All practice questions

Question 201Question →

The quadratic equation $x^2 - 6x + a = 0$ has two distinct real roots $\alpha$ and $\gamma$ , and the quadratic equation $y^2 - by + 21 = 0$ has two distinct real roots $\beta$ and $\delta$ . If $\alpha < \beta < \gamma < \delta$ and the four roots form an arithmetic progression in that order, what is the value of $a + b$ ?

Show answer & explanation

Answer: 15

Answer

The value of

a + b

is 15.

By representing the four ordered roots as

r, r+d, r+2d, r+3d

, Vieta's formula for the sum of roots of the first equation yields

r + (r+2d) = 2(r+d) = 6

, which implies

r+d = 3

. Thus, the second root is

\beta = 3

. Using the product of roots for the second equation,

3\delta = 21

gives

\delta = 7

. The common difference is

d = (7-3)/2 = 2

, which gives the roots

1, 3, 5, 7

. Finally,

a = 1 \times 5 = 5

and

b = 3 + 7 = 10

, so

a + b = 15

.

Step-by-Step Solution

1

Apply Vieta's formulas to both quadratic equations.

\alpha + \gamma = 6

,

\alpha\gamma = a

,

\beta + \delta = b

, and

\beta\delta = 21

.

Vieta's relations connect the coefficients of a quadratic polynomial to the sum and product of its roots.

2

Set up the arithmetic progression representation for the roots.

\alpha = r

,

\beta = r + d

,

\gamma = r + 2d

, and

\delta = r + 3d

, where

d > 0

.

The roots form an increasing arithmetic progression in the order

\alpha, \beta, \gamma, \delta

.

3

Substitute the expressions for

\alpha

and

\gamma

into

\alpha + \gamma = 6

.

r + (r + 2d) = 2r + 2d = 2(r + d) = 6 \implies r + d = 3

.

Combining terms simplifies the sum of the first and third terms of the arithmetic progression.

4

Identify the value of

\beta

and solve for

\delta

.

\beta = r + d = 3

, so

\beta\delta = 21 \implies 3\delta = 21 \implies \delta = 7

.

Since

\beta = r + d

, its value is directly determined as 3, allowing

\delta

to be solved from the product relation.

5

Calculate the common difference

d

and the first term

r

.

\delta - \beta = 2d = 7 - 3 = 4 \implies d = 2

, and

r = 3 - 2 = 1

.

The difference between the fourth and second terms of an AP is equal to

2d

.

6

Find

a

,

b

, and their sum

a + b

.

a = \alpha\gamma = 1 \times 5 = 5

,

b = \beta + \delta = 3 + 7 = 10

, so

a + b = 5 + 10 = 15

.

With all four roots determined (

1, 3, 5, 7

), the missing coefficients are calculated using Vieta's formulas.

Key Concept

Combining Vieta's Formulas with Arithmetic Progressions to Solve Quadratic Systems

Question 202Question →

At a corporate law firm, the ratio of Partners to Associates was initially $3 : 8$ , and the ratio of Associates to Paralegals was initially $4 : 5$ . During an internal restructuring, $15$ Associates were promoted to Partners, and $2$ Paralegals resigned. As a result of these two changes, the ratio of Partners to Paralegals became $5 : 8$ . Shortly thereafter, the firm conducted a hiring drive, recruiting additional Associates until the ratio of Associates to Paralegals became $7 : 6$ , while the numbers of Partners and Paralegals remained unchanged. What was the total number of staff members (Partners, Associates, and Paralegals) at the firm after the hiring drive?

Show answer & explanation

Answer: 134

Answer

134

The solution unifies the initial ratios into 3 : 8 : 10, sets up the algebraic equations reflecting both promotions and resignations to find the multiplier k = 5, updates the staff counts, and applies the final ratio to find 56 Associates, yielding a final total of 134 staff members.

Step-by-Step Solution

1

Unify initial two-variable ratios into a continuous three-part ratio.

Partners : Associates : Paralegals = 3 : 8 : 10

Associates is the common term. Scaling 4 : 5 by 2 gives 8 : 10, matching the 8 parts in the Partners-to-Associates ratio.

2

Formulate algebraic expressions for post-restructuring staff counts.

Partners = 3k + 15, Associates = 8k - 15, Paralegals = 10k - 2

Promotions transfer 15 from Associates to Partners, and 2 Paralegals leave.

3

Solve for the ratio multiplier k using the new Partner-to-Paralegal ratio.

k = 5

Setting (3k + 15) / (10k - 2) = 5 / 8 yields 24k + 120 = 50k - 10, so 26k = 130.

4

Calculate exact staff counts after restructuring.

Partners = 30, Associates = 25, Paralegals = 48

Substitute k = 5 into the expressions from Step 2.

5

Determine final Associate count and total staff after hiring drive.

Final Associates = 56; Total Staff = 134

Associates = (7/6) * 48 = 56. Summing all roles gives 30 + 56 + 48 = 134.

Key Concept

Multi-part ratio unification, internal transfer ratio alteration, and sequential proportion scaling

Question 203Question →

Scanner Unit Alpha and Scanner Unit Beta, operating independently at their respective constant rates, can complete a document digitization project together in $12$ hours. Scanner Unit Alpha operates at a rate that is $50\%$ faster than that of Scanner Unit Beta. The project is carried out in three consecutive stages: first, Scanner Unit Alpha operates alone for $4$ hours; second, Scanner Unit Beta joins Alpha and both operate together until $70\%$ of the entire project is completed; finally, Scanner Unit Beta finishes the remaining portion of the project alone. How many total hours does it take to complete the entire digitization project?

Show answer & explanation

Answer: 19

Answer

The total time required to complete the entire digitization project is 19 hours.

To find the total time needed, calculate the individual rates first. With a combined rate of

r_A + r_B = \frac{1}{12}

and

r_A = 1.5 r_B

, solving

2.5 r_B = \frac{1}{12}

yields

r_B = \frac{1}{30}

and

r_A = \frac{1}{20}

. During Stage 1 (

4

hours), Alpha completes

4 \times \frac{1}{20} = 0.20

of the job. In Stage 2, both units work together to bring completion from

20\%

to

70\%

(

0.50

work), taking

\frac{0.50}{1/12} = 6

hours. In Stage 3, Beta completes the remaining

0.30

work alone, taking

\frac{0.30}{1/30} = 9

hours. Summing all stage durations yields

4 + 6 + 9 = 19

hours.

Step-by-Step Solution

1

Determine the individual work rates of Scanner Unit Alpha (

r_A

) and Scanner Unit Beta (

r_B

).

r_B = \frac{1}{30}

project per hour, and

r_A = \frac{1}{20}

project per hour.

Since their combined rate is

\frac{1}{12}

project per hour and

r_A = 1.5 r_B

, we solve

2.5 r_B = \frac{1}{12}

to find

r_B = \frac{1}{30}

and

r_A = \frac{1}{20}

.

2

Calculate the fraction of work completed during Stage 1.

Scanner Unit Alpha completes

0.20

(

20\%

) of the project in

4

hours.

Alpha works alone for

4

hours at a rate of

\frac{1}{20}

project per hour:

4 \times \frac{1}{20} = 0.20

.

3

Calculate the duration of Stage 2 where both units work together.

Stage 2 takes

6

hours.

The combined units must complete the portion from

20\%

to

70\%

, which represents

0.70 - 0.20 = 0.50

of the project. At a combined rate of

\frac{1}{12}

project per hour, the time required is

\frac{0.50}{1/12} = 6

hours.

4

Calculate the duration of Stage 3 where Scanner Unit Beta works alone.

Stage 3 takes

9

hours.

Beta must complete the remaining

30\%

(

0.30

) of the project alone. At a rate of

\frac{1}{30}

project per hour, the time required is

\frac{0.30}{1/30} = 9

hours.

5

Sum the durations of all three stages to determine total project time.

Total time =

19

hours.

Adding the duration of each stage:

4 \text{ hours} + 6 \text{ hours} + 9 \text{ hours} = 19 \text{ hours}

.

Key Concept

Work Rate and Combined Work

Question 204Question →

A box contains 20 cards numbered sequentially from 1 through 20, inclusive. If one card is drawn at random from the box, what is the probability that the number on the drawn card is a prime number? Express your answer as a decimal.

Show answer & explanation

Answer: 0.4

Answer

0.4

The total number of possible outcomes when selecting one card from 20 is 20. The prime numbers between 1 and 20 inclusive are 2, 3, 5, 7, 11, 13, 17, and 19, giving 8 favorable outcomes (remembering that 1 is not prime). The single-event probability is calculated by dividing the number of favorable outcomes by the total number of outcomes, yielding

8 / 20 = 0.4

.

Step-by-Step Solution

1

Determine the total number of possible outcomes in the sample space.

The sample space consists of 20 equally likely outcomes (integers 1 through 20).

Calculating single-event probability requires establishing the size of the total outcome space

N

.

2

Count the number of prime numbers in the set

\{1, 2, \dots, 20\}

.

There are 8 prime numbers:

\{2, 3, 5, 7, 11, 13, 17, 19\}

.

By definition, a prime number is an integer greater than 1 with exactly two positive divisors: 1 and itself. Thus, 1 is excluded.

3

Compute the probability using

P(E) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}

.

P = \frac{8}{20} = 0.4

.

Directly apply the basic single-event probability formula.

Key Concept

Basic Single-Event Probability and Prime Number Identification

Estimated Time:45s

Question 205Question →

A specialized mapping drone, Model X, can complete an aerial survey of a solar farm in $20$ hours operating alone at a constant rate. A second drone, Model Y, can complete the exact same survey in $30$ hours operating alone at a constant rate. If both drones operate simultaneously at their respective constant rates, how many hours will it take them to complete the aerial survey together?

Show answer & explanation

Answer: 12

Answer

It will take 12 hours for both drones operating together to complete the survey.

Model X completes

\frac{1}{20}

of the job per hour and Model Y completes

\frac{1}{30}

of the job per hour. Combined, their rate is

\frac{1}{20} + \frac{1}{30} = \frac{5}{60} = \frac{1}{12}

of the job per hour. Taking the reciprocal yields a total time of 12 hours.

Step-by-Step Solution

1

Calculate individual work rates

Model X rate =

\frac{1}{20}

job/hr, Model Y rate =

\frac{1}{30}

job/hr

Work rate is the reciprocal of the total time required to complete one full job.

2

Calculate the combined rate of work

Combined rate =

\frac{1}{20} + \frac{1}{30} = \frac{5}{60} = \frac{1}{12}

job/hr

When working simultaneously, individual rates add together.

3

Compute total time required for combined work

Total time =

\frac{1}{1/12} = 12

hours

Time is equal to total work (1 job) divided by the combined work rate.

Key Concept

Combined Work Rate formula:

\text{Rate}_{\text{total}} = \text{Rate}_1 + \text{Rate}_2

, and

\text{Time}_{\text{total}} = \frac{1}{\text{Rate}_{\text{total}}}

.

Question 206Question →

A rectangular garden has an area of $48$ square meters. If the length of the garden is $2$ meters greater than its width, what is the perimeter of the garden, in meters?

Show answer & explanation

Answer: 28

Answer

The perimeter of the garden is 28 meters.

By setting up the area equation

w(w + 2) = 48

, we obtain the quadratic equation

w^2 + 2w - 48 = 0

. Factoring gives

(w + 8)(w - 6) = 0

. Since width must be positive,

w = 6

meters. The length is

6 + 2 = 8

meters. Thus, the perimeter is

2(6 + 8) = 28

meters.

Step-by-Step Solution

1

Define variables for width and length.

Width =

w

, Length =

w + 2

.

The length is given as 2 meters greater than the width.

2

Formulate and rearrange the quadratic equation for area.

w^2 + 2w - 48 = 0

Area is length multiplied by width, set equal to 48.

3

Factor the quadratic polynomial.

(w + 8)(w - 6) = 0

Find two numbers that multiply to -48 and add up to +2.

4

Determine the valid physical dimension.

w = 6

meters and length

l = 8

meters.

A physical dimension cannot be negative, so

w = -8

is discarded.

5

Compute the perimeter.

Perimeter =

2(6 + 8) = 28

meters.

Perimeter of a rectangle is twice the sum of its length and width.

Key Concept

Solving quadratic equations by factoring in word problem contexts.

Question 207Question →

For any real number $x$ , the function $h$ is defined by $h(x) = 3x^2 - 5x + 4$ . What is the value of $h(3)$ ?

Show answer & explanation

Answer: 16

Answer

16

To find

h(3)

, substitute

x = 3

into the function

h(x) = 3x^2 - 5x + 4

. First, compute

3^2 = 9

. Multiplying by 3 yields 27. Subtracting

5(3) = 15

gives 12, and adding 4 results in the final value of 16.

Step-by-Step Solution

1

Substitute the value

x = 3

into the definition of

h(x)

.

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

To evaluate a function at a specific value, replace all instances of the input variable with that value.

2

Evaluate the exponent and carry out the multiplications.

h(3) = 3(9) - 15 + 4 = 27 - 15 + 4

Follow standard order of operations: compute powers before performing multiplication.

3

Perform the remaining addition and subtraction.

h(3) = 16

Complete the calculation from left to right.

Key Concept

Function Evaluation

Question 208Question →

A graphic design department has senior designers and junior designers in a ratio of $3 : 4$ . If there are $24$ junior designers in the department, what is the total number of senior and junior designers in the department?

Show answer & explanation

Answer: 42

Answer

The total number of senior and junior designers in the department is 42.

The ratio of senior to junior designers is

3 : 4

. Given that there are

24

junior designers,

4

ratio parts equal

24

, making

1

ratio part equal to

6

. The total department consists of

3 + 4 = 7

ratio parts. Therefore, the total number of designers is

7 \times 6 = 42

.

Step-by-Step Solution

1

Determine the value of one ratio unit using the given number of junior designers.

Since

4

ratio parts correspond to

24

junior designers,

1

ratio part

= 24 / 4 = 6

.

The given count of

24

aligns with the

4

parts of junior designers in the

3 : 4

ratio.

2

Calculate the total ratio parts and find the total count of designers.

Total ratio parts

= 3 + 4 = 7

. Total designers

= 7 \times 6 = 42

.

The total department size is represented by the sum of the ratio components.

Key Concept

Ratio and Proportion Word Problems

Question 209Question →

If $x^2 - 9x + 18 = 0$ , and $x_1$ and $x_2$ are the real roots of the equation such that $x_1 > x_2$ , what is the value of $2x_1 - x_2$ ?

Show answer & explanation

Answer: 9

Answer

The value of

2x_1 - x_2

is 9.

Factoring the quadratic equation

x^2 - 9x + 18 = 0

yields

(x - 6)(x - 3) = 0

. The roots are

x = 6

and

x = 3

. Given that

x_1 > x_2

, we assign

x_1 = 6

and

x_2 = 3

. Evaluating

2x_1 - x_2

gives

2(6) - 3 = 9

.

Step-by-Step Solution

1

Factor the quadratic expression

(x - 6)(x - 3) = 0

Find two numbers that multiply to

18

and add to

-9

, which are

-6

and

-3

.

2

Solve for the roots of the equation

x = 6

or

x = 3

Set each factor equal to zero:

x - 6 = 0 \implies x = 6

and

x - 3 = 0 \implies x = 3

.

3

Assign root values based on the condition

x_1 > x_2

x_1 = 6

and

x_2 = 3

Since

6 > 3

, the larger root is assigned to

x_1

and the smaller root to

x_2

.

4

Evaluate the target expression

9

Substitute

x_1 = 6

and

x_2 = 3

into

2x_1 - x_2 = 2(6) - 3 = 12 - 3 = 9

.

Key Concept

Solving Quadratic Equations by Factoring

Question 210Question →

At a textile weaving mill, high-grade cotton, synthetic polyester, and natural silk threads are blended to produce a specialized fabric. Initially, the weights of cotton, polyester, and silk in the blending vat are in the ratio of $5 : 3 : 2$ , respectively. To meet a custom order requirement, $12$ kilograms of silk are added to the vat, and $4$ kilograms of polyester are removed, while the amount of cotton remains unchanged. If the new ratio of cotton weight to silk weight is $5 : 4$ , how many kilograms of synthetic polyester are in the blending vat after these adjustments?

Show answer & explanation

Answer: 14

Answer

14 kg

Representing the initial weights as

5x

,

3x

, and

2x

, the addition of

12

kg of silk updates the silk quantity to

2x + 12

. Setting the unchanged cotton quantity (

5x

) over the new silk quantity equal to

\frac{5}{4}

yields

x = 6

. The initial weight of polyester was

3(6) = 18

kg, and removing

4

kg gives a final polyester weight of

14

kg.

Step-by-Step Solution

1

Express initial component weights using a common ratio multiplier

x

Cotton =

5x

, Polyester =

3x

, Silk =

2x

The given initial ratio of cotton to polyester to silk is

5 : 3 : 2

.

2

Set up a proportion for the updated Cotton-to-Silk ratio

\frac{5x}{2x + 12} = \frac{5}{4}

Adding

12

kg of silk increases its weight to

(2x + 12)

kg, while cotton remains at

5x

kg.

3

Solve the algebraic proportion for

x

x = 6

Cross-multiplying gives

20x = 5(2x + 12) = 10x + 60

, which simplifies to

10x = 60

, so

x = 6

.

4

Calculate the final weight of synthetic polyester

14 kg

Initial polyester was

3(6) = 18

kg. Subtracting the

4

kg removed leaves

18 - 4 = 14

kg.

Key Concept

Multi-part ratios and algebraic setting of updated proportional quantities

Question 211Question →

At an architectural firm, the ratio of senior architects to junior architects to structural engineers was initially $3 : 7 : 4$ , respectively. After the firm hired $6$ additional senior architects and $4$ structural engineers left the firm, the ratio of senior architects to structural engineers became $3 : 2$ , while the number of junior architects remained unchanged. What was the total number of senior architects, junior architects, and structural engineers combined initially at the firm?

Show answer & explanation

Answer: 56

Answer

The total number of senior architects, junior architects, and structural engineers combined initially at the firm was 56.

Let the initial numbers of senior architects, junior architects, and structural engineers be

3x

,

7x

, and

4x

, respectively. After adding

6

senior architects and subtracting

4

structural engineers, the updated counts are

3x + 6

and

4x - 4

. Setting their ratio to

3 : 2

yields

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

. Cross-multiplying gives

6x + 12 = 12x - 12

, which simplifies to

6x = 24

, so

x = 4

. The total number of initial employees is

3x + 7x + 4x = 14x = 14(4) = 56

.

Step-by-Step Solution

1

Define initial quantities using ratio multiplier x

Senior Architects

= 3x

, Junior Architects

= 7x

, Structural Engineers

= 4x

Ratios specify relative proportions, so multiplying each term by

x

gives actual counts.

2

Formulate new quantities after staff changes

Senior Architects

= 3x + 6

, Structural Engineers

= 4x - 4

6 senior architects joined and 4 structural engineers left.

3

Set up ratio equation and solve for x

\frac{3x + 6}{4x - 4} = \frac{3}{2} \implies 2(3x + 6) = 3(4x - 4) \implies 6x + 12 = 12x - 12 \implies 6x = 24 \implies x = 4

Cross-multiplication converts the proportion into a linear equation.

4

Compute total initial staff count

Initial Total

= 3x + 7x + 4x = 14x = 14 \times 4 = 56

Summing all 3 initial ratio parts and substituting

x = 4

yields the total initial count.

Key Concept

Multi-part ratio formulation and linear equation setup based on additions and subtractions

Estimated Time:2m 0s

Question 212Question →

For all real numbers $x$ and $y$ , the custom operation $\Delta$ is defined by $x \Delta y = (x + 1)(y - 2)$ . If the function $g$ is defined by $g(n) = n^2 - 3$ for all real numbers $n$ , what is the value of $g(3) \Delta g(2)$ ?

Show answer & explanation

Answer: -7

Answer

The value of

g(3) \Delta g(2)

is

-7

.

Evaluating

g(3)

gives

3^2 - 3 = 6

, and evaluating

g(2)

gives

2^2 - 3 = 1

. Substituting these values as

x = 6

and

y = 1

into the custom operator definition

x \Delta y = (x + 1)(y - 2)

yields

(6 + 1)(1 - 2) = 7 \times (-1) = -7

.

Step-by-Step Solution

1

Evaluate the inner function

g(n)

at

n = 3

g(3) = 3^2 - 3 = 9 - 3 = 6

Before evaluating the custom operator, the inputs

g(3)

and

g(2)

must be explicitly calculated.

2

Evaluate the inner function

g(n)

at

n = 2

g(2) = 2^2 - 3 = 4 - 3 = 1

Determine the second operand for the custom operation.

3

Apply the custom operation

\Delta

to the results from Step 1 and Step 2

6 \Delta 1 = (6 + 1)(1 - 2) = 7 \times (-1) = -7

Substitute

x = 6

and

y = 1

into the formula

x \Delta y = (x + 1)(y - 2)

.

Key Concept

Evaluating functions and applying non-commutative custom operators

Question 213Question →

At an international conference of $200$ delegates, every delegate speaks at least one of three languages: French, German, or Spanish. Exactly $115$ delegates speak French, $85$ speak German, and $67$ speak Spanish. Furthermore, exactly $15$ delegates speak all three languages. If the number of delegates who speak only one language is four times the number of delegates who speak exactly two languages, how many delegates speak exactly two of these three languages?

Show answer & explanation

Answer: 37

Answer

The number of delegates who speak exactly two of the three languages is 37.

By dividing the 200 delegates into three disjoint groups—those speaking exactly 1 language (

E_1

), exactly 2 languages (

E_2

), and all 3 languages (

E_3 = 15

)—we establish that

E_1 + E_2 + 15 = 200

, giving

E_1 + E_2 = 185

. Given

E_1 = 4E_2

, substituting yields

5E_2 = 185

, which gives

E_2 = 37

.

Step-by-Step Solution

1

Define exact-count variables for the overlapping sets

Let

E_1

be the number of delegates speaking exactly one language,

E_2

be the number speaking exactly two languages, and

E_3 = 15

be the number speaking all three languages.

Categorizing elements into mutually exclusive subsets (exactly 1, exactly 2, exactly 3) simplifies 3-set inclusion-exclusion problems.

2

Set up the total population equation

E_1 + E_2 + E_3 = 200 \implies E_1 + E_2 + 15 = 200 \implies E_1 + E_2 = 185

Since every delegate speaks at least one language, the sum of all mutually exclusive regions equals the total number of delegates.

3

Apply the given relationship between

E_1

and

E_2

Substitute

E_1 = 4E_2

into

E_1 + E_2 = 185

to get

4E_2 + E_2 = 185 \implies 5E_2 = 185

The problem establishes that the number of single-language speakers is four times the number of dual-language speakers.

4

Solve for the target quantity

E_2

E_2 = 37

Dividing 185 by 5 yields the exact count of delegates speaking exactly two languages.

Key Concept

Three-Set Inclusion-Exclusion Principle using Exact-Overlap Subsets

Question 214Question →

A software engineering team uses three automated refactoring scripts—Script A, Script B, and Script C—to update a large codebase. Working alone at their respective constant rates, Script A can finish the entire codebase in $10$ hours, Script B in $15$ hours, and Script C in $30$ hours. All three scripts begin running simultaneously. After $2$ hours, Script A encounters an error and stops running. Script B and Script C continue running together without interruption until the codebase is completely refactored. How many additional hours will it take for Script B and Script C to complete the remaining work?

Show answer & explanation

Answer: 6

Answer

6 hours

The rate of Script A is

\frac{1}{10}

of the codebase per hour, Script B is

\frac{1}{15}

per hour, and Script C is

\frac{1}{30}

per hour. Working together for

2

hours, all three scripts complete

2 \times \left(\frac{1}{10} + \frac{1}{15} + \frac{1}{30}\right) = 2 \times \frac{1}{5} = \frac{2}{5}

of the codebase, leaving

1 - \frac{2}{5} = \frac{3}{5}

of the task remaining. With Script A offline, the combined rate of Script B and Script C is

\frac{1}{15} + \frac{1}{30} = \frac{1}{10}

per hour. The additional time required to complete the remaining work is

\frac{3/5}{1/10} = 6

hours.

Step-by-Step Solution

1

Determine the individual work rates of Script A, Script B, and Script C.

Rate of Script A =

\frac{1}{10}

codebase/hour, Rate of Script B =

\frac{1}{15}

codebase/hour, Rate of Script C =

\frac{1}{30}

codebase/hour.

Work rate is the reciprocal of the total time required to complete one entire job working alone.

2

Calculate the combined work rate of all three scripts during the initial 2-hour period.

Combined Rate(A+B+C) =

\frac{1}{10} + \frac{1}{15} + \frac{1}{30} = \frac{3 + 2 + 1}{30} = \frac{6}{30} = \frac{1}{5}

codebase/hour.

When multiple agents work together simultaneously, their individual work rates are added.

3

Determine the fraction of the codebase refactored during the first 2 hours.

Work Completed =

2 \text{ hours} \times \frac{1}{5} \text{ codebase/hour} = \frac{2}{5}

of the total codebase.

Total work done equals rate multiplied by time.

4

Calculate the remaining fraction of the codebase left to be processed.

Remaining Work =

1 - \frac{2}{5} = \frac{3}{5}

of the total codebase.

Subtracting the completed fraction from 1 gives the uncompleted portion of the job.

5

Calculate the combined rate of Script B and Script C after Script A stops.

Rate(B+C) =

\frac{1}{15} + \frac{1}{30} = \frac{2 + 1}{30} = \frac{3}{30} = \frac{1}{10}

codebase/hour.

Only Script B and Script C contribute to the remaining work.

6

Calculate the additional time required for Script B and Script C to finish the remaining work.

Additional Time =

\frac{3/5}{1/10} = \frac{3}{5} \times 10 = 6

hours.

Time equals remaining work divided by the active combined rate.

Key Concept

Work Rate and Combined Work

Estimated Time:1m 30s

Question 215Question →

If $x = 1$ is a root of the cubic polynomial $P(x) = x^3 + ax^2 - 11x + 12$ , where $a$ is a constant, what is the value of the largest real root of $P(x)$ ?

Show answer & explanation

Answer: 4

Answer

The largest real root of the polynomial

P(x)

is

4

.

Evaluating

P(1) = 0

yields

1 + a - 11 + 12 = 0

, giving

a = -2

. Factoring the polynomial

x^3 - 2x^2 - 11x + 12

by

(x - 1)

produces the quadratic expression

x^2 - x - 12

. Factoring this quadratic gives

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

. The roots of the polynomial are

-3

,

1

, and

4

. Comparing these values, the maximum root is

4

.

Step-by-Step Solution

1

Find the constant

a

using the given root

x = 1

.

a = -2

Since

x = 1

is a root,

P(1) = 1^3 + a(1)^2 - 11(1) + 12 = 0

, which simplifies to

a + 2 = 0

.

2

Factor

(x - 1)

out of

P(x) = x^3 - 2x^2 - 11x + 12

.

P(x) = (x - 1)(x^2 - x - 12)

Dividing

x^3 - 2x^2 - 11x + 12

by

(x - 1)

yields the quotient

x^2 - x - 12

.

3

Factor the quadratic factor

x^2 - x - 12

.

(x - 4)(x + 3)

The quadratic factors into two linear terms whose roots are

4

and

-3

.

4

Determine the largest of all real roots of

P(x)

.

4

The complete set of real roots is

\{-3, 1, 4\}

, of which

4

is the greatest.

Key Concept

Polynomial Factoring and Remainder/Factor Theorem

Question 216Question →

An agricultural processing facility uses three independent conveyor systems—Conveyor $X$ , Conveyor $Y$ , and Conveyor $Z$ —to fill a grain storage silo. Working together at their respective constant rates, Conveyors $X$ and $Y$ can fill the empty silo in $12$ hours, while Conveyors $Y$ and $Z$ working together can fill the empty silo in $20$ hours. Initially, Conveyors $X$ and $Z$ work together for $5$ hours, completing exactly $\frac{1}{3}$ of the silo. How many hours would it take Conveyor $Y$ operating alone to fill the remaining $\frac{2}{3}$ of the silo?

Show answer & explanation

Answer: 20

Answer

It would take Conveyor

Y

operating alone

20

hours to fill the remaining

\frac{2}{3}

of the silo.

By representing the rate of each pair of conveyors as a fraction of the total job per hour, we find

R_X + R_Y = 1/12

,

R_Y + R_Z = 1/20

, and

R_X + R_Z = 1/15

. Summing these three equations yields

2(R_X + R_Y + R_Z) = 1/5

, so

R_X + R_Y + R_Z = 1/10

. Subtracting

R_X + R_Z = 1/15

gives

R_Y = 1/30

silo per hour. To fill the remaining

2/3

of the silo, Conveyor

Y

requires

(2/3) / (1/30) = 20

hours.

Step-by-Step Solution

1

Express the combined rate of each pair of conveyors as a fraction of the silo filled per hour.

R_X + R_Y = \frac{1}{12}

,

R_Y + R_Z = \frac{1}{20}

, and

R_X + R_Z = \frac{1/3}{5} = \frac{1}{15}

.

Work rate equals work done divided by time taken.

2

Sum the three pairwise rates to determine the combined rate of all three conveyors working together.

2(R_X + R_Y + R_Z) = \frac{1}{12} + \frac{1}{20} + \frac{1}{15} = \frac{12}{60} = \frac{1}{5} \implies R_X + R_Y + R_Z = \frac{1}{10}

.

Adding pairwise rates counts each conveyor's individual rate twice.

3

Subtract the combined rate of Conveyors

X

and

Z

from the total rate of all three conveyors to isolate the rate of Conveyor

Y

.

R_Y = \frac{1}{10} - \frac{1}{15} = \frac{3 - 2}{30} = \frac{1}{30}

of the silo per hour.

Subtracting

(R_X + R_Z)

from

(R_X + R_Y + R_Z)

yields

R_Y

directly.

4

Divide the remaining fraction of work by Conveyor

Y

's individual rate to find the required time.

\text{Time} = \frac{2/3}{1/30} = 20

hours.

Time required equals remaining work divided by the individual work rate.

Key Concept

Solving systems of simultaneous work rate equations by summing pairwise rates.

Question 217Question →

A specialized express freight facility uses three automated sorting systems—System 1, System 2, and System 3—to process incoming shipments. Operating continuously at their respective constant rates, System 1 and System 2 working together can process 1 full shipment in 6 hours; System 2 and System 3 working together can process the same shipment in 10 hours; and System 1 and System 3 working together can process the shipment in 7.5 hours. Processing begins with System 1 and System 2 working together. After 3 hours, System 1 breaks down and stops operating, at which point System 3 is immediately activated to work alongside System 2 until the entire shipment is completed. How many total hours does it take, from the start of processing, to complete the entire shipment?

Show answer & explanation

Answer: 8

Answer

The total time required from start to completion is 8 hours.

Converting completion times to work rates per hour gives paired rates of 1/6, 1/10, and 2/15. Summing these and dividing by 2 yields a combined three-system rate of 1/5 shipment per hour. In the first 3 hours, System 1 and System 2 complete 3 * (1/6) = 1/2 of the shipment. System 2 and System 3 then finish the remaining 1/2 at their combined rate of 1/10 per hour, requiring (1/2) / (1/10) = 5 hours. Total elapsed time is 3 + 5 = 8 hours.

Step-by-Step Solution

1

Set up equations for the rate of work done per hour by each pair of systems

Let

r_1, r_2, r_3

be the individual work rates in shipments per hour. Then

r_1 + r_2 = \frac{1}{6}

,

r_2 + r_3 = \frac{1}{10}

, and

r_1 + r_3 = \frac{1}{7.5} = \frac{2}{15}

.

Work rate is inversely proportional to completion time (

Rate = \frac{Work}{Time}

).

2

Calculate the combined processing rate of all three systems

2(r_1 + r_2 + r_3) = \frac{1}{6} + \frac{1}{10} + \frac{2}{15} = \frac{5 + 3 + 4}{30} = \frac{12}{30} = \frac{2}{5}

, which simplifies to

r_1 + r_2 + r_3 = \frac{1}{5}

shipment per hour.

Adding the three paired rates accounts for each individual system's rate exactly twice.

3

Determine the amount of work completed during the initial 3-hour period

Work completed =

3 \text{ hours} \times (r_1 + r_2) = 3 \times \frac{1}{6} = \frac{1}{2}

of the total shipment.

System 1 and System 2 operate together at a combined rate of

\frac{1}{6}

shipment per hour for 3 hours.

4

Calculate the time needed for System 2 and System 3 to complete the remaining shipment

Remaining work =

1 - \frac{1}{2} = \frac{1}{2}

. Time required =

\frac{1/2}{r_2 + r_3} = \frac{1/2}{1/10} = 5

hours.

System 2 and System 3 work together at a combined rate of

\frac{1}{10}

shipment per hour to finish the remaining half of the job.

5

Calculate the total time elapsed from start to finish

Total time =

3 \text{ hours} + 5 \text{ hours} = 8 \text{ hours}

.

The question asks for the total duration of the process from the beginning.

Key Concept

Solving systems of simultaneous work rate equations and analyzing multi-stage combined work.

Question 218Question →

A jar contains $3$ green marbles and $2$ red marbles. If two marbles are selected at random one after another without replacement, what is the probability that both selected marbles are green?

Show answer & explanation

Answer: 0.3

Answer

The probability that both selected marbles are green is

0.3

.

Because selection is done without replacement, the outcome of the first draw changes the composition of the jar for the second draw. The probability of selecting a green marble first is

3/5

. With one green marble removed,

2

green marbles remain among

4

total marbles, so the probability of selecting green on the second draw is

2/4

. Multiplying these sequential probabilities gives

(3/5) \times (2/4) = 6/20 = 0.3

.

Step-by-Step Solution

1

Calculate the probability of drawing a green marble on the first selection.

P(\text{First Green}) = \frac{3}{5}

There are

3

green marbles out of

5

total marbles in the jar.

2

Calculate the conditional probability of drawing a green marble on the second selection.

P(\text{Second Green} \mid \text{First Green}) = \frac{2}{4} = \frac{1}{2}

Because the first marble was not replaced,

2

green marbles remain out of a total of

4

remaining marbles.

3

Multiply the probabilities of the dependent sequential events.

P(\text{Both Green}) = \frac{3}{5} \times \frac{1}{2} = \frac{3}{10} = 0.3

The probability of both dependent events occurring is the product of the first event's probability and the conditional probability of the second event.

Key Concept

Probability of Dependent Events Without Replacement

Question 219Question →

In a market research study of $100$ consumers, $65$ consumers stated that they regularly purchase Brand A, and $45$ consumers stated that they regularly purchase Brand B. If $20$ consumers purchase neither Brand A nor Brand B, how many consumers regularly purchase both Brand A and Brand B?

Show answer & explanation

Answer: 30

Answer

The number of consumers who regularly purchase both Brand A and Brand B is

30

.

According to the principle of inclusion-exclusion for two sets,

\text{Total} = \text{Brand A} + \text{Brand B} - \text{Both} + \text{Neither}

. Substituting the given numbers into this relationship gives

100 = 65 + 45 - \text{Both} + 20

, which simplifies to

100 = 130 - \text{Both}

. Subtracting

100

from

130

yields

\text{Both} = 30

.

Step-by-Step Solution

1

Identify the given values from the problem statement.

\text{Total} = 100

,

\text{Brand A} = 65

,

\text{Brand B} = 45

,

\text{Neither} = 20

.

Extracting all known set quantities to set up the overlapping sets equation.

2

Apply the inclusion-exclusion formula for two overlapping sets.

\text{Total} = \text{Brand A} + \text{Brand B} - \text{Both} + \text{Neither}

.

Subtracting the overlap prevents double-counting elements belonging to both groups.

3

Substitute the values and solve for the unknown overlap.

100 = 65 + 45 - \text{Both} + 20 \implies 100 = 130 - \text{Both} \implies \text{Both} = 30

.

Algebraically determining the number of consumers in the intersection.

Key Concept

Two-Set Inclusion-Exclusion Principle

Question 220Question →

If $x$ is a real number such that $\sqrt[3]{\frac{9^{2x+1} \cdot 27^{x-1}}{3^{x+2}}} = 243$ , what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The value of

x

is 3.

Converting all terms inside the radical to powers of 3 yields

\sqrt[3]{\frac{3^{4x+2} \cdot 3^{3x-3}}{3^{x+2}}} = \sqrt[3]{3^{6x-3}} = 3^{2x-1}

. Setting

3^{2x-1} = 243 = 3^5

gives

2x - 1 = 5

, which solves to

x = 3

.

Step-by-Step Solution

1

Express all exponential terms inside the radical using a common base of 3.

9^{2x+1} = 3^{4x+2}

and

27^{x-1} = 3^{3x-3}

.

Unifying bases allows the application of exponent multiplication and division rules.

2

Simplify the algebraic expression inside the cube root.

\frac{3^{4x+2} \cdot 3^{3x-3}}{3^{x+2}} = 3^{(4x+2) + (3x-3) - (x+2)} = 3^{6x-3}

.

Adding exponents for multiplication of like bases and subtracting for division.

3

Simplify the radical expression using fractional exponent rules.

\sqrt[3]{3^{6x-3}} = (3^{6x-3})^{1/3} = 3^{2x-1}

.

Taking the

n

-th root of a power corresponds to dividing the exponent by

n

.

4

Solve for

x

by equating the exponent to the prime factorized form of 243.

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

.

Since the bases are identical and non-zero, the exponents must be equal.

Key Concept

Exponent and radical rules with common base conversion

Answer

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