All practice questions

Question 281Question →

An investor holds $x$ shares of Stock X and $y$ shares of Stock Y in a financial portfolio. The total monetary value of these holdings is $\$6,300$ , modeled by the linear equation $40x + 20y = 6,300$ . If the ratio of the number of shares of Stock X to the total number of shares held is $2$ to $5$ , what is the total number of shares of Stock X and Stock Y in the portfolio?

90

135

210

225

245

Show answer & explanation

Answer: 225

Answer

225

The correct answer is 225. Expressing the part-to-whole ratio

\frac{x}{x+y} = \frac{2}{5}

algebraically gives

3x = 2y

, or

y = 1.5x

. Substituting

1.5x

for

y

in the value equation

40x + 20y = 6,300

yields

70x = 6,300

, so

x = 90

. Then

y = 135

, and the total number of shares is

90 + 135 = 225

.

Step-by-Step Solution

1

Set up the linear relationship between

x

and

y

using the given ratio.

\frac{x}{x + y} = \frac{2}{5} \implies 5x = 2x + 2y \implies 3x = 2y \implies y = 1.5x

The problem specifies that the ratio of Stock X shares (

x

) to total shares (

x + y

) is

2:5

.

2

Substitute

y = 1.5x

into the portfolio value equation

40x + 20y = 6,300

.

40x + 20(1.5x) = 6,300 \implies 40x + 30x = 6,300 \implies 70x = 6,300 \implies x = 90

Replacing

y

with an equivalent expression in terms of

x

reduces the system to a single linear equation in one variable.

3

Calculate

y

and determine the total number of shares

x + y

.

y = 1.5(90) = 135 \implies x + y = 90 + 135 = 225

The question asks for the total combined number of shares of both stocks held in the portfolio.

Key Concept

Solving systems of linear equations formed by combining a linear value equation with a ratio relationship.

Question 282Question →

Machine X, working alone at a constant rate, can complete a manufacturing order in $4$ hours. Machine Y, working alone at a constant rate, can complete the same manufacturing order in $6$ hours. If Machine X and Machine Y work simultaneously at their respective constant rates, how many hours will it take them to complete $\frac{5}{6}$ of the manufacturing order?

2

hours

2.4

hours

4.17

hours

5

hours

8.33

hours

Show answer & explanation

Answer: $2$ hours

Answer

2

hours

Machine X completes

\frac{1}{4}

of the order per hour and Machine Y completes

\frac{1}{6}

of the order per hour. Working together, their combined rate is

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

of the order per hour. To determine the time required to finish

\frac{5}{6}

of the order, divide the targeted work amount by the combined rate:

\frac{5/6}{5/12} = 2

hours.

Step-by-Step Solution

1

Determine the individual work rates for Machine X and Machine Y

Rate of Machine X =

\frac{1}{4}

order per hour; Rate of Machine Y =

\frac{1}{6}

order per hour.

Work rate is defined as

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

.

2

Calculate the combined work rate when both machines work together

Combined Rate =

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

order per hour.

When machines work simultaneously, their individual rates add up.

3

Calculate the time required to complete

\frac{5}{6}

of the order using the combined rate

\text{Time} = \frac{\text{Work}}{\text{Combined Rate}} = \frac{5/6}{5/12} = \frac{5}{6} \times \frac{12}{5} = 2

hours.

Rearranging

\text{Work} = \text{Rate} \times \text{Time}

gives

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

.

Key Concept

Combined Work Rates

Estimated Time:2m 0s

Question 283Question →

A boutique jewelry designer crafts customized gold necklaces. The total cost to produce each necklace is the sum of a fixed material cost of $\$800$ and a variable design cost. To determine the list price, the designer marks up the total production cost by $60\%$ . During a seasonal promotion, the designer offers a $25\%$ discount off the list price. If the designer earns a net profit of $\$360$ on each necklace sold during the promotion, what is the variable design cost, in dollars, per necklace?

Show answer & explanation

Answer: 1000

Answer

The variable design cost per necklace is $1,000.

The total production cost per necklace is

C = 800 + L

. Marking up by

60\%

yields a list price of

1.60C

. A

25\%

discount reduces the price to

0.75 \times 1.60C = 1.20C

. The net profit per necklace is

1.20C - C = 0.20C

. Given that the net profit is

\$360

, we set

0.20C = 360

, which gives

C = 1,800

. Subtracting the fixed material cost of

\$800

yields the variable design cost of

\$1,000

.

Step-by-Step Solution

1

Define total production cost in terms of variable cost

C = 800 + L

, where

C

is total production cost and

L

is variable design cost

Total cost is the sum of fixed material cost and variable design cost.

2

Determine the selling price after markup and discount

Selling Price

S = 0.75 \times (1.60 C) = 1.20 C

A

60\%

markup increases total cost by a factor of

1.60

, and a

25\%

discount reduces that marked price to

75\%

(

0.75

).

3

Formulate the profit equation and solve for total production cost

C

Profit

= S - C = 1.20 C - C = 0.20 C = 360 \implies C = 1,800

Net profit is the selling price minus the total production cost.

4

Calculate variable design cost

L

L = 1,800 - 800 = 1,000

Subtract fixed material cost from total production cost.

Key Concept

Profit, Loss, and Markup with Successive Percentage Adjustments

Question 284Question →

A laboratory technician has $40$ liters of a saline solution that is $15\%$ salt by volume. How many liters of pure water must be evaporated from the solution so that the remaining solution is $25\%$ salt by volume?

Show answer & explanation

Answer: 16

Answer

The correct answer is

16

liters.

Because evaporation removes only pure water, the quantity of salt remains constant at

6

liters (

40 \times 0.15

). For

6

liters of salt to constitute

25\%

of the final mixture volume

V

, we set up

0.25V = 6

, which yields

V = 24

liters. Subtracting the final volume of

24

liters from the initial volume of

40

liters gives

16

liters of evaporated water.

Step-by-Step Solution

1

Calculate the volume of pure salt in the initial solution.

The initial salt volume is

40 \times 0.15 = 6

liters.

The solute amount is determined by multiplying total volume by concentration.

2

Determine the required total solution volume after evaporation to achieve a 25% concentration.

The required total final volume is

\frac{6}{0.25} = 24

liters.

Since evaporation removes only water, the volume of salt remains 6 liters, which must equal 25% of the new total volume.

3

Calculate the amount of water evaporated by taking the difference between the initial and final total volumes.

The volume of water evaporated is

40 - 24 = 16

liters.

The decrease in total solution volume equals the volume of pure water removed by evaporation.

Key Concept

Concentration change via evaporation (solute mass conservation)

Question 285Question →

A quality assurance technician randomly selects 2 microprocessors from a batch of $16$ microprocessors, of which $6$ were manufactured at Facility A and $10$ were manufactured at Facility B. The selections are made one after another without replacement. What is the probability that both selected microprocessors were manufactured at Facility A?

Show answer & explanation

Answer: 0.125

Answer

The probability that both selected microprocessors were manufactured at Facility A is 0.125 (or 1/8).

Because the selections are made without replacement, the outcome of the first draw changes the pool of available microprocessors for the second draw. The probability of choosing a Facility A microprocessor first is

\frac{6}{16} = \frac{3}{8}

. Following that selection,

5

Facility A microprocessors remain among

15

total microprocessors, giving a conditional probability of

\frac{5}{15} = \frac{1}{3}

for the second selection. Multiplying these dependent probabilities yields

\frac{3}{8} \times \frac{1}{3} = \frac{1}{8} = 0.125

.

Step-by-Step Solution

1

Determine the probability of selecting a Facility A microprocessor on the first draw.

P(\text{First is Facility A}) = \frac{6}{16} = \frac{3}{8}

There are 6 microprocessors from Facility A out of 16 total microprocessors.

2

Determine the conditional probability of selecting a Facility A microprocessor on the second draw.

P(\text{Second is Facility A} \mid \text{First is Facility A}) = \frac{5}{15} = \frac{1}{3}

Since the selection is made without replacement, 5 Facility A microprocessors remain out of a reduced total of 15 microprocessors.

3

Multiply the dependent probabilities to find the combined probability.

P(\text{Both are Facility A}) = \frac{3}{8} \times \frac{1}{3} = \frac{3}{24} = \frac{1}{8} = 0.125

The joint probability of sequential dependent events is the product of the initial probability and the conditional probability.

Key Concept

Probability of Dependent Events (Sampling without Replacement)

Estimated Time:1m 30s

Question 286Question →

A panel of 6 distinct experts—3 scientists, 2 economists, and 1 moderator—are to sit in a single row of 6 chairs for a discussion. If the 3 scientists must all sit in adjacent chairs and the 2 economists cannot sit in adjacent chairs, how many different seating arrangements are possible?

12

36

72

108

144

Show answer & explanation

Answer: 72

Answer

72 seating arrangements

To find the number of valid seating arrangements, we first group the 3 scientists together as 1 unit, leaving us with 2 non-economist entities (the scientist block and the moderator). These 2 entities can be arranged in 2! = 2 ways. Placing the 2 distinct economists into the 3 available gaps around these entities ensures they are not adjacent, yielding P(3, 2) = 6 choices. Finally, multiplying by the 3! = 6 internal arrangements of the scientists yields a total of 2 * 6 * 6 = 72 valid arrangements.

Step-by-Step Solution

1

Group the 3 scientists into a single block

S

, and treat the moderator

M

as an individual unit.

There are 2 non-economist units: block

S

and moderator

M

.

Grouping elements that must be adjacent allows us to treat them temporarily as a single entity.

2

Calculate the arrangements of the non-economist units.

The 2 non-economist units can be arranged in

2! = 2

ways.

Linear arrangement of 2 distinct entities.

3

Insert the 2 economists into the available gaps created by the non-economist units.

For any arrangement of

S

and

M

(e.g., _

S

_

M

_), there are 3 available gaps. The 2 distinct economists can be placed in these gaps in

P(3,2) = 3 \times 2 = 6

ways.

To ensure no two economists sit together, each economist must occupy a separate gap.

4

Account for the internal arrangements of the 3 scientists within block

S

.

The 3 distinct scientists can be arranged among themselves in

3! = 6

ways.

Order matters among distinct individuals within a grouped block.

5

Apply the fundamental counting principle to compute total arrangements.

2 \times 6 \times 6 = 72

total arrangements.

Multiply the independent choices made in steps 2, 3, and 4.

Key Concept

Permutations with Adjacency and Non-Adjacency Restrictions

Question 287Question →

A venture capital firm allocated its initial investment fund between two startups, Enterprise X and Enterprise Y. Over a two-year period, the investment in Enterprise X earned simple interest at an annual rate of $15\%$ . During the same period, the value of the investment in Enterprise Y increased by $25\%$ in the first year and then decreased by $20\%$ of its new value in the second year. If the total combined value of the two investments at the end of the two years was $12\%$ greater than the initial investment fund, what percentage of the initial fund was invested in Enterprise X?

28\%

40\%

50\%

60\%

80\%

Show answer & explanation

Answer: $40\%$

Answer

40\%

of the initial fund was invested in Enterprise X.

The correct answer is

40\%

. Enterprise X earns simple interest of

15\%

per year for 2 years, yielding a growth factor of

1 + 2(0.15) = 1.30

. Enterprise Y increases by

25\%

and then decreases by

20\%

, yielding a net growth factor of

1.25 \times 0.80 = 1.00

. Setting the overall weighted multiplier equal to

1.12

gives

1.30x + 1.00(1-x) = 1.12

, which yields

x = 0.40

, or

40\%

.

Step-by-Step Solution

1

Define variables and determine the growth factor for Enterprise X.

Let

x

be the fraction of the fund invested in Enterprise X, so

(1-x)

is the fraction invested in Enterprise Y. Over 2 years at an annual simple interest rate of

15\%

, the growth multiplier for Enterprise X is

1 + 2 \times 0.15 = 1.30

.

Simple interest accumulates linearly over time.

2

Calculate the net successive percentage multiplier for Enterprise Y.

A

25\%

increase followed by a

20\%

decrease gives a net multiplier of

(1 + 0.25) \times (1 - 0.20) = 1.25 \times 0.80 = 1.00

.

Successive percentage changes must be multiplied sequentially based on the intermediate values.

3

Set up the weighted average equation for the total investment.

1.30x + 1.00(1-x) = 1.12

.

The total combined value at the end of 2 years is

12\%

greater than the initial fund, corresponding to a total multiplier of

1.12

.

4

Solve for

x

.

1.30x + 1.00 - 1.00x = 1.12 \Rightarrow 0.30x = 0.12 \Rightarrow x = 0.40 = 40\%

.

Isolating

x

gives the proportion of the fund allocated to Enterprise X.

Key Concept

Combining simple interest and successive percentage change in weighted portfolio problems.

Question 288Question →

An educational foundation set an annual fundraising goal for a given year. During the first quarter, the foundation raised $30\%$ of its annual goal. In the second quarter, the amount raised was $25\%$ greater than the amount raised in the first quarter. In the third quarter, the amount raised was $40\%$ less than the total amount raised in the first two quarters combined. If the total amount raised across the first three quarters combined was $\$129,600$ , what was the foundation's annual fundraising goal, in dollars?

Show answer & explanation

Answer: 120000

Answer

The foundation's annual fundraising goal was $120,000.

To determine the annual goal

G

, calculate the raised amount in each quarter relative to

G

. The first quarter yields

Q_1 = 0.30G

. The second quarter amount is

25\%

greater than

Q_1

, which equals

1.25 \times 0.30G = 0.375G

. The sum of the first two quarters is

0.30G + 0.375G = 0.675G

. The third quarter amount is

40\%

less than this combined sum, giving

0.60 \times 0.675G = 0.405G

. Summing the three quarters gives a total of

0.30G + 0.375G + 0.405G = 1.08G

. Equating

1.08G = 129,600

and dividing by

1.08

gives

G = 120,000

.

Step-by-Step Solution

1

Define the variable for the unknown quantity

Let

G

represent the foundation's annual fundraising goal in dollars.

Establishing a variable allows all quarterly contributions to be expressed as a linear algebraic function of the goal.

2

Express the amount raised in the first quarter in terms of

G

Q_1 = 0.30G

The foundation raised

30\%

of its annual goal during the first quarter.

3

Express the amount raised in the second quarter in terms of

G

Q_2 = 1.25 \times 0.30G = 0.375G

An increase of

25\%

over

Q_1

means

Q_2 = (1 + 0.25) Q_1 = 1.25 \times 0.30G

.

4

Calculate the combined total raised in the first two quarters

Q_1 + Q_2 = 0.30G + 0.375G = 0.675G

This combined sum serves as the base value for calculating the third quarter's contribution.

5

Express the amount raised in the third quarter in terms of

G

Q_3 = (1 - 0.40) \times 0.675G = 0.60 \times 0.675G = 0.405G

The third quarter raised

40\%

less than the combined amount of the first two quarters, meaning it equaled

60\%

of

(Q_1 + Q_2)

.

6

Sum the contributions of all three quarters and solve for

G

Total

= 0.30G + 0.375G + 0.405G = 1.08G = 129,600 \implies G = 120,000

Setting the sum equal to the total dollar amount raised (

\$129,600

) yields

1.08G = 129,600

, so

G = \frac{129,600}{1.08} = 120,000

.

Key Concept

Successive Percent Change and Base Identification

Question 289Question →

If $x$ is a real number that satisfies the equation $|2x - 1| = 3x + 11$ , what is the value of $x^2 + 2x$ ?

0

3

8

35

120

Show answer & explanation

Answer: $0$

Answer

The value of

x^2 + 2x

is

0

.

Solving the equation

|2x - 1| = 3x + 11

produces two algebraic candidates:

x = -12

and

x = -2

. Substituting

x = -12

into the right side yields

3(-12) + 11 = -25

. Since absolute value expressions cannot be negative,

x = -12

is an extraneous solution. Substituting

x = -2

yields

|-5| = 5

, which is valid. Evaluating

x^2 + 2x

at

x = -2

gives

(-2)^2 + 2(-2) = 0

.

Step-by-Step Solution

1

Set up the two linear cases for the absolute value equation

|2x - 1| = 3x + 11

.

Case 1:

2x - 1 = 3x + 11

; Case 2:

2x - 1 = -(3x + 11)

.

By definition,

|u| = c

implies

u = c

or

u = -c

(provided

c \geq 0

).

2

Solve each linear equation for candidate values of

x

.

From Case 1:

x = -12

. From Case 2:

2x - 1 = -3x - 11 \implies 5x = -10 \implies x = -2

.

Isolate

x

algebraically in both equations.

3

Check candidate solutions in the original equation to eliminate extraneous roots.

For

x = -12

:

|2(-12) - 1| = |-25| = 25

, but

3(-12) + 11 = -25 \neq 25

(extraneous). For

x = -2

:

|2(-2) - 1| = |-5| = 5

, and

3(-2) + 11 = 5

(valid solution).

An absolute value cannot equal a negative number; substituting back is mandatory.

4

Evaluate the target expression

x^2 + 2x

using the valid root

x = -2

.

(-2)^2 + 2(-2) = 4 - 4 = 0

.

Substitute the verified real solution into the given expression.

Key Concept

Solving Absolute Value Linear Equations and Checking for Extraneous Solutions

Estimated Time:1m 30s

Question 290Question →

Set $S$ consists of $n$ consecutive integers. The arithmetic mean of the $5$ smallest integers in Set $S$ is $-12$ , and the arithmetic mean of the $5$ largest integers in Set $S$ is $24$ . What is the value of $n$ ?

36

37

40

41

42

Show answer & explanation

Answer: 41

Answer

41

For an evenly spaced set of 5 consecutive integers, the arithmetic mean equals the middle term. Therefore, the 3rd smallest term of Set

S

is

-12

, making the 1st term

x = -14

. Similarly, the 3rd term from the end of Set

S

is

24

. Representing the 3rd term from the end as

x + n - 3

and substituting

x = -14

yields

-14 + n - 3 = 24

, which simplifies to

n = 41

.

Step-by-Step Solution

1

Express the 5 smallest integers and find the first term of the set.

Let the set

S

be represented as

\{x, x+1, x+2, \dots, x+n-1\}

. The 5 smallest integers are

x, x+1, x+2, x+3, x+4

. Their arithmetic mean is the middle term,

x+2

. Setting

x+2 = -12

yields

x = -14

.

In any set of consecutive integers with an odd number of elements, the arithmetic mean equals the median (middle term).

2

Express the 5 largest integers and set up an equation for

n

.

The 5 largest integers in Set

S

are

x+n-5, x+n-4, x+n-3, x+n-2, x+n-1

. Their arithmetic mean is the middle term,

x+n-3

. Setting

x+n-3 = 24

and substituting

x = -14

gives

-14 + n - 3 = 24

.

The 5 largest elements also form an evenly spaced set whose mean is the middle of those 5 terms.

3

Solve for

n

.

n - 17 = 24 \implies n = 41

.

Simplifying the linear equation gives the total number of consecutive integers in Set

S

.

Key Concept

Arithmetic Mean and Median Equivalence in Consecutive Integer Subsets

Question 291Question →

An integer is selected at random from the set of all positive integers less than or equal to 40. Given that the selected integer is a multiple of 3, what is the probability that it is also a multiple of 4?

\frac{3}{40}

\frac{3}{13}

\frac{3}{10}

\frac{1}{4}

\frac{4}{13}

Show answer & explanation

Answer: $\frac{3}{13}$

Answer

The probability that the selected integer is a multiple of 4, given that it is a multiple of 3, is

\frac{3}{13}

.

The condition 'given that the selected integer is a multiple of 3' restricts the sample space to the 13 positive integers up to 40 that are divisible by 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, and 39. Among these 13 numbers, those that are also multiples of 4 are multiples of 12, namely 12, 24, and 36 (3 numbers). Therefore, the conditional probability is

\frac{3}{13}

.

Step-by-Step Solution

1

Identify the restricted sample space defined by the condition.

The positive integers less than or equal to 40 that are multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, and 39. There are 13 such integers.

For conditional probability

P(A \mid B)

, the sample space must be restricted to all outcomes satisfying condition

B

(being a multiple of 3).

2

Identify the favorable outcomes within the restricted sample space.

Integers that are multiples of both 3 and 4 must be multiples of 12. Among the 13 multiples of 3, those that are also multiples of 12 are: 12, 24, and 36. There are 3 such integers.

The numerator of

P(A \mid B)

counts the elements in the intersection

A \cap B

.

3

Calculate the conditional probability.

P(\text{Multiple of 4} \mid \text{Multiple of 3}) = \frac{3}{13}

.

Divide the number of favorable outcomes (3) by the total number of outcomes in the restricted sample space (13).

Key Concept

Conditional Probability with Restricted Sample Space

Estimated Time:1m 30s

Question 292Question →

If $n$ is an integer such that $|2n + 1| \le |n - 4|$ , how many distinct integer values of $n$ satisfy the inequality?

5

6

7

8

9

Show answer & explanation

Answer: 7

Answer

There are 7 distinct integer values of n that satisfy the inequality.

Squaring both non-negative sides of the absolute value inequality

|2n + 1| \le |n - 4|

eliminates the absolute value bars without requiring multiple case splits. Expanding

(2n + 1)^2 \le (n - 4)^2

yields

4n^2 + 4n + 1 \le n^2 - 8n + 16

, which simplifies to

3n^2 + 12n - 15 \le 0

. Dividing the inequality by 3 gives

n^2 + 4n - 5 \le 0

, which factors as

(n + 5)(n - 1) \le 0

. The inequality is satisfied for all values in the closed interval

[-5, 1]

. Listing the integers in this range yields

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

, which comprises exactly 7 integers.

Step-by-Step Solution

1

Square both sides of the inequality since absolute values are non-negative.

(2n + 1)^2 \le (n - 4)^2

Since

|A| \le |B|

is equivalent to

A^2 \le B^2

for all real numbers.

2

Expand both algebraic expressions and move all terms to one side.

4n^2 + 4n + 1 \le n^2 - 8n + 16 \implies 3n^2 + 12n - 15 \le 0

Standard quadratic inequality form requires comparing the quadratic polynomial to zero.

3

Simplify by dividing by 3 and factor the quadratic expression.

n^2 + 4n - 5 \le 0 \implies (n + 5)(n - 1) \le 0

Dividing by a positive constant preserves the inequality sign and allows factoring into linear binomials.

4

Determine the solution set for the inequality and count the integer solutions.

The inequality holds for

-5 \le n \le 1

. The integer solutions are

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

, giving a total of

1 - (-5) + 1 = 7

integers.

The quadratic expression is non-positive between its two real roots, inclusive of the endpoints.

Key Concept

Solving Absolute Value Inequalities via Squaring and Quadratic Factoring

Estimated Time:1m 30s

Question 293Question →

A laboratory tested $200$ synthetic compound samples for two properties: thermal stability and chemical resistance. Among the samples, $120$ exhibited thermal stability, $90$ exhibited chemical resistance, and $50$ exhibited neither property. If a sample is selected at random from those that exhibited thermal stability, what is the probability that it also exhibited chemical resistance?

Show answer & explanation

Answer: 0.5

Answer

0.5

The conditional probability of selecting a sample with chemical resistance given that it has thermal stability is found by dividing the number of samples with both properties (

60

) by the total number of samples with thermal stability (

120

), giving

\frac{60}{120} = 0.5

.

Step-by-Step Solution

1

Find the total number of samples exhibiting at least one of the two properties.

Since

50

out of

200

samples exhibited neither property, the number of samples exhibiting at least one property is

200 - 50 = 150

.

The total population consists of samples exhibiting at least one property plus samples exhibiting neither property.

2

Calculate the number of samples exhibiting both thermal stability (

T

) and chemical resistance (

C

).

Using the inclusion-exclusion principle

|T \cup C| = |T| + |C| - |T \cap C|

, we have

150 = 120 + 90 - |T \cap C|

, which yields

|T \cap C| = 60

.

Overlapping sets require subtracting the intersection to avoid double-counting elements.

3

Compute the conditional probability

P(C|T)

.

P(C|T) = \frac{|T \cap C|}{|T|} = \frac{60}{120} = 0.5

.

The given condition restricts the sample space to only the

120

samples exhibiting thermal stability.

Key Concept

Conditional probability restricts the sample space to the given condition's outcome space:

P(A|B) = \frac{|A \cap B|}{|B|}

.

Estimated Time:1m 30s

Question 294Question →

A store owner purchased a shipment of coats. She marked up the cost price of each coat by $40\%$ to set the regular selling price. During a clearance sale, she offered a discount of $15\%$ off the regular selling price on all remaining coats. If a coat was sold at the clearance price for $\$238$ , what was the cost price of the coat?

\$168.00

\$170.00

\$190.40

\$200.00

\$212.50

Show answer & explanation

Answer: $\$200.00$

Answer

The cost price of the coat was

\$200.00

.

The correct answer is

\$200.00

. The regular price is

140\%

of the cost price (

1.40C

). A

15\%

discount on the regular price means the clearance price is

85\%

of the regular price, which translates to

0.85 \times 1.40C = 1.19C

. Setting

1.19C = \$238

yields

C = \$200.00

.

Step-by-Step Solution

1

Define variables and express the regular selling price in terms of the cost price.

Let

C

be the cost price of a coat. The regular selling price is

R = C + 0.40C = 1.40C

.

Markup is calculated as a percentage of the cost price.

2

Express the clearance price in terms of the regular selling price and cost price.

The clearance price is

R \times (1 - 0.15) = 1.40C \times 0.85 = 1.19C

.

The discount of

15\%

is applied to the regular selling price, resulting in

85\%

of the regular selling price.

3

Solve for the cost price

C

using the given clearance price.

1.19C = 238 \implies C = \frac{238}{1.19} = 200

.

Equating the algebraic expression for the clearance price to the given numerical value gives the original cost price.

Key Concept

Successive Percentage Changes in Profit and Loss

Question 295Question →

A boutique bakery sells custom gift baskets containing two types of pastries: almond tarts and chocolate croissants. Basket A contains 4 almond tarts and 3 chocolate croissants and costs $62. Basket B contains 3 almond tarts and 4 chocolate croissants and costs$ 57. What is the combined cost of 1 almond tart and 1 chocolate croissant?

$15

$17

$19

$30

$34

Show answer & explanation

Answer: $17

Answer

The combined cost of 1 almond tart and 1 chocolate croissant is $17.

By representing the prices of an almond tart and a chocolate croissant as

t

and

c

, we form the system

4t + 3c = 62

and

3t + 4c = 57

. Adding both equations yields

7t + 7c = 119

. Dividing both sides by 7 gives

t + c = 17

, which directly provides the combined price of 1 almond tart and 1 chocolate croissant.

Step-by-Step Solution

1

Set up a system of two linear equations using variables for the prices of the pastries.

Let

t

be the price of one almond tart and

c

be the price of one chocolate croissant.
Equation 1:

4t + 3c = 62

Equation 2:

3t + 4c = 57

Translating the problem statement into standard linear algebraic equations.

2

Add the two linear equations together.

(4t + 3c) + (3t + 4c) = 62 + 57 \implies 7t + 7c = 119

Symmetric coefficients allow finding the sum of

t + c

without needing to solve for individual variables first.

3

Divide the combined equation by 7 to solve for

(t + c)

.

t + c = \frac{119}{7} = 17

Factoring out 7 gives

7(t + c) = 119

, which simplifies directly to the requested sum.

Key Concept

Solving Systems of Linear Equations via Symmetric Coefficient Addition

Alternative Method

Alternatively, solve for one variable first: multiply Equation 1 by 3 (

12t + 9c = 186

) and Equation 2 by 4 (

12t + 16c = 228

). Subtracting the equations gives

7c = 42 \implies c = 6

. Substituting

c = 6

into Equation 1 gives

4t + 18 = 62 \implies 4t = 44 \implies t = 11

. Thus,

t + c = 11 + 6 = 17

.

Estimated Time:1m 15s

Question 296Question →

A municipal water treatment facility operates three types of filtration units: Model X, Model Y, and Model Z.

\bullet \text{ 2 Model X units, 3 Model Y units, and 1 Model Z unit together process 134,000 gallons per hour.}

\bullet \text{ 1 Model X unit, 4 Model Y units, and 2 Model Z units together process 156,000 gallons per hour.}

\bullet \text{ 3 Model X units, 1 Model Y unit, and 4 Model Z units together process 196,000 gallons per hour.}

What is the processing capacity, in thousands of gallons per hour, of a single Model Y filtration unit?

Show answer & explanation

Answer: 20

Answer

The processing capacity of a single Model Y filtration unit is 20 thousand gallons per hour.

By setting up the system of three linear equations

2x + 3y + z = 134

,

x + 4y + 2z = 156

, and

3x + y + 4z = 196

, and eliminating variables systematically through substitution and combination, we find

y = 20

. Thus, a single Model Y filtration unit processes 20 thousand gallons per hour.

Step-by-Step Solution

1

Set up a system of linear equations in three variables.

Let

x

,

y

, and

z

represent the hourly capacities (in thousands of gallons) of Model X, Model Y, and Model Z respectively.

\begin{aligned} (1) \quad 2x + 3y + z &= 134 \\ (2) \quad x + 4y + 2z &= 156 \\ (3) \quad 3x + y + 4z &= 196 \end{aligned}

Expressing the given conditions algebraically translates the word problem into a solvable linear system.

2

Express

x

in terms of

y

and

z

using Equation (2).

x = 156 - 4y - 2z

Equation (2) has a coefficient of 1 for

x

, making it ideal for algebraic substitution.

3

Substitute

x

into Equations (1) and (3) to eliminate

x

.

Substituting into Equation (1):

2(156 - 4y - 2z) + 3y + z = 134 \implies 312 - 8y - 4z + 3y + z = 134

312 - 5y - 3z = 134 \implies (4) \quad 5y + 3z = 178

Substituting into Equation (3):

3(156 - 4y - 2z) + y + 4z = 196 \implies 468 - 12y - 6z + y + 4z = 196

468 - 11y - 2z = 196 \implies (5) \quad 11y + 2z = 272

Reducing a 3-variable system to a 2-variable system simplifies the calculation.

4

Eliminate variable

z

from Equations (4) and (5) to solve for

y

.

Multiply Equation (4) by 2:

10y + 6z = 356

Multiply Equation (5) by 3:

33y + 6z = 816

Subtract the first result from the second:

(33y + 6z) - (10y + 6z) = 816 - 356

23y = 460 \implies y = 20

Eliminating

z

directly yields the required value of

y

, which is the capacity of Model Y.

Key Concept

Solving systems of three linear equations in three variables via substitution and elimination.

Question 297Question →

An investor allocated a total initial capital of $\$60,000$ between two funds, Account X and Account Y. During the first year, Account X increased in value by $40\%$ , while Account Y increased in value by $20\%$ . During the second year, Account X decreased in value by $15\%$ relative to its value at the end of the first year, while Account Y increased in value by $10\%$ relative to its value at the end of the first year. If the total combined value of both accounts at the end of the second year was $\$74,000$ , what was the initial amount invested in Account X?

\$20,000

\$30,000

\$40,000

\$45,000

\$50,000

Show answer & explanation

Answer: $\$40,000$

Answer

The initial amount invested in Account X was

\$40,000

.

The correct answer is

\$40,000

. Account X increases by

40\%

in Year 1 to

1.40X

and decreases by

15\%

in Year 2, yielding

1.40X \times 0.85 = 1.19X

. Account Y increases by

20\%

in Year 1 and

10\%

in Year 2, yielding

(60,000 - X) \times 1.20 \times 1.10 = 1.32(60,000 - X)

. Setting

1.19X + 1.32(60,000 - X) = 74,000

gives

-0.13X = -5,200

, so

X = 40,000

.

Step-by-Step Solution

1

Calculate the net multiplier for Account X over the two-year period.

Multiplier for Account X =

(1 + 0.40) \times (1 - 0.15) = 1.40 \times 0.85 = 1.19

Successive percentage changes must be applied sequentially to updated base values.

2

Calculate the net multiplier for Account Y over the two-year period.

Multiplier for Account Y =

(1 + 0.20) \times (1 + 0.10) = 1.20 \times 1.10 = 1.32

Successive percentage increases compound on the value at the end of the first year.

3

Set up an algebraic equation for the total combined final value.

Let

X

be the initial capital in Account X. Then

60,000 - X

is the initial capital in Account Y.

1.19X + 1.32(60,000 - X) = 74,000

The sum of the final values of both accounts equals the total combined value of

\$74,000

.

4

Solve the linear equation for

X

.

1.19X + 79,200 - 1.32X = 74,000 \implies -0.13X = -5,200 \implies X = 40,000

Dividing

-5,200

by

-0.13

yields the initial amount invested in Account X.

Key Concept

Successive Percent Change and Base Value Determination

Question 298Question →

A technology company conducted a survey of its $200$ software engineers regarding their proficiency in three programming languages: Python, Java, and C++. The survey revealed the following data:
- $110$ engineers are proficient in Python.
- $85$ engineers are proficient in Java.
- $75$ engineers are proficient in C++.
- $40$ engineers are proficient in both Python and Java.
- $35$ engineers are proficient in both Java and C++.
- $30$ engineers are proficient in both Python and C++.
- $15$ engineers are proficient in all three programming languages.

How many of the surveyed engineers are proficient in none of these three programming languages?

15

20

25

35

50

Show answer & explanation

Answer: 20

Answer

The number of software engineers proficient in none of the three programming languages is 20.

Using the 3-set inclusion-exclusion formula, the total number of engineers proficient in at least one programming language is

110 + 85 + 75 - (40 + 35 + 30) + 15 = 180

. Subtracting this from the total sample size of

200

yields

200 - 180 = 20

engineers proficient in none of the three languages.

Step-by-Step Solution

1

State the Principle of Inclusion-Exclusion formula for three sets.

N(\text{At least one}) = N(P) + N(J) + N(C) - [N(P \cap J) + N(J \cap C) + N(P \cap C)] + N(P \cap J \cap C)

To find the total number of engineers in at least one category without double-counting overlapping groups.

2

Substitute the given values into the formula.

N(\text{At least one}) = 110 + 85 + 75 - (40 + 35 + 30) + 15 = 270 - 105 + 15 = 180

Calculate the total number of unique engineers who know at least one language.

3

Subtract the number of engineers proficient in at least one language from the total population.

N(\text{None}) = 200 - 180 = 20

The total population consists of engineers proficient in at least one language plus those proficient in none.

Key Concept

Three-Set Overlapping Venn Diagrams and the Principle of Inclusion-Exclusion

Estimated Time:2m 0s

Question 299Question →

In a medical study involving $300$ clinical trial participants, researchers recorded the occurrence of three common side effects: fatigue, nausea, and insomnia. A total of $160$ participants reported fatigue, $140$ reported nausea, and $120$ reported insomnia. Additionally, $60$ participants reported both fatigue and nausea, $50$ reported both fatigue and insomnia, and $55$ reported both nausea and insomnia. If $30$ participants reported none of these three side effects, how many participants reported exactly two of these side effects?

Show answer & explanation

Answer: 120

Answer

120 participants reported exactly two of the side effects.

To calculate the number of participants with exactly two side effects, first find the total number of participants experiencing at least one side effect (

300 - 30 = 270

). Next, using the inclusion-exclusion formula

270 = 160 + 140 + 120 - (60 + 50 + 55) + x

, we find that

x = 15

participants experienced all three side effects. Finally, subtracting

15

from each pairwise overlap gives the counts for exactly two side effects:

45

(Fatigue & Nausea only),

35

(Fatigue & Insomnia only), and

40

(Nausea & Insomnia only). Summing these regions yields

45 + 35 + 40 = 120

.

Step-by-Step Solution

1

Calculate the total number of participants who experienced at least one side effect

270 participants

Subtracting the 30 participants who reported no side effects from the total study size of 300 gives

|F \cup N \cup I| = 300 - 30 = 270

.

2

Determine the number of participants who experienced all three side effects

15 participants

Applying inclusion-exclusion gives

270 = 160 + 140 + 120 - (60 + 50 + 55) + x

, simplifying to

270 = 255 + x

, so

x = 15

.

3

Calculate the sum of participants experiencing exactly two side effects

120 participants

The sum of pairwise intersections (

60 + 50 + 55 = 165

) counts individuals with all three side effects three times. Subtracting

3 \times 15 = 45

from

165

yields

120

.

Key Concept

Three-Set Overlapping Sets and Inclusion-Exclusion Principle

Estimated Time:2m 0s

Question 300Question →

Historians analyzing early twentieth-century iron gall ink manuscripts observed that documents stored in wooden cabinets suffered significantly less paper degradation than identical manuscripts stored in metal cabinets within the same archive room. The archivists concluded that volatile organic compounds emitted by the untreated cedar wood reacted with atmospheric moisture to form a protective barrier on the paper surface, thereby inhibiting acid hydrolysis of the cellulose fibers.

Which of the following, if true, provides the strongest support for the archivists' conclusion?

Manuscripts stored in cedar cabinets whose interior surfaces had been coated with an impermeable synthetic sealant experienced paper degradation rates identical to those of manuscripts stored in metal cabinets.

Chemical assays demonstrate that the volatile organic compounds emitted by untreated cedar wood increase the overall acidity of surrounding air over prolonged storage periods.

Iron gall ink was the primary writing medium utilized by government offices and academic institutions throughout Western Europe during the early twentieth century.

Historical documents written with carbon-based inks show minimal cellulose breakdown regardless of whether they are housed in metal or wooden storage units.

High ambient humidity levels within archival storage rooms generally correlate with accelerated rates of paper cellulose breakdown across diverse document collections.

Show answer & explanation

Answer: Manuscripts stored in cedar cabinets whose interior surfaces had been coated with an impermeable synthetic sealant experienced paper degradation rates identical to those of manuscripts stored in metal cabinets.

Answer

The argument is most strengthened by evidence showing that when cedar cabinets are treated with an impermeable sealant that prevents volatile organic compound emissions, the preservation benefit disappears and degradation matches that of metal cabinets.

The archivists claim that the protective effect is specifically caused by volatile organic compounds emitted by cedar wood. The option establishing that sealing the interior of cedar cabinets—preventing those emissions while retaining the cabinet's physical material and structure—eliminates the preservation advantage serves as a classic control experiment. It confirms that the emissions are the active protective agent, strongly supporting the archivists' conclusion.

Step-by-Step Solution

1

Deconstruct the argument structure into premise and conclusion.

Premise: Iron gall manuscripts stored in untreated cedar cabinets degraded less than identical manuscripts in metal cabinets in the same room. Conclusion: Volatile organic compounds emitted by cedar wood react with moisture to form a protective barrier that inhibits acid hydrolysis.

Identifying the explicit causal mechanism proposed by the author is essential for evaluating strengthening options.

2

Identify potential vulnerabilities or assumptions in the causal leap.

The author assumes that the chemical emissions themselves are the cause of preservation, rather than secondary physical attributes of wooden cabinets (such as light blockage, temperature insulation, or structural shielding).

A strong strengthening statement will validate the proposed cause by eliminating alternative explanations or testing the mechanism directly.

3

Evaluate the choices to find an option that confirms the proposed cause-and-effect relationship.

The choice showing that blocking cedar emissions with an impermeable sealant results in degradation rates identical to metal cabinets proves that the emissions are necessary for the preservation effect.

Eliminating the hypothesized cause while keeping the physical cabinet structure identical demonstrates a direct causal link.

Key Concept

Strengthening Arguments via Causal Mechanism Isolation

Estimated Time:2m 0s

Answer

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