All practice questions

Question 941Question →

Car 1 departs from City A at 8:00 AM and travels along a straight highway toward City B at a constant speed of $50$ miles per hour. At 8:30 AM, Car 2 departs from City B and travels along the same highway toward City A at a constant speed of $60$ miles per hour. At 9:15 AM, Car 1 encounters heavy traffic and reduces its speed by $20$ percent, maintaining this reduced speed for the rest of its journey, while Car 2 continues at its original speed. If the two cars pass each other at exactly 10:00 AM, what is the distance, in miles, between City A and City B?

Show answer & explanation

Answer: 182.5

Answer

The distance between City A and City B is 182.5 miles.

The total distance between the two cities is the sum of the distances traveled by both cars prior to their meeting point at 10:00 AM. Car 1 travels for 1.25 hours at 50 mph (62.5 miles) and 0.75 hours at 40 mph (30 miles), totaling 92.5 miles. Car 2 travels for 1.5 hours at 60 mph (90 miles). Summing these gives 92.5 + 90 = 182.5 miles.

Step-by-Step Solution

1

Determine the distance traveled by Car 1 prior to its speed reduction

Car 1 traveled 62.5 miles between 8:00 AM and 9:15 AM

Car 1 drove for 1 hour and 15 minutes (1.25 hours) at 50 mph, so distance = 50 × 1.25 = 62.5 miles.

2

Determine the distance traveled by Car 1 after its speed reduction

Car 1 traveled 30 miles between 9:15 AM and 10:00 AM

Car 1 reduced its speed by 20%, bringing its rate to 50 × 0.80 = 40 mph. It traveled at 40 mph for 45 minutes (0.75 hours), so distance = 40 × 0.75 = 30 miles.

3

Determine the distance traveled by Car 2

Car 2 traveled 90 miles between 8:30 AM and 10:00 AM

Car 2 drove from 8:30 AM to 10:00 AM, which is 1 hour and 30 minutes (1.5 hours), at a constant speed of 60 mph, so distance = 60 × 1.5 = 90 miles.

4

Calculate the total distance between City A and City B

The total distance is 182.5 miles

Because the two cars traveled toward each other from opposite cities until they met, the total distance between the cities equals the sum of the individual distances: 62.5 + 30 + 90 = 182.5 miles.

Key Concept

Converging motion with variable rates and time offsets

Question 942Question →

A chemical processing plant blends Solution X, which contains $25\%$ acid by volume, with Solution Y, which contains $65\%$ acid by volume, to create a $100$ -liter mixture that is $40\%$ acid by volume. If Solution X costs $\$3$ per liter and Solution Y costs $\$7$ per liter, what is the total cost of the solutions used to produce the mixture?

\$450

\$500

\$530

\$540

\$550

Show answer & explanation

Answer: $\$450$

Answer

The total cost of the solutions used to produce the mixture is

\$450

.

To determine the total cost, first solve for the exact volume of each solution needed. Let

x

be the volume of Solution X in liters and

y

be the volume of Solution Y in liters. Since the total mixture volume is

100

liters,

x + y = 100

, so

y = 100 - x

. Setting up the acid balance equation gives

0.25x + 0.65y = 0.40(100)

. Substituting

y = 100 - x

yields

0.25x + 0.65(100 - x) = 40

, which simplifies to

-0.40x + 65 = 40

, leading to

0.40x = 25

and

x = 62.5

. Thus,

62.5

liters of Solution X and

37.5

liters of Solution Y are required. Multiplying each volume by its price per liter gives

3(62.5) + 7(37.5) = 187.50 + 262.50 = 450

. The total cost is

\$450

.

Step-by-Step Solution

1

Define variables and write the equation for total volume.

Let

x

be the volume of Solution X in liters and

y

be the volume of Solution Y in liters. Then

x + y = 100

, which implies

y = 100 - x

.

The total required volume of the mixture is

100

liters.

2

Write the acid balance equation and solve for

x

and

y

.

Equation:

0.25x + 0.65(100 - x) = 0.40(100) \implies 0.25x + 65 - 0.65x = 40 \implies -0.40x = -25 \implies x = 62.5

liters. Thus,

y = 100 - 62.5 = 37.5

liters.

The total volume of pure acid in the final mixture must equal the sum of the pure acid contributed by Solution X and Solution Y.

3

Calculate the total cost of the mixture components.

Total Cost =

3(62.5) + 7(37.5) = 187.5 + 262.5 = 450

.

Multiply the volume of each solution by its respective price per liter and sum the results.

Key Concept

Algebraic Modeling of Mixture Problems and Systems of Linear Equations

Estimated Time:2m 0s

Question 943Question →

An event catering service charges a fixed setup fee plus a constant hourly rate for event staff. For a 4-hour event, the total charge is $680. For a 7-hour event with the same staff requirements, the total charge is$ 1,070. What is the fixed setup fee, in dollars, charged by the catering service?

Show answer & explanation

Answer: 160

Answer

The fixed setup fee charged by the catering service is 160 dollars.

Modeling the total cost as

C = S + rt

, where

S

is the fixed setup fee and

r

is the hourly rate, yields the system of equations

S + 4r = 680

and

S + 7r = 1070

. Subtracting the first equation from the second yields

3r = 390

, so

r = 130

. Substituting

r = 130

back into the first equation yields

S + 520 = 680

, which gives

S = 160

.

Step-by-Step Solution

1

Define the algebraic model for total cost.

Let

S

be the fixed setup fee in dollars and

r

be the hourly rate in dollars per hour. The total cost

C

for

t

hours is modeled by

C = S + rt

.

The problem presents a linear relationship between time and total cost, consisting of a fixed component and a variable component.

2

Construct a system of linear equations.

S + 4r = 680

and

S + 7r = 1070

.

Substitute the two given combinations of time (

t = 4

and

t = 7

) and total cost (

C = 680

and

C = 1070

) into the algebraic model.

3

Solve for the variable hourly rate

r

.

(S + 7r) - (S + 4r) = 1070 - 680 \implies 3r = 390 \implies r = 130

.

Subtracting the first equation from the second eliminates the fixed setup fee variable

S

.

4

Calculate the fixed setup fee

S

.

S + 4(130) = 680 \implies S + 520 = 680 \implies S = 160

.

Substitute

r = 130

back into the first equation to find the value of

S

.

Key Concept

Linear Equation Modeling and Systems of Equations

Question 944Question →

An airline creates a $5$ -character identification code for each of its cargo routes. The characters in the code must satisfy the following sequential rules from left to right:

1. The first character must be a non-zero even digit.
2. The second and third characters must be distinct uppercase vowels chosen from the set $\{A, E, I, O, U\}$ .
3. The fourth character must be a single-digit prime number.
4. The fifth character must be a single digit ( $0$ through $9$ ) such that the sum of the first digit and the fifth digit is odd.

How many such unique identification codes can the airline create?

Show answer & explanation

Answer: 1600

Answer

1,600 unique identification codes

To find the total number of unique identification codes, we evaluate the number of available options for each character position and multiply them together using the Fundamental Counting Principle:

1. First position (non-zero even digit): The even digits from 0 to 9 are

0, 2, 4, 6, 8

. Excluding 0 leaves

4

options (

2, 4, 6, 8

).
2. Second and third positions (distinct vowels): There are

5

vowels (

A, E, I, O, U

). The second position has

5

options, and the third position has

4

remaining options, yielding

5 \times 4 = 20

combinations.
3. Fourth position (prime digit): The single-digit primes are

2, 3, 5, 7

, giving

4

options.
4. Fifth position (parity condition): The sum of the first digit and fifth digit must be odd. Since the first digit is an even number, the fifth digit must be an odd number to make the sum odd. The single-digit odd numbers are

1, 3, 5, 7, 9

, giving

5

options.

Multiplying the choices for all positions gives:

4 \times 20 \times 4 \times 5 = 1,600

Step-by-Step Solution

1

Determine the number of valid choices for the 1st position (non-zero even digit).

4 choices (

2, 4, 6, 8

).

The digit 0 is excluded because the first character must be non-zero.

2

Determine the number of valid choices for the 2nd and 3rd positions (distinct vowels).

5 \times 4 = 20

choices.

There are 5 vowels available for the 2nd position, and 4 remaining vowels for the 3rd position since they must be distinct.

3

Determine the number of valid choices for the 4th position (prime digit).

4 choices (

2, 3, 5, 7

).

The single-digit prime numbers are 2, 3, 5, and 7.

4

Determine the number of valid choices for the 5th position (parity constraint).

5 choices (

1, 3, 5, 7, 9

).

The sum of an even integer and another integer is odd if and only if the second integer is odd. Since the 1st digit is guaranteed to be even, the 5th digit must be odd.

5

Apply the Fundamental Counting Principle across all 5 sequential stages.

4 \times 20 \times 4 \times 5 = 1,600

unique codes.

Each stage represents an independent decision stage with a fixed number of valid choices.

Key Concept

Fundamental Counting Principle with Restricted Positions and Parity Rules

Estimated Time:2m 0s

Question 945Question →

If $x$ is a real number that satisfies the equation $|5 - 2x| = 3x - 10$ , what is the value of $|x - 4| + 2x$ ?

5

7

11

14

18

Show answer & explanation

Answer: 11

Answer

11

Solving the equation

|5 - 2x| = 3x - 10

yields two potential values:

x = 3

and

x = 5

. Substituting

x = 3

into the original equation yields

|-1| = -1

, which is impossible because an absolute value cannot be negative. Therefore,

x = 3

is an extraneous root. Substituting

x = 5

yields

|-5| = 5

, which is true. Evaluating the requested expression

|x - 4| + 2x

at

x = 5

gives

|5 - 4| + 2(5) = 1 + 10 = 11

.

Step-by-Step Solution

1

Set up the two algebraic cases for the absolute value equation

|5 - 2x| = 3x - 10

.

Case 1:

5 - 2x = 3x - 10

when

5 - 2x \ge 0

(

x \le 2.5

).
Case 2:

-(5 - 2x) = 3x - 10

when

5 - 2x < 0

(

x > 2.5

).

An absolute value expression

|u|

equals

u

when

u \ge 0

and

-u

when

u < 0

.

2

Solve Case 1 algebraically.

5 - 2x = 3x - 10 \implies 15 = 5x \implies x = 3

.

Isolate the variable

x

on one side of the equation.

3

Solve Case 2 algebraically.

2x - 5 = 3x - 10 \implies 5 = x \implies x = 5$.

Simplify and isolate

x

.

4

Check both candidate solutions in the original equation to filter out extraneous roots.

For

x = 3

:

|5 - 2(3)| = 3(3) - 10 \implies |-1| = -1 \implies 1 = -1

(False, extraneous).
For

x = 5

:

|5 - 2(5)| = 3(5) - 10 \implies |-5| = 5 \implies 5 = 5

(True, valid).

Since the right-hand side

3x - 10

must be non-negative for the absolute value to hold, candidate solutions must be tested in the original equation.

5

Substitute the valid solution

x = 5

into the targeted expression

|x - 4| + 2x

.

|5 - 4| + 2(5) = |1| + 10 = 1 + 10 = 11

.

Evaluate the target expression using the single real value

x = 5

that satisfies the given linear absolute value equation.

Key Concept

Solving absolute value linear equations and identifying extraneous solutions

Estimated Time:2m 0s

Question 946Question →

A market research firm surveyed $200$ smartphone owners regarding two security features enabled on their devices: Facial Recognition and Fingerprint Scanning. Exactly $140$ owners enabled Facial Recognition, $110$ owners enabled Fingerprint Scanning, and $30$ owners enabled neither feature. How many of the surveyed smartphone owners enabled Facial Recognition ONLY?

30

60

80

90

140

Show answer & explanation

Answer: 60

Answer

60 smartphone owners enabled Facial Recognition only.

Out of

200

owners,

30

used neither feature, leaving

170

owners who enabled at least one feature. Using the overlapping set formula

\text{At Least One} = \text{Facial} + \text{Fingerprint} - \text{Both}

, we get

170 = 140 + 110 - \text{Both}

, so

\text{Both} = 80

. To find the number of owners who enabled Facial Recognition ONLY, we subtract the overlap from the total Facial Recognition count:

140 - 80 = 60

.

Step-by-Step Solution

1

Calculate the total number of owners who enabled at least one security feature.

200 - 30 = 170

owners.

Owners who used neither feature must be subtracted from the total universe of surveyed owners.

2

Apply the Overlapping Sets formula to find the number of owners who enabled both features.

\text{Total At Least One} = \text{Facial} + \text{Fingerprint} - \text{Both}

170 = 140 + 110 - \text{Both} \implies \text{Both} = 250 - 170 = 80

.

Summing individual sets double-counts the intersection area.

3

Subtract the intersection from the total Facial Recognition set to find Facial Recognition only.

140 - 80 = 60

owners.

The Facial Recognition group consists of those who enabled Facial Recognition only plus those who enabled both features.

Key Concept

Two-Set Overlapping Sets Formula: Total = Set A + Set B - Both + Neither

Question 947Question →

A positive integer $n$ can be expressed in the form $p^3 q^2$ , where $p$ and $q$ are distinct prime numbers. If $n$ is divisible by 45 and is a factor of 4,050, what is the value of $n$ ?

Show answer & explanation

Answer: 675

Answer

675

Prime factorizing

45 = 3^2 \cdot 5

and

4,050 = 2 \cdot 3^4 \cdot 5^2

shows that

n

must be composed of the prime factors 3 and 5. The two candidate values for

p^3 q^2

are

3^3 \cdot 5^2 = 675

and

5^3 \cdot 3^2 = 1,125

. Both are multiples of 45, but only 675 is a factor of 4,050 because its power of 5 does not exceed

5^2

.

Step-by-Step Solution

1

Determine the prime factorizations of 45 and 4,050

45 = 3^2 \cdot 5^1

and

4,050 = 2^1 \cdot 3^4 \cdot 5^2

Decomposing the given numbers into prime factorizations determines the prime building blocks for

n

.

2

Identify the distinct prime factors

p

and

q

The primes

p

and

q

must be 3 and 5

Since

n

is divisible by 45, its prime factorization must contain at least

3^2

and

5^1

. Since

n = p^3 q^2

has exactly two distinct prime factors, those factors must be 3 and 5.

3

Evaluate both possible assignments for

p

and

q

n = 3^3 \cdot 5^2 = 675

or

n = 5^3 \cdot 3^2 = 1,125

Assigning 3 to

p

and 5 to

q

yields 675, while assigning 5 to

p

and 3 to

q

yields 1,125.

4

Filter using the constraint that

n

is a factor of 4,050

675 is a factor of 4,050; 1,125 is not a factor of 4,050

For a number to be a factor of

4,050 = 2^1 \cdot 3^4 \cdot 5^2

, the exponent of prime 5 cannot exceed 2. In 1,125, the exponent of 5 is 3, which makes 1,125 invalid.

Key Concept

Prime Factorization, Divisibility Rules, and Factors of Integers

Question 948Question →

If $y$ is a real number satisfying the inequality $\frac{|3y - 6|}{y + 2} < 1$ , which of the following represents the complete set of all possible values of $y$ ?

y < -2

or

1 < y < 4

1 < y < 4

-2 < y < 4

y < 1

or

y > 4

y < -2

or

y > 1

Show answer & explanation

Answer: $y < -2$ or $1 < y < 4$

Answer

y < -2

or

1 < y < 4

To evaluate

\frac{|3y - 6|}{y + 2} < 1

, analyze the sign of the denominator

y + 2

. If

y < -2

, the denominator is negative, so multiplying both sides by

y + 2

reverses the inequality to

|3y - 6| > y + 2

. Because absolute values are always non-negative and

y + 2

is negative for

y < -2

, this inequality holds for all

y < -2

. If

y > -2

, the denominator is positive, yielding

|3y - 6| < y + 2

, which expands to

-y - 2 < 3y - 6 < y + 2

. Solving the left inequality gives

y > 1

, and solving the right gives

y < 4

, producing

1 < y < 4

. Combining both valid cases gives

y < -2

or

1 < y < 4

.

Step-by-Step Solution

1

Determine the domain restriction and split into cases based on the denominator's sign.

The expression is undefined when

y = -2

. We analyze Case 1 (

y > -2

) and Case 2 (

y < -2

).

Multiplying an inequality by an algebraic expression requires knowing its sign to preserve or reverse the inequality direction.

2

Solve Case 1 where

y > -2

(positive denominator).

Multiplying by

y + 2

gives

|3y - 6| < y + 2

, which expands to

-(y + 2) < 3y - 6 < y + 2

. Solving

-y - 2 < 3y - 6

gives

4 < 4y \implies y > 1

. Solving

3y - 6 < y + 2

gives

2y < 8 \implies y < 4

. Combining gives

1 < y < 4

.

Since

y + 2 > 0

, multiplying preserves the inequality sign.

3

Solve Case 2 where

y < -2

(negative denominator).

Multiplying by

y + 2

flips the inequality sign to

|3y - 6| > y + 2

. Since

|3y - 6| \ge 0

for all real

y

and

y + 2 < 0

when

y < -2

, a non-negative number is always strictly greater than a negative number. Thus, all

y < -2

are valid solutions.

Any non-negative real value is strictly greater than any negative value.

4

Combine the valid intervals from both cases.

y < -2

or

1 < y < 4

.

The complete solution set is the union of solutions from Case 1 and Case 2.

Key Concept

Solving Rational Absolute Value Inequalities via Denominator Sign Case Analysis

Estimated Time:2m 0s

Question 949Question →

A passenger train and a cargo train both travel along parallel tracks on a $120$ -mile route between Station P and Station Q. The passenger train travels at a constant speed of $50$ miles per hour for the entire trip. The cargo train completes the first $60\%$ of the total distance at a constant speed of $30$ miles per hour and the remaining $40\%$ of the distance at a constant speed of $60$ miles per hour. By what percentage is the average speed of the passenger train for the entire trip greater than the average speed of the cargo train for the entire trip?

11\frac{1}{9}\%

20\%

25\%

33\frac{1}{3}\%

40\%

Show answer & explanation

Answer: $33\frac{1}{3}\%$

Answer

The average speed of the passenger train is

33\frac{1}{3}\%

greater than the average speed of the cargo train.

The correct answer accurately determines the total travel time of the cargo train (

3.2

hours) to find its true average speed (

37.5

mph). Comparing the passenger train's speed (

50

mph) against this baseline gives

\frac{50 - 37.5}{37.5} = \frac{1}{3}

, which corresponds to

33\frac{1}{3}\%

.

Step-by-Step Solution

1

Calculate the travel time and average speed of the passenger train.

The passenger train travels

120

miles at

50

mph, so its average speed is

50

mph.

The passenger train maintains a single constant speed throughout the trip.

2

Calculate the distance and travel time for each leg of the cargo train's trip.

Leg 1 distance:

60\%

of

120 = 72

miles; Leg 1 time:

\frac{72}{30} = 2.4

hours. Leg 2 distance:

40\%

of

120 = 48

miles; Leg 2 time:

\frac{48}{60} = 0.8

hours.

Time equals distance divided by speed for each specific segment.

3

Determine the total travel time and overall average speed of the cargo train.

Total time =

2.4 + 0.8 = 3.2

hours. Cargo train average speed =

\frac{120\text{ miles}}{3.2\text{ hours}} = 37.5

mph.

Average speed for a multi-leg journey is strictly defined as total distance divided by total time.

4

Calculate by what percentage the passenger train's average speed is greater than the cargo train's average speed.

Percentage increase =

\frac{50 - 37.5}{37.5} \times 100\% = \frac{12.5}{37.5} \times 100\% = \frac{1}{3} \times 100\% = 33\frac{1}{3}\%

.

The baseline for 'percent greater than the cargo train' is the cargo train's speed (

37.5

mph).

Key Concept

Average Speed for Multi-Leg Journeys and Percentage Difference Baseline

Question 950Question →

A commercial retailer purchased 100 identical smartwatches for a total cost of $\$10,000$ . Before any sales were made, 10 smartwatches were damaged and could not be sold. The retailer marked up the unit cost of each remaining smartwatch by $P$ percent to determine its list price. During a promotional period, 50 smartwatches were sold at the full list price. The remaining 40 undamaged smartwatches were subsequently sold during a clearance sale at a 20 percent discount off the list price. If the retailer earned a total net profit of 23 percent on the initial investment of $\$10,000$ , what is the value of $P$ ?

35

40

45

50

60

Show answer & explanation

Answer: 50

Answer

50

The unit cost of each smartwatch is

\$10,000 / 100 = \$100

. A markup of

P\%

sets the list price to

L = 100 + P

. The retailer sells 50 units at full list price

L

and 40 units at

0.80L

, earning total revenue of

50L + 32L = 82L = 82(100 + P) = 8,200 + 82P

. To earn a 23% profit on the entire

\$10,000

investment, total revenue must equal

\$12,300

. Solving

8,200 + 82P = 12,300

yields

82P = 4,100

, so

P = 50

.

Step-by-Step Solution

1

Calculate the unit cost and total required revenue.

Unit cost

c = \frac{\$10,000}{100} = \$100

. Required revenue

R = \$10,000 \times (1 + 0.23) = \$12,300

.

Net profit is 23% of the total initial investment of

\$10,000

.

2

Express the list price

L

and clearance price in terms of

P

.

List price

L = 100 \times \left(1 + \frac{P}{100}\right) = 100 + P

. Clearance price

= 0.80 \times L = 0.80(100 + P) = 80 + 0.8P

.

Markup

P\%

is calculated relative to the unit cost of

\$100

, and clearance discount is 20% off list price.

3

Set up the total revenue equation based on sold units.

Total Revenue

= 50L + 40(0.80L) = 50L + 32L = 82L = 82(100 + P) = 8,200 + 82P

.

10 units are unsellable ($0 revenue), 50 units generate full list price, and 40 units generate 80% of list price.

4

Solve for

P

.

8,200 + 82P = 12,300 \implies 82P = 4,100 \implies P = 50

.

Equating total generated revenue to the required revenue of

\$12,300

.

Key Concept

Profit, Loss, and Markup with Damaged Inventory and Tiered Discounts

Question 951Question →

Container X initially contains $30$ liters of a solution that is $20\%$ acid by volume. Container Y contains $V$ liters of a solution that is $50\%$ acid by volume. If $10$ liters of pure water are added to Container X and the entire contents of Container X are then mixed with Container Y, the resulting mixture is $30\%$ acid by volume. What is the value of $V$ , in liters?

12

15

20

30

40

Show answer & explanation

Answer: $30$

Answer

30

liters

The option stating

30

liters is correct because Container X initially contains

6

liters of acid (

0.20 \times 30

). After adding

10

liters of water, Container X holds

40

liters of solution with

6

liters of acid. Container Y contributes

0.50V

liters of acid out of

V

liters. Equating the total acid ratio to

30\%

yields

(6 + 0.50V) / (40 + V) = 0.30

, which solves cleanly to

V = 30

.

Step-by-Step Solution

1

Calculate the volume of pure acid initially present in Container X.

\text{Acid in X} = 20\% \text{ of } 30 = 0.20 \times 30 = 6 \text{ liters}

.

Determining the absolute amount of solute is required to set up the mixture equation.

2

Determine the new total volume of Container X after adding pure water.

\text{New Volume of X} = 30 + 10 = 40 \text{ liters}

. The acid amount remains

6

liters.

Adding pure water increases the solution's total volume while leaving the acid quantity unchanged.

3

Express the amount of acid in Container Y in terms of

V

.

\text{Acid in Y} = 50\% \text{ of } V = 0.50V \text{ liters}

.

Container Y contains a

50\%

concentration of acid across

V

liters.

4

Formulate and solve the concentration equation for the final combined mixture.

\frac{6 + 0.50V}{40 + V} = 0.30 \implies 6 + 0.50V = 0.30(40 + V) \implies 6 + 0.50V = 12 + 0.30V \implies 0.20V = 6 \implies V = 30

.

Setting total combined acid over total combined volume equal to the final concentration of

30\%

gives the unknown volume

V

.

Key Concept

Two-stage dilution and mixture concentration using weighted average equations

Estimated Time:2m 0s

Question 952Question →

A tech company generates a 5-digit security badge number, represented as $d_1 d_2 d_3 d_4 d_5$ , using digits from $0$ to $9$ , inclusive. The badge numbers must satisfy the following constraints:
- The first digit, $d_1$ , cannot be $0$ or $1$ .
- The second and third digits, $d_2$ and $d_3$ , must be distinct even digits.
- The fourth digit, $d_4$ , must be an odd digit strictly greater than $3$ .
- The fifth digit, $d_5$ , can be any digit except that it cannot be equal to $d_1$ .

How many different 5-digit badge numbers can be created under these rules?

4,320

4,860

5,400

5,760

6,000

Show answer & explanation

Answer: 4,320

Answer

4,320

According to the Fundamental Counting Principle, the total number of configurations is the product of the number of options available at each stage. For the first position, excluding 0 and 1 leaves 8 possible digits. For the second position, any of the 5 even digits can be chosen. For the third position, one of the remaining 4 even digits must be selected to preserve distinctness. For the fourth position, the odd digits strictly greater than 3 are 5, 7, and 9, providing 3 options. Finally, for the fifth position, 9 digits remain available after excluding the specific digit chosen for the first position. Multiplying these independent counts gives 8 × 5 × 4 × 3 × 9 = 4,320.

Step-by-Step Solution

1

Determine the number of available choices for the first digit,

d_1

.

8 choices

The available digits are

0

through

9

(10 total). Excluding

0

and

1

leaves

\{2, 3, 4, 5, 6, 7, 8, 9\}

, which gives

10 - 2 = 8

choices.

2

Determine the number of available choices for the second and third digits,

d_2

and

d_3

.

20 choices total for

(d_2, d_3)

The set of even digits is

\{0, 2, 4, 6, 8\}

, containing 5 digits. There are 5 choices for

d_2

. Since

d_3

must be distinct from

d_2

, there are

5 - 1 = 4

choices for

d_3

. Thus, there are

5 \times 4 = 20

choices.

3

Determine the number of available choices for the fourth digit,

d_4

.

3 choices

The set of odd digits is

\{1, 3, 5, 7, 9\}

. Odd digits strictly greater than

3

are

\{5, 7, 9\}

, which gives 3 choices.

4

Determine the number of available choices for the fifth digit,

d_5

.

9 choices

d_5

can be any digit from

0

through

9

except

d_1

. Out of 10 total digits, excluding 1 digit leaves

10 - 1 = 9

choices.

5

Apply the Fundamental Counting Principle to compute total combinations.

4,320

Multiply the number of independent choices across all 5 stages:

8 \times 5 \times 4 \times 3 \times 9 = 4,320

.

Key Concept

Fundamental Counting Principle with Multi-Stage Positional Restrictions

Question 953Question →

A publishing firm evaluated $160$ manuscript submissions based on three quality criteria: Plot Originality ( $P$ ), Character Development ( $C$ ), and Dialogue Quality ( $D$ ). Each manuscript was evaluated against all three criteria.

- $75$ manuscripts met the criteria for Plot Originality ( $P$ ).
- $80$ manuscripts met the criteria for Character Development ( $C$ ).
- $70$ manuscripts met the criteria for Dialogue Quality ( $D$ ).
- $30$ manuscripts met the criteria for both Plot Originality and Character Development ( $P \cap C$ ).
- $25$ manuscripts met the criteria for both Character Development and Dialogue Quality ( $C \cap D$ ).
- $20$ manuscripts met the criteria for both Plot Originality and Dialogue Quality ( $P \cap D$ ).
- $10$ manuscripts met all three criteria ( $P \cap C \cap D$ ).

How many manuscripts met exactly one of the three criteria?

Show answer & explanation

Answer: 105

Answer

The number of manuscripts that met exactly one of the three criteria is 105.

To find the number of manuscripts meeting exactly one criterion, we isolate the unique regions for each set. Subtracting the overlapping intersections from the total of each set yields 35 manuscripts for Plot Originality only, 35 for Character Development only, and 35 for Dialogue Quality only. Combining these gives 105.

Step-by-Step Solution

1

Find the number of elements in the double-intersection regions only.

Exactly (P and C) = 20; Exactly (C and D) = 15; Exactly (P and D) = 10.

The given two-set intersections include manuscripts that satisfied all three criteria, so the triple intersection of 10 must be subtracted from each.

2

Calculate the count of manuscripts meeting only P, only C, and only D.

Only P = 35, Only C = 35, Only D = 35.

Subtract all shared regions (two-set-only overlaps plus the three-set overlap) from each individual total.

3

Sum the three single-criterion regions.

35 + 35 + 35 = 105.

The question asks specifically for the total number of manuscripts that met exactly one criterion.

Key Concept

3-Set Overlapping Venn Diagrams and Region Separation

Question 954Question →

An artisan workshop produces custom wooden chairs and tables. Each chair requires $3$ hours of carving and $2$ hours of finishing, while each table requires $5$ hours of carving and $4$ hours of finishing. If the workshop logged a total of $110$ hours of carving and $84$ hours of finishing last week, how many tables were produced?

10

14

16

18

22

Show answer & explanation

Answer: 16

Answer

The workshop produced 16 tables.

By defining

c

as the number of chairs and

t

as the number of tables, we can set up two linear equations representing total hours:

3c + 5t = 110

for carving and

2c + 4t = 84

for finishing. Multiplying the second equation by

1.5

yields

3c + 6t = 126

. Subtracting

3c + 5t = 110

from

3c + 6t = 126

leaves

t = 16

. Thus, the workshop produced 16 tables.

Step-by-Step Solution

1

Define variables and translate the word problem into a system of linear equations.

Let

c

be the number of chairs and

t

be the number of tables. Carving equation:

3c + 5t = 110

. Finishing equation:

2c + 4t = 84

.

The total hours for each activity equal the sum of hours spent on chairs and tables.

2

Eliminate variable

c

to solve for

t

.

Multiply the finishing equation by

1.5

to get

3c + 6t = 126

. Subtract the carving equation (

3c + 5t = 110

) from this equation:

(3c + 6t) - (3c + 5t) = 126 - 110 \implies t = 16

.

Aligning the coefficient of

c

in both equations allows direct elimination of

c

to isolate

t

.

3

Verify the solution by calculating

c

and checking both original equations.

Substitute

t = 16

into

2c + 4(16) = 84 \implies 2c + 64 = 84 \implies 2c = 20 \implies c = 10

. Check carving:

3(10) + 5(16) = 30 + 80 = 110

.

Ensures that

t = 16

and

c = 10

satisfy both resource constraints without calculation errors.

Key Concept

Setting up and solving a system of two linear equations in two variables

Hints

1

Set up two separate linear equations: one for total carving hours and one for total finishing hours.

2

Let

c

be the number of chairs and

t

be the number of tables. Your system is

3c + 5t = 110

and

2c + 4t = 84

.

3

Multiply

2c + 4t = 84

by

1.5

to get

3c + 6t = 126

, then subtract

3c + 5t = 110

to find

t

directly.

Practice More

Try solving a similar problem where the total revenue and total unit count are given to practice standard linear system modeling.

Alternative Method

Divide the finishing equation

2c + 4t = 84

by

2

to get

c + 2t = 42 \implies c = 42 - 2t

. Substitute this into the carving equation:

3(42 - 2t) + 5t = 110 \implies 126 - 6t + 5t = 110 \implies -t = -16 \implies t = 16

.

Estimated Time:1m 30s

Question 955Question →

A medical research laboratory creates a 4-digit identification code, $d_1 d_2 d_3 d_4$ , for sample trays using digits selected from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ . The digits in each code must satisfy the following conditions:
- The first digit, $d_1$ , must be an odd prime number.
- The second digit, $d_2$ , must be an even integer.
- All four digits in the code must be distinct.

How many different identification codes can be created under these rules?

18

360

480

504

768

Show answer & explanation

Answer: 360

Answer

360

To form the 4-digit code, we calculate the number of options at each sequential stage. First,

d_1

must be an odd prime from

\{1, 2, 3, 4, 5, 6, 7, 8\}

, which gives

3

options (

3, 5, 7

). Second,

d_2

must be an even integer from the set, giving

4

options (

2, 4, 6, 8

). Third, because all digits must be distinct, picking

d_1

and

d_2

uses

2

digits, leaving

8 - 2 = 6

choices for

d_3

and

5

choices for

d_4

. By the Fundamental Counting Principle, the total number of codes is

3 \times 4 \times 6 \times 5 = 360

.

Step-by-Step Solution

1

Determine the number of valid choices for the first digit (

d_1

)

3 choices (

3, 5, 7

)

The odd prime numbers in the given set

\{1, 2, 3, 4, 5, 6, 7, 8\}

are

3, 5,

and

7

. Note that

1

is not prime and

2

is an even prime.

2

Determine the number of valid choices for the second digit (

d_2

)

4 choices (

2, 4, 6, 8

)

The even integers in the given set are

2, 4, 6,

and

8

. Notice that the set of choices for

d_1

and

d_2

are disjoint.

3

Determine the number of valid choices for the third digit (

d_3

) and fourth digit (

d_4

)

6 choices for

d_3

and 5 choices for

d_4

Since all 4 digits must be distinct, 2 distinct digits have already been used for

d_1

and

d_2

. From the original set of 8 digits,

8 - 2 = 6

digits remain for

d_3

, and then

5

digits remain for

d_4

.

4

Apply the Fundamental Counting Principle to calculate the total number of codes

3 \times 4 \times 6 \times 5 = 360

The overall number of distinct multi-stage decisions is found by multiplying the number of choices available at each stage.

Key Concept

Fundamental Counting Principle with Restricted and Non-Repeating Selections

Estimated Time:2m 0s

Question 956Question →

If $x$ is a positive real number such that $x^{\frac{1}{2}} + x^{-\frac{1}{2}} = 4$ , what is the value of $x^{\frac{3}{2}} + x^{-\frac{3}{2}}$ ?

48

52

56

61

64

Show answer & explanation

Answer: 52

Answer

52

Cubing both sides of

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

yields

(x^{\frac{1}{2}} + x^{-\frac{1}{2}})^3 = 64

. By the identity

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

, the left side expands to

(x^{\frac{3}{2}} + x^{-\frac{3}{2}}) + 3(x^{\frac{1}{2}} \cdot x^{-\frac{1}{2}})(x^{\frac{1}{2}} + x^{-\frac{1}{2}})

. Substituting

x^{\frac{1}{2}} \cdot x^{-\frac{1}{2}} = 1

and

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

gives

(x^{\frac{3}{2}} + x^{-\frac{3}{2}}) + 12 = 64

. Subtracting 12 yields 52.

Step-by-Step Solution

1

Set up the cubic identity for the sum of fractional exponents.

Let

a = x^{\frac{1}{2}}

and

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

. Then

a + b = 4

and

ab = x^{\frac{1}{2}} \cdot x^{-\frac{1}{2}} = 1

.

Recognizing that

x^{\frac{3}{2}} = a^3

and

x^{-\frac{3}{2}} = b^3

allows the use of binomial expansion.

2

Cube both sides of the given equation

a + b = 4

.

(a + b)^3 = a^3 + b^3 + 3ab(a + b) \implies 4^3 = (x^{\frac{3}{2}} + x^{-\frac{3}{2}}) + 3(1)(4)

.

Expanding the cube retains the desired expression

a^3 + b^3

alongside simpler terms.

3

Simplify and solve for

x^{\frac{3}{2}} + x^{-\frac{3}{2}}

.

64 = (x^{\frac{3}{2}} + x^{-\frac{3}{2}}) + 12 \implies x^{\frac{3}{2}} + x^{-\frac{3}{2}} = 64 - 12 = 52.

Subtracting 12 from 64 gives the exact value of the expression.

Key Concept

Algebraic identities with fractional exponents

Estimated Time:2m 0s

Question 957Question →

Set $S$ consists of $n$ consecutive odd integers, ordered from least to greatest. The arithmetic mean of the integers in set $S$ is $42$ . If the sum of the smallest integer and the largest integer in set $S$ divided by $n$ equals $7$ , what is the largest integer in set $S$ ?

31

48

53

55

61

Show answer & explanation

Answer: 53

Answer

The largest integer in set

S

is 53.

For an evenly spaced set of numbers, the arithmetic mean is the average of the first and last terms. Therefore,

(a_1 + a_n) / 2 = 42

, which means

a_1 + a_n = 84

. Using the given condition

(a_1 + a_n) / n = 7

, we substitute

84 / n = 7

to find

n = 12

. Since consecutive odd integers increase by 2, the difference between the 12th term and the 1st term is

(12 - 1) \times 2 = 22

. Solving the system

a_n + a_1 = 84

and

a_n - a_1 = 22

yields

2a_n = 106

, so the largest integer

a_n

is 53.

Step-by-Step Solution

1

Express the sum of the smallest and largest integers using the mean.

Sum of smallest and largest integers

= 84

In any set of consecutive integers (an arithmetic progression), the mean is equal to the average of the first term

a_1

and the last term

a_n

:

\frac{a_1 + a_n}{2} = 42 \implies a_1 + a_n = 84

.

2

Determine the number of elements

n

in set

S

.

n = 12

The stem specifies that

\frac{a_1 + a_n}{n} = 7

. Substituting

a_1 + a_n = 84

gives

\frac{84}{n} = 7 \implies n = 12

.

3

Calculate the difference between the largest and smallest integers.

a_n - a_1 = 22

In a sequence of

n

consecutive odd integers, the common difference between terms is

2

. The number of gaps between the 1st term and the

n

-th term is

n - 1 = 11

. Thus,

a_n - a_1 = 11 \times 2 = 22

.

4

Solve for the largest integer

a_n

.

a_n = 53

Adding the equations

a_n + a_1 = 84

and

a_n - a_1 = 22

yields

2a_n = 106 \implies a_n = 53

.

Key Concept

Properties of consecutive odd integers and evenly spaced sets (mean-median equivalence and indexing)

Estimated Time:1m 30s

Question 958Question →

A corporate compliance archive contains $10$ confidential files and $5$ public files. An auditor randomly selects two files from the archive, one after another without replacement. What is the probability that at least one of the selected files is public?

\frac{2}{21}

\frac{10}{21}

\frac{11}{21}

\frac{5}{9}

\frac{4}{7}

Show answer & explanation

Answer: $\frac{4}{7}$

Answer

The probability that at least one of the selected files is public is

\frac{4}{7}

.

To find the probability that at least one file is public, subtract the probability that no public files are selected (meaning both files are confidential) from 1. The probability of choosing a confidential file on the first draw is

\frac{10}{15} = \frac{2}{3}

. Since the selection is without replacement,

9

confidential files remain out of

14

total files. The probability of choosing a confidential file on the second draw is

\frac{9}{14}

. The joint probability of drawing two confidential files is

\frac{2}{3} \times \frac{9}{14} = \frac{3}{7}

. Subtracting this from 1 yields

1 - \frac{3}{7} = \frac{4}{7}

.

Step-by-Step Solution

1

Identify the complementary event

The complementary event to 'at least one selected file is public' is 'both selected files are confidential'.

Calculating the probability of the complement is simpler than summing the probabilities of all positive individual outcomes.

2

Calculate the probability of drawing two confidential files sequentially without replacement

The probability of drawing a confidential file on the first selection is

\frac{10}{15} = \frac{2}{3}

. After removing one confidential file, 9 confidential files remain out of 14 total files. The probability on the second selection is

\frac{9}{14}

. Thus,

P(\text{both confidential}) = \frac{10}{15} \times \frac{9}{14} = \frac{2}{3} \times \frac{9}{14} = \frac{3}{7}

.

Because the drawing is done without replacement, the outcome of the first draw affects the total sample space and event space for the second draw.

3

Subtract the complementary probability from 1

P(\text{at least one public}) = 1 - P(\text{both confidential}) = 1 - \frac{3}{7} = \frac{4}{7}

.

The sum of the probabilities of an event and its complement equals 1.

Key Concept

Probability of Dependent Events and Complementary Probability

Estimated Time:1m 30s

Question 959Question →

If $x$ is an integer such that $|x - 3| + |x + 5| \le 12$ , how many possible values of $x$ exist?

Show answer & explanation

Answer: 13

Answer

13

Evaluating the inequality

|x - 3| + |x + 5| \le 12

across the three intervals defined by critical points

x = -5

and

x = 3

yields the continuous real solution set

[-7, 5]

. Counting all integers in this inclusive range gives

5 - (-7) + 1 = 13

possible values.

Step-by-Step Solution

1

Identify the critical points of the absolute value expressions.

The critical points where the expressions inside the absolute values change sign are

x = 3

and

x = -5

.

Setting

x - 3 = 0

gives

x = 3

, and setting

x + 5 = 0

gives

x = -5

.

2

Solve the inequality for the region

x < -5

.

-(x - 3) - (x + 5) \le 12 \implies -2x - 2 \le 12 \implies -2x \le 14 \implies x \ge -7

. Thus,

-7 \le x < -5

.

When

x < -5

, both

x - 3 < 0

and

x + 5 < 0

, so

|x - 3| = -(x - 3)

and

|x + 5| = -(x + 5)

.

3

Solve the inequality for the region

-5 \le x \le 3

.

-(x - 3) + (x + 5) \le 12 \implies 8 \le 12

, which is universally true for all

x

in this interval.

When

-5 \le x \le 3

,

x - 3 \le 0

and

x + 5 \ge 0

, so

|x - 3| = -(x - 3)

and

|x + 5| = x + 5

.

4

Solve the inequality for the region

x > 3

.

(x - 3) + (x + 5) \le 12 \implies 2x + 2 \le 12 \implies 2x \le 10 \implies x \le 5

. Thus,

3 < x \le 5

.

When

x > 3

, both

x - 3 > 0

and

x + 5 > 0

.

5

Combine the valid intervals and count the integer solutions.

The full solution range is

[-7, 5]

. The number of integer values is

5 - (-7) + 1 = 13

.

The number of integers in an inclusive range

[a, b]

is given by

b - a + 1

.

Key Concept

Solving multi-term absolute value inequalities by dividing the domain at critical points into distinct cases.

Question 960Question →

An athlete completes a three-leg training course consisting of cycling, running, and swimming, covering a total distance of $60$ miles in a total time of $4$ hours. The athlete cycles at a constant speed of $24$ miles per hour, runs at a constant speed of $10$ miles per hour, and swims at a constant speed of $2$ miles per hour. If the time spent cycling is equal to the combined time spent running and swimming, how many miles did the athlete run?

Show answer & explanation

Answer: 10

Answer

The athlete ran 10 miles.

Because cycling time equals combined running and swimming time, the 4-hour total splits evenly into 2 hours for cycling and 2 hours for the remaining legs. Cycling at 24 mph covers 48 miles, leaving 12 miles and 2 hours for running and swimming. Writing the total time equation as (d_running / 10) + ((12 - d_running) / 2) = 2 and solving yields d_running = 10 miles.

Step-by-Step Solution

1

Determine the time spent cycling

2 hours

Since the total time is 4 hours and cycling time equals the sum of running and swimming times, 2 * t_cycling = 4.

2

Calculate the distance covered while cycling

48 miles

Distance = Speed * Time = 24 mph * 2 hours = 48 miles.

3

Determine the combined distance and time for running and swimming

Combined distance = 12 miles, Combined time = 2 hours

Total distance minus cycling distance is 60 - 48 = 12 miles; total time minus cycling time is 4 - 2 = 2 hours.

4

Set up and solve the system of equations for running distance

Running distance = 10 miles

Using time = distance / speed gives (d_running / 10) + ((12 - d_running) / 2) = 2. Multiplying by 10 yields d_running + 60 - 5 * d_running = 20, leading to d_running = 10 miles.

Key Concept

Multi-leg rate, time, and distance problems using systems of linear equations.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Hints

Practice More

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

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Common Mistakes

Answer

Step-by-Step Solution

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Common Mistakes

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Common Mistakes

Answer

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Common Mistakes