All practice questions

Question 221Question →

A commercial facility prepares a fruit blend by mixing fruit concentrate with water in a ratio of $3:7$ by volume. After $15\text{ liters}$ of water evaporate from the mixture during processing, the ratio of fruit concentrate to water in the remaining mixture becomes $1:2$ . What was the total volume, in liters, of the original mixture before evaporation?

Show answer & explanation

Answer: 150

Answer

The total volume of the original mixture before evaporation was 150 liters.

Represent the initial concentrate volume as

3x

liters and the initial water volume as

7x

liters, making the initial total volume

10x

liters. Evaporating

15

liters of water leaves

7x - 15

liters of water while the concentrate remains

3x

liters. Setting the ratio

\frac{3x}{7x - 15}

equal to

\frac{1}{2}

yields

6x = 7x - 15

, so

x = 15

. Substituting

x = 15

into the total volume expression

10x

gives

10(15) = 150

liters.

Step-by-Step Solution

1

Define initial component volumes using ratio multiplier x

Concentrate volume =

3x

, Water volume =

7x

, Total volume =

10x

The given initial ratio of concentrate to water is

3:7

.

2

Formulate equation based on water evaporation and the new ratio

\frac{3x}{7x - 15} = \frac{1}{2}

Evaporation reduces only the water volume by 15 liters, establishing a new ratio of

1:2

.

3

Solve the algebraic equation for x

6x = 7x - 15 \implies x = 15

Cross-multiplication simplifies the proportional relationship into a linear equation.

4

Calculate original total volume

10 \times 15 = 150

liters

The original total volume is represented by

10x

.

Key Concept

Solving component adjustment problems using ratio multipliers

Question 222Question →

A commercial print shop operates two types of high-speed printers, Printer M and Printer N. Operating simultaneously for $5$ hours, $2$ units of Printer M and $3$ units of Printer N print a total of $5,500$ pages. Operating simultaneously for $4$ hours, $5$ units of Printer M and $2$ units of Printer N print a total of $6,600$ pages. What is the hourly page output of a single Printer M?

Show answer & explanation

Answer: 250

Answer

The hourly page output of a single Printer M is 250 pages per hour.

Dividing each total page output by the corresponding number of hours produces the simplified linear system:

2m + 3n = 1100

and

5m + 2n = 1650

, where

m

and

n

are the hourly rates of Printer M and Printer N. Multiplying the first equation by

2

gives

4m + 6n = 2200

, and multiplying the second equation by

3

gives

15m + 6n = 4950

. Subtracting the two equations eliminates

n

, giving

11m = 2750

, which simplifies to

m = 250

pages per hour.

Step-by-Step Solution

1

Define variables and write initial algebraic equations based on time and rate.

Let

m

represent the hourly page output of Printer M and

n

represent the hourly page output of Printer N. The total outputs give

5(2m + 3n) = 5500

and

4(5m + 2n) = 6600

.

Total page output equals total operating time multiplied by the combined hourly output rate.

2

Simplify the system by dividing each equation by its respective number of hours.

First equation:

2m + 3n = 1100

. Second equation:

5m + 2n = 1650

.

Simplifying yields a standard system of linear equations representing the combined hourly rate.

3

Eliminate variable

n

to solve for

m

.

Multiply

2m + 3n = 1100

by

2

to get

4m + 6n = 2200

. Multiply

5m + 2n = 1650

by

3

to get

15m + 6n = 4950

. Subtracting the first modified equation from the second yields

11m = 2750

, so

m = 250

.

Equating the coefficients of

n

allows direct elimination of

n

when subtracting the equations.

Key Concept

Solving a 2x2 system of linear equations using the method of elimination

Estimated Time:1m 30s

Question 223Question →

For all real numbers

x \neq -5

, the algebraic expression

\frac{x^3 + 125}{x^2 - 5x + 25} - \frac{x^2 - 25}{x + 5}

simplifies to a single constant value. What is the value of this constant?

Show answer & explanation

Answer: 10

Answer

The simplified value of the expression for all valid real numbers x is 10.

Factoring the numerators reveals that the first term reduces to

x + 5

and the second term reduces to

x - 5

. Subtracting

(x - 5)

from

(x + 5)

yields

(x + 5) - (x - 5) = 10

, which is constant for all valid values of

x

.

Step-by-Step Solution

1

Factor the numerator of the first rational term using the sum of cubes identity.

x^3 + 125 = (x + 5)(x^2 - 5x + 25)

The sum of cubes formula

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

applies with

a = x

and

b = 5

.

2

Cancel the non-zero factor

(x^2 - 5x + 25)

from the first fraction.

\frac{(x + 5)(x^2 - 5x + 25)}{x^2 - 5x + 25} = x + 5

The quadratic factor

x^2 - 5x + 25

has a negative discriminant (

(-5)^2 - 4(1)(25) = -75 < 0

), so it is never zero for any real number

x

.

3

Factor the numerator of the second rational term using the difference of squares identity.

x^2 - 25 = (x - 5)(x + 5)

The difference of squares formula

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

applies with

a = x

and

b = 5

.

4

Cancel the common factor

(x + 5)

from the second fraction.

\frac{(x - 5)(x + 5)}{x + 5} = x - 5

Given

x \neq -5

, the factor

x + 5

is non-zero and can be canceled.

5

Subtract the two simplified terms.

(x + 5) - (x - 5) = x + 5 - x + 5 = 10

Distribute the negative sign to both terms in

(x - 5)

and combine like terms.

Key Concept

Factoring sum of cubes and difference of squares to simplify rational expressions

Question 224Question →

For all real numbers

x

, the algebraic expression

\frac{x^6 - 27}{x^4 + 3x^2 + 9}

can be simplified to the polynomial form

ax^2 + bx + c

, where

a

,

b

, and

c

are real constants. What is the value of

a + b + c

?

Show answer & explanation

Answer: -2

Answer

The simplified expression is

x^2 - 3

, which corresponds to polynomial coefficients

a = 1

,

b = 0

, and

c = -3

. The sum

a + b + c

equals

-2

.

Factoring the numerator

x^6 - 27

as a difference of cubes

(x^2)^3 - 3^3

produces

(x^2 - 3)(x^4 + 3x^2 + 9)

. Canceling the non-zero factor

(x^4 + 3x^2 + 9)

from the numerator and denominator simplifies the expression to

x^2 - 3

. In standard form

ax^2 + bx + c

,

a = 1

,

b = 0

, and

c = -3

. Adding these coefficients gives

1 + 0 + (-3) = -2

.

Step-by-Step Solution

1

Factor the numerator using the difference of cubes formula

x^6 - 27 = (x^2)^3 - 3^3 = (x^2 - 3)(x^4 + 3x^2 + 9)

The expression

x^6 - 27

matches the pattern

u^3 - v^3

with

u = x^2

and

v = 3

.

2

Simplify the rational expression by canceling the common quadratic-biquadratic factor

\frac{(x^2 - 3)(x^4 + 3x^2 + 9)}{x^4 + 3x^2 + 9} = x^2 - 3

Since

x^4 + 3x^2 + 9 > 0

for all real numbers

x

, the denominator is never zero, allowing direct cancellation of the common factor.

3

Match coefficients with

ax^2 + bx + c

and calculate

a + b + c

a = 1, b = 0, c = -3 \implies a + b + c = 1 + 0 + (-3) = -2

Comparing

x^2 - 3 = 1x^2 + 0x - 3

to

ax^2 + bx + c

determines the values of constants

a

,

b

, and

c

.

Key Concept

Difference of Cubes Factoring Identity

Estimated Time:1m 30s

Question 225Question →

The length of a rectangular plot of land is $3$ meters less than twice its width. If the area of the plot is $90$ square meters, what is the perimeter of the plot, in meters?

Show answer & explanation

Answer: 39

Answer

The perimeter of the plot of land is 39 meters.

Setting the length to

2w - 3

gives an area equation of

w(2w - 3) = 90

, which expands and rearranges to

2w^2 - 3w - 90 = 0

. Factoring this quadratic equation yields

(2w - 15)(w + 6) = 0

. Since width must be positive,

w = 7.5

meters, which means the length is

12

meters. The perimeter is

2(12 + 7.5) = 39

meters.

Step-by-Step Solution

1

Express the length in terms of width and set up the area equation.

Let

w

be the width of the rectangle. Length

l = 2w - 3

. Area equation:

w(2w - 3) = 90

.

The area of a rectangle is equal to length multiplied by width.

2

Rearrange into standard quadratic form

ax^2 + bx + c = 0

.

2w^2 - 3w - 90 = 0

Distributing

w

and subtracting

90

from both sides puts the equation in standard quadratic form.

3

Factor the quadratic equation.

(2w - 15)(w + 6) = 0

Finding two numbers with a product of

2 \times (-90) = -180

and a sum of

-3

gives

-15

and

12

.

4

Determine the valid physical dimensions.

w = 7.5

meters and

l = 12

meters.

The root

w = -6

is discarded because physical length cannot be negative. Thus

w = \frac{15}{2} = 7.5

meters.

5

Calculate the perimeter.

Perimeter

= 2(l + w) = 2(12 + 7.5) = 39

meters.

The perimeter of a rectangle is given by

2 \times (\text{length} + \text{width})

.

Key Concept

Solving quadratic word problems via factoring

Estimated Time:1m 30s

Question 226Question →

Pumps $A$ and $B$ work independently at their respective constant rates to empty a water storage reservoir. Working together, Pump $A$ and Pump $B$ can empty the reservoir in $12$ hours. If Pump $A$ 's rate of pumping is $1.5$ times the rate of Pump $B$ , how many hours would it take Pump $B$ working alone at its constant rate to empty the entire reservoir?

Show answer & explanation

Answer: 30

Answer

It would take Pump B working alone 30 hours to empty the entire reservoir.

Since Pump A's rate is

1.5

times Pump B's rate, their combined rate is

1.5 r_B + r_B = 2.5 r_B

reservoirs per hour. Working together for

12

hours empties

12 \times 2.5 r_B = 30 r_B

reservoirs. Setting

30 r_B = 1

reservoir yields

r_B = \frac{1}{30}

reservoir per hour. Therefore, Pump B working alone requires

30

hours to empty the reservoir.

Step-by-Step Solution

1

Express Pump A's rate in terms of Pump B's rate.

Let

r_B

be the rate of Pump B in reservoirs per hour. Then the rate of Pump A is

r_A = 1.5 r_B

.

Expressing both rates in terms of a single variable simplifies setting up the combined rate equation.

2

Calculate the combined pumping rate.

Combined rate =

r_A + r_B = 1.5 r_B + r_B = 2.5 r_B

.

When two machines work simultaneously, their individual rates add together.

3

Set up the work equation using the given combined time of 12 hours.

\text{Total Work} = \text{Combined Rate} \times 12 = 2.5 r_B \times 12 = 30 r_B = 1 \text{ reservoir}

.

The total work performed equals the combined rate multiplied by the total time taken.

4

Solve for the time required for Pump B to complete the job alone.

\text{Time}_B = \frac{1}{r_B} = 30 \text{ hours}

.

The time needed for Pump B alone to empty

1

reservoir at rate

r_B

is the reciprocal of

r_B

.

Key Concept

Combined Work Rates and Rate Ratios

Alternative Method

Choose a convenient hypothetical capacity for the reservoir. Let Pump B's rate be

2

units per hour. Then Pump A's rate is

1.5 \times 2 = 3

units per hour. Working together, they empty

2 + 3 = 5

units per hour. In

12

hours, the reservoir holds

5 \times 12 = 60

units. Pump B alone at

2

units per hour takes

60 / 2 = 30

hours.

Estimated Time:1m 30s

Question 227Question →

For what positive value of the constant $k$ does the quadratic equation $x^2 - 2kx + 3k + 10 = 0$ have exactly one real solution?

Show answer & explanation

Answer: 5

Answer

The positive value of

k

is 5.

A quadratic equation has exactly one real solution when its discriminant

b^2 - 4ac

equals

0

. For

x^2 - 2kx + (3k + 10) = 0

, substituting

a = 1

,

b = -2k

, and

c = 3k + 10

gives

(-2k)^2 - 4(1)(3k + 10) = 0

, which simplifies to

4k^2 - 12k - 40 = 0

. Dividing the entire equation by

4

gives

k^2 - 3k - 10 = 0

. Factoring this quadratic yields

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

, giving solutions

k = 5

and

k = -2

. Since the question requires the positive value of

k

, the correct answer is

5

.

Step-by-Step Solution

1

Identify the condition for a quadratic equation to have exactly one real solution.

The discriminant must equal zero (

b^2 - 4ac = 0

).

A quadratic equation

ax^2 + bx + c = 0

has a single repeated real root if and only if its discriminant is zero.

2

Set up the discriminant equation using the coefficients from

x^2 - 2kx + (3k + 10) = 0

.

Discriminant

= (-2k)^2 - 4(1)(3k + 10) = 4k^2 - 12k - 40 = 0

.

Here

a = 1

,

b = -2k

, and

c = 3k + 10

.

3

Divide by 4 and solve the quadratic in terms of

k

by factoring.

k^2 - 3k - 10 = 0 \Rightarrow (k - 5)(k + 2) = 0

, so

k = 5

or

k = -2

.

Factoring the simplified quadratic yields the two potential values of

k

.

4

Choose the value of

k

that satisfies the constraint in the stem.

k = 5

.

The question asks specifically for the positive value of

k

, so

k = -2

is rejected.

Key Concept

Quadratic Discriminant and Factoring

Estimated Time:1m 30s

Question 228Question →

A digital publishing company uses two high-speed printing presses, Press P and Press Q. Press P prints pages at a constant rate that is $40\%$ faster than the rate of Press Q. If Press P and Press Q work simultaneously at their respective constant rates, they can complete a printing job of $18,000$ pages in $5$ hours. How many hours would it take Press Q, working alone at its constant rate, to complete a job of $15,000$ pages?

Show answer & explanation

Answer: 10

Answer

10

Working together, Press P and Press Q complete

18,000

pages in

5

hours, which corresponds to a combined rate of

3,600

pages per hour. Because Press P's rate is

1.4

times Press Q's rate, their combined rate is

2.4

times Press Q's rate. Dividing

3,600

by

2.4

gives Press Q's individual rate of

1,500

pages per hour. Finally, dividing

15,000

pages by

1,500

pages per hour yields

10

hours.

Step-by-Step Solution

1

Relate the rate of Press P to Press Q

r_P = 1.4 r_Q

Press P is 40% faster than Press Q, so its rate is

1 + 0.40 = 1.4

times the rate of Press Q.

2

Calculate the combined rate expression

r_{\text{combined}} = 2.4 r_Q

When working together, their rates add:

r_P + r_Q = 1.4 r_Q + r_Q = 2.4 r_Q

.

3

Solve for the rate of Press Q (

r_Q

)

r_Q = 1,500

pages per hour

Using

\text{Work} = \text{Rate} \times \text{Time}

, we have

18,000 = (2.4 r_Q) \times 5 = 12 r_Q

. Dividing

18,000

by

12

yields

r_Q = 1,500

.

4

Determine the time required for Press Q to complete

15,000

pages

10

hours

Dividing the target workload by Press Q's rate gives

\frac{15,000\text{ pages}}{1,500\text{ pages/hour}} = 10

hours.

Key Concept

Combined Work Rates and Direct Proportions

Estimated Time:1m 30s

Question 229Question →

When the algebraic expression $\frac{4x^3 - 16x}{2x^2 + 4x}$ is simplified for all $x \neq 0$ and $x \neq -2$ , it reduces to the linear polynomial $ax + b$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Show answer & explanation

Answer: -2

Answer

The correct numerical answer is -2.

Factoring the numerator yields

4x(x - 2)(x + 2)

and factoring the denominator yields

2x(x + 2)

. Canceling the common factors

2x

and

(x + 2)

leaves

2(x - 2) = 2x - 4

. Comparing

2x - 4

to

ax + b

gives

a = 2

and

b = -4

. Summing these values gives

a + b = 2 + (-4) = -2

.

Step-by-Step Solution

1

Factor out the greatest common factor and apply the difference of squares formula to the numerator.

4x^3 - 16x = 4x(x^2 - 4) = 4x(x - 2)(x + 2)

Fully factoring the numerator allows identification of all linear factors.

2

Factor out the greatest common factor from the denominator.

2x^2 + 4x = 2x(x + 2)

Extracting

2x

reveals the common terms shared with the numerator.

3

Divide the numerator by the denominator by canceling identical non-zero factors

2x

and

(x + 2)

.

\frac{4x(x - 2)(x + 2)}{2x(x + 2)} = \frac{4}{2}(x - 2) = 2(x - 2) = 2x - 4

Simplifying rational expressions requires canceling common factors present in both numerator and denominator.

4

Compare the simplified expression

2x - 4

to the form

ax + b

to find

a

and

b

, then compute their sum.

a = 2

and

b = -4 \implies a + b = 2 + (-4) = -2

Matching corresponding terms identifies the values of the target constants.

Key Concept

Simplifying rational expressions by factoring out common terms and applying the difference of squares identity.

Question 230Question →

If $a$ and $b$ are the two real solutions to the quadratic equation $2x^2 - 11x + 12 = 0$ , such that $a > b$ , what is the value of $a - 2b$ ?

Show answer & explanation

Answer: 1

Answer

The value of

a - 2b

is 1.

Factoring

2x^2 - 11x + 12 = 0

yields

(2x - 3)(x - 4) = 0

, giving solutions

x = 1.5

and

x = 4

. Given that

a > b

, we must set

a = 4

and

b = 1.5

. Substituting these values into

a - 2b

gives

4 - 2(1.5) = 1

.

Step-by-Step Solution

1

Factor the quadratic equation

2x^2 - 11x + 12 = 0

(2x - 3)(x - 4) = 0

Find two linear factors whose product expands to

2x^2 - 11x + 12

.

2

Find the roots of the equation

x = \frac{3}{2} = 1.5

and

x = 4

Apply the zero product property:

2x - 3 = 0 \Rightarrow x = 1.5

and

x - 4 = 0 \Rightarrow x = 4

.

3

Assign values to

a

and

b

based on the inequality

a > b

a = 4

and

b = 1.5

Since

4 > 1.5

,

a

must be 4 and

b

must be 1.5.

4

Evaluate the targeted expression

a - 2b

4 - 2(1.5) = 1

Substitute

a = 4

and

b = 1.5

into

a - 2b

.

Key Concept

Factoring Quadratic Equations

Question 231Question →

If $x$ and $y$ are non-zero real numbers such that $x \neq y$ , $\frac{x^2 - y^2}{x - y} = 12$ , and $\frac{x^2y - xy^2}{xy} = 4$ , what is the value of $x^2 + y^2$ ?

Show answer & explanation

Answer: 80

Answer

The value of

x^2 + y^2

is 80.

Simplifying the first equation by factoring the difference of squares

\frac{(x-y)(x+y)}{x-y} = 12

gives

x + y = 12

. Simplifying the second equation by factoring out

xy

yields

\frac{xy(x-y)}{xy} = 4

, which gives

x - y = 4

. Solving this system yields

x = 8

and

y = 4

. Squaring and summing these values gives

8^2 + 4^2 = 64 + 16 = 80

. Alternatively, using the algebraic identity

\frac{(x+y)^2 + (x-y)^2}{2} = \frac{12^2 + 4^2}{2} = \frac{144 + 16}{2} = 80

directly yields the correct answer.

Step-by-Step Solution

1

Simplify the first given algebraic expression using the difference of squares identity.

x + y = 12

Factoring

x^2 - y^2

gives

(x - y)(x + y)

. Since

x \neq y

, we can cancel the non-zero common factor

(x - y)

from the numerator and denominator.

2

Simplify the second given algebraic expression by factoring out the greatest common factor.

x - y = 4

Factoring

xy

from

x^2y - xy^2

yields

xy(x - y)

. Since

x, y \neq 0

, we cancel

xy

from the numerator and denominator.

3

Solve for the individual values of

x

and

y

.

x = 8

and

y = 4

Adding

(x + y = 12)

and

(x - y = 4)

gives

2x = 16

, so

x = 8

. Subtracting the equations gives

2y = 8

, so

y = 4

.

4

Calculate the target expression

x^2 + y^2

.

80

x^2 + y^2 = 8^2 + 4^2 = 64 + 16 = 80

.

Key Concept

Factoring Algebraic Expressions (Difference of Squares and GCF Extraction)

Answer

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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