All practice questions

Question 341Question →

A jogger runs from point A to point B at a constant speed of $8$ miles per hour and then immediately returns from point B to point A along the exact same route at a constant speed of $12$ miles per hour. If the total time for the entire round trip is $5$ hours, what is the distance, in miles, from point A to point B?

Show answer & explanation

Answer: 24

Answer

The distance from point A to point B is 24 miles.

Let

d

be the distance in miles between point A and point B. The time taken for the outbound journey is

\frac{d}{8}

hours, and the time taken for the return journey is

\frac{d}{12}

hours. Since the total time for the round trip is

5

hours, we set up the equation

\frac{d}{8} + \frac{d}{12} = 5

. Finding a common denominator of

24

, the equation becomes

\frac{3d + 2d}{24} = 5

, which simplifies to

\frac{5d}{24} = 5

. Multiplying both sides by

24

gives

5d = 120

, and dividing by

5

yields

d = 24

miles.

Step-by-Step Solution

1

Define the variable

d

as the distance in miles between point A and point B, and express the time for each leg of the trip.

Outbound time =

\frac{d}{8}

hours, Return time =

\frac{d}{12}

hours.

Using the fundamental rate formula

t = \frac{d}{r}

, time equals distance divided by speed.

2

Sum the time spent on the outbound leg and the return leg to match the total round-trip time.

\frac{d}{8} + \frac{d}{12} = 5

.

The total time for the trip is given as

5

hours.

3

Find a common denominator to add the algebraic fractions and solve for

d

.

\frac{3d + 2d}{24} = 5 \implies \frac{5d}{24} = 5 \implies 5d = 120 \implies d = 24

.

The common denominator of

8

and

12

is

24

. Simplifying the resulting linear equation yields the one-way distance.

Key Concept

Rate, Time, and Distance Problems:

d = r \cdot t

. Total trip time is the sum of individual leg times (

t_{\text{total}} = t_1 + t_2

).

Estimated Time:1m 30s

Question 342Question →

A container holds $6$ red tokens and $n$ blue tokens, where $n$ is a positive integer. Two tokens are drawn at random from the container one after another without replacement. If the probability that the first token drawn is red and the second token drawn is blue is equal to $\frac{9}{35}$ , what is the value of $n$ ?

Show answer & explanation

Answer: 9

Answer

The value of

n

is

9

.

By applying the multiplication rule for dependent events without replacement, the probability is

\frac{6}{6+n} \times \frac{n}{5+n} = \frac{9}{35}

. Simplifying and solving the resulting quadratic equation

3n^2 - 37n + 90 = 0

yields

n = 9

as the only integer solution.

Step-by-Step Solution

1

Formulate the joint probability for dependent events.

The probability of drawing a red token first then a blue token second is

\frac{6}{6+n} \times \frac{n}{5+n}

.

Because sampling is done without replacement, the total number of remaining tokens decreases from

6+n

to

5+n

for the second draw.

2

Set up the algebraic equation with the given probability.

\frac{6n}{(6+n)(5+n)} = \frac{9}{35}

Equate the theoretical probability expression to the given value

\frac{9}{35}

.

3

Simplify and expand the equation.

Divide both sides of the numerator by

3

: \frac{2n}{n^2 + 11n + 30} = \frac{3}{35}. Cross-multiplying yields

70n = 3n^2 + 33n + 90

.

Simplifying ratios makes polynomial expansion easier.

4

Solve the quadratic equation for

n

.

Rearrange to

3n^2 - 37n + 90 = 0

, which factors into

(3n - 10)(n - 9) = 0

.

Find the roots of the quadratic equation.

5

Select the valid integer solution.

Since

n

must be an integer count of tokens,

n = 9

(rejecting

n = \frac{10}{3}

).

The number of tokens

n

must be a whole number.

Key Concept

Probability of Dependent Events (Sampling Without Replacement)

Question 343Question →

A researcher recorded six test measurements arranged in ascending order: $1, 2, 8, x, y, 15$ . If the unique mode of the data set is $8$ and the arithmetic mean of the measurements is equal to their median, what is the value of $y$ ?

Show answer & explanation

Answer: 14

Answer

14

Because the data set is arranged in ascending order, the elements satisfy

1 \le 2 \le 8 \le x \le y \le 15

. For

8

to be the unique mode,

8

must appear more times than any other number, which forces

x = 8

. The median of a

6

-element set is the average of the

3^{\text{rd}}

and

4^{\text{th}}

elements, which are both

8

, making the median

8

. Setting the mean

\frac{1 + 2 + 8 + 8 + y + 15}{6} = \frac{34 + y}{6}

equal to

8

gives

34 + y = 48

, which yields

y = 14

. Since

8 \le 14 \le 15

, this maintains the ascending order and unique mode condition.

Step-by-Step Solution

1

Determine the value of x

x = 8

The set is given in ascending order (1 <= 2 <= 8 <= x <= y <= 15). For 8 to be the unique mode, 8 must appear at least twice, which requires x = 8.

2

Calculate the median of the data set

Median = 8

With 6 elements, the median is the average of the 3rd and 4th elements: (8 + 8) / 2 = 8.

3

Set up the equation for the arithmetic mean

(34 + y) / 6 = 8

The sum of the six numbers is 1 + 2 + 8 + 8 + y + 15 = 34 + y. Since the mean equals the median, the mean must equal 8.

4

Solve for y

y = 14

Multiplying both sides by 6 gives 34 + y = 48, so y = 48 - 34 = 14.

Key Concept

Using statistical properties (mean, median, mode) of an ordered set to determine unknown values

Estimated Time:1m 15s

Question 344Question →

An architectural firm offers custom townhouse design packages created by making sequential choices across three stages: Exterior Facade Material (4 choices), Floor Plan Layout (5 choices), and Interior Lighting Scheme (6 choices).

However, two design constraints apply:
1. If the first exterior facade material option is chosen, the fifth floor plan layout option cannot be selected.
2. The sixth interior lighting scheme option can only be selected if the fifth floor plan layout option is chosen.

How many different valid townhouse design packages can a client create?

Show answer & explanation

Answer: 98

Answer

The total number of valid townhouse design packages is 98.

The solution splits the decision process into two mutually exclusive scenarios based on whether the 5th layout option is chosen. In Case 1 (5th layout chosen), 3 facade choices × 1 layout choice × 6 lighting choices yields 18 valid packages. In Case 2 (1st through 4th layout chosen), 4 facade choices × 4 layout choices × 5 lighting choices yields 80 valid packages. Adding these mutually exclusive totals gives 18 + 80 = 98 valid design packages.

Step-by-Step Solution

1

Analyze the decision tree by splitting into mutually exclusive cases based on whether Floor Plan Layout Option 5 is selected.

Two distinct cases are established: Case 1 (Layout Option 5 selected) and Case 2 (Layout Options 1–4 selected).

The constraints depend directly on whether Floor Plan Layout Option 5 is chosen.

2

Calculate valid combinations for Case 1 (Layout Option 5 selected).

Facade choices = 3 (excluding Option 1), Layout choices = 1 (Option 5), Lighting choices = 6 (including Option 6). Total = 3 × 1 × 6 = 18 packages.

Constraint 1 eliminates 1 facade option, while Constraint 2 allows all 6 lighting options.

3

Calculate valid combinations for Case 2 (Layout Option 5 NOT selected).

Facade choices = 4, Layout choices = 4 (Options 1–4), Lighting choices = 5 (excluding Option 6). Total = 4 × 4 × 5 = 80 packages.

Constraint 1 does not apply, but Constraint 2 prevents selecting Lighting Option 6.

4

Sum the valid outcomes from both mutually exclusive cases.

18 + 80 = 98 valid packages.

By the Addition Principle of Counting, non-overlapping cases are added together.

Key Concept

Fundamental Counting Principle with Constrained Cases

Question 345Question →

A software consultant charges an enterprise client a fixed setup fee of $200$ dollars plus an hourly rate of $r$ dollars for 1-on-1 technical training. For team training workshops, the consultant charges a fixed setup fee of $400$ dollars plus an hourly rate that is $50\%$ higher than the 1-on-1 rate. If the client was billed a total of $2,400$ dollars for $12$ hours of 1-on-1 training and $8$ hours of team training workshops, what is the value of $r$ ?

Show answer & explanation

Answer: 75

Answer

The individual hourly rate

r

is

75

dollars per hour.

The total cost consists of fixed setup fees (

200 + 400 = 600

) and variable hourly charges (

12r + 8 \times 1.5r = 24r

). Setting the total cost

600 + 24r

equal to

2,400

and solving for

r

yields

r = 75

.

Step-by-Step Solution

1

Formulate the total cost equation for the 1-on-1 technical training.

\text{Cost}_{\text{1-on-1}} = 200 + 12r

The client pays a

200

dollar setup fee plus

r

dollars for each of the

12

hours.

2

Determine the hourly rate and total cost equation for the team training workshops.

\text{Hourly Rate}_{\text{team}} = 1.5r

, so

\text{Cost}_{\text{team}} = 400 + 8(1.5r) = 400 + 12r

The team hourly rate is

50\%

greater than

r

, which equals

r + 0.5r = 1.5r

. Multiplying by

8

hours yields

12r

in hourly fees, plus the

400

dollar setup fee.

3

Combine the total costs to create a single linear equation in terms of

r

.

(200 + 12r) + (400 + 12r) = 2400 \implies 600 + 24r = 2400

The total amount billed across both services is given as

2,400

dollars.

4

Isolate the variable

r

and calculate its numerical value.

24r = 1800 \implies r = 75

Subtracting

600

from both sides gives

24r = 1800

, and dividing by

24

yields

r = 75

.

Key Concept

Linear Equations in One and Two Variables

Estimated Time:1m 30s

Question 346Question →

If $2^{a+b} = 12$ and $2^{a-b} = 3$ , what is the value of $2^{2a} - 2^{2b}$ ?

Show answer & explanation

Answer: 32

Answer

32

Multiplying the equations gives

(2^{a+b})(2^{a-b}) = 2^{2a} = 12 \times 3 = 36

. Dividing the first equation by the second gives

\frac{2^{a+b}}{2^{a-b}} = 2^{2b} = \frac{12}{3} = 4

. Subtracting the two values gives

2^{2a} - 2^{2b} = 36 - 4 = 32

.

Step-by-Step Solution

1

Multiply the given exponent expressions to evaluate

2^{2a}

.

2^{2a} = (2^{a+b})(2^{a-b}) = 12 \times 3 = 36

.

According to the product rule of exponents,

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

. Adding

(a+b) + (a-b)

yields

2a

.

2

Divide the first exponent expression by the second to evaluate

2^{2b}

.

2^{2b} = \frac{2^{a+b}}{2^{a-b}} = \frac{12}{3} = 4

.

According to the quotient rule of exponents,

\frac{2^{x}}{2^{y}} = 2^{x-y}

. Subtracting

(a+b) - (a-b)

yields

2b

.

3

Calculate the value of

2^{2a} - 2^{2b}

.

36 - 4 = 32

.

Substitute the evaluated components

2^{2a} = 36

and

2^{2b} = 4

into the expression.

Key Concept

Exponent Operations (Product and Quotient Rules)

Estimated Time:1m 30s

Question 347Question →

In January, a logistics company's total monthly operational expenditure for its electric delivery fleet was $\$C$ . In February, the number of vehicles in the fleet increased by $25\%$ , while the monthly operational cost per vehicle decreased by $10\%$ due to optimized charging schedules. If the fleet's total operational expenditure in February was $\$33,750$ , what was the total operational expenditure, in dollars, in January?

Show answer & explanation

Answer: 30000

Answer

The total operational expenditure in January was $30,000.

The total expenditure is the product of the number of vehicles and the cost per vehicle. An increase of

25\%

in fleet size introduces a multiplier of

1.25

, while a

10\%

decrease in unit cost introduces a multiplier of

0.90

. The net multiplier for the total expenditure is

1.25 \times 0.90 = 1.125

(or

\frac{9}{8}

). Dividing the February total of

\$33,750

by

1.125

yields the January expenditure of

\$30,000

.

Step-by-Step Solution

1

Define variables for January's components and express January's total cost

Let

N

be the number of vehicles in January and

V

be the operational cost per vehicle in January. Total January expenditure

C = N \times V

.

Total cost is the product of quantity and unit cost.

2

Determine the updated fleet size and unit cost for February

February fleet size

= N \times (1 + 0.25) = 1.25 N = \frac{5}{4} N

. February unit cost

= V \times (1 - 0.10) = 0.90 V = \frac{9}{10} V

.

A

25\%

increase corresponds to a factor of

1.25

, and a

10\%

decrease corresponds to a factor of

0.90

.

3

Calculate February's total expenditure in terms of

C

February total cost

= \left(\frac{5}{4} N\right) \times \left(\frac{9}{10} V\right) = \frac{45}{40} NV = 1.125 C = \frac{9}{8} C

.

The total cost in February is the product of the updated fleet size and the updated unit cost.

4

Solve for January's total cost

C

\frac{9}{8} C = 33,750 \implies C = 33,750 \times \frac{8}{9} = 3,750 \times 8 = 30,000

.

Divide February's known cost by the net multiplier

1.125

(or

\frac{9}{8}

) to reverse the percent change.

Key Concept

Net percentage change on a product of variables and reverse calculation

Question 348Question →

An express cargo train leaves Station A heading toward Station B at a constant speed of $50$ miles per hour. At the exact same time, a local freight train leaves Station B heading toward Station A along a parallel track at a constant speed of $30$ miles per hour. If the total distance between Station A and Station B is $360$ miles, how many miles from Station A will the two trains meet?

Show answer & explanation

Answer: 225

Answer

The two trains will meet 225 miles from Station A.

When two objects move directly toward each other, their rate of approach is the sum of their speeds (

50 + 30 = 80

mph). Covering the total distance of

360

miles at a rate of

80

mph takes

4.5

hours. In

4.5

hours, the train traveling from Station A at

50

mph covers a distance of

50 \times 4.5 = 225

miles.

Step-by-Step Solution

1

Calculate the combined relative speed of both trains.

Relative speed =

50 + 30 = 80

miles per hour.

Since the two trains move toward each other, their relative rate of convergence is the sum of their individual speeds.

2

Calculate the time required for the two trains to meet.

Time

t = \frac{360 \text{ miles}}{80 \text{ mph}} = 4.5

hours.

Time equals total distance divided by combined relative speed.

3

Calculate the distance covered by the express train from Station A.

Distance =

50 \text{ mph} \times 4.5 \text{ hours} = 225

miles.

Distance equals the rate of the train departing Station A multiplied by the elapsed time.

Key Concept

Relative speed of converging objects moving toward each other

Estimated Time:1m 30s

Question 349Question →

Let $a$ , $b$ , and $c$ be integers such that $-5 \le a \le 3$ , $-4 \le b \le 6$ , and $-3 \le c \le 4$ . If $a b < 0$ and $b c^2 > 0$ , what is the maximum possible value of the expression $\frac{a - c}{b}$ ?

Show answer & explanation

Answer: 2

Answer

The maximum possible value of the expression is 2.

From

b c^2 > 0

, since

c^2 > 0

for any non-zero real number

c

, it follows that

b > 0

and

c \neq 0

. Given

a b < 0

and

b > 0

,

a

must be strictly negative (

a < 0

). To maximize

\frac{a - c}{b}

where

b > 0

, we maximize the numerator

(a - c)

and minimize the denominator

b

. The maximum negative integer for

a

is

-1

, and the minimum integer for

c

is

-3

, giving a maximum numerator of

-1 - (-3) = 2

. The minimum positive integer for

b

is

1

. Hence, the maximum value is

\frac{2}{1} = 2

.

Step-by-Step Solution

1

Analyze the sign of

b c^2 > 0

b > 0

and

c \neq 0

The square of any non-zero real number is strictly positive (

c^2 > 0

), so

b

must be positive for the product

b c^2

to be positive.

2

Analyze the sign of

a b < 0

a < 0

Since

b > 0

,

a

must be negative for their product to be less than zero.

3

Determine candidate integer sets

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

,

b \in \{1, 2, 3, 4, 5, 6\}

,

c \in \{-3, -2, -1, 1, 2, 3, 4\}

Filter the given intervals by the sign restrictions (

a < 0

,

b > 0

,

c \neq 0

).

4

Optimize numerator

(a - c)

and denominator

b

a = -1

,

c = -3

,

b = 1

To maximize a quotient with a positive denominator, make the numerator as large as possible (

a - c = -1 - (-3) = 2

) and the denominator as small as possible (

b = 1

).

5

Evaluate the maximum value

2

\frac{-1 - (-3)}{1} = \frac{2}{1} = 2

.

Key Concept

Deducing sign constraints from products and powers of variables and optimizing algebraic quotients under sign and domain restrictions.

Estimated Time:1m 30s

Question 350Question →

A logistics company operates two types of delivery trucks: Model A and Model B. Model A consumes $0.25$ liters of fuel per mile and incurs $\$1.50$ in maintenance costs per mile. Model B consumes $0.40$ liters of fuel per mile and incurs $\$1.00$ in maintenance costs per mile. On a given day, the fleet consumed a total of $300$ liters of fuel and incurred a total of $\$1,100$ in maintenance costs. What was the total number of miles traveled by the Model A trucks on that day?

Show answer & explanation

Answer: 400

Answer

The Model A trucks traveled a total of 400 miles on that day.

Let

x

be the number of miles driven by Model A trucks and

y

be the number of miles driven by Model B trucks. Set up two equations:

0.25x + 0.40y = 300

(fuel) and

1.50x + 1.00y = 1100

(maintenance). Substituting

y = 1100 - 1.5x

into the fuel equation yields

0.25x + 0.40(1100 - 1.5x) = 300

, which simplifies to

-0.35x + 440 = 300

, leading directly to

x = 400

.

Step-by-Step Solution

1

Formulate the linear system

0.25x + 0.40y = 300

and

1.50x + 1.00y = 1100

, where

x

is miles driven by Model A and

y

is miles driven by Model B.

Translates the rates of fuel consumption and maintenance per mile into total daily figures.

2

Express

y

in terms of

x

y = 1100 - 1.5x

Isolating

y

makes substitution into the fuel equation straightforward.

3

Substitute

y

into the first equation and solve for

x

0.25x + 0.40(1100 - 1.5x) = 300 \implies -0.35x + 440 = 300 \implies -0.35x = -140 \implies x = 400

Reduces the system to a single linear equation in

x

to find the distance for Model A.

Key Concept

Setting up and solving a two-variable system of linear equations via substitution or elimination.

Question 351Question →

A dataset consists of 7 integers listed in ascending order: $3, 4, 4, x, 9, y, 15$ . If the unique mode of the dataset is $4$ and the arithmetic mean of the dataset is equal to its median, what is the value of $y$ ?

Show answer & explanation

Answer: 13

Answer

The value of

y

is 13.

Because the dataset is in ascending order, the median of the 7 terms is the 4th term,

x

. Setting the mean

\frac{35 + x + y}{7}

equal to

x

yields

y = 6x - 35

. Since

y

lies between

9

and

15

inclusive, solving

9 \le 6x - 35 \le 15

gives

x = 8

. Substituting

x = 8

into

y = 6x - 35

gives

y = 13

.

Step-by-Step Solution

1

Determine the median of the 7-element ordered set.

The median is the 4th term,

x

.

For an odd number of ordered terms (

n = 7

), the median is the

\frac{7+1}{2} = 4\text{th}

term.

2

Set up the equation equating the arithmetic mean to the median.

\frac{35 + x + y}{7} = x

, which simplifies to

y = 6x - 35

.

The mean of a dataset is the sum of all elements divided by the total number of elements.

3

Apply the ascending order constraint to find the value of

x

.

Since

9 \le y \le 15

, we have

9 \le 6x - 35 \le 15

, giving

44 \le 6x \le 50

. The only integer value for

x

in this range is

8

.

The dataset is listed in ascending order, so the element

y

after

9

and before or equal to

15

must satisfy

9 \le y \le 15

.

4

Calculate the value of

y

.

y = 6(8) - 35 = 13

.

Substitute

x = 8

back into the simplified linear equation relating

x

and

y

.

Key Concept

Relating arithmetic mean and median in an ordered set using inequalities derived from ascending order constraints.

Estimated Time:1m 40s

Question 352Question →

An event organizer is scheduling a conference featuring $7$ distinct speakers: $3$ technology experts, $3$ finance experts, and $1$ keynote presenter. The $7$ speakers will give back-to-back presentations on a single main stage. If all $3$ technology experts must speak consecutively in any order, and no two finance experts may speak consecutively, how many different presentation schedules are possible?

Show answer & explanation

Answer: 72

Answer

The total number of possible presentation schedules satisfying all conditions is 72.

To find the number of valid schedules, we combine the block method and the gap method. First, bundle the 3 technology experts into a single block. This block can be ordered internally in

3! = 6

ways. Next, arrange the non-finance items (the technology block and the keynote presenter), which can be ordered in

2! = 2

ways. These 2 items create 3 available gaps (one at each end and one in the middle) for the 3 finance experts. To ensure no two finance experts are consecutive, place exactly one finance expert into each of the 3 gaps, which can be done in

3! = 6

ways. Multiplying these independent choices yields

6 \times 2 \times 6 = 72

total valid schedules.

Step-by-Step Solution

1

Group the 3 technology experts into a single unit and calculate internal permutations

3! = 6 internal arrangements for the technology block

The technology experts must appear consecutively, so treating them as a single block ensures they remain together.

2

Arrange the non-finance elements (the technology block and the keynote presenter)

2! = 2 linear arrangements

There are 2 distinct items that act as separators for the finance experts.

3

Apply the gap method to place the 3 finance experts into non-adjacent slots

3 available gaps for 3 finance experts gives 3! = 6 ways

Placing at most one finance expert in each gap created by the non-finance items guarantees that no two finance experts are seated or scheduled consecutively.

4

Calculate total permutations using the Fundamental Counting Principle

6 × 2 × 6 = 72 total schedules

The decisions for internal block order, base item order, and gap assignments are independent.

Key Concept

Permutations with Combined Block and Separation (Gap Method) Restrictions

Question 353Question →

If $m$ and $n$ are real numbers such that $m < 0 < n$ and $m^2 - 3mn - 4n^2 = 0$ , what is the value of $\frac{m + 2n}{m - n}$ ?

Show answer & explanation

Answer: -0.5

Answer

The value of the expression is -0.5.

Factoring

m^2 - 3mn - 4n^2 = 0

yields

(m - 4n)(m + n) = 0

, giving solutions

m = 4n

or

m = -n

. Given

m < 0 < n

,

m

is negative and

n

is positive, so

m

and

n

must have opposite signs. The solution

m = 4n

implies that

m

and

n

have the same sign (both positive), which violates

m < 0

. Therefore,

m

must equal

-n

. Substituting

m = -n

into

\frac{m + 2n}{m - n}

results in

\frac{-n + 2n}{-n - n} = \frac{n}{-2n} = -0.5

.

Step-by-Step Solution

1

Factor the quadratic equation

m^2 - 3mn - 4n^2 = 0

.

(m - 4n)(m + n) = 0

Factoring allows determination of the possible linear relationships between

m

and

n

.

2

Apply the sign condition

m < 0 < n

to eliminate the invalid case.

m = -n

Since

n > 0

,

4n

must be positive. Setting

m = 4n

would make

m > 0

, contradicting

m < 0

. Thus,

m + n = 0

, giving

m = -n

.

3

Substitute

m = -n

into

\frac{m + 2n}{m - n}

.

\frac{-n + 2n}{-n - n} = \frac{n}{-2n} = -0.5

Replacing

m

with

-n

simplifies the ratio to a constant value.

Key Concept

Positive and Negative Number Properties in Quadratic Constraints

Question 354Question →

A commercial bakery produces two batches of bread dough. Batch X consists of $30$ kilograms of dough with a sugar concentration of $12\%$ by weight. Batch Y consists of $45$ kilograms of dough with a sugar concentration of $20\%$ by weight. If Batch X and Batch Y are mixed together completely, what is the sugar concentration, as a percentage by weight, of the combined mixture?

Show answer & explanation

Answer: 16.8

Answer

The sugar concentration of the combined mixture is 16.8 percent by weight.

The overall concentration is calculated by taking the weighted average of the two concentrations. Since Batch X and Batch Y have weights in the ratio of 2 to 3, the overall percentage is [2(12%) + 3(20%)] / (2 + 3) = 84 / 5 = 16.8%. Alternatively, the total amount of sugar is 3.6 kg + 9.0 kg = 12.6 kg out of a total mass of 75 kg, yielding 12.6 / 75 = 16.8%.

Step-by-Step Solution

1

Calculate the total mass of sugar contributed by Batch X

3.6 kg of sugar (

30 \times 0.12 = 3.6

)

To find the total sugar in the combined mixture, compute the sugar content from each component batch.

2

Calculate the total mass of sugar contributed by Batch Y

9.0 kg of sugar (

45 \times 0.20 = 9.0

)

Compute the sugar mass for the second batch.

3

Sum the total sugar mass and total batch mass

Total sugar =

12.6

kg (

3.6 + 9.0

); Total mass =

75

kg (

30 + 45

)

The combined concentration equals total sugar divided by total mixture mass.

4

Compute the combined percentage concentration

16.8% (

\frac{12.6}{75} \times 100\% = 16.8\%

)

Divide total sugar mass by total mixture mass and multiply by 100.

Key Concept

Weighted Average of Combined Sets

Estimated Time:1m 30s

Question 355Question →

A cyclist travels along a scenic mountain road from a base lodge to a summit point at a constant speed of $12$ miles per hour. The cyclist then returns along the exact same route from the summit point back to the base lodge at a constant speed of $24$ miles per hour. If the entire round-trip ride takes a total of $3$ hours, what is the total distance, in miles, for the entire round trip?

Show answer & explanation

Answer: 48

Answer

The total distance for the entire round trip is 48 miles.

To find the total round-trip distance, let

d

be the one-way distance. The time spent going uphill is

\frac{d}{12}

hours and downhill is

\frac{d}{24}

hours. Combining these gives

\frac{d}{12} + \frac{d}{24} = 3

, which simplifies to

\frac{3d}{24} = 3

or

d = 24

miles. Since the trip covers both legs, the total distance is

2d = 48

miles.

Step-by-Step Solution

1

Define the variable for distance.

Let

d

represent the one-way distance in miles between the base lodge and the summit point.

Establishing a variable for the unknown leg distance allows setting up rate-time-distance relationships.

2

Express time in terms of distance and rate for both legs.

Uphill time

t_1 = \frac{d}{12}

hours; Downhill time

t_2 = \frac{d}{24}

hours.

Using

\text{Time} = \frac{\text{Distance}}{\text{Rate}}

converts given speeds into expressions for duration.

3

Formulate and solve the total time equation.

\frac{d}{12} + \frac{d}{24} = 3 \implies \frac{3d}{24} = 3 \implies d = 24 \text{ miles}.

The sum of the time taken for both legs equals the given total round-trip time of 3 hours.

4

Compute the total round-trip distance.

\text{Total distance} = 2 \times d = 2 \times 24 = 48 \text{ miles}.

The question specifically asks for the total distance of the entire round trip, which consists of two equal legs of length

d

.

Key Concept

Harmonic Mean and Round-Trip Rate Problems

Estimated Time:2m 0s

Question 356Question →

At the beginning of 2023, a manufacturing plant had an initial annual production capacity of $C$ units. Over the next three years, its annual production capacity experienced the following changes: it increased by $10\%$ in 2023, decreased by $20\%$ in 2024, and increased by $25\%$ in 2025. If the plant's production capacity at the end of 2025 was $22,000$ units, what was the initial production capacity $C$ , in units, at the beginning of 2023?

Show answer & explanation

Answer: 20000

Answer

The initial production capacity at the beginning of 2023 was 20,000 units.

Each annual percentage change applies to the value at the end of the previous period. Representing the changes as multipliers gives

1.10

,

0.80

, and

1.25

. Multiplying these together gives a net change factor of

1.10 \times 0.80 \times 1.25 = 1.10

. Setting

1.10 C = 22,000

and solving for

C

yields

C = 20,000

.

Step-by-Step Solution

1

Convert each percentage change into a decimal multiplier

Increase of

10\% \rightarrow 1.10

; decrease of

20\% \rightarrow 0.80

; increase of

25\% \rightarrow 1.25

.

Successive percentage changes compound sequentially by multiplying their respective growth or decay factors.

2

Compute the overall compound multiplier factor

Overall multiplier =

1.10 \times 0.80 \times 1.25 = 1.10 \times (0.80 \times 1.25) = 1.10 \times 1 = 1.10

.

Multiplying

0.80 \times 1.25

yields

1

, simplifying the combined factor to

1.10

times the original capacity.

3

Solve for the initial capacity

C

1.10 \times C = 22,000 \implies C = \frac{22,000}{1.10} = 20,000

.

Dividing the final capacity by the combined multiplier yields the initial capacity.

Key Concept

Successive Percent Change

Question 357Question →

A boutique hotel offers two types of rooms: Executive Suites and Deluxe Suites. During a convention weekend, booking $12$ Executive Suites and $18$ Deluxe Suites generated total revenue of $\$7{}800$ . The following weekend, booking $15$ Executive Suites and $10$ Deluxe Suites generated total revenue of $\$7{}250$ . If the nightly rate for each type of suite remained constant, what was the nightly rental rate, in dollars, of an Executive Suite?

Show answer & explanation

Answer: 350

Answer

The nightly rental rate of an Executive Suite was 350 dollars.

Formulating the revenue statements as linear equations gives

12x + 18y = 7,800

and

15x + 10y = 7,250

. Simplifying these by dividing by

6

and

5

yields

2x + 3y = 1,300

and

3x + 2y = 1,450

. Multiplying the first simplified equation by

2

(

4x + 6y = 2,600

) and the second by

3

(

9x + 6y = 4,350

) aligns the

y

-coefficients. Subtracting the equations leads to

5x = 1,750

, giving

x = 350

.

Step-by-Step Solution

1

Define variables and construct the system of equations.

12x + 18y = 7,800

and

15x + 10y = 7,250

, where

x

represents the Executive Suite rate and

y

represents the Deluxe Suite rate.

Translating word problem information into algebraic equations.

2

Simplify the system equations.

Dividing the first equation by

6

gives

2x + 3y = 1,300

. Dividing the second equation by

5

gives

3x + 2y = 1,450

.

Reducing the coefficients minimizes computation steps and errors.

3

Eliminate variable

y

to solve for

x

.

Multiply

2x + 3y = 1,300

by

2

to get

4x + 6y = 2,600

. Multiply

3x + 2y = 1,450

by

3

to get

9x + 6y = 4,350

. Subtracting gives

5x = 1,750

, so

x = 350

.

Eliminating

y

yields the requested value of

x

directly.

Key Concept

Solving Systems of Two Linear Equations via Elimination

Estimated Time:2m 0s

Question 358Question →

A university research department must select a 4-member committee from a group of 6 biologists and 4 chemists. If the committee must contain at least 1 biologist and at least 1 chemist, how many different 4-member committees can be formed?

Show answer & explanation

Answer: 194

Answer

194

Using complementary counting, the total number of ways to pick 4 members out of 10 is 210. The scenarios that break the rule are committees with all biologists (15 ways) or all chemists (1 way). Subtracting these 16 restricted selections from 210 results in 194 valid committees.

Step-by-Step Solution

1

Find total unrestricted combinations

210 total committees

Choosing any 4 people out of the pool of 10 without restrictions yields 10C4 = 210.

2

Calculate restricted/invalid cases

16 invalid committees

All-biologist groups account for 6C4 = 15 combinations, and all-chemist groups account for 4C4 = 1 combination.

3

Apply complementary counting

194 valid committees

Subtracting the 16 invalid combinations from the total 210 combinations gives 194.

Key Concept

Combinations with Restrictions (Complementary Counting)

Question 359Question →

A technology company purchased a total of $35$ computers for its new office, consisting of standard desktop units costing $\$800$ each and high-performance workstation units costing $\$1,400$ each. If the total expenditure for all $35$ computers was $\$34,000$ , how many standard desktop units did the company purchase?

Show answer & explanation

Answer: 25

Answer

The company purchased 25 standard desktop units.

Let

x

be the number of standard desktop units and

y

be the number of workstation units. The situation yields two equations:

x + y = 35

and

800x + 1400y = 34000

. Simplifying the second equation gives

4x + 7y = 170

. Multiplying the first equation by

7

gives

7x + 7y = 245

. Subtracting the simplified cost equation from this product yields

3x = 75

, so

x = 25

.

Step-by-Step Solution

1

Define variables and establish equations from the problem statement.

System of equations:

x + y = 35

and

800x + 1400y = 34000

, where

x

represents standard desktop units and

y

represents workstation units.

Linear systems represent total quantity and total cost constraints simultaneously.

2

Simplify the linear cost equation by dividing by the greatest common divisor (

200

).

Simplified equation:

4x + 7y = 170

.

Simplifying coefficients reduces arithmetic complexity and potential errors.

3

Solve for

x

using substitution or elimination.

-3x = -75 \implies x = 25

.

Isolating

x

directly calculates the required count of standard desktop units.

Key Concept

Solving a system of two linear equations in two variables using substitution or elimination

Estimated Time:1m 30s

Question 360Question →

An initial investment of $\$8,000$ was deposited into an account earning an annual interest rate of $r\%$ , compounded annually. If no additional deposits or withdrawals were made and the total balance in the account after $2$ years was $\$11,520$ , what is the value of $r$ ?

Show answer & explanation

Answer: 20

Answer

The value of

r

is 20.

The compound interest formula for annual compounding is

A = P\left(1 + \frac{r}{100}\right)^t

. Substituting

P = 8,000

,

A = 11,520

, and

t = 2

gives

11,520 = 8,000\left(1 + \frac{r}{100}\right)^2

. Dividing both sides by

8,000

yields

\left(1 + \frac{r}{100}\right)^2 = 1.44

. Taking the square root gives

1 + \frac{r}{100} = 1.2

, so

\frac{r}{100} = 0.20

, which corresponds to

r = 20

.

Step-by-Step Solution

1

Set up the compound interest equation using given principal and accumulated balance.

11,520 = 8,000 \left(1 + \frac{r}{100}\right)^2

Interest compounded annually over

2

years scales the initial principal by a factor of

\left(1 + \frac{r}{100}\right)^2

.

2

Divide both sides by

8,000

to isolate the annual growth factor squared.

\left(1 + \frac{r}{100}\right)^2 = \frac{11,520}{8,000} = 1.44

Finding the ratio of final balance to initial principal gives the overall multiplier over the two-year period.

3

Take the square root of both sides to obtain the single-year multiplier.

1 + \frac{r}{100} = \sqrt{1.44} = 1.2

Since the interest rate

r > 0

, taking the positive square root gives the multiplier for one year.

4

Solve for

r

.

r = (1.2 - 1) \times 100 = 20

Subtracting

1

yields the annual decimal interest rate of

0.20

, which corresponds to

20\%

.

Key Concept

Annual Compound Interest Rate Calculation

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept