All practice questions

Question 321Question →

A specialized coffee roastery produces three custom blends—Roast Alpha, Roast Beta, and Roast Gamma—using three varieties of single-origin beans: Grade A, Grade B, and Grade C.

- One batch of Roast Alpha requires 3 kg of Grade A, 1 kg of Grade B, and 2 kg of Grade C beans, and has a total raw material cost of $64. - One batch of Roast Beta requires 1 kg of Grade A, 4 kg of Grade B, and 2 kg of Grade C beans, and has a total raw material cost of$ 64.
- One batch of Roast Gamma requires 2 kg of Grade A, 2 kg of Grade B, and 5 kg of Grade C beans, and has a total raw material cost of $90.

What is the cost, in dollars, of 1 kg of Grade A beans?

Show answer & explanation

Answer: 12

Answer

The cost of 1 kg of Grade A beans is 12 dollars.

Setting up the 3-variable linear system

3x + y + 2z = 64

,

x + 4y + 2z = 64

, and

2x + 2y + 5z = 90

allows us to eliminate

z

by subtracting the second equation from the first, yielding

2x - 3y = 0

, or

x = 1.5y

. Substituting this relationship back into the system leads to

y = 8

and

x = 12

. Thus, 1 kg of Grade A beans costs 12 dollars.

Step-by-Step Solution

1

Formulate linear equations representing the total cost of each coffee blend batch.

Let

x

be the cost per kg of Grade A beans,

y

be the cost per kg of Grade B beans, and

z

be the cost per kg of Grade C beans:

\begin{aligned} (1)\quad 3x + y + 2z &= 64 \\ (2)\quad x + 4y + 2z &= 64 \\ (3)\quad 2x + 2y + 5z &= 90 \end{aligned}

Translating the word problem into a system of 3 linear equations with 3 variables.

2

Eliminate variable

z

by subtracting Equation (2) from Equation (1).

(3x + y + 2z) - (x + 4y + 2z) = 64 - 64 \implies 2x - 3y = 0 \implies x = 1.5y

Since both equations (1) and (2) contain the term

+2z

, subtracting them removes

z

directly and provides a simple relation between

x

and

y

.

3

Substitute

x = 1.5y

into Equation (1) and Equation (3) to obtain a system in terms of

y

and

z

.

From Equation (1):

3(1.5y) + y + 2z = 64 \implies 5.5y + 2z = 64 \implies 11y + 4z = 128 \quad (4)

From Equation (3):

2(1.5y) + 2y + 5z = 90 \implies 5y + 5z = 90 \implies y + z = 18 \implies z = 18 - y

Reducing the system from 3 variables down to 2 variables.

4

Substitute

z = 18 - y

into Equation (4) to solve for

y

, and subsequently calculate

x

.

11y + 4(18 - y) = 128 \implies 7y + 72 = 128 \implies 7y = 56 \implies y = 8

Using

x = 1.5y

:

x = 1.5 \times 8 = 12

Solving the single-variable linear equation for

y

, then substituting back to find the required cost

x

for Grade A beans.

Key Concept

Solving a 3-Variable System of Linear Equations via Variable Elimination

Question 322Question →

A specialty coffee roaster creates a signature espresso blend by combining Arabica coffee beans costing $18.00 per kilogram with Robusta coffee beans costing$ 12.00 per kilogram. The final blend weighs 50 kilograms and has an overall average cost of $15.60 per kilogram. How many kilograms of Arabica beans are included in the blend?

Show answer & explanation

Answer: 30

Answer

The blend contains 30 kilograms of Arabica coffee beans.

To determine the required quantity of Arabica beans, establish the total cost equation:

18A + 12(50 - A) = 50(15.60)

. Expanding the terms gives

18A + 600 - 12A = 780

, which simplifies to

6A = 180

, yielding

A = 30

kilograms.

Step-by-Step Solution

1

Define variables and relate component weights

Let

A

represent the mass of Arabica beans in kilograms. The mass of Robusta beans is

(50 - A)

kilograms.

The sum of the individual component weights must equal the total blend weight of 50 kilograms.

2

Formulate the weighted average total cost equation

18A + 12(50 - A) = 15.60 \times 50 = 780

The combined monetary cost of both bean types equals the total value of the 50 kg blend.

3

Solve for the unknown variable

A

6A + 600 = 780 \implies 6A = 180 \implies A = 30

Simplifying the algebraic linear equation determines the exact weight of Arabica beans.

Key Concept

Weighted Averages in Applied Contexts

Question 323Question →

At the beginning of 2021, a biotechnology firm allocated a fixed annual budget to its principal research laboratory. In 2022, the laboratory's budget was increased by $20\%$ relative to its 2021 budget. In 2023, the budget was reduced by $15\%$ from its 2022 level. In 2024, the budget was increased by $25\%$ over its 2023 level. If the budget in 2024 exceeded the budget in 2021 by $\$55,000$ , what was the laboratory's budget in 2021, in dollars?

Show answer & explanation

Answer: 200000

Answer

200,000 dollars

The initial 2021 budget

B

undergoes successive percentage changes over three years. A

20\%

increase in 2022 results in

1.20B

. A

15\%

decrease in 2023 yields

1.20B \times 0.85 = 1.02B

. A

25\%

increase in 2024 yields

1.02B \times 1.25 = 1.275B

. The net increase over the initial budget is

1.275B - B = 0.275B

. Setting

0.275B = 55,000

gives

B = \frac{55,000}{0.275} = 200,000

dollars.

Step-by-Step Solution

1

Represent the annual budgets sequentially in terms of the initial 2021 budget

B

2022 budget =

1.20B

, 2023 budget =

1.20B \times 0.85 = 1.02B

, 2024 budget =

1.02B \times 1.25 = 1.275B

Calculate successive percentage changes sequentially by multiplying the respective multipliers for each period

2

Calculate the net change in budget from 2021 to 2024

1.275B - B = 0.275B

Determine how much the final year's budget exceeds the initial base budget

3

Set up and solve the linear equation for

B

0.275B = 55,000 \implies B = \frac{55,000}{0.275} = 200,000

Equate the net algebraic difference to the given dollar amount to solve for the initial 2021 budget

Key Concept

Successive Percent Change and Base Value Tracking

Question 324Question →

Let $N = 504$ . If $M$ is the smallest positive integer such that $N \times M$ is a perfect cube, what is the value of $M$ ?

Show answer & explanation

Answer: 147

Answer

The smallest positive integer

M

such that

504 \times M

is a perfect cube is 147.

For an integer to be a perfect cube, the exponent of each prime factor in its prime factorization must be a multiple of 3. Prime factorizing 504 yields

2^3 \times 3^2 \times 7^1

. The exponent of 2 is 3 (already a multiple of 3). The exponent of 3 is 2, which requires 1 additional factor of 3 to reach 3. The exponent of 7 is 1, which requires 2 additional factors of 7 (

7^2

) to reach 3. Therefore, the minimum value for

M

is

3^1 \times 7^2 = 3 \times 49 = 147

.

Step-by-Step Solution

1

Express 504 as a product of its prime factors.

504 = 2^3 \times 3^2 \times 7^1

Decomposing

N

into prime factors allows analysis of the exponents required for perfect power conditions.

2

Apply the prime exponent rule for perfect cubes.

Every prime factor in

N \times M

must have an exponent that is a multiple of 3.

A number

K

is a perfect cube if and only if

K = p_1^{3a} \times p_2^{3b} \times \dots

3

Calculate the missing prime factors needed to complete the cube.

M = 3^{3-2} \times 7^{3-1} = 3^1 \times 7^2

To minimize

M

, we raise each prime to the smallest non-negative power that rounds the existing exponent up to the nearest multiple of 3.

4

Evaluate the value of

M

.

M = 3 \times 49 = 147

Direct arithmetic computation.

Key Concept

Prime Factorization and Exponent Requirements for Perfect Powers

Estimated Time:1m 30s

Question 325Question →

An executive board consisting of 8 distinct members needs to form a subcommittee of 4 members. However, board members Alex and Blair refuse to serve on the subcommittee together unless board member Morgan is also selected. How many different 4-member subcommittees can be formed under these conditions?

Show answer & explanation

Answer: 60

Answer

60

The solution uses complementary counting. First, calculate the total possible 4-member subcommittees from 8 members without restrictions, which is

\binom{8}{4} = 70

. Second, identify the restricted scenario that is not allowed: Alex and Blair are both selected, but Morgan is excluded. In this invalid scenario, 2 spots are taken by Alex and Blair, Morgan is excluded from consideration, leaving 2 spots to be filled from the remaining 5 board members, which equals

\binom{5}{2} = 10

invalid subcommittees. Subtracting the invalid subcommittees from the total gives

70 - 10 = 60

valid subcommittees.

Step-by-Step Solution

1

Calculate the total number of ways to choose a 4-member subcommittee from 8 members without restrictions.

The total unrestricted combinations is

\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70

.

Since the order of selecting members into a subcommittee does not matter, use the combination formula

\binom{n}{k}

.

2

Determine the condition under which a subcommittee selection is invalid.

A subcommittee is invalid if and only if both Alex and Blair are selected AND Morgan is excluded.

Alex and Blair agree to serve together only if Morgan is also present. Thus, having Alex and Blair together without Morgan violates the condition.

3

Calculate the number of invalid subcommittees.

The number of invalid subcommittees is

\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10

.

Alex and Blair take 2 of the 4 spots, and Morgan cannot take any spot. The remaining 2 spots must be filled from the remaining

8 - 3 = 5

members.

4

Subtract the invalid subcommittees from the total unrestricted subcommittees using complementary counting.

70 - 10 = 60

valid subcommittees.

Complementary counting yields the total number of subcommittees that satisfy the restriction.

Key Concept

Combinations with Restrictions and Complementary Counting

Estimated Time:2m 0s

Question 326Question →

A financial firm has a team of 10 auditors consisting of 6 certified public accountants (CPAs) and 4 audit assistants. A special audit task force of 3 auditors is to be selected at random from the team. If the probability that the task force contains at least one CPA is expressed as a fraction $\frac{a}{b}$ in simplest form, what is the value of $a + b$ ?

Show answer & explanation

Answer: 59

Answer

59

To find the probability of selecting at least one CPA, it is most efficient to use the complementary probability formula:

P(\text{at least 1 CPA}) = 1 - P(\text{no CPAs})

. The total number of ways to select any 3 auditors from the team of 10 is given by

\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120

. The number of ways to select 3 auditors such that none are CPAs (i.e., all 3 are audit assistants) is

\binom{4}{3} = 4

. Therefore, the probability of choosing zero CPAs is

\frac{4}{120} = \frac{1}{30}

. Subtracting from 1 gives

1 - \frac{1}{30} = \frac{29}{30}

. The fraction

\frac{29}{30}

is in simplest form because 29 is a prime number and does not divide 30. Thus,

a = 29

and

b = 30

, making

a + b = 59

.

Step-by-Step Solution

1

Calculate total possible combinations for selecting 3 members from 10.

\binom{10}{3} = 120

Selection is made without replacement and order of selection does not matter.

2

Calculate combinations of selecting 3 members with no CPAs.

\binom{4}{3} = 4

All 3 selected members must come from the pool of 4 audit assistants.

3

Determine probability of the complement event (no CPAs).

P(\text{no CPA}) = \frac{4}{120} = \frac{1}{30}

Probability is favorable outcomes over total outcomes.

4

Compute probability of at least one CPA.

P(\text{at least 1 CPA}) = 1 - \frac{1}{30} = \frac{29}{30}

P(\text{at least one}) = 1 - P(\text{none}).

5

Find the sum of numerator a and denominator b in simplest form.

a + b = 29 + 30 = 59

29/30 cannot be simplified further as 29 is a prime number.

Key Concept

Complementary Probability and At-Least-One Scenarios

Question 327Question →

An express delivery drone flies from Station Alpha to Station Beta at a constant speed of $60$ miles per hour against a headwind. On the return flight along the exact same path from Station Beta to Station Alpha, with the wind acting as a tailwind of identical strength, the drone travels at a constant speed of $90$ miles per hour. If the entire round-trip flight took a total of $5$ hours, what is the distance, in miles, between Station Alpha and Station Beta?

Show answer & explanation

Answer: 180

Answer

The distance between Station Alpha and Station Beta is 180 miles.

To find the distance

D

, set up the total time equation:

\frac{D}{60} + \frac{D}{90} = 5

. Finding the common denominator yields

\frac{5D}{180} = 5

, leading directly to

D = 180

miles.

Step-by-Step Solution

1

Define the unknown variable and write time expressions for each leg.

Let

D

be the distance between Station Alpha and Station Beta. Outbound time is

\frac{D}{60}

hours and return time is

\frac{D}{90}

hours.

Using the rate-time-distance formula

t = \frac{d}{r}

allows expressing unknown time components using distance.

2

Set up an equation for total round-trip time.

\frac{D}{60} + \frac{D}{90} = 5

The sum of the time spent on the outbound leg and the return leg equals the total given flight time of

5

hours.

3

Solve the algebraic equation for

D

.

Finding a common denominator of

180

gives

\frac{3D + 2D}{180} = 5 \implies \frac{5D}{180} = 5 \implies 5D = 900 \implies D = 180

.

Multiplying both sides by

180

clears denominators and isolates

D

.

Key Concept

Rate, Time, and Distance Relationship

Question 328Question →

An investor deposits $\$5,000$ into Account B, which earns interest at a rate of $20\%$ per year compounded annually for $2$ years. The same investor deposits another $\$5,000$ into Account A, which earns simple annual interest at a rate of $r\%$ per year for $3$ years. If the total interest earned from Account B exceeds the total interest earned from Account A by $\$400$ , what is the value of $r$ ?

Show answer & explanation

Answer: 12

Answer

The annual simple interest rate r is 12.

To find

r

, calculate the interest from each account. Account B grows compounded annually to

\$5,000 \times (1.20)^2 = \$7,200

, producing

\$2,200

in interest. Account A produces

5,000 \times \frac{r}{100} \times 3 = 150r

in simple interest. The problem states that

2,200 - 150r = 400

. Solving for

r

gives

150r = 1,800

, so

r = 12

.

Step-by-Step Solution

1

Calculate the compound interest earned from Account B

Interest from Account B = $2,200

Using the compound interest formula

A = P(1 + i)^n

, the balance after 2 years is

\$5,000 \times (1.20)^2 = \$7,200

. Subtracting the principal gives

\$7,200 - \$5,000 = \$2,200

.

2

Express the simple interest earned from Account A in terms of r

Interest from Account A = 150r

Simple interest is calculated as

I = P \times \frac{r}{100} \times t = 5,000 \times \frac{r}{100} \times 3 = 150r

.

3

Formulate and solve the linear equation relating the two interest amounts

r = 12

Subtracting the simple interest from the compound interest gives

2,200 - 150r = 400

. Solving yields

150r = 1,800

, which gives

r = 12

.

Key Concept

Comparing simple interest and compound interest expressions to solve for an unknown rate

Question 329Question →

Set $S$ consists of consecutive integers. The sum of all positive integers in set $S$ is $300$ , and the median of set $S$ is $-4.5$ . How many negative integers are in set $S$ ?

Show answer & explanation

Answer: 33

Answer

The set S contains 33 negative integers.

To find the number of negative integers in set S, first determine the maximum positive integer N in the set using the sum formula for consecutive positive integers: \frac{N(N+1)}{2} = 300, which yields N = 24. Next, use the property that for an evenly spaced set, the median is the average of the smallest and largest numbers: -4.5 = \frac{\text{Smallest} + 24}{2}. Solving for the smallest integer gives -33. Finally, count the negative integers in set S, which run from -33 to -1 inclusive: -1 - (-33) + 1 = 33.

Step-by-Step Solution

1

Find the largest integer in Set S

The largest integer is 24.

The positive integers in set S must form a sequence from 1 to N. The sum of positive integers is given by \frac{N(N + 1)}{2} = 300, which simplifies to N(N + 1) = 600. Since 24 \times 25 = 600, N = 24.

2

Find the smallest integer in Set S using the median formula

The smallest integer is -33.

In any set of consecutive integers, the median is equal to the arithmetic mean of the smallest and largest terms: \text{Median} = \frac{\text{Smallest} + \text{Largest}}{2}. Substituting -4.5 for the median and 24 for the largest term gives -4.5 = \frac{\text{Smallest} + 24}{2} \implies \text{Smallest} + 24 = -9 \implies \text{Smallest} = -33.

3

Count the number of negative integers in Set S

There are 33 negative integers.

The negative integers in set S range from -33 to -1, inclusive. The count is calculated as -1 - (-33) + 1 = 33.

Key Concept

Properties of consecutive integer sets, median-mean equivalence, and inclusive range counting

Estimated Time:1m 30s

Question 330Question →

A market research study examined the streaming service subscriptions of $250$ households. Every surveyed household subscribes to at least one of three streaming services: Service X, Service Y, or Service Z.

- $130$ households subscribe to Service X.
- $110$ households subscribe to Service Y.
- $120$ households subscribe to Service Z.
- $40$ households subscribe to both Service X and Service Y.
- $35$ households subscribe to both Service Y and Service Z.
- $45$ households subscribe to both Service X and Service Z.

How many households subscribe to all three streaming services?

Show answer & explanation

Answer: 10

Answer

10 households subscribe to all three streaming services.

According to the Principle of Inclusion-Exclusion for three sets,

|X \cup Y \cup Z| = |X| + |Y| + |Z| - (|X \cap Y| + |Y \cap Z| + |X \cap Z|) + |X \cap Y \cap Z|

. Since every household subscribes to at least one service,

|X \cup Y \cup Z| = 250

. Substituting the values yields

250 = 130 + 110 + 120 - (40 + 35 + 45) + |X \cap Y \cap Z|

, which simplifies to

250 = 240 + |X \cap Y \cap Z|

. Solving for

|X \cap Y \cap Z|

gives

10

.

Step-by-Step Solution

1

Set up the Inclusion-Exclusion formula for three overlapping sets

Total = |X| + |Y| + |Z| - (|X ∩ Y| + |Y ∩ Z| + |X ∩ Z|) + |X ∩ Y ∩ Z|

Because every household subscribes to at least one service, the total number of households equals the union of all three sets.

2

Sum the individual set sizes and the pairwise intersection sizes

Sum of individual sets = 130 + 110 + 120 = 360; Sum of pairwise intersections = 40 + 35 + 45 = 120

Aggregating individual and overlapping counts simplifies substitution.

3

Substitute the values into the formula and solve for the intersection of all three sets

250 = 360 - 120 + |X ∩ Y ∩ Z| => 250 = 240 + |X ∩ Y ∩ Z| => |X ∩ Y ∩ Z| = 10

Subtracting 240 from 250 yields the exact count of households subscribing to all three services.

Key Concept

Inclusion-Exclusion Principle for Three Sets

Estimated Time:1m 30s

Question 331Question →

A furniture manufacturer allows customers to customize a dining set by selecting options from four categories:
- Tabletop shape: rectangular, oval, or round (3 choices)
- Wood finish: oak, walnut, cherry, or maple (4 choices)
- Leg design: tapered, hairpin, or turned (3 choices)
- Number of chairs: 4, 6, or 8 (3 choices)

However, due to space constraints, a round tabletop cannot be paired with a set of 8 chairs. How many distinct dining set configurations can a customer assemble?

Show answer & explanation

Answer: 96

Answer

96 distinct dining set configurations can be assembled.

To find the number of valid dining set configurations, calculate total unrestricted choices using the Fundamental Counting Principle: multiplying 3 tabletop shapes, 4 wood finishes, 3 leg designs, and 3 chair count options gives 108 total combinations. Next, count the prohibited combinations consisting of a round tabletop (1 option) paired with 8 chairs (1 option) across all 4 finishes and 3 leg designs, giving 1 × 4 × 3 × 1 = 12 restricted combinations. Subtracting 12 restricted combinations from 108 total combinations yields 96 valid configurations.

Step-by-Step Solution

1

Calculate total possible combinations without restrictions

3 × 4 × 3 × 3 = 108 combinations

By the Fundamental Counting Principle, multiplying the number of available options across all independent decision stages gives the total number of unrestricted arrangements.

2

Calculate the number of invalid configurations violating the space constraint

1 × 4 × 3 × 1 = 12 invalid combinations

Restricted configurations consist of 1 tabletop shape (round), 4 wood finishes, 3 leg designs, and 1 chair quantity selection (8 chairs).

3

Subtract the invalid combinations from the total unrestricted combinations

108 - 12 = 96 valid configurations

Subtracting the prohibited configurations from the total possible combinations yields the count of permissible configurations.

Key Concept

Fundamental Counting Principle with Subtraction of Restricted Cases

Question 332Question →

If $9^{x+1} - 9^x = 216$ , what is the value of $4^x$ ?

Show answer & explanation

Answer: 8

Answer

The value of

4^x

is 8.

Factoring

9^x

from

9^{x+1} - 9^x

gives

9^x(9 - 1) = 216

, or

8 \cdot 9^x = 216

. Dividing by 8 yields

9^x = 27

. Expressing both sides with base 3 gives

3^{2x} = 3^3

, so

2x = 3

and

x = \frac{3}{2}

. Substituting this value into

4^x

results in

4^{3/2} = (\sqrt{4})^3 = 2^3 = 8

.

Step-by-Step Solution

1

Factor out common exponent terms from the left side of the equation

9^x(9^1 - 1) = 216 \implies 8 \cdot 9^x = 216

Using the exponent property

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

, rewrite

9^{x+1}

as

9^x \cdot 9^1

to factor out

9^x

.

2

Isolate the exponential term

9^x = 27

Dividing both sides of

8 \cdot 9^x = 216

by 8 yields

27

.

3

Convert both sides to a common prime base of 3

(3^2)^x = 3^3 \implies 3^{2x} = 3^3 \implies 2x = 3 \implies x = \frac{3}{2}

Since bases are equal, exponents must be equal.

4

Evaluate the target expression

4^x

4^{3/2} = (4^{1/2})^3 = 2^3 = 8

Substitute

x = \frac{3}{2}

into

4^x

and apply the rule

a^{m/n} = (\sqrt[n]{a})^m

.

Key Concept

Solving exponential equations by factoring and equating powers with a common base.

Question 333Question →

A panel of $150$ sommeliers evaluated three vintages of wine: Cabernet, Pinot Noir, and Syrah. Every sommelier rated at least one vintage as Exceptional. Overall, $79$ sommeliers rated Cabernet as Exceptional, $71$ rated Pinot Noir as Exceptional, and $84$ rated Syrah as Exceptional. Exactly $18$ sommeliers rated all three vintages as Exceptional, and exactly $27$ rated ONLY Cabernet as Exceptional. If the total number of sommeliers who rated BOTH Cabernet and Pinot Noir as Exceptional is twice the number of sommeliers who rated ONLY Pinot Noir and Syrah as Exceptional, how many sommeliers rated ONLY Cabernet and Syrah as Exceptional?

Show answer & explanation

Answer: 24

Answer

24

Applying the 3-set Inclusion-Exclusion Principle determines that

48

sommeliers rated exactly two vintages. Combining the Cabernet set total (

79

) with the given ONLY Cabernet count (

27

) establishes that

cp + cs = 34

. Substituting

cp = 2ps - 18

into these two relationships forms a system of linear equations (

3ps + cs = 66

and

2ps + cs = 52

). Solving this system yields

ps = 14

and

cs = 24

sommeliers who rated ONLY Cabernet and Syrah as Exceptional.

Step-by-Step Solution

1

Calculate the sum of all pairwise set overlaps using the Inclusion-Exclusion Principle.

|C \cap P| + |P \cap S| + |C \cap S| = 102

The total union is equal to the sum of individual set sizes minus the sum of pairwise intersections plus the three-set intersection.

2

Determine the sum of regions representing sommeliers who rated exactly two vintages.

cp + ps + cs = 48

Each pairwise intersection consists of an 'exactly two' region plus the 'all three' region (

18

). Subtracting

3 \times 18 = 54

from

102

leaves

48

.

3

Set up linear equations using the given ratio and Cabernet set total.

3ps + cs = 66

and

2ps + cs = 52

Expressing

cp = 2ps - 18

and substituting it into

cp + ps + cs = 48

gives the first equation; substituting into the Cabernet total

27 + cp + cs + 18 = 79

gives the second equation.

4

Solve the system of equations for the target region

cs

.

ps = 14

and

cs = 24

Subtracting the two equations yields

ps = 14

, and substituting back gives

cs = 24

.

Key Concept

Three-set Venn diagram algebraic modeling and inclusion-exclusion principle

Question 334Question →

A cyclist travels from City A to City B at a constant speed of $20$ miles per hour. On the return trip along the exact same route, the cyclist increases their speed by $25\%$ . If the total time taken for the round trip is $9$ hours, what is the distance, in miles, between City A and City B?

Show answer & explanation

Answer: 100

Answer

The distance between City A and City B is 100 miles.

To find the one-way distance

d

, determine the return speed as

20 \times 1.25 = 25

miles per hour. The outbound time is

\frac{d}{20}

hours and the return time is

\frac{d}{25}

hours. Adding these gives

\frac{d}{20} + \frac{d}{25} = 9

, which simplifies to

\frac{9d}{100} = 9

, so

d = 100

miles.

Step-by-Step Solution

1

Calculate the speed on the return trip.

The return speed is

20 \times 1.25 = 25

miles per hour.

The return speed increases by

25\%

over the outbound speed of

20

miles per hour.

2

Express outbound and inbound travel times using distance

d

.

Outbound time

t_1 = \frac{d}{20}

hours, Inbound time

t_2 = \frac{d}{25}

hours.

Using the rate-time-distance formula

t = \frac{d}{r}

.

3

Set up and solve the total time equation.

\frac{d}{20} + \frac{d}{25} = 9 \implies \frac{9d}{100} = 9 \implies d = 100

miles.

The total duration of the round trip is given as

9

hours.

Key Concept

Rate, Time, and Distance Relationship in Round Trips

Question 335Question →

If $x$ and $y$ are integers such that $-8 \le x \le 4$ and $-2 \le y \le 5$ , $x y^2 > 0$ , and $x^3 y < 0$ , how many distinct integer values are possible for the product $x y$ ?

Show answer & explanation

Answer: 6

Answer

The total number of distinct integer values for the product

x y

is 6.

Analyzing the given inequalities reveals sign constraints on both variables:

x y^2 > 0

forces

x > 0

(since

y^2 > 0

for non-zero

y

), limiting

x

to integer values

1, 2, 3, 4

. Then

x^3 y < 0

forces

y < 0

(since

x^3 > 0

), limiting

y

to integer values

-2, -1

. Multiplying each value of

x

by each value of

y

gives the products

-1, -2, -3, -4, -6, -8

, which constitutes exactly 6 distinct values.

Step-by-Step Solution

1

Determine the sign and allowable integer values of

x

x \in \{1, 2, 3, 4\}

Since

y^2

is strictly positive for any non-zero real number

y

,

x y^2 > 0

implies

x > 0

and

y \neq 0

.

2

Determine the sign and allowable integer values of

y

y \in \{-2, -1\}

Because

x > 0

,

x^3

is also positive. For

x^3 y < 0

,

y

must be negative.

3

Calculate products for all valid pairs of

(x, y)

and remove duplicates

The distinct product values are

-1, -2, -3, -4, -6, -8

, giving 6 values in total.

Evaluating

x y

for

x \in \{1, 2, 3, 4\}

and

y \in \{-2, -1\}

yields

-1, -2, -3, -4

when

y = -1

and

-2, -4, -6, -8

when

y = -2

.

Key Concept

Even powers of non-zero numbers are always positive, while odd powers preserve the sign of the base number.

Question 336Question →

A 5-digit passcode is to be created using five distinct digits chosen from the set $\{1, 2, 3, 4, 5, 6, 7\}$ . The passcode must include both the digit $3$ and the digit $5$ , and the digit $3$ must appear somewhere to the left of the digit $5$ in the passcode. How many such 5-digit passcodes can be formed?

Show answer & explanation

Answer: 600

Answer

600

To form a valid passcode, first select 3 digits from the 5 available digits

\{1, 2, 4, 6, 7\}

, which can be done in

\binom{5}{3} = 10

ways. Each chosen set of 5 distinct digits (including 3 and 5) can be arranged in

5! = 120

ways. Because the digits 3 and 5 are distinct, the digit 3 appears before the digit 5 in exactly half of these arrangements (

120 / 2 = 60

). Multiplying the 10 combinations of digits by the 60 valid arrangements gives a total of

600

valid passcodes.

Step-by-Step Solution

1

Determine the number of ways to choose the remaining digits.

\binom{5}{3} = 10

ways to choose 3 additional digits from

\{1, 2, 4, 6, 7\}

.

Since the passcode must contain both 3 and 5, 3 additional distinct digits must be selected from the 5 available remaining digits.

2

Calculate the total permutations of the 5 chosen digits.

5! = 120

total permutations.

Any set of 5 distinct digits can be arranged into a 5-digit sequence in

5!

ways.

3

Apply the positional restriction using symmetry.

\frac{120}{2} = 60

valid arrangements per set of digits.

In exactly half of all permutations containing both 3 and 5, the digit 3 appears before the digit 5.

4

Compute the total number of valid passcodes.

10 \times 60 = 600

passcodes.

Multiply the number of digit selections by the number of valid orderings per selection.

Key Concept

Combining selection (combinations) with symmetry-restricted arrangements (permutations)

Estimated Time:2m 0s

Question 337Question →

A security system generates 4-digit passcodes using distinct digits selected from the set $\{1, 2, 3, 4, 5, 6, 7\}$ . How many different 4-digit passcodes can be formed if the first digit must be odd and the last digit must be even?

Show answer & explanation

Answer: 240

Answer

240

To form a valid passcode under the given constraints, we analyze each position step-by-step. The first digit has 4 possible choices (odd numbers: 1, 3, 5, 7), and the fourth digit has 3 possible choices (even numbers: 2, 4, 6). Since the problem specifies that the digits in the passcode must be distinct, choosing the first and last digits consumes 2 of the 7 available digits, leaving 5 digits for the second position and 4 digits for the third position. Applying the Fundamental Counting Principle gives a total of

4 \times 5 \times 4 \times 3 = 240

unique passcodes.

Step-by-Step Solution

1

Determine choices for the first digit

4 choices

The set contains 4 odd digits: 1, 3, 5, and 7.

2

Determine choices for the fourth (last) digit

3 choices

The set contains 3 even digits: 2, 4, and 6.

3

Determine choices for the second and third digits

5 choices for the second digit, 4 choices for the third digit

All digits in the passcode must be distinct. Having used 2 digits for the first and last positions, 5 digits remain out of 7 for the second slot, and 4 digits remain for the third slot.

4

Apply the Fundamental Counting Principle

4 × 5 × 4 × 3 = 240

Multiply the number of available options for each position to find the total number of distinct passcodes.

Key Concept

Permutations with Position Restrictions

Question 338Question →

Let $m$ and $n$ be integers such that $-4 \le m \le 6$ and $-8 \le n \le 3$ . If $m^2 n < 0$ and $m + n > 0$ , what is the minimum possible value of the product $m \cdot n$ ?

Show answer & explanation

Answer: -30

Answer

The minimum possible value of

m \cdot n

is

-30

.

The correct answer is -30. Analyzing

m^2 n < 0

shows that

n

must be negative and

m

cannot be zero. The condition

m + n > 0

implies

m > -n = |n|

, making

m

positive. Since

m \le 6

, the maximum possible value for

|n|

is 5, which corresponds to

n = -5

. When

n = -5

,

m

must be strictly greater than 5, leaving

m = 6

as the only valid value. The product is

6 \times (-5) = -30

, which is the minimum value achievable under all constraints.

Step-by-Step Solution

1

Determine the signs of

m

and

n

using the given inequalities.

n < 0

and

m > 0

with

m \ne 0

.

Since

m^2

is strictly positive for any non-zero integer

m

,

m^2 n < 0

forces

n < 0

. Then

m + n > 0

requires

m > -n > 0

.

2

Find the range of valid integer values for

n

.

n \in \{-5, -4, -3, -2, -1\}

.

Since

m \le 6

and

m > -n

, we must have

-n \le 5

, which gives

n \ge -5

.

3

Evaluate the minimum product

m \cdot n

across all allowed values of

n

.

The minimum product is

-30

, occurring when

n = -5

and

m = 6

.

To minimize a negative product, maximize the absolute product

m \cdot |n|

. When

n = -5

,

m

must be

6

, yielding

6 \cdot (-5) = -30

.

Key Concept

Positive and Negative Number Properties with Inequalities

Question 339Question →

A jewelry store window displays $6$ distinct luxury watches— $3$ gold watches and $3$ silver watches—in a single straight line from left to right. If no two gold watches can be placed adjacent to each other, how many different linear arrangements of the $6$ watches are possible?

Show answer & explanation

Answer: 144

Answer

144

To place items such that no two gold watches are adjacent, we use the gap method. First, arrange the 3 distinct silver watches, which can be done in

3! = 6

ways. Placing these 3 silver watches creates 4 potential spaces (one at each end and two between the silver watches). Next, select 3 of these 4 spaces to place the gold watches, which can be done in

\binom{4}{3} = 4

ways. Finally, arrange the 3 distinct gold watches within the selected spaces in

3! = 6

ways. Multiplying these possibilities gives

6 \times 4 \times 6 = 144

total distinct arrangements.

Step-by-Step Solution

1

Calculate the arrangements of the 3 distinct silver watches.

3! = 6 ways

The 3 silver watches are distinct and can be arranged among themselves in 3! ways.

2

Determine the available positions (gaps) for the gold watches.

4 available gaps

Placing 3 silver watches in a line creates 4 potential spaces (before the first, between adjacent pairs, and after the last) to ensure non-adjacency.

3

Select positions and arrange the 3 distinct gold watches.

C(4, 3) × 3! = 4 × 6 = 24 ways

Choosing 3 out of 4 gaps gives C(4, 3) = 4 combinations, and ordering the 3 distinct gold watches in those selected gaps gives 3! = 6 arrangements.

4

Multiply the possibilities from all independent steps using the Fundamental Counting Principle.

6 × 24 = 144

The total number of valid linear arrangements is the product of the number of ways to complete each step.

Key Concept

Linear Permutations with Non-Adjacent Restrictions (Gap Method)

Estimated Time:2m 0s

Question 340Question →

If $\frac{8^x \cdot 4^{x-1}}{16^{x+1}} = 32$ , what is the value of $x$ ?

Show answer & explanation

Answer: 11

Answer

The value of

x

is 11.

Rewriting all bases as powers of 2 yields

\frac{2^{3x} \cdot 2^{2x-2}}{2^{4x+4}} = 2^5

. Simplifying the left side gives

2^{x-6} = 2^5

. Equating exponents results in

x - 6 = 5

, giving

x = 11

.

Step-by-Step Solution

1

Express all bases as powers of 2.

8 = 2^3

,

4 = 2^2

,

16 = 2^4

, and

32 = 2^5

.

Converting to a common base allows exponents to be combined using standard exponent laws.

2

Simplify the numerator and denominator using the power rule

(a^m)^n = a^{mn}

and product rule

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

.

Numerator:

2^{3x} \cdot 2^{2x-2} = 2^{5x-2}

. Denominator:

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

.

Distribute exponents carefully when multiplying powers with equal bases.

3

Apply the quotient rule

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

to combine the left side into a single power of 2.

\frac{2^{5x-2}}{2^{4x+4}} = 2^{(5x-2) - (4x+4)} = 2^{x-6}

.

Subtract the denominator's exponent from the numerator's exponent.

4

Equate exponents and solve for

x

.

2^{x-6} = 2^5 \implies x - 6 = 5 \implies x = 11

.

Since the bases are identical (

2

), the exponent expressions must be equal.

Key Concept

Solving exponential equations by expressing all numbers with a common base and applying exponent rules.

Answer

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