All practice questions

Question 741Question →

An artisan chocolate factory produces a signature dark chocolate blend using cocoa mass, cocoa butter, and cane sugar in the initial ratio of $2 : 3 : 4$ by weight, respectively. To modify the flavor profile for a new batch, the master chocolatier adds $10$ kilograms of cocoa mass and $20$ kilograms of cocoa butter to the mix, while removing $5$ kilograms of cane sugar. As a result, the ratio of cocoa mass to cocoa butter in the modified mixture becomes $4 : 7$ . What was the total weight, in kilograms, of the initial dark chocolate blend?

25

35

45

60

70

Show answer & explanation

Answer: $45$

Answer

The total weight of the initial dark chocolate blend was

45

kilograms.

By setting the initial weights of cocoa mass, cocoa butter, and cane sugar as

2x

,

3x

, and

4x

, the initial total weight is

9x

. The modified weights for cocoa mass and cocoa butter are

2x + 10

and

3x + 20

, respectively. Equating their ratio to

\frac{4}{7}

gives

\frac{2x + 10}{3x + 20} = \frac{4}{7}

, which simplifies to

14x + 70 = 12x + 80

, so

2x = 10

and

x = 5

. Substituting

x = 5

into the total weight expression

9x

yields

9(5) = 45

kilograms.

Step-by-Step Solution

1

Define initial ingredient weights using a multiplier variable.

Cocoa mass =

2x

, Cocoa butter =

3x

, Cane sugar =

4x

. Total initial weight =

2x + 3x + 4x = 9x

.

Expressing quantities in terms of a common ratio multiplier

x

enables setting up algebraic equations after alterations.

2

Express the modified weights of cocoa mass and cocoa butter.

New cocoa mass =

2x + 10

; New cocoa butter =

3x + 20

.

The problem states

10

kg of cocoa mass and

20

kg of cocoa butter were added.

3

Set up the proportion equation using the new ratio of cocoa mass to cocoa butter.

\frac{2x + 10}{3x + 20} = \frac{4}{7}

The modified ratio of cocoa mass to cocoa butter is given as

4 : 7

.

4

Cross-multiply and solve for

x

.

7(2x + 10) = 4(3x + 20) \implies 14x + 70 = 12x + 80 \implies 2x = 10 \implies x = 5

.

Solving the linear equation determines the common ratio multiplier value.

5

Calculate the initial total weight of the blend.

Initial total weight =

9x = 9(5) = 45

kg.

Multiplying the sum of all initial ratio parts by the multiplier gives the requested initial total weight.

Key Concept

Ratio setup and algebraic scaling in multi-part word problems with changing quantities

Estimated Time:2m 0s

Question 742Question →

A specialty coffee shop sells two types of coffee beans: Type A for $\$8$ per pound and Type B for $\$12$ per pound. A customer buys a total of $x$ pounds of coffee, consisting of $a$ pounds of Type A and $b$ pounds of Type B, for a total cost of $C$ dollars. Which of the following equations correctly model the relationship between these quantities? Select all that apply.

Select all that apply

a = x - b

C = 12x - 4a

b = \frac{C - 8a}{12}

C = 20x

a = \frac{C - 12}{8x}

Show answer & explanation

Answer: $a = x - b$ ; $C = 12x - 4a$ ; $b = \frac{C - 8a}{12}$

Answer

The correct formulations are

a = x - b

,

C = 12x - 4a

, and

b = \frac{C - 8a}{12}

.

The system is defined by two fundamental linear relationships: weight total

a + b = x

and cost total

8a + 12b = C

. Isolating

a

from the weight total gives

a = x - b

. Substituting

b = x - a

into the cost equation yields

C = 8a + 12(x - a) = 12x - 4a

. Isolating

b

from the cost equation gives

12b = C - 8a \implies b = \frac{C - 8a}{12}

. All three expressions are mathematically equivalent and correct.

Step-by-Step Solution

1

Set up the basic linear system from the word problem.

Total weight equation:

a + b = x

; Total cost equation:

8a + 12b = C

.

Word problems with total quantity and total monetary value translate into two distinct linear equations.

2

Rearrange the total weight equation to express

a

in terms of

x

and

b

.

a = x - b

.

Subtracting

b

from both sides isolates

a

.

3

Substitute

b = x - a

into the total cost equation.

C = 8a + 12(x - a) = 8a + 12x - 12a = 12x - 4a

.

Expressing cost in terms of a single variable

a

and total weight

x

simplifies multi-variable models.

4

Isolate

b

in the total cost equation.

12b = C - 8a \implies b = \frac{C - 8a}{12}

.

Subtracting the cost contribution of Type A (

8a

) and dividing by the unit cost of Type B (

12

) isolates

b

.

Key Concept

Linear Equation Modeling in Word Problems

Question 743Question →

A market research firm surveyed $300$ consumers regarding their subscriptions to three streaming services: Service X, Service Y, and Service Z.

- $40$ consumers subscribed to none of the three services.
- $160$ consumers subscribed to Service X.
- $140$ consumers subscribed to Service Y.
- $130$ consumers subscribed to Service Z.
- Exactly $60$ consumers subscribed to exactly two of the services.

How many consumers subscribed to all three services?

35

50

55

65

110

Show answer & explanation

Answer: 55

Answer

55 consumers subscribed to all three services.

The total number of consumers subscribing to at least one service is

300 - 40 = 260

. Expressing the total union in terms of non-overlapping region counts gives

e_1 + e_2 + e_3 = 260

, where

e_1

is the number subscribing to exactly one service,

e_2 = 60

is the number subscribing to exactly two, and

e_3

is the number subscribing to all three. Thus,

e_1 + e_3 = 200

. Summing the individual totals gives

|X| + |Y| + |Z| = e_1 + 2e_2 + 3e_3 = 430

, which simplifies to

e_1 + 3e_3 = 310

. Subtracting

e_1 + e_3 = 200

from

e_1 + 3e_3 = 310

yields

2e_3 = 110

, so

e_3 = 55

.

Step-by-Step Solution

1

Calculate the total number of consumers who subscribed to at least one streaming service.

Total in union

|X \cup Y \cup Z| = 300 - 40 = 260

.

Consumers subscribing to at least one service represent the total sample minus those subscribing to none.

2

Set up the region equations for 3-set inclusion-exclusion in terms of elements in exactly 1 set (

e_1

), exactly 2 sets (

e_2

), and all 3 sets (

e_3

).

Equation 1:

e_1 + e_2 + e_3 = 260

; Equation 2:

|X| + |Y| + |Z| = e_1 + 2e_2 + 3e_3 = 160 + 140 + 130 = 430

.

Summing individual set sizes counts elements in exactly one set once, elements in exactly two sets twice, and elements in all three sets three times.

3

Substitute

e_2 = 60

into both equations.

Equation 1:

e_1 + 60 + e_3 = 260 \implies e_1 + e_3 = 200

; Equation 2:

e_1 + 2(60) + 3e_3 = 430 \implies e_1 + 3e_3 = 310

.

Simplifying both equations yields a system of two linear equations in terms of

e_1

and

e_3

.

4

Subtract the simplified Equation 1 from Equation 2 to solve for

e_3

.

(e_1 + 3e_3) - (e_1 + e_3) = 310 - 200 \implies 2e_3 = 110 \implies e_3 = 55

.

Subtracting eliminates

e_1

, directly yielding the number of consumers subscribing to all three services.

Key Concept

3-Set Inclusion-Exclusion Principle using Exact Region Counts

Question 744Question →

If $x^2(2x - 3) = 9(2x - 3)$ , what is the sum of all distinct real solutions for $x$ ?

-1.5

0

1.5

3

4.5

Show answer & explanation

Answer: $1.5$

Answer

The sum of all distinct real solutions for

x

is

1.5

.

To find all solutions without losing roots, set the equation to zero:

x^2(2x - 3) - 9(2x - 3) = 0

. Factoring out

(2x - 3)

gives

(x^2 - 9)(2x - 3) = 0

, which fully factors into

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

. The distinct real roots are

3

,

-3

, and

1.5

. Adding these values together yields

3 + (-3) + 1.5 = 1.5

.

Step-by-Step Solution

1

Move all terms to one side of the equation to avoid losing solutions by dividing by a variable expression.

x^2(2x - 3) - 9(2x - 3) = 0

Dividing by an expression containing variables can eliminate valid roots.

2

Factor out the common term

(2x - 3)

.

(x^2 - 9)(2x - 3) = 0

Factoring allows application of the zero-product property.

3

Factor the quadratic term

x^2 - 9

as a difference of squares.

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

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

.

4

Set each factor to zero to find all real solutions.

x - 3 = 0 \implies x = 3

,

x + 3 = 0 \implies x = -3

, and

2x - 3 = 0 \implies x = 1.5

.

If the product of factors is zero, at least one factor must equal zero.

5

Sum all distinct real solutions.

3 + (-3) + 1.5 = 1.5

The question asks for the sum of all distinct real solutions.

Key Concept

Factoring Polynomial Equations and Avoiding Variable Cancellation

Estimated Time:1m 30s

Question 745Question →

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 746Question →

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 747Question →

An industrial semiconductor fabrication facility operates three specialized chemical vapor deposition (CVD) chambers—Chamber 1, Chamber 2, and Chamber 3—to process a large batch of silicon wafers.

When Chamber 1 and Chamber 2 operate together at their respective constant rates, they can complete the entire batch in $12$ hours. When Chamber 2 and Chamber 3 operate together at their respective constant rates, they can complete the entire batch in $20$ hours.

To process a new batch, Chamber 1 operates alone for $4$ hours. Next, Chamber 2 is turned on to assist Chamber 1, and both chambers operate together for an additional $6$ hours. Finally, Chambers 1 and 2 are shut down, and Chamber 3 operates alone for $18$ hours to finish the remaining portion of the batch.

How many hours would it take Chamber 3 to process the entire batch working alone at its constant rate?

40 hours

48 hours

54 hours

60 hours

72 hours

Show answer & explanation

Answer: 60 hours

Answer

60 hours

The correct answer is 60 hours. By subtracting the combined rate of Chambers 2 and 3 (

\frac{1}{20}

) from the combined rate of Chambers 1 and 2 (

\frac{1}{12}

), we find that Chamber 1's rate exceeds Chamber 3's rate by

\frac{1}{30}

batch per hour. Substituting

r_1 = r_3 + \frac{1}{30}

into the total work equation for the three stages (

4 r_1 + 6(r_1 + r_2) + 18 r_3 = 1

) yields

22 r_3 + \frac{2}{15} = \frac{1}{2}

, giving

r_3 = \frac{1}{60}

. Therefore, Chamber 3 requires 60 hours alone.

Step-by-Step Solution

1

Define rate variables and establish equations for the combined pairs.

Let

r_1, r_2, r_3

represent the work rates of Chambers 1, 2, and 3 in fraction of batch per hour. We are given

r_1 + r_2 = \frac{1}{12}

and

r_2 + r_3 = \frac{1}{20}

.

Work rate is the reciprocal of completion time for a full job.

2

Find the difference between the rates of Chamber 1 and Chamber 3.

(r_1 + r_2) - (r_2 + r_3) = \frac{1}{12} - \frac{1}{20} \implies r_1 - r_3 = \frac{5}{60} - \frac{3}{60} = \frac{2}{60} = \frac{1}{30}

. Thus,

r_1 = r_3 + \frac{1}{30}

.

Eliminating

r_2

allows us to express

r_1

strictly in terms of

r_3

.

3

Calculate the work completed in each stage of the multi-stage schedule.

Stage 1 work =

4 r_1

. Stage 2 work =

6(r_1 + r_2) = 6 \left(\frac{1}{12}\right) = \frac{1}{2}

. Stage 3 work =

18 r_3

.

Work completed equals rate multiplied by time operating.

4

Set up the total work equation and solve for Chamber 3's rate

r_3

.

4 r_1 + \frac{1}{2} + 18 r_3 = 1 \implies 4 \left(r_3 + \frac{1}{30}\right) + 18 r_3 = \frac{1}{2} \implies 22 r_3 + \frac{2}{15} = \frac{1}{2} \implies 22 r_3 = \frac{15 - 4}{30} = \frac{11}{30} \implies r_3 = \frac{11}{30 \times 22} = \frac{1}{60}

.

Sum of work across all stages equals

1

complete batch.

5

Determine the time required for Chamber 3 to complete the entire batch alone.

Time

= \frac{1}{r_3} = 60

hours.

Total time for a single entity is the reciprocal of its individual work rate.

Key Concept

Multi-stage work rate modeling with system of linear equations

Estimated Time:2m 30s

Question 748Question →

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 749Question →

A commercial coffee roasting facility operates two industrial roasters, Roaster $X$ and Roaster $Y$ . Working alone at its constant rate, Roaster $X$ can process a standard batch of coffee beans in $12$ hours. Working alone at its constant rate, Roaster $Y$ can process the exact same batch in $8$ hours.

Roaster $X$ begins processing a standard batch alone. After $3$ hours, Roaster $Y$ is turned on, and both roasters work together at their respective constant rates until the batch is completely processed. What is the total time, in hours, required to process the entire batch of coffee beans?

3.6

hours

6.6

hours

7.8

hours

9.0

hours

10.5

hours

Show answer & explanation

Answer: $6.6$ hours

Answer

6.6 hours

In the first

3

hours, Roaster

X

completes

3 \times \frac{1}{12} = \frac{1}{4}

of the total batch, leaving

\frac{3}{4}

of the batch remaining. When both roasters work together, their combined rate is

\frac{1}{12} + \frac{1}{8} = \frac{5}{24}

batch per hour. The time required for both roasters to finish the remaining

\frac{3}{4}

batch is

\frac{3/4}{5/24} = 3.6

hours. Adding the initial

3

hours yields a total elapsed time of

6.6

hours.

Step-by-Step Solution

1

Calculate individual work rates per hour.

Roaster

X

rate =

\frac{1}{12}

batch/hr; Roaster

Y

rate =

\frac{1}{8}

batch/hr.

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

2

Determine the amount of work completed by Roaster

X

in the first

3

hours and the remaining work.

Work done =

3 \times \frac{1}{12} = \frac{1}{4}

batch. Remaining work =

1 - \frac{1}{4} = \frac{3}{4}

batch.

Roaster

X

worked alone for the first

3

hours.

3

Calculate the combined work rate of both roasters working together.

Combined rate =

\frac{1}{12} + \frac{1}{8} = \frac{2}{24} + \frac{3}{24} = \frac{5}{24}

batch/hr.

Simultaneous work rates are additive.

4

Find the time taken to complete the remaining work together.

Combined time =

\frac{3/4}{5/24} = \frac{3}{4} \times \frac{24}{5} = \frac{18}{5} = 3.6

hours.

Time equals remaining work divided by the combined work rate.

5

Add the initial single-roaster phase to the combined work phase to find total time.

Total time =

3 + 3.6 = 6.6

hours.

The question asks for the total time to process the entire batch.

Key Concept

Work Rate and Combined Work

Question 750Question →

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

Question 751Question →

A merchant combines $10$ kilograms of Brand A tea costing $\$20$ per kilogram with $30$ kilograms of Brand B tea costing $\$40$ per kilogram. What is the average cost per kilogram of the combined mixture?

\$30

\$35

\$25

\$28

\$60

Show answer & explanation

Answer: $\$35$

Answer

The average cost per kilogram of the combined tea mixture is

\$35

.

The correct option of

\$35

properly weights each unit price by its respective quantity. Brand A contributes

10 \times 20 = \$200

and Brand B contributes

30 \times 40 = \$1200

, giving a combined total cost of

\$1400

for

40

kg. Dividing

\$1400

by

40

yields

\$35

per kilogram.

Step-by-Step Solution

1

Calculate the total cost of Brand A tea

10 \text{ kg} \times \$20/\text{kg} = \$200

To find the overall weighted cost, we first determine the total dollar amount spent on Brand A.

2

Calculate the total cost of Brand B tea

30 \text{ kg} \times \$40/\text{kg} = \$1200

Next, we calculate the total dollar amount spent on Brand B.

3

Find total combined cost and total combined weight

\text{Total cost} = \$200 + \$1200 = \$1400; \text{Total weight} = 10 + 30 = 40 \text{ kg}

Sum the total cost of both components and their combined weights.

4

Divide total combined cost by total combined weight

\text{Weighted Average Cost} = \frac{\$1400}{40 \text{ kg}} = \$35/\text{kg}

The weighted average formula is Total Cost divided by Total Quantity.

Key Concept

Weighted Average in Applied Contexts

Estimated Time:1m 0s

Question 752Question →

For all non-zero real numbers $a$ and $b$ , the custom binary operator $\star$ is defined by $a \star b = \frac{a^2 - b^2}{ab}$ . A function $f$ is defined for all positive numbers $x$ by $f(x) = (x + 1) \star x$ . What is the value of $f(3) \star f(2)$ ?

-\frac{3}{70}

\frac{51}{70}

-\frac{51}{70}

-\frac{56}{45}

-6

Show answer & explanation

Answer: $-\frac{51}{70}$

Answer

-\frac{51}{70}

First, evaluate the inner function values:

f(3) = 4 \star 3 = \frac{4^2 - 3^2}{4 \cdot 3} = \frac{7}{12}

and

f(2) = 3 \star 2 = \frac{3^2 - 2^2}{3 \cdot 2} = \frac{5}{6} = \frac{10}{12}

. Next, evaluate the outer operation

\left(\frac{7}{12}\right) \star \left(\frac{10}{12}\right) = \frac{(7/12)^2 - (10/12)^2}{(7/12)(10/12)} = \frac{49 - 100}{70} = -\frac{51}{70}

. Hence, the correct response is

-\frac{51}{70}

.

Step-by-Step Solution

1

Evaluate

f(3)

using the definition of

f(x)

and the custom operator

\star

.

f(3) = (3 + 1) \star 3 = 4 \star 3 = \frac{4^2 - 3^2}{4 \times 3} = \frac{16 - 9}{12} = \frac{7}{12}

By definition,

f(x) = (x+1) \star x

. Setting

x=3

yields

4 \star 3

.

2

Evaluate

f(2)

using the definition of

f(x)

and the custom operator

\star

.

f(2) = (2 + 1) \star 2 = 3 \star 2 = \frac{3^2 - 2^2}{3 \times 2} = \frac{9 - 4}{6} = \frac{5}{6}

Setting

x=2

in

f(x)

yields

3 \star 2

.

3

Evaluate

f(3) \star f(2) = \left(\frac{7}{12}\right) \star \left(\frac{5}{6}\right)

.

\left(\frac{7}{12}\right) \star \left(\frac{10}{12}\right) = \frac{\left(\frac{7}{12}\right)^2 - \left(\frac{10}{12}\right)^2}{\left(\frac{7}{12}\right)\left(\frac{10}{12}\right)} = \frac{\frac{49 - 100}{144}}{\frac{70}{144}} = -\frac{51}{70}

Apply the rule

u \star v = \frac{u^2 - v^2}{uv}

with

u = \frac{7}{12}

and

v = \frac{10}{12}

.

Key Concept

Custom non-commutative binary operators and function evaluation

Estimated Time:2m 0s

Question 753Question →

Working together at their respective constant rates, Machine X and Machine Y can complete a specialized production order in $12$ hours. If Machine X works alone for $4$ hours and Machine Y works alone for $9$ hours, they complete $\frac{7}{12}$ of the total order. Machine Z operates at a constant rate that is $50\%$ greater than the rate of Machine X. How many hours would it take Machine Y and Machine Z working together to complete the entire order?

10

12

15

20

40

Show answer & explanation

Answer: $10$

Answer

10

hours

The correct answer is

10

hours. Solving the system of equations yields individual rates of

\frac{1}{30}

order/hour for Machine X and

\frac{1}{20}

order/hour for Machine Y. A

50\%

rate increase gives Machine Z a rate of

\frac{1}{20}

order/hour. Adding the rates of Machine Y and Machine Z yields a combined rate of

\frac{1}{10}

order/hour, which requires

10

hours to complete one job.

Step-by-Step Solution

1

Define variables and model the joint rate equation.

Let

x

be the rate of Machine X (order/hour) and

y

be the rate of Machine Y (order/hour). Their combined rate equation is

x + y = \frac{1}{12}

.

Working together for

12

hours completes

1

full order.

2

Set up the second equation based on the partial work completed.

4x + 9y = \frac{7}{12}

.

Machine X works for

4

hours and Machine Y works for

9

hours to complete

\frac{7}{12}

of the order.

3

Solve the system of linear equations for

x

and

y

.

Express

4x + 9y

as

4(x + y) + 5y = \frac{7}{12}

. Substituting

x + y = \frac{1}{12}

gives

4\left(\frac{1}{12}\right) + 5y = \frac{7}{12} \implies \frac{4}{12} + 5y = \frac{7}{12} \implies 5y = \frac{3}{12} = \frac{1}{4} \implies y = \frac{1}{20}

. Subsequently,

x = \frac{1}{12} - \frac{1}{20} = \frac{5 - 3}{60} = \frac{2}{60} = \frac{1}{30}

.

Determines the individual work rates of Machine X and Machine Y.

4

Calculate the work rate of Machine Z.

Rate of Machine Z =

1.5 \times x = 1.5 \times \frac{1}{30} = \frac{3}{2} \times \frac{1}{30} = \frac{1}{20}

.

Machine Z is

50\%

faster than Machine X.

5

Calculate the combined rate and total time for Machine Y and Machine Z.

Combined rate =

y + z = \frac{1}{20} + \frac{1}{20} = \frac{2}{20} = \frac{1}{10}

. Total time =

\frac{1}{\frac{1}{10}} = 10

hours.

The inverse of the combined rate gives the total time required to finish one job.

Key Concept

Work Rates and Simultaneous Linear Modeling

Estimated Time:2m 30s

Question 754Question →

For all real numbers $a$ and $b$ , the custom binary operation $\star$ is defined by $a \star b = a^2 - 2b^2 + ab$ . The function $g$ is defined by $g(x) = x \star 3$ . What is the positive value of $x$ for which $g(x) = 10$ ?

4

1

7

\sqrt{28}

3

Show answer & explanation

Answer: 4

Answer

The positive value of

x

for which

g(x) = 10

is

4

.

Evaluating

g(x) = x \star 3

gives

x^2 - 2(3)^2 + x(3) = x^2 + 3x - 18

. Setting this expression equal to

10

produces the quadratic equation

x^2 + 3x - 28 = 0

. Factoring yields

(x + 7)(x - 4) = 0

, giving solutions

x = -7

and

x = 4

. Since

x

must be positive, the correct value is

4

.

Step-by-Step Solution

1

Apply the definition of the custom binary operator

\star

to find

g(x)

.

g(x) = x \star 3 = x^2 - 2(3)^2 + x(3) = x^2 - 18 + 3x = x^2 + 3x - 18

Substitute

a = x

and

b = 3

into the given formula

a \star b = a^2 - 2b^2 + ab

.

2

Set the function expression equal to

10

and rearrange into standard quadratic form.

x^2 + 3x - 18 = 10 \implies x^2 + 3x - 28 = 0

Subtract

10

from both sides to form a quadratic equation equal to zero.

3

Factor the quadratic equation to find the values of

x

.

(x + 7)(x - 4) = 0 \implies x = -7 \text{ or } x = 4

Find two numbers that multiply to

-28

and add to

3

, which are

7

and

-4

.

4

Select the positive value of

x

as specified in the stem.

x = 4

The question asks specifically for the positive value of

x

.

Key Concept

Evaluating non-commutative custom operators and solving polynomial equations resulting from function evaluations.

Estimated Time:2m 0s

Question 755Question →

A local theater sold a total of $120$ tickets for a weekend performance, consisting only of adult tickets and student tickets. Adult tickets were sold for $\$15$ each, and student tickets were sold for $\$10$ each. If the total revenue from the ticket sales was $\$1{}500$ , how many adult tickets were sold?

Show answer & explanation

Answer: 60

Answer

The correct answer is 60 adult tickets.

By setting up the total ticket equation

a + s = 120

and total revenue equation

15a + 10s = 1500

, substituting

s = 120 - a

gives

15a + 10(120 - a) = 1500

. Simplifying yields

5a = 300

, so

a = 60

. Thus, 60 adult tickets were sold.

Step-by-Step Solution

1

Set up the linear equation in one variable.

15a + 10(120 - a) = 1500

Express the number of student tickets as

(120 - a)

so the revenue equation relies on a single variable.

2

Expand and combine like terms.

5a + 1200 = 1500

Distribute

10

across

(120 - a)

to simplify the equation.

3

Isolate the variable

a

.

a = 60

Subtract

1200

from both sides and divide by

5

to find the number of adult tickets.

Key Concept

Algebraic Modeling of Linear Systems in Two Variables

Estimated Time:1m 0s

Question 756Question →

How many integer values of $x$ satisfy the inequality $|x - 3| \leq 4$ ?

Show answer & explanation

Answer: 9

Answer

There are 9 integer values of

x

that satisfy the inequality.

Unfolding

|x - 3| \leq 4

yields

-4 \leq x - 3 \leq 4

. Adding 3 across all terms gives

-1 \leq x \leq 7

. The integer values satisfying this range are

-1, 0, 1, 2, 3, 4, 5, 6, 7

, which totals 9 values.

Step-by-Step Solution

1

Convert the absolute value inequality into a compound linear inequality.

-4 \leq x - 3 \leq 4

The absolute value inequality

|k| \leq c

(for

c \geq 0

) is equivalent to

-c \leq k \leq c

.

2

Isolate

x

by adding 3 across all parts of the inequality.

-1 \leq x \leq 7

Adding the same constant to all parts of an inequality maintains the inequality relationships.

3

Count the total number of integer values within the inclusive range

[-1, 7]

.

9

The number of integers in an inclusive range

[a, b]

is calculated as

b - a + 1

, which gives

7 - (-1) + 1 = 9

.

Key Concept

Solving absolute value inequalities and counting integer solutions within a bounded interval

Question 757Question →

For all real numbers $a$ and $b$ , the custom binary operator $\star$ is defined by $a \star b = 2a - b$ . The function $f$ is defined for all real numbers by $f(x) = x^2 - 1$ . If $(x \star 3) \star f(3) = 6$ , what is the value of $x$ ?

-2

4

5

6

3

Show answer & explanation

Answer: 5

Answer

5

First, evaluate the function

f(3) = 3^2 - 1 = 8

. Next, evaluate the first custom operation

x \star 3 = 2x - 3

. Substituting these into the overall expression yields

(2x - 3) \star 8 = 2(2x - 3) - 8 = 4x - 14

. Setting this equal to 6 gives

4x - 14 = 6

, which simplifies to

4x = 20

, so

x = 5

.

Step-by-Step Solution

1

Evaluate the inner function

f(3)

f(3) = 3^2 - 1 = 9 - 1 = 8

Substitute

x = 3

into the given definition

f(x) = x^2 - 1

.

2

Evaluate the inner custom operation

x \star 3

x \star 3 = 2x - 3

Apply the custom operator rule

a \star b = 2a - b

with

a = x

and

b = 3

.

3

Evaluate the outer custom operation

(x \star 3) \star f(3)

(2x - 3) \star 8 = 2(2x - 3) - 8 = 4x - 6 - 8 = 4x - 14

Substitute the results from Step 1 and Step 2 into the custom operator definition.

4

Solve the algebraic equation for

x

4x - 14 = 6 \implies 4x = 20 \implies x = 5

Set the simplified expression equal to 6 as given in the problem stem and isolate

x

.

Key Concept

Custom Operator Evaluation and Nested Functions

Estimated Time:1m 30s

Question 758Question →

A commercial bakery prepares a specialty grain blend using Organic Wheat, Spelt, and Rye flour. Initially, the ratio of Organic Wheat to Spelt to Rye flour in the batch is $4 : 3 : 2$ by weight. To meet specific recipe requirements, $30$ kilograms of Organic Wheat and $15$ kilograms of Spelt are added to the batch, while $10$ kilograms of Rye are removed. After these adjustments, the ratio of Organic Wheat to Rye flour in the batch becomes $3 : 1$ . What was the total weight, in kilograms, of the initial batch of flour before any adjustments were made?

120

135

210

270

305

Show answer & explanation

Answer: 270

Answer

270 kilograms

By representing the initial amounts of Organic Wheat, Spelt, and Rye as

4x

,

3x

, and

2x

kilograms respectively, the initial total weight is

9x

kilograms. Following the adjustments, the new amount of Wheat is

4x + 30

kg and the new amount of Rye is

2x - 10

kg. Equating their ratio to

3 : 1

gives

\frac{4x + 30}{2x - 10} = \frac{3}{1}

. Solving for

x

gives

4x + 30 = 6x - 30

, leading to

2x = 60

or

x = 30

. Therefore, the initial total weight of the flour batch was

9 \times 30 = 270

kilograms.

Step-by-Step Solution

1

Define initial quantities using a common multiplier

x

.

Organic Wheat

= 4x

, Spelt

= 3x

, Rye

= 2x

. Initial total weight

= 4x + 3x + 2x = 9x

.

The initial ratio

4 : 3 : 2

specifies the relative proportions of the three components in the batch.

2

Express the updated amounts after additions and removals.

New Organic Wheat

= 4x + 30

, New Spelt

= 3x + 15

, New Rye

= 2x - 10

.

Incorporating the specified additions (

+30

kg Wheat,

+15

kg Spelt) and removal (

-10

kg Rye).

3

Set up an equation using the new ratio of Organic Wheat to Rye (

3 : 1

).

\frac{4x + 30}{2x - 10} = \frac{3}{1}

The problem states that the updated ratio of Wheat to Rye is

3 : 1

.

4

Solve for the multiplier

x

.

4x + 30 = 3(2x - 10) \implies 4x + 30 = 6x - 30 \implies 2x = 60 \implies x = 30

.

Cross-multiplying and solving the linear equation yields

x = 30

.

5

Calculate the total weight of the initial batch.

Initial total

= 9x = 9 \times 30 = 270

kilograms.

Substituting

x = 30

back into the initial total expression

9x

gives the required initial batch mass.

Key Concept

Ratio Modification and Multi-Part Proportions

Estimated Time:2m 0s

Question 759Question →

Worker A can complete a specific publishing task alone in $4$ hours, while Worker B can complete the exact same task alone in $6$ hours. Working together continuously at their respective constant individual rates, how many hours will it take both workers to complete one such task?

2.4

5.0

10.0

12.0

2.0

Show answer & explanation

Answer: $2.4$

Answer

The combined time required to complete the task is

2.4

hours.

The correct response calculates each worker's hourly rate of completion (one-fourth and one-sixth of the job per hour), sums them to find a combined rate of five-twelfths of the job per hour, and takes the reciprocal to find that the total time required is

2.4

hours.

Step-by-Step Solution

1

Determine individual work rates per hour.

Worker A's rate is

\frac{1}{4}

of the task per hour; Worker B's rate is

\frac{1}{6}

of the task per hour.

Work rate is defined as the reciprocal of total time required to complete one unit of work.

2

Sum the individual rates to find the combined work rate.

Combined Rate =

\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}

task per hour.

When working together, rates add together linearly.

3

Calculate total time by taking the reciprocal of the combined rate.

Time =

\frac{1}{\text{Combined Rate}} = \frac{12}{5} = 2.4

hours.

Time equals total work (1 unit) divided by the combined rate.

Key Concept

Combined Work Rates

Estimated Time:1m 0s

Question 760Question →

At a technology company, a team of $60$ software engineers works on product development. Of these engineers, $35$ work on the mobile application, $28$ work on the web platform, and $10$ work on neither project. How many engineers work on both the mobile application and the web platform?

3

13

18

25

63

Show answer & explanation

Answer: $13$

Answer

13 engineers work on both the mobile application and the web platform.

According to the fundamental principle of overlapping sets for two groups, the total number of individuals in the universe is equal to the number of individuals in Group A plus the number in Group B, minus those counted in both groups (to prevent double counting), plus those in neither group:

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

. Substituting the given values gives

60 = 35 + 28 - \text{Both} + 10

. Simplifying the right side yields

60 = 73 - \text{Both}

, so

\text{Both} = 13

. Thus, the value

13

is correct.

Step-by-Step Solution

1

Identify the given set values and formula

\text{Total} = 60

,

\text{Mobile} = 35

,

\text{Web} = 28

,

\text{Neither} = 10

Define all components of the standard two-set overlapping formula:

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

.

2

Substitute the values into the overlapping sets formula

60 = 35 + 28 - \text{Both} + 10

Set up the linear equation to solve for the unknown intersection (Both).

3

Simplify and solve for Both

60 = 73 - \text{Both} \implies \text{Both} = 73 - 60 = 13

Combine like terms (

35 + 28 + 10 = 73

) and isolate the variable to find the number of engineers in both groups.

Key Concept

Two-Set Overlapping Venn Diagram Principle

Estimated Time:1m 0s

Answer

Step-by-Step Solution

Key Concept

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Common Mistakes

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Common Mistakes

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Common Mistakes

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