs.

15w + 25 = 10w + 45 + 30

The problem states the cost of Canine Care is $30 more than the cost of Paws & Claws.

4

Simplify the equation and solve for

w

.

5w = 50 \Rightarrow w = 10

Combine like terms and isolate the variable

w

to find the number of dog walks.

Anahtar Kavram

Translating and Solving Algebraic Word Problems

Soru 224Soru →

For all positive real numbers $x$ and $y$ , consider the algebraic expression:

\frac{\left((x + y)y^2\right)^{1/2} (x + y)^{2/3}}{(x + y)^{-1/2}y}

Which of the following is equivalent to this expression?

(x + y)^{5/6}

x^{5/3} + y^{5/3}

(x + y)^{5/3}

(x + y)^{11/10}

(x + y)^{5/3} y

Cevabı ve açıklamayı göster

Cevap: $(x + y)^{5/3}$

Cevap

(x + y)^{5/3}

The correct answer is obtained by applying the properties of exponents step-by-step. First, apply the power of a product rule to the term

((x+y)y^2)^{1/2}

to get

(x+y)^{1/2}y

. Next, combine the terms in the numerator using the product rule of exponents to get

(x+y)^{7/6}y

. Finally, divide by the denominator using the quotient rule of exponents, which cancels the

y

terms and results in

(x+y)^{5/3}

.

Adım Adım Çözüm

1

Apply the power of a product property to simplify the first term in the numerator.

\left((x + y)y^2\right)^{1/2} = (x + y)^{1/2}(y^2)^{1/2} = (x + y)^{1/2}y

To distribute the exponent

1/2

to each factor inside the parentheses.

2

Multiply the simplified term by the other factor in the numerator.

(x + y)^{1/2}y \cdot (x + y)^{2/3} = (x + y)^{1/2 + 2/3}y = (x + y)^{7/6}y

To combine bases of the same value by adding their exponents:

\frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6}

.

3

Divide the simplified numerator by the denominator.

\frac{(x + y)^{7/6}y}{(x + y)^{-1/2}y} = (x + y)^{7/6 - (-1/2)} = (x + y)^{7/6 + 3/6} = (x + y)^{10/6} = (x + y)^{5/3}

To subtract the exponent in the denominator from the exponent in the numerator and cancel the common factor

y

.

Anahtar Kavram

Properties of Exponents in Algebraic Expressions

Alternatif Yöntem

Instead of simplifying the numerator first, we can divide the terms inside the parentheses first if we rewrite the expression, but the standard method of simplifying the numerator and then performing division is the most direct.

Tahmini Süre:1m 30s

Soru 225Soru →

The length $L$ , in centimeters, of a copper rod at a temperature of $T$ degrees Celsius can be modeled by the linear equation $L = 150.04 + 0.012T$ . If the length of the rod is measured to be $150.40$ centimeters, what is its temperature in degrees Celsius?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The temperature of the rod is 30 degrees Celsius.

The correct answer is 30. By substituting the given length of

150.40

centimeters into the equation for

L

, we get

150.40 = 150.04 + 0.012T

. Subtracting

150.04

from both sides gives

0.36 = 0.012T

. Finally, dividing both sides by

0.012

yields the temperature

T = 30

degrees Celsius.

Adım Adım Çözüm

1

Substitute the measured length

L = 150.40

into the equation.

150.40 = 150.04 + 0.012T

To set up the equation with the given value for length.

2

Subtract 150.04 from both sides of the equation.

0.36 = 0.012T

To isolate the variable term containing

T

.

3

Divide both sides of the equation by 0.012.

T = 30

To find the temperature

T

.

Anahtar Kavram

Solving a multi-step linear equation involving decimals

Soru 226Soru →

An online retail store charges a flat shipping fee of $10.00$ for any order, plus $5.00$ per pound of the package's weight. A customer places an order and uses a coupon that gives them a $20\%$ discount on the total cost (the flat shipping fee plus the weight cost). If the customer's total bill after the discount is $36.00$ , what would their total bill have been, in dollars, if they had not used the coupon?

30.00

38.00

42.50

43.20

45.00

Cevabı ve açıklamayı göster

Cevap: $45.00$

Cevap

The total bill without the coupon would have been

45.00

.

The correct answer represents the total cost before the coupon discount. Since a

20\%

discount was applied to the entire bill, the final bill of

36.00

represents

80\%

of the original total bill. Setting up the equation

0.80T = 36.00

and dividing

36.00

by

0.80

yields the original total bill of

45.00

.

Adım Adım Çözüm

1

Define the variable for the original total bill and set up the equation using the discount percentage.

Let

T

represent the total bill before the coupon discount. A

20\%

discount means the customer pays

100\% - 20\% = 80\%

of the original total bill. This gives the linear equation

0.80T = 36.00

.

Establishing a direct relationship between the original price and the discounted price allows for a direct solution without needing to calculate the weight of the package first.

2

Solve the equation for

T

by dividing the discounted cost by the remaining percentage factor.

T = \frac{36.00}{0.80} = 45.00

Dividing both sides of the equation by

0.80

isolates the variable

T

, representing the original total bill in dollars.

Anahtar Kavram

Translating percentage changes and linear cost models into algebraic equations.

Alternatif Yöntem

You can also solve this by finding the weight of the package first. Setting up the equation with the weight

w

gives

0.80(10.00 + 5.00w) = 36.00

. Dividing by

0.80

simplifies the expression to

10.00 + 5.00w = 45.00

, which is the original bill. Solving further gives

5.00w = 35.00

and

w = 7

pounds. Substituting

7

back into the original cost formula yields

10.00 + 5.00(7) = 45.00

.

Tahmini Süre:1m 30s

Soru 227Soru →

For all non-zero real numbers $x$ and $y$ , the expression $\frac{(x^2 + y^{-1})^2}{x^4}$ is equivalent to which of the following?

1 + x^{-4}y^{-2}

1 + 2x^{-2}y + x^{-4}y^2

1 + 2x^{-2}y^{-1} + x^{-4}y^{-2}

1 + 2x^2 y^{-1} + y^{-2}

y^{-2} + 2x^{-2}y^{-2} + x^{-4}y^{-2}

Cevabı ve açıklamayı göster

Cevap: $1 + 2x^{-2}y^{-1} + x^{-4}y^{-2}$

Cevap

1 + 2x^{-2}y^{-1} + x^{-4}y^{-2}

The correct expression is obtained by expanding the binomial in the numerator using the perfect square formula

(a + b)^2 = a^2 + 2ab + b^2

. Substituting

a = x^2

and

b = y^{-1}

yields

(x^2 + y^{-1})^2 = (x^2)^2 + 2(x^2)(y^{-1}) + (y^{-1})^2 = x^4 + 2x^2 y^{-1} + y^{-2}

. Dividing each term by the denominator

x^4

and applying the quotient rule for exponents

\frac{x^m}{x^n} = x^{m-n}

results in

1 + 2x^{-2}y^{-1} + x^{-4}y^{-2}

.

Adım Adım Çözüm

1

Expand the binomial in the numerator.

(x^2 + y^{-1})^2 = (x^2)^2 + 2(x^2)(y^{-1}) + (y^{-1})^2 = x^4 + 2x^2 y^{-1} + y^{-2}

Apply the binomial expansion formula

(a + b)^2 = a^2 + 2ab + b^2

where

a = x^2

and

b = y^{-1}

.

2

Divide each term in the expanded numerator by the denominator.

\frac{x^4}{x^4} + \frac{2x^2 y^{-1}}{x^4} + \frac{y^{-2}}{x^4}

Distribute the division over the terms in the numerator.

3

Simplify each fraction using exponent properties.

1 + 2x^{-2}y^{-1} + x^{-4}y^{-2}

Use the quotient rule

\frac{x^m}{x^n} = x^{m-n}

to simplify the variables.

Anahtar Kavram

Properties of Exponents in Algebraic Expressions

Soru 228Soru →

A landscaping company charges a one-time equipment mobilization fee of $45$ plus an hourly labor rate of $32.50$ per worker. A homeowner hires a crew of $3$ workers to clear their yard. If the total bill for the job is $435$ dollars, for how many hours did the crew work?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

The total cost of

435

dollars is the sum of the one-time

45

dollar mobilization fee and the hourly labor cost of

32.50

dollars per worker for

3

workers over

h

hours. This translates to the equation

45 + 3(32.50)h = 435

, which simplifies to

45 + 97.50h = 435

. Subtracting

45

from both sides gives

97.50h = 390

. Dividing by

97.50

yields

h = 4

hours.

Adım Adım Çözüm

1

Set up the algebraic expression representing the total cost based on the number of hours worked,

h

. The total cost consists of a fixed mobilization fee of

45

dollars and a variable labor cost of

32.50

dollars per worker per hour.

45 + 3(32.50)h

To represent the relation between the hours worked and the total charge.

2

Equate the expression for the total cost to the actual total bill of

435

dollars and simplify the labor rate coefficient.

45 + 97.50h = 435

To form a solvable linear equation in terms of the unknown number of hours,

h

.

3

Subtract the fixed mobilization fee of

45

from both sides of the equation.

97.50h = 390

To isolate the term containing the variable

h

.

4

Divide both sides of the equation by

97.50

to solve for

h

.

h = 4

To determine the number of hours the crew worked.

Anahtar Kavram

Formulating and solving linear equations from real-world contexts

Tahmini Süre:1m 30s

Soru 229Soru →

A marketing firm charges a flat setup fee of $160$ plus $3.20$ for each promotional brochure printed. For a large order, the firm applies a $15\%$ discount to the total cost (the sum of the setup fee and the printing cost). If the final discounted bill is $680$ , how many brochures were printed?

170

20

200

250

300

Cevabı ve açıklamayı göster

Cevap: 200

Cevap

200

The correct answer of 200 brochures is found by formulating the equation

0.85(160 + 3.2b) = 680

. Dividing both sides by

0.85

yields

160 + 3.2b = 800

. Subtracting

160

gives

3.2b = 640

, and dividing by

3.2

results in

b = 200

.

Adım Adım Çözüm

1

Translate the word problem into a linear equation where

b

represents the number of brochures. The total cost before discount is

160 + 3.2b

. Applying a

15\%

discount means paying

85\%

of this total.

0.85(160 + 3.2b) = 680

This sets up the relationship between the total discounted bill and the number of brochures.

2

Divide both sides of the equation by

0.85

to isolate the expression within parentheses.

160 + 3.2b = 800

Dividing by

0.85

simplifies the equation by calculating the total cost before the discount was applied.

3

Subtract

160

from both sides to isolate the term containing the variable

b

.

3.2b = 640

Subtracting the flat setup fee isolates the printing cost of the brochures.

4

Divide both sides by

3.2

to solve for

b

.

b = 200

Dividing by the per-brochure printing cost yields the total number of brochures printed.

Anahtar Kavram

Solving multi-step linear equations involving percentage discounts and decimal coefficients.

Alternatif Yöntem

Instead of dividing by

0.85

first, you can distribute

0.85

to both terms inside the parentheses to get

136 + 2.72b = 680

. Subtract

136

from both sides to get

2.72b = 544

, then divide by

2.72

to find

b = 200

.

Tahmini Süre:1m 30s

Soru 230Soru →

If $u$ and $v$ are positive real numbers such that $\frac{(u^{-2} v^3)^3}{u^k v^{-1}} = \frac{v^{10}}{u^{10}}$ , what is the value of the exponent $k$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of the exponent is 4.

By applying the exponent rules systematically, the expression on the left simplifies to

u^{-6-k} v^{10}

, and the expression on the right is

u^{-10} v^{10}

. Equating the exponents of

u

gives

-6-k = -10

, which solves to

k = 4

.

Adım Adım Çözüm

1

Apply the power of a product and power of a power properties to the numerator

(u^{-2} v^3)^3

.

u^{-6} v^9

According to the power of a product rule,

(xy)^a = x^a y^a

, and the power of a power rule,

(x^a)^b = x^{ab}

.

2

Use the quotient of powers property to simplify the left side of the equation.

u^{-6-k} v^{10}

The quotient of powers rule states that

\frac{x^a}{x^b} = x^{a-b}

, so the exponents of like bases are subtracted:

-6 - k

for

u

and

9 - (-1) = 10

for

v

.

3

Rewrite the right side of the equation,

\frac{v^{10}}{u^{10}}

, using a negative exponent.

u^{-10} v^{10}

Applying the negative exponent rule,

\frac{1}{x^a} = x^{-a}

.

4

Set the simplified expressions equal and solve for

k

.

k = 4

Since

u^{-6-k} v^{10} = u^{-10} v^{10}

, the exponents of the base

u

must be equal. Therefore,

-6 - k = -10

, which simplifies to

k = 4

.

Anahtar Kavram

Properties of Exponents in Algebraic Expressions

Tahmini Süre:1m 30s

Soru 231Soru →

A closed cardboard box has a height of $x$ inches, a width of $3x - 2$ inches, and a length of $2x + 5$ inches. When the volume of the box, in cubic inches, is written as a polynomial in standard form, what is the coefficient of the $x^2$ term?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The coefficient of the

x^2

term is 11.

Expanding the volume expression

V(x) = x(3x - 2)(2x + 5)

yields

6x^3 + 11x^2 - 10x

. The coefficient of the

x^2

term is the numerical value associated with

x^2

, which is 11.

Adım Adım Çözüm

1

Set up the polynomial expression for the volume.

V(x) = x(3x - 2)(2x + 5)

The volume of a rectangular prism is the product of its length, width, and height.

2

Multiply the binomials

(3x - 2)

and

(2x + 5)

by distributing terms.

(3x - 2)(2x + 5) = 6x^2 + 15x - 4x - 10 = 6x^2 + 11x - 10

To find the product of two binomials, multiply each term of the first binomial by each term of the second binomial and combine like terms.

3

Distribute the monomial

x

to each term in the simplified trinomial.

x(6x^2 + 11x - 10) = 6x^3 + 11x^2 - 10x

The height

x

must scale the entire base area polynomial.

4

Identify the coefficient of the quadratic term

x^2

.

11

The coefficient of a term is the numerical factor multiplied by the variable part.

Anahtar Kavram

Multiplying polynomials and identifying coefficients of specific terms in the resulting standard form polynomial.

Soru 232Soru →

If $x$ is a real number that satisfies the equation $\frac{3}{5}(2x - 7) + 0.4 = 0.2(x + 3)$ , what is the value of $5x - 2$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The value of the expression

5x - 2

is

20

.

First, convert the fraction

\frac{3}{5}

to the decimal

0.6

. The equation becomes

0.6(2x - 7) + 0.4 = 0.2(x + 3)

. Distribute on both sides to get

1.2x - 4.2 + 0.4 = 0.2x + 0.6

. Combine constant terms on the left side to get

1.2x - 3.8 = 0.2x + 0.6

. Subtract

0.2x

and add

3.8

to both sides to isolate the variable, resulting in

x = 4.4

. Finally, substitute

x = 4.4

into the expression

5x - 2

to get

5(4.4) - 2 = 22 - 2 = 20

.

Adım Adım Çözüm

1

Convert the fraction and distribute the coefficients

1.2x - 4.2 + 0.4 = 0.2x + 0.6

To clear parentheses and align terms using decimals.

2

Combine constants on the left side

1.2x - 3.8 = 0.2x + 0.6

To simplify the left-hand side of the linear equation.

3

Isolate the variable

x

x = 4.4

To determine the value of the unknown variable.

4

Evaluate the target expression

20

To compute the final value of the expression

5x - 2

.

Anahtar Kavram

Solving linear equations with fractional and decimal coefficients

Soru 233Soru →

A smartphone's battery is fully charged to $100\%$ . When the phone is in sleep mode, the battery drains at a constant rate of $1\%$ per hour. When the phone is in active use, the battery drains at a constant rate of $6\%$ per hour. Over a $20$ -hour period, the phone is either in sleep mode or in active use, and the battery level decreases to $45\%$ . For how many hours was the phone in active use during this period?

5

7

9

11

13

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The phone was in active use for 7 hours.

To find the active hours, first calculate the total percentage of the battery consumed:

100\% - 45\% = 55\%

. Let

x

represent the number of hours in active use. The remaining time,

20 - x

, represents the hours in sleep mode. Set up the equation representing the total battery drain:

6x + 1(20 - x) = 55

. Simplifying this equation gives

5x + 20 = 55

, which reduces to

5x = 35

, resulting in

x = 7

hours.

Adım Adım Çözüm

1

Calculate the total percentage of the battery that drained during the period.

100\% - 45\% = 55\%

To determine the exact battery percentage consumed by both active use and sleep mode combined.

2

Set up a system of equations using

x

for hours in active use and

y

for hours in sleep mode.

x + y = 20

and

6x + 1y = 55

To represent the constraints on the total duration (20 hours) and the total battery drain (55%).

3

Substitute

y = 20 - x

into the drain equation and solve for

x

.

6x + (20 - x) = 55 \implies 5x + 20 = 55 \implies 5x = 35 \implies x = 7

To find the number of hours the phone was actively used.

Anahtar Kavram

Translating and solving a system of linear equations from a real-world word problem.

Soru 234Soru →

Which of the following expressions is equivalent to $\frac{a^{-2} - b^{-2}}{a^{-1} - b^{-1}}$ for all non-zero real numbers $a$ and $b$ where $a \neq b$ ?

a^2 - b^2

\frac{1}{a+b}

\frac{a+b}{ab}

\frac{1}{a-b}

a+b

Cevabı ve açıklamayı göster

Cevap: $\frac{a+b}{ab}$

Cevap

The correct answer is

\frac{a+b}{ab}

because simplifying the original expression using the difference of squares or by rewriting negative exponents as fractions leads directly to this value.

By applying the difference of squares factorization to the numerator, the expression simplifies to

a^{-1} + b^{-1}

. Converting these to standard fractions and finding a common denominator yields

\frac{a+b}{ab}

.

Adım Adım Çözüm

1

Factor the numerator as a difference of squares.

a^{-2} - b^{-2} = (a^{-1})^2 - (b^{-1})^2 = (a^{-1} - b^{-1})(a^{-1} + b^{-1})

Since the terms are squared with negative exponents, we can use the difference of squares identity

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

where

x = a^{-1}

and

y = b^{-1}

.

2

Substitute the factored numerator back into the expression and cancel the common factor.

\frac{(a^{-1} - b^{-1})(a^{-1} + b^{-1})}{a^{-1} - b^{-1}} = a^{-1} + b^{-1}

Since

a \neq b

, we know

a^{-1} - b^{-1} \neq 0

, allowing us to divide out the common factor in the numerator and denominator.

3

Rewrite the expression using positive exponents and find a common denominator.

a^{-1} + b^{-1} = \frac{1}{a} + \frac{1}{b} = \frac{b}{ab} + \frac{a}{ab} = \frac{a+b}{ab}

A negative exponent

x^{-n}

represents the reciprocal

\frac{1}{x^n}

. Finding the common denominator

ab

allows us to combine the two fractions into a single expression.

Anahtar Kavram

Properties of negative exponents and difference of squares factorization

Alternatif Yöntem

Convert all terms to fractions with positive exponents first: write the numerator as

\frac{1}{a^2} - \frac{1}{b^2} = \frac{b^2 - a^2}{a^2b^2}

and the denominator as

\frac{1}{a} - \frac{1}{b} = \frac{b-a}{ab}

. Then, divide the two complex fractions by multiplying the numerator by the reciprocal of the denominator:

\frac{(b-a)(b+a)}{a^2b^2} \times \frac{ab}{b-a}

. After canceling the common factors

(b-a)

and

ab

, the expression simplifies directly to

\frac{a+b}{ab}

.

Tahmini Süre:1m 30s

Soru 235Soru →

For all positive real numbers $w$ , the expression $\left(\frac{w^4 \cdot \sqrt{w}}{w^{-1.5}}\right)^{2/3}$ is equivalent to $w^k$ , where $k$ is a constant. What is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

k

is

4

.

Applying the exponent rules in sequence: first, rewrite the square root as a fractional exponent to get

w^{0.5}

. Next, multiply the terms in the numerator by adding their exponents:

4 + 0.5 = 4.5

. Then, divide the numerator by the denominator by subtracting the denominator's exponent from the numerator's exponent:

4.5 - (-1.5) = 6

. Finally, raise this result to the

2/3

power by multiplying the exponents:

6 \times (2/3) = 4

. This yields

w^4

, so the constant exponent is

4

.

Adım Adım Çözüm

1

Convert the radical expression to an exponential expression.

\sqrt{w} = w^{0.5}

Converting all terms to base

w

with decimal or fractional exponents makes it easier to apply exponent properties.

2

Apply the product rule of exponents to the numerator.

w^4 \cdot w^{0.5} = w^{4 + 0.5} = w^{4.5}

When multiplying exponential terms with the same base, add their exponents:

w^a \cdot w^b = w^{a+b}

.

3

Apply the quotient rule of exponents to the fraction.

\frac{w^{4.5}}{w^{-1.5}} = w^{4.5 - (-1.5)} = w^6

When dividing exponential terms with the same base, subtract the exponent of the denominator from the exponent of the numerator:

\frac{w^a}{w^b} = w^{a-b}

.

4

Apply the power rule of exponents to the simplified term.

(w^6)^{2/3} = w^{6 \cdot \frac{2}{3}} = w^4

When raising a power to another power, multiply the exponents:

(w^a)^b = w^{a \cdot b}

.

Anahtar Kavram

Properties of Exponents in Algebraic Expressions

Soru 236Soru →

A food truck selling gourmet grilled cheese sandwiches has a daily fixed operating cost of $120$ . Each sandwich costs $2.50$ to make and is sold for $6.50$ . What is the minimum number of sandwiches the food truck must sell in one day to make a net profit of at least $180$ ?

Cevabı ve açıklamayı göster

Cevap: 75

Cevap

The food truck must sell a minimum of 75 sandwiches to make a net profit of at least $180.

Representing the number of sandwiches sold as

x

, the total revenue is

6.50x

and the total cost is

120 + 2.50x

. The net profit is the difference between revenue and cost:

6.50x - (120 + 2.50x)

, which simplifies to

4x - 120

. Setting up the inequality for a profit of at least

180

gives

4x - 120 \geq 180

. Solving for

x

gives

4x \geq 300

, which simplifies to

x \geq 75

. Therefore, the minimum number of sandwiches that must be sold is

75

.

Adım Adım Çözüm

1

Define the variable and write expressions for revenue and cost.

Let

x

represent the number of sandwiches sold. Total Revenue =

6.50x

and Total Cost =

120 + 2.50x

.

Defining the variable and translating the verbal descriptions of revenue and cost into algebraic expressions is necessary to model the profit.

2

Formulate the net profit expression.

Net Profit = Total Revenue - Total Cost =

6.50x - (120 + 2.50x) = 4x - 120

.

Net profit is calculated by subtracting all fixed and variable costs from the total revenue.

3

Set up and solve the linear inequality.

4x - 120 \geq 180 \Rightarrow 4x \geq 300 \Rightarrow x \geq 75

.

To find the minimum number of sandwiches needed to reach a target profit of at least

180

, we solve the inequality

4x - 120 \geq 180

for

x

.

Anahtar Kavram

Translating a real-world scenario into a linear inequality and solving for the unknown variable.

Soru 237Soru →

A laboratory technician mixes two solutions. The volume of the first solution is represented by $\frac{1}{2}(x - 5)$ liters, and the volume of the second solution is represented by $\frac{1}{3}(2x + 1)$ liters, where $x$ is a positive real number. If the sum of the volumes of these two solutions is $6$ liters, what is the value of $3x - 4$ ?

6

9

17

30

53

Cevabı ve açıklamayı göster

Cevap: $17$

Cevap

The value of the expression is

17

.

The sum of the volumes of the two solutions is

6

liters, which translates to the linear equation

\frac{1}{2}(x - 5) + \frac{1}{3}(2x + 1) = 6

. Multiplying the entire equation by the least common multiple of the denominators,

6

, yields

3(x - 5) + 2(2x + 1) = 36

. Distributing the constants leads to

3x - 15 + 4x + 2 = 36

. Combining like terms gives

7x - 13 = 36

. Adding

13

to both sides results in

7x = 49

, which gives

x = 7

. Substituting

x = 7

into the expression

3x - 4

yields

3(7) - 4 = 17

, which is the correct value.

Adım Adım Çözüm

1

Set up the linear equation based on the word problem context.

\frac{1}{2}(x - 5) + \frac{1}{3}(2x + 1) = 6

The sum of the volumes of the two solutions is given as

6

liters.

2

Clear the fractions by multiplying both sides of the equation by the least common multiple of the denominators.

3(x - 5) + 2(2x + 1) = 36

Multiplying by

6

eliminates the fractions and simplifies the equation for solving.

3

Distribute the coefficients to eliminate the parentheses.

3x - 15 + 4x + 2 = 36

Applying the distributive property allows like terms to be grouped.

4

Combine like terms and isolate the variable

x

.

7x - 13 = 36 \implies 7x = 49 \implies x = 7

Adding

13

to both sides and dividing by

7

solves for the value of the variable

x

.

5

Evaluate the expression requested in the question using the solved value of

x

.

3(7) - 4 = 21 - 4 = 17

Substituting

x = 7

into

3x - 4

yields the final requested value.

Anahtar Kavram

Solving linear equations with fractional coefficients by clearing denominators and evaluating variable expressions.

Alternatif Yöntem

Instead of clearing the fractions immediately, we can distribute the fractions first:

\frac{1}{2}x - 2.5 + \frac{2}{3}x + \frac{1}{3} = 6

. Converting to fractions with a common denominator of 6, we get

\frac{3}{6}x - \frac{15}{6} + \frac{4}{6}x + \frac{2}{6} = 6

, which simplifies to

\frac{7}{6}x - \frac{13}{6} = 6

. Adding

\frac{13}{6}

to both sides yields

\frac{7}{6}x = \frac{49}{6}

, so

7x = 49

, and

x = 7

. Substituting

x = 7

into the expression

3x - 4

yields

17

.

Tahmini Süre:1m 30s

Soru 238Soru →

A business analyst uses the linear equation $\frac{2}{5}(3p - 10) - 1.2 = 0.4(p + 5)$ to estimate the equilibrium price $p$ , in dollars, of a new product. What is the equilibrium price, in dollars, of the product?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The equilibrium price of the product is

9

dollars.

Converting the fraction to a decimal gives

0.4(3p - 10) - 1.2 = 0.4(p + 5)

. Dividing both sides by

0.4

results in

3p - 10 - 3 = p + 5

, which simplifies to

3p - 13 = p + 5

. Subtracting

p

and adding

13

to both sides gives

2p = 18

, and dividing by

2

results in

9

.

Adım Adım Çözüm

1

Convert the fraction to decimal form

0.4(3p - 10) - 1.2 = 0.4(p + 5)

Converting

\frac{2}{5}

to

0.4

makes all terms decimals, simplifying further operations.

2

Divide both sides of the equation by

0.4

(3p - 10) - 3 = p + 5

Since

0.4

is a common factor and

1.2 / 0.4 = 3

, dividing both sides by

0.4

simplifies the coefficients.

3

Simplify the left side of the equation

3p - 13 = p + 5

Combine the constant terms

-10

and

-3

to simplify the expression.

4

Isolate the variable terms on one side

2p = 18

Subtract

p

from both sides and add

13

to both sides.

5

Solve for

p

p = 9

Divide both sides by

2

to find the final value.

Anahtar Kavram

Solving Linear Equations

Soru 239Soru →

A company's weekly revenue, $R(x)$ , and weekly cost, $C(x)$ , in dollars, are modeled by the functions $R(x) = (2x^2 + 3)^2$ and $C(x) = x^2(x^3 - 5)$ , where $x$ represents the number of units produced and sold. Which of the following expressions represents the company's weekly profit, $P(x) = R(x) - C(x)$ ?

-x^5 + 4x^4 + 7x^2 + 9

-x^6 + 4x^4 + 17x^2 + 9

-x^5 + 4x^4 + 17x^2 + 9

-x^5 + 4x^4 + 5x^2 + 9

-x^6 + 4x^4 + 7x^2 + 9

Cevabı ve açıklamayı göster

Cevap: $-x^5 + 4x^4 + 17x^2 + 9$

Cevap

The expression representing the weekly profit is

-x^5 + 4x^4 + 17x^2 + 9

.

To find the profit function, we subtract the cost function from the revenue function. First, expanding

(2x^2 + 3)^2

yields

4x^4 + 12x^2 + 9

. Second, distributing

x^2

over

(x^3 - 5)

yields

x^5 - 5x^2

. Subtracting these two functions gives

(4x^4 + 12x^2 + 9) - (x^5 - 5x^2)

. Distributing the subtraction sign changes the signs of the cost function to

-x^5 + 5x^2

. Combining like terms (

12x^2 + 5x^2 = 17x^2

) and organizing the expression in descending order yields

-x^5 + 4x^4 + 17x^2 + 9

.

Adım Adım Çözüm

1

Expand the revenue function

R(x) = (2x^2 + 3)^2

using the binomial square identity

(a+b)^2 = a^2 + 2ab + b^2

.

R(x) = (2x^2)^2 + 2(2x^2)(3) + 3^2 = 4x^4 + 12x^2 + 9

This simplifies the revenue expression into a standard polynomial form.

2

Expand the cost function

C(x) = x^2(x^3 - 5)

by distributing

x^2

to both terms inside the parentheses and applying the product rule for exponents

x^a \cdot x^b = x^{a+b}

.

C(x) = x^2(x^3) - x^2(5) = x^5 - 5x^2

This simplifies the cost expression into a standard polynomial form.

3

Subtract

C(x)

from

R(x)

by setting up

P(x) = R(x) - C(x)

and distributing the negative sign to all terms of

C(x)

.

P(x) = (4x^4 + 12x^2 + 9) - (x^5 - 5x^2) = 4x^4 + 12x^2 + 9 - x^5 + 5x^2

Profit is revenue minus cost, and distributing the negative sign correctly is essential.

4

Combine like terms and write the final expression in descending order of exponents.

P(x) = -x^5 + 4x^4 + 17x^2 + 9

To present the polynomial in standard form.

Anahtar Kavram

Polynomial operations, including expanding binomial squares, distributing monomial terms, and subtracting polynomials with careful attention to sign distribution.

Soru 240Soru →

The ratio of the number of students in a school's science club to the number of students in its art club is $3:5$ . There are $8$ more students in the art club than in the science club. If $x$ students from the art club leave to join the science club, the ratio of the number of science club members to the number of art club members becomes $3:1$ . What is the value of $x$ ?

4

6

12

24

48

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

The correct answer is 12. Let the initial number of students in the science and art clubs be

3k

and

5k

, respectively. Since there are 8 more students in the art club, we set up the equation

5k = 3k + 8

, which simplifies to

2k = 8

, giving

k = 4

. This means there are initially 12 science club members and 20 art club members. When

x

students transfer from the art club to the science club, the new sizes are

12 + x

and

20 - x

. Setting up the new ratio gives

\frac{12 + x}{20 - x} = 3

. Solving this equation yields

12 + x = 60 - 3x

, which simplifies to

4x = 48

, resulting in

x = 12

.

Adım Adım Çözüm

1

Represent the initial number of students in each club using the given ratio.

Let the number of science club members be

3k

and the number of art club members be

5k

, where

k

is a positive constant.

The ratio of science club members to art club members is

3:5

.

2

Set up and solve a linear equation to find the value of

k

and the starting number of members in each club.

5k = 3k + 8 \Rightarrow 2k = 8 \Rightarrow k = 4

. Thus, the science club has

3(4) = 12

members and the art club has

5(4) = 20

members.

There are

8

more students in the art club than in the science club, so the difference between the two groups is

8

.

3

Write the new member counts after

x

students transfer and set up the ratio equation.

The new science club size is

12 + x

and the new art club size is

20 - x

. The new ratio equation is

\frac{12 + x}{20 - x} = \frac{3}{1}

.

The

x

students leave the art club (subtracting

x

) and join the science club (adding

x

), resulting in a new ratio of

3:1

.

4

Solve the rational equation for

x

.

12 + x = 3(20 - x) \Rightarrow 12 + x = 60 - 3x \Rightarrow 4x = 48 \Rightarrow x = 12

.

Cross-multiplying and isolating

x

yields the number of students who transferred.

Anahtar Kavram

Translating verbal statements involving ratios and changes in quantities into solvable linear algebraic equations.

Tahmini Süre:1m 30s

Elementary Algebra

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Alternatif Yöntem

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