All practice questions

Question 201Question →

A cyclist completes a journey consisting of three distinct segments: an uphill segment, a flat segment, and a downhill segment. The ratio of the distances of the uphill, flat, and downhill segments is $2 : 3 : 5$ , respectively. The cyclist's average speed on the flat segment is twice her average speed on the uphill segment, and her average speed on the downhill segment is three times her average speed on the uphill segment. If the cyclist's overall average speed for the entire journey is $30$ miles per hour, what is her average speed, in miles per hour, on the flat segment?

Show answer & explanation

Answer: 31

Answer

31

The correct average speed on the flat segment is 31 miles per hour. Setting up segment distances as

2x

,

3x

, and

5x

(total distance

10x

) and segment speeds as

v

,

2v

, and

3v

, the segment times are

t_1 = \frac{2x}{v}

,

t_2 = \frac{3x}{2v}

, and

t_3 = \frac{5x}{3v}

. The total travel time is

T = \frac{31x}{6v}

. Dividing total distance

10x

by total time

T

yields an overall average speed of

\frac{60v}{31} = 30

. Solving for

v

gives

v = 15.5

miles per hour. Thus, the average speed on the flat segment is

2v = 31

miles per hour.

Step-by-Step Solution

1

Define segment distances using ratio multipliers.

Distances are

d_1 = 2x

,

d_2 = 3x

, and

d_3 = 5x

, giving total distance

D = 10x

.

The distances of the three segments are in the ratio

2 : 3 : 5

.

2

Express segment speeds relative to the uphill speed

v

.

Uphill speed is

v

, flat speed is

2v

, and downhill speed is

3v

.

The problem states flat speed is twice uphill speed, and downhill speed is three times uphill speed.

3

Calculate the time spent on each segment.

t_1 = \frac{2x}{v}

,

t_2 = \frac{3x}{2v}

, and

t_3 = \frac{5x}{3v}

.

Time equals distance divided by speed (

t = \frac{d}{v}

).

4

Calculate total travel time by summing individual segment times.

T = \frac{x}{v} \left(2 + \frac{3}{2} + \frac{5}{3}\right) = \frac{31x}{6v}

.

Combining fractions with a common denominator of 6 gives

\frac{12 + 9 + 10}{6} = \frac{31}{6}

.

5

Relate total distance and total time to the overall average speed.

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{10x}{\frac{31x}{6v}} = \frac{60v}{31} = 30

.

Overall average speed is defined as total distance divided by total time.

6

Solve for

v

and calculate the flat segment speed

2v

.

v = 15.5

mph, so flat segment speed

= 2(15.5) = 31

mph.

Solving

\frac{60v}{31} = 30

yields

v = 15.5

, making

2v = 31

.

Key Concept

Weighted Average Speed and Multi-Segment Distance-Rate-Time Ratios

Question 202Question →

For all real numbers

x

such that

x \neq 3

and

x \neq -3

, the algebraic expression

\frac{\frac{x^4 - 81}{x^2 - 9} \cdot \frac{x^3 - 27}{x^2 + 3x + 9} - x(x - 3)^2}{x^2 - 9}

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

Show answer & explanation

Answer: 3

Answer

The simplified expression evaluates to the constant value

3

.

Factoring the numerator components using difference of squares and difference of cubes simplifies the product term to

x^3 - 3x^2 + 9x - 27

. Subtracting

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

simplifies the entire numerator to

3x^2 - 27 = 3(x^2 - 9)

. Dividing by the denominator

(x^2 - 9)

cancels out the variable terms entirely, yielding the constant value 3.

Step-by-Step Solution

1

Simplify the first rational component using the difference of squares identity

\frac{x^4 - 81}{x^2 - 9} = \frac{(x^2 - 9)(x^2 + 9)}{x^2 - 9} = x^2 + 9

Since

x \neq \pm 3

,

x^2 - 9 \neq 0

, allowing direct cancellation of

(x^2 - 9)

.

2

Simplify the second rational component using the difference of cubes identity

\frac{x^3 - 27}{x^2 + 3x + 9} = \frac{(x - 3)(x^2 + 3x + 9)}{x^2 + 3x + 9} = x - 3

Applying

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

where

a = x

and

b = 3

allows cancellation of the quadratic factor.

3

Multiply the simplified expressions

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

Distribute each term of the binomials to get the expanded polynomial.

4

Expand the subtracted term in the numerator

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

Expand

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

and distribute

x

.

5

Subtract the two expanded expressions to simplify the entire numerator

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

Combine like terms;

x^3

and

9x

terms cancel out, leaving

3x^2 - 27

.

6

Divide the simplified numerator by the main denominator

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

Cancel the common factor

(x^2 - 9)

from numerator and denominator.

Key Concept

Simplifying complex algebraic expressions via polynomial factoring (difference of squares and difference of cubes) and combining like terms.

Question 203Question →

If $x > 0$ and $x^2 - 6x - 16 = 0$ , what is the value of $x$ ?

Show answer & explanation

Answer: 8

Answer

The correct value of

x

is 8.

Factoring

x^2 - 6x - 16 = 0

gives

(x - 8)(x + 2) = 0

. Setting each factor to zero yields

x = 8

and

x = -2

. Given the constraint

x > 0

, the only valid answer is 8.

Step-by-Step Solution

1

Factor the quadratic expression

(x - 8)(x + 2) = 0

Find two numbers that multiply to

-16

and add to

-6

, which are

-8

and

2

.

2

Solve for possible values of

x

x = 8

or

x = -2

By the zero-product property, if

(x - 8)(x + 2) = 0

, then either

x - 8 = 0

or

x + 2 = 0

.

3

Apply the given constraint

x > 0

x = 8

The solution

x = -2

is rejected because

x

must be positive.

Key Concept

Factoring Quadratic Equations

Estimated Time:45s

Question 204Question →

In a regular polygon, the ratio of the measure of an interior angle to the measure of an exterior angle is $7:2$ . What is the total number of diagonals of this polygon?

Show answer & explanation

Answer: 27

Answer

The total number of diagonals of the regular polygon is

27

.

The correct answer is

27

. An interior angle and an exterior angle of a polygon are supplementary (

180^\circ

). Given the ratio

7:2

, the exterior angle is

\frac{2}{9} \times 180^\circ = 40^\circ

. Since the sum of exterior angles of any convex polygon is

360^\circ

, the number of sides is

n = \frac{360^\circ}{40^\circ} = 9

. Using the formula for the number of diagonals,

\frac{n(n-3)}{2}

, we obtain

\frac{9(9-3)}{2} = 27

.

Step-by-Step Solution

1

Determine the measure of the exterior angle using the given interior-to-exterior ratio.

Exterior angle measure =

40^\circ

At any vertex of a polygon, the interior angle and exterior angle sum to

180^\circ

. With a ratio of

7:2

, the exterior angle represents

\frac{2}{7+2} = \frac{2}{9}

of the total

180^\circ

.

2

Calculate the number of sides (

n

) of the regular polygon.

n = 9

The sum of the exterior angles of any convex polygon is

360^\circ

. Since all exterior angles in a regular polygon are equal,

n = \frac{360^\circ}{40^\circ} = 9

.

3

Calculate the total number of diagonals using the formula

\frac{n(n-3)}{2}

.

Number of diagonals =

27

Substituting

n = 9

into

\frac{n(n-3)}{2}

yields

\frac{9 \times 6}{2} = 27

.

Key Concept

Interior and exterior angle properties of regular polygons, and the diagonal counting formula for convex polygons.

Question 205Question →

On a number line, point $A$ is located at $10$ and point $B$ is located at $2$ . If point $P$ , with coordinate $x > 0$ , is three times as far from point $A$ as it is from point $B$ , what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

4

The distance from

P(x)

to

A(10)

is

|x - 10|

and to

B(2)

is

|x - 2|

. Setting

|x - 10| = 3|x - 2|

gives two equations:

x - 10 = 3x - 6

, which yields

x = -2

, and

x - 10 = -3x + 6

, which yields

x = 4

. Since

x > 0

, the correct value is 4.

Step-by-Step Solution

1

Formulate the distance relationship using absolute value notation.

|x - 10| = 3|x - 2|

The distance between two points

u

and

v

on the real number line is given by

|u - v|

.

2

Split the absolute value equation into two linear equations representing possible cases.

x - 10 = 3(x - 2)

or

x - 10 = -3(x - 2)

The equality

|a| = |b|

implies

a = b

or

a = -b

.

3

Solve each case algebraically.

Case 1 gives

x - 10 = 3x - 6 \implies 2x = -4 \implies x = -2

. Case 2 gives

x - 10 = -3x + 6 \implies 4x = 16 \implies x = 4

.

Standard linear equation solving.

4

Apply the given domain condition

x > 0

.

x = 4

The solution

x = -2

is negative and therefore violates the condition

x > 0

.

Key Concept

Distance on a number line represented by absolute value equations

Question 206Question →

A research laboratory conducted $6$ experimental trials to measure the duration, in milliseconds, of a specific chemical reaction. The durations recorded for $5$ of the trials were $240$ , $215$ , $260$ , $225$ , and $245$ . If the median duration of all $6$ trials was $235$ milliseconds, what was the duration, in milliseconds, of the 6th trial?

Show answer & explanation

Answer: 230

Answer

230

For a dataset with 6 numbers, the median is the arithmetic mean of the 3rd and 4th numbers in ascending order. Arranging the 5 given numbers gives

215, 225, 240, 245, 260

. Since the target median is

235

, the sum of the two middle numbers must be

235 \times 2 = 470

. Placing

x = 230

into the dataset yields the ordered set

215, 225, 230, 240, 245, 260

, where the 3rd and 4th numbers are

230

and

240

. Their mean is

(230 + 240) / 2 = 235

, which matches the given median.

Step-by-Step Solution

1

Order the 5 given numbers from least to greatest

The sorted list of known values is

215, 225, 240, 245, 260

.

Calculating or using median requires ordering the data points.

2

Express the median condition for an even number of data points (

n = 6

)

\text{Median} = \frac{\text{3rd value} + \text{4th value}}{2} = 235

, so

\text{3rd value} + \text{4th value} = 470

.

For an even number of values, the median is the average of the two middle numbers.

3

Determine the position and value of the unknown 6th trial

x

If

x \le 225

, the 3rd and 4th values would be

225

and

240

(median

232.5

). If

x \ge 245

, the 3rd and 4th values would be

240

and

245

(median

242.5

). Thus,

x

must lie between

225

and

240

.

Analyzing boundary conditions places

x

as the 3rd value and

240

as the 4th value.

4

Solve for

x

x + 240 = 470 \implies x = 230

.

The sum of the two middle values must equal

470

to yield a median of

235

.

Key Concept

Finding a missing value in a dataset given the median of an even number of observations.

Question 207Question →

A manufacturing plant operates two assembly lines, Line X and Line Y. Line X produces 15 units per hour, and Line Y produces 22 units per hour. On a certain day, Line X operated for 3 hours longer than Line Y did, and the two lines produced a total of 415 units. How many hours did Line Y operate?

Show answer & explanation

Answer: 10

Answer

10

Let

h

represent the number of hours Line Y operated. Because Line X operated for 3 hours longer than Line Y, Line X operated for

h + 3

hours. The total number of units produced by both lines is the sum of their individual outputs:

15(h + 3) + 22h = 415

. Distributing 15 yields

15h + 45 + 22h = 415

. Combining like terms gives

37h + 45 = 415

. Subtracting 45 from both sides yields

37h = 370

, and dividing by 37 gives

h = 10

. Therefore, Line Y operated for 10 hours.

Step-by-Step Solution

1

Define the variable representing Line Y's operating time in hours.

Let

h

represent the number of hours Line Y operated. Line X's operating time is

h + 3

hours.

Line X operated for 3 hours longer than Line Y.

2

Formulate a linear equation in one variable for total units produced.

15(h + 3) + 22h = 415

Total production is the sum of production from Line X (

15 \times (h + 3)

) and Line Y (

22 \times h

).

3

Distribute and combine like terms on the left side of the equation.

15h + 45 + 22h = 415 \implies 37h + 45 = 415

Apply the distributive property and combine variable terms.

4

Isolate the variable

h

.

37h = 370 \implies h = 10

Subtract 45 from both sides of the equation and divide by 37.

Key Concept

Linear Equations in One Variable

Estimated Time:1m 30s

Question 208Question →

The prime factorizations of two positive integers $A$ and $B$ are given by $A = 2^a \times 3^5 \times 5^b$ and $B = 2^4 \times 3^c \times 7^2$ , where $a$ , $b$ , and $c$ are positive integers. If the greatest common divisor of $A$ and $B$ is $\gcd(A, B) = 2^2 \times 3^3$ , and their least common multiple is \text{lcm} $(A, B) = 2^4 \times 3^5 \times 5^3 \times 7^2$ , what is the value of $a + b + c$ ?

Show answer & explanation

Answer: 8

Answer

The correct answer is 8.

For any two positive integers expressed in prime factorized form, the greatest common divisor contains each prime factor raised to the minimum of its exponents in the two numbers, while the least common multiple contains each prime factor raised to the maximum of its exponents. For prime factor 2, the GCD has exponent 2, so min(a, 4) = 2, giving a = 2. For prime factor 3, the GCD has exponent 3, so min(5, c) = 3, giving c = 3. For prime factor 5, the LCM has exponent 3, so max(b, 0) = 3, giving b = 3. Adding these values together yields a + b + c = 2 + 3 + 3 = 8.

Step-by-Step Solution

1

Analyze the prime factor

2

min(a, 4) = 2, so a = 2

The greatest common divisor takes the minimum exponent for each prime factor shared between A and B.

2

Analyze the prime factor

3

min(5, c) = 3, so c = 3

The exponent of 3 in the GCD is 3, which must equal the smaller of the two exponents 5 and c.

3

Analyze the prime factor

5

max(b, 0) = 3, so b = 3

The least common multiple takes the maximum exponent for each prime factor present in either number.

4

Calculate the sum

a + b + c

2 + 3 + 3 = 8

Add the solved values of the three unknown prime exponents.

Key Concept

Relating prime factor exponents to GCD (minimum exponent) and LCM (maximum exponent)

Question 209Question →

If

x

is a positive integer such that

\frac{4^x + 4^x + 4^x + 4^x}{2^x + 2^x} = 512

what is the value of

x

?

Show answer & explanation

Answer: 8

Answer

The value of

x

is 8.

Combining four terms of

4^x

yields

4 \cdot 4^x = 4^{x+1} = 2^{2x+2}

. Combining two terms of

2^x

yields

2 \cdot 2^x = 2^{x+1}

. Dividing the numerator by the denominator gives

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

. Since

512 = 2^9

, setting

2^{x+1} = 2^9

gives

x + 1 = 9

, which leads directly to

x = 8

.

Step-by-Step Solution

1

Simplify the numerator by combining identical added terms.

The numerator

4^x + 4^x + 4^x + 4^x

equals

4 \cdot 4^x

, which simplifies to

4^{x+1}

.

Adding four identical quantities is equivalent to multiplying that quantity by 4.

2

Simplify the denominator by combining identical added terms.

The denominator

2^x + 2^x

equals

2 \cdot 2^x

, which simplifies to

2^{x+1}

.

Adding two identical quantities is equivalent to multiplying that quantity by 2.

3

Convert the numerator to base 2 and simplify the fraction.

Since

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

, the fraction becomes

\frac{2^{2x+2}}{2^{x+1}} = 2^{(2x+2)-(x+1)} = 2^{x+1}

.

Converting all terms to a common base allows using the exponent quotient rule

a^m / a^n = a^{m-n}

.

4

Solve for

x

by equating the simplified power to 512.

Setting

2^{x+1} = 512 = 2^9

yields

x + 1 = 9

, so

x = 8

.

When exponential expressions with the same positive base (other than 1) are equal, their exponents must be equal.

Key Concept

Combining repeated addition of exponential terms and converting powers to a common base using exponent laws (

a^m \cdot a^n = a^{m+n}

and

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

).

Question 210Question →

A chemical processing tank receives two liquid solutions, Solution $X$ and Solution $Y$ , from separate inlet pipes.

- Solution $X$ contains chemical $A$ and water in a volume ratio of $3:2$ and enters the tank at a constant rate of $150$ liters per hour.
- Solution $Y$ contains chemical $A$ and water in a volume ratio of $1:4$ and enters the tank at a constant rate of $250$ liters per hour.

Both inlet pipes run simultaneously into an initially empty tank for $4$ hours. After $4$ hours, the inlet pipes are shut off. To adjust the mixture, pure chemical $A$ is added to the tank at a constant rate of $50$ liters per hour, while water is continuously drained from the tank at a constant rate of $30$ liters per hour.

How many hours must this adjustment process run until the volume of chemical $A$ in the tank is equal to the volume of water in the tank?

Show answer & explanation

Answer: 6

Answer

6

To find the time

t

when the volumes of chemical A and water in the tank are equal, first determine the initial quantities contributed by both solutions during the 4-hour filling period. Solution X provides

150 \times 4 = 600

liters total, containing

\frac{3}{5} \times 600 = 360

liters of chemical A and

\frac{2}{5} \times 600 = 240

liters of water. Solution Y provides

250 \times 4 = 1000

liters total, containing

\frac{1}{5} \times 1000 = 200

liters of chemical A and

\frac{4}{5} \times 1000 = 800

liters of water. Adding these amounts yields

360 + 200 = 560

liters of chemical A and

240 + 800 = 1040

liters of water. In the adjustment phase of

t

hours, chemical A increases at

50

L/hr to

560 + 50t

, while water decreases at

30

L/hr to

1040 - 30t

. Equating the two expressions gives

560 + 50t = 1040 - 30t

, which simplifies to

80t = 480

, resulting in

t = 6

hours.

Step-by-Step Solution

1

Determine the volumes of chemical A and water supplied by Solution X during the first 4 hours.

Solution X delivers 600 liters in total, consisting of 360 liters of chemical A and 240 liters of water.

Solution X flows at 150 L/hr for 4 hours (150 * 4 = 600 L) with a 3:2 chemical A to water ratio, meaning chemical A represents 3/5 of the total volume and water represents 2/5.

2

Determine the volumes of chemical A and water supplied by Solution Y during the first 4 hours.

Solution Y delivers 1000 liters in total, consisting of 200 liters of chemical A and 800 liters of water.

Solution Y flows at 250 L/hr for 4 hours (250 * 4 = 1000 L) with a 1:4 chemical A to water ratio, meaning chemical A represents 1/5 of the total volume and water represents 4/5.

3

Calculate the total initial quantities of chemical A and water present in the tank prior to the adjustment phase.

Total chemical A = 560 liters; Total water = 1040 liters.

Sum the quantities from both solutions: Chemical A = 360 + 200 = 560 L; Water = 240 + 800 = 1040 L.

4

Formulate linear expressions representing the total volume of chemical A and water after t hours of adjustment.

Chemical A volume = 560 + 50t; Water volume = 1040 - 30t.

Pure chemical A is added at 50 L/hr, increasing its total volume, while water is drained at 30 L/hr, reducing its total volume.

5

Set the two component volume expressions equal to each other and solve for t.

t = 6 hours.

Solving 560 + 50t = 1040 - 30t leads to 80t = 480, which yields t = 6.

Key Concept

Multi-stream mixture rate integration and ratio equality modeling

Question 211Question →

Two software security tools, Tool X and Tool Y, operate independently to scan code repositories for vulnerabilities. The probability that Tool X detects a specific type of security flaw is $0.80$ , and the probability that Tool Y detects the same flaw is $0.75$ . What is the probability that exactly one of the two tools detects the flaw?

Show answer & explanation

Answer: 0.35

Answer

The probability that exactly one of the two tools detects the flaw is

0.35

.

The scenario requires finding the probability that exactly one tool detects the flaw. For independent events

X

and

Y

, 'exactly one' consists of two mutually exclusive events: (1) Tool X succeeds while Tool Y fails, which has probability

0.80 \times (1 - 0.75) = 0.80 \times 0.25 = 0.20

, and (2) Tool Y succeeds while Tool X fails, which has probability

0.75 \times (1 - 0.80) = 0.75 \times 0.20 = 0.15

. Summing these mutually exclusive probabilities gives

0.20 + 0.15 = 0.35

. Alternatively, one can subtract the probability of both tools succeeding (

0.80 \times 0.75 = 0.60

) from the probability of at least one tool succeeding (

0.80 + 0.75 - 0.60 = 0.95

), yielding

0.95 - 0.60 = 0.35

.

Step-by-Step Solution

1

Determine the complement probabilities for each independent tool failing to detect the flaw.

P(\text{Not X}) = 1 - 0.80 = 0.20

and

P(\text{Not Y}) = 1 - 0.75 = 0.25

.

The probability of an event not occurring is equal to 1 minus the probability that it occurs.

2

Calculate the joint probability of Tool X detecting the flaw and Tool Y failing to detect it.

P(\text{X and Not Y}) = 0.80 \times 0.25 = 0.20

.

Because the tools operate independently, the joint probability is the product of their individual probabilities.

3

Calculate the joint probability of Tool Y detecting the flaw and Tool X failing to detect it.

P(\text{Y and Not X}) = 0.75 \times 0.20 = 0.15

.

Tool independence allows multiplying the individual probabilities of detection and non-detection.

4

Sum the probabilities of the two mutually exclusive scenarios representing 'exactly one tool detects the flaw'.

P(\text{Exactly One}) = 0.20 + 0.15 = 0.35

.

The events 'X only' and 'Y only' cannot happen simultaneously, so their probabilities add directly.

Key Concept

Independence and Mutual Exclusivity Rules in Compound Probability

Question 212Question →

A venue offers two types of event packages: Standard and Deluxe. The total cost of 3 Standard packages and 2 Deluxe packages is $410. The total cost of 2 Standard packages and 5 Deluxe packages is$ 640. What is the cost, in dollars, of 1 Deluxe package?

Show answer & explanation

Answer: 100

Answer

100

Setting up the linear system

3S + 2D = 410

and

2S + 5D = 640

allows us to multiply the equations by 2 and 3 respectively, obtaining

6S + 4D = 820

and

6S + 15D = 1920

. Subtracting the two equations eliminates

S

and gives

11D = 1100

, leading to

D = 100

.

Step-by-Step Solution

1

Define variables and establish the system of linear equations.

Let

S

represent the cost of a Standard package and

D

represent the cost of a Deluxe package.
Equation 1:

3S + 2D = 410

Equation 2:

2S + 5D = 640

Translating the scenario into mathematical equations forms a 2x2 system of linear equations.

2

Use elimination to eliminate variable

S

.

Multiply Equation 1 by 2:

6S + 4D = 820

Multiply Equation 2 by 3:

6S + 15D = 1920

Creating matching coefficients for

S

allows elimination by subtraction.

3

Subtract the transformed equations and solve for

D

.

(6S + 15D) - (6S + 4D) = 1920 - 820 \implies 11D = 1100 \implies D = 100

Subtracting cancels out

S

, leaving a single linear equation in terms of

D

.

Key Concept

Solving 2x2 Systems of Linear Equations via Elimination

Alternative Method

Use the substitution method: Solve for

S

in terms of

D

from the first equation (

S = \frac{410 - 2D}{3}

) and substitute this into the second equation (

2\left(\frac{410 - 2D}{3}\right) + 5D = 640

). Multiplying both sides by 3 yields

820 - 4D + 15D = 1920

, which simplifies to

11D = 1100

, so

D = 100

.

Estimated Time:1m 30s

Question 213Question →

A data processing center uses two server clusters, Cluster $X$ and Cluster $Y$ , operating at constant individual processing rates. The ratio of the rate of Cluster $X$ to the rate of Cluster $Y$ is $3 : 5$ . Cluster $X$ alone can process a standard dataset of size $D$ gigabytes in 20 hours.

If Cluster $X$ and Cluster $Y$ work together for 4 hours at their initial rates, and then Cluster $X$ 's rate is increased by $33\frac{1}{3}\%$ while Cluster $Y$ 's rate is increased by $20\%$ , how many additional hours will it take for the two clusters working together at their new rates to complete the remaining portion of dataset $D$ ?

Show answer & explanation

Answer: 2.8

Answer

It will take 2.8 additional hours for the two clusters working together at their new rates to complete the remaining portion of dataset

D

.

Representing Cluster

X

's rate as

3k

and Cluster

Y

's rate as

5k

establishes the dataset size

D = 20 \times 3k = 60k

. During the first 4 hours, both clusters process

4 \times (3k + 5k) = 32k

GB, leaving

28k

GB remaining. After rate increases, Cluster

X

's rate becomes

4k

and Cluster

Y

's rate becomes

6k

, resulting in a new combined rate of

10k

. Dividing the remaining

28k

GB by

10k

GB/hr yields

2.8

hours.

Step-by-Step Solution

1

Define variables for the initial rates and dataset size based on the given ratio.

Let the processing rate of Cluster

X

be

r_X = 3k

GB/hr and Cluster

Y

be

r_Y = 5k

GB/hr for some constant

k > 0

. Since Cluster

X

alone completes dataset

D

in 20 hours,

D = 3k \times 20 = 60k

GB.

Relating the ratio of individual rates to total work defines all quantities in terms of a single parameter

k

.

2

Calculate the amount of work finished during the initial joint operation.

The combined initial rate is

r_X + r_Y = 3k + 5k = 8k

GB/hr. Working together for 4 hours completes

8k \times 4 = 32k

GB.

Working simultaneously means their processing rates add together.

3

Find the remaining work and the updated processing rates after adjustments.

Remaining dataset volume =

60k - 32k = 28k

GB. Cluster

X

's new rate =

3k \times \left(1 + \frac{1}{3}\right) = 4k

GB/hr. Cluster

Y

's new rate =

5k \times 1.20 = 6k

GB/hr. New combined rate =

4k + 6k = 10k

GB/hr.

Modifying the individual rates changes the combined throughput for the remaining task.

4

Compute the additional time required to process the remaining dataset.

Additional time =

\frac{28k \text{ GB}}{10k \text{ GB/hr}} = 2.8

hours.

Dividing the remaining work volume by the new combined rate gives the exact required time.

Key Concept

Combined work rates, ratio proportionality, and percentage rate adjustments.

Question 214Question →

Three consecutive integers $a$ , $b$ , and $c$ satisfy $a < b < c$ . If $a + b + c = -9$ and $a \cdot b \cdot c < 0$ , what is the value of $(-1)^a + (-1)^b + (-1)^c$ ?

Show answer & explanation

Answer: 1

Answer

The value of the expression is 1.

The sum of three consecutive integers

a + b + c = 3b = -9

determines

b = -3

, making

a = -4

and

c = -2

. The product

(-4)(-3)(-2) = -24

is negative, confirming the given condition. Applying exponent sign rules, raising

-1

to an even integer power yields

1

, while raising

-1

to an odd integer power yields

-1

. Thus,

(-1)^{-4} = 1

,

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

, and

(-1)^{-2} = 1

. Summing these three terms gives

1 + (-1) + 1 = 1

.

Step-by-Step Solution

1

Find the values of integers

a

,

b

, and

c

.

a = -4

,

b = -3

,

c = -2

Three consecutive integers centered at

b

sum to

3b = -9

, so

b = -3

.

2

Check the sign condition of the product

a \cdot b \cdot c

.

(-4)(-3)(-2) = -24 < 0

The product of three negative numbers is negative.

3

Evaluate

(-1)^n

for each integer power.

(-1)^{-4} = 1

,

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

,

(-1)^{-2} = 1

Negative one raised to an even integer power is 1; raised to an odd integer power is -1.

4

Sum the three evaluated terms.

1 + (-1) + 1 = 1

Addition of the resulting values.

Key Concept

Even-odd exponent rules for negative bases and sign rules for product of signed integers.

Question 215Question →

The annual snowfall totals in a high-altitude meteorological district are normally distributed with a mean of $140$ inches and a standard deviation of $12$ inches. According to the 68–95–99.7 empirical rule for normal distributions, what percent of the years have an annual snowfall between $116$ inches and $152$ inches?

Show answer & explanation

Answer: 81.5

Answer

81.5%

To find the percentage of data between

116

inches and

152

inches, first calculate the standard deviation distances (

z

-scores) from the mean of

140

inches.

116

inches is

24

inches below the mean, which corresponds to

z = -2

.

152

inches is

12

inches above the mean, which corresponds to

z = +1

. According to the 68–95–99.7 empirical rule,

95\%

of the data lies within

2

standard deviations of the mean, meaning

47.5\%

lies between

z = -2

and

z = 0

. Similarly,

68\%

of the data lies within

1

standard deviation of the mean, meaning

34\%

lies between

z = 0

and

z = +1

. Summing these two symmetric halves gives

47.5\% + 34\% = 81.5\%

.

Step-by-Step Solution

1

Convert the boundary values (

116

inches and

152

inches) into standard z-scores.

z_{lower} = \frac{116 - 140}{12} = -2

and

z_{upper} = \frac{152 - 140}{12} = +1

Standardizing raw values into z-scores allows the application of standard normal distribution properties.

2

Apply the 68–95–99.7 empirical rule to split the area relative to the mean (

z = 0

).

Area from

z = -2

to

z = 0

is

47.5\%

; Area from

z = 0

to

z = +1

is

34\%

.

The normal curve is symmetrical around the mean. Thus,

95\%

between

-2\sigma

and

+2\sigma

yields

47.5\%

below the mean, and

68\%

between

-1\sigma

and

+1\sigma

yields

34\%

above the mean.

3

Sum the percentages of the two disjoint regions bounded by

z = -2

and

z = +1

.

47.5\% + 34\% = 81.5\%

Combining the area below the mean and the area above the mean gives the total percentage of values falling within the specified interval.

Key Concept

Normal distribution empirical rule (68–95–99.7 rule) with asymmetric standard deviation boundaries

Estimated Time:1m 30s

Question 216Question →

A startup hired two freelance software developers, Developer A and Developer B, for a combined total of $45$ hours on a single project. Developer A charges $\$65$ per hour, and Developer B charges $\$80$ per hour. If the total amount paid to both developers was $\$3{}225$ , how many hours did Developer A work on the project?

Show answer & explanation

Answer: 25

Answer

Developer A worked on the project for 25 hours.

Let

x

be the hours Developer A worked and

y

be the hours Developer B worked. From

x + y = 45

, we get

y = 45 - x

. Substituting into

65x + 80y = 3225

yields

65x + 80(45 - x) = 3225

. Expanding gives

65x + 3600 - 80x = 3225

, which simplifies to

-15x = -375

, so

x = 25

.

Step-by-Step Solution

1

Set up a system of linear equations representing the total hours and total cost.

x + y = 45

and

65x + 80y = 3225

, where

x

is Developer A's hours and

y

is Developer B's hours.

Translating the verbal conditions into mathematical equations.

2

Express

y

in terms of

x

from the hours equation.

y = 45 - x

Prepares the linear system for substitution.

3

Substitute

y = 45 - x

into the total cost equation and solve for

x

.

65x + 80(45 - x) = 3225 \implies -15x = -375 \implies x = 25

Solves for the requested variable

x

directly.

Key Concept

Solving word problems using 2x2 systems of linear equations via substitution or elimination.

Question 217Question →

A bakery packages two types of gift baskets containing gourmet croissants and blueberry muffins. Basket X contains $4$ croissants and $3$ muffins, with a total production cost of $\$19.00$ . Basket Y contains $2$ croissants and $5$ muffins, with a total production cost of $\$16.50$ . Assuming the cost per croissant and the cost per muffin are constant across all baskets, what is the production cost, in dollars, of a single croissant?

Show answer & explanation

Answer: 3.25

Answer

The production cost of a single croissant is 3.25 dollars.

Let

c

represent the cost of a croissant and

m

represent the cost of a muffin. The given situation translates to the system of equations

4c + 3m = 19.00

and

2c + 5m = 16.50

. Multiplying the second equation by

2

gives

4c + 10m = 33.00

. Subtracting

4c + 3m = 19.00

from

4c + 10m = 33.00

results in

7m = 14.00

, which gives

m = 2.00

. Substituting

m = 2.00

into

2c + 5(2.00) = 16.50

yields

2c + 10 = 16.50

, so

2c = 6.50

and

c = 3.25

. Thus, a single croissant costs $3.25 dollars.

Step-by-Step Solution

1

Set up a system of linear equations

4c + 3m = 19.00

and

2c + 5m = 16.50

Translate the contents and costs of Basket X and Basket Y into algebraic equations where

c

is the price of a croissant and

m

is the price of a muffin.

2

Eliminate variable c

4c + 10m = 33.00

, then subtracting

4c + 3m = 19.00

yields

7m = 14.00

, so

m = 2.00

Multiplying the second equation by 2 aligns the coefficients of

c

, allowing elimination by subtraction.

3

Solve for variable c

2c + 5(2.00) = 16.50 \implies 2c = 6.50 \implies c = 3.25

Substitute the value found for

m

back into one of the original linear equations to calculate the cost of a croissant.

Key Concept

Solving Systems of Linear Equations via Elimination

Alternative Method

Express

c

in terms of

m

using the second equation:

c = 8.25 - 2.5m

. Substitute this expression into the first equation:

4(8.25 - 2.5m) + 3m = 19.00 \implies 33 - 10m + 3m = 19.00 \implies -7m = -14.00 \implies m = 2.00

. Finally, calculate

c = 8.25 - 2.5(2.00) = 3.25

.

Estimated Time:1m 30s

Question 218Question →

A logistics facility utilizes two automated sorting lines, Line A and Line B, to process incoming shipments. Line A processes shipments at a constant rate that is 20 percent greater than the constant rate of Line B. When both lines operate simultaneously, they process a combined total of 3,300 shipments in 3 hours. Working alone at its constant rate, how many hours will it take Line B to process 2,250 shipments?

Show answer & explanation

Answer: 4.5

Answer

4.5 hours

Let

r

represent the rate of Line B in shipments per hour. Because Line A processes 20% faster than Line B, Line A's rate is

1.20r

. Together, their combined processing rate is

r + 1.20r = 2.20r

shipments per hour. Operating for 3 hours, the total shipments processed is

3 \times 2.20r = 6.60r = 3,300

. Solving for

r

gives

r = 500

shipments per hour. To process 2,250 shipments alone, Line B requires

\frac{2,250}{500} = 4.5

hours.

Step-by-Step Solution

1

Define the relationship between the individual processing rates.

If Line B processes at rate

r

shipments per hour, Line A processes at

1.20r

shipments per hour.

Line A's rate is 20 percent greater than Line B's rate.

2

Find the combined processing rate.

Combined rate =

r + 1.20r = 2.20r

shipments per hour.

When working together, individual rates add up.

3

Determine Line B's rate (

r

).

3 \times 2.20r = 3,300 \implies 6.60r = 3,300 \implies r = 500

shipments per hour.

Total work equals combined rate multiplied by time.

4

Calculate the time required for Line B to complete 2,250 shipments.

\text{Time} = \frac{2,250}{500} = 4.5

hours.

Time taken equals total work divided by the individual rate.

Key Concept

Combined Rates and Ratio Relationships

Question 219Question →

For pairwise distinct real numbers $x$ , $y$ , and $z$ , consider the algebraic expression:

E(x, y, z) = \frac{(x^2 - y^2)^3 + (y^2 - z^2)^3 + (z^2 - x^2)^3}{(x - y)^3 + (y - z)^3 + (z - x)^3}

If $x = 5$ , $y = 3$ , and $z = 1$ , what is the numerical value of $E(5, 3, 1)$ ?

Show answer & explanation

Answer: 192

Answer

192

Using the identity that

a^3 + b^3 + c^3 = 3abc

whenever

a + b + c = 0

, both the numerator and denominator can be factored directly. Factoring the difference of squares in the numerator yields

3(x - y)(x + y)(y - z)(y + z)(z - x)(z + x)

. Dividing this by the factored denominator

3(x - y)(y - z)(z - x)

simplifies the expression to

(x + y)(y + z)(z + x)

. Substituting

x = 5

,

y = 3

, and

z = 1

yields

(8)(4)(6) = 192

.

Step-by-Step Solution

1

Use the conditional cubic identity

a + b + c = 0 \implies a^3 + b^3 + c^3 = 3abc

on the denominator.

Denominator becomes

3(x - y)(y - z)(z - x)

.

The sum of the three terms

(x - y) + (y - z) + (z - x)

equals 0.

2

Apply the same identity to the numerator.

Numerator becomes

3(x^2 - y^2)(y^2 - z^2)(z^2 - x^2)

.

The sum of the squared difference terms

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

also equals 0.

3

Factor each difference of squares in the numerator.

Numerator becomes

3(x - y)(x + y)(y - z)(y + z)(z - x)(z + x)

.

Using the difference of squares identity

u^2 - v^2 = (u - v)(u + v)

on each term.

4

Simplify the fraction by dividing the common factors in the numerator and denominator.

E(x, y, z) = (x + y)(y + z)(z + x)

.

The factors

3

,

(x - y)

,

(y - z)

, and

(z - x)

cancel out completely.

5

Evaluate the simplified product for

x = 5

,

y = 3

, and

z = 1

.

(5 + 3)(3 + 1)(1 + 5) = 8 \times 4 \times 6 = 192

.

Direct evaluation after algebraic simplification.

Key Concept

Simplifying rational expressions involving sum of cubes identity

a^3 + b^3 + c^3 = 3abc

when

a + b + c = 0

and difference of squares factoring.

Question 220Question →

If $n$ is a real number such that $\frac{27^n + 27^n + 27^n}{3^{n+2}} = 243$ , what is the value of $n$ ?

Show answer & explanation

Answer: 3

Answer

3

Rewriting

27^n + 27^n + 27^n

as

3 \cdot (3^3)^n = 3^{3n+1}

allows the left-hand side to simplify to

\frac{3^{3n+1}}{3^{n+2}} = 3^{2n-1}

. Equating this to

243 = 3^5

gives

2n - 1 = 5

, which solves to

n = 3

.

Step-by-Step Solution

1

Express repeated addition in the numerator as multiplication.

27^n + 27^n + 27^n = 3 \cdot 27^n

Adding three identical quantities is equivalent to multiplying one quantity by 3.

2

Convert base 27 to base 3 and apply exponent multiplication.

3 \cdot (3^3)^n = 3^1 \cdot 3^{3n} = 3^{3n+1}

Since

27 = 3^3

, using the power rule

(a^b)^c = a^{bc}

and product rule

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

converts the numerator to a single power of 3.

3

Simplify the fraction using the quotient rule of exponents.

\frac{3^{3n+1}}{3^{n+2}} = 3^{(3n+1) - (n+2)} = 3^{2n-1}

Dividing exponential terms with the same base requires subtracting the exponent in the denominator from the exponent in the numerator.

4

Rewrite 243 with base 3 and equate exponents across the equal sign.

3^{2n-1} = 3^5 \implies 2n - 1 = 5

Since

243 = 3^5

, two exponential expressions with the same base are equal if and only if their exponents are equal.

5

Solve the linear equation for

n

.

2n = 6 \implies n = 3

Adding 1 to both sides yields

2n = 6

, and dividing by 2 yields

n = 3

.

Key Concept

Combining repeated addition of exponential terms and converting expressions to a common base using exponent rules.

Answer

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