All practice questions

Question 881Question →

An investment fund allocated its total capital into three distinct asset classes: Class X, Class Y, and Class Z. Class X yielded an annual return of $12\%$ , Class Y yielded an annual return of $18\%$ , and Class Z yielded an annual return of $6\%$ . The capital invested in Class X was exactly twice the capital invested in Class Y. If the overall weighted annual return for the entire fund was $11\%$ , what percentage of the total capital was invested in Class Z?

Show answer & explanation

Answer: 37.5

Answer

37.5%

Using the given relationship

W_X = 2W_Y

and total weight constraint

W_X + W_Y + W_Z = 1

, Class Z's weight can be written as

1 - 3W_Y

. Plugging these into the overall weighted return formula yields

12(2W_Y) + 18W_Y + 6(1 - 3W_Y) = 11

, which solves to

W_Y = 5/24

. Substituting back gives

W_Z = 9/24 = 37.5\%

.

Step-by-Step Solution

1

Define fractional weight variables for each asset class.

Let

W_X

,

W_Y

, and

W_Z

be the proportions of total capital invested in Class X, Class Y, and Class Z, such that

W_X + W_Y + W_Z = 1

.

Establishing a standard framework for weighted average calculation.

2

Express

W_X

and

W_Z

in terms of

W_Y

.

W_X = 2W_Y

, so

2W_Y + W_Y + W_Z = 1 \implies W_Z = 1 - 3W_Y

.

Reducing the number of unknown variables to one.

3

Formulate the weighted average equation using component returns.

12W_X + 18W_Y + 6W_Z = 11 \implies 12(2W_Y) + 18W_Y + 6(1 - 3W_Y) = 11

.

The fund's overall return equals the weighted sum of individual returns.

4

Solve the algebraic equation for

W_Y

.

24W_Y + 18W_Y + 6 - 18W_Y = 11 \implies 24W_Y + 6 = 11 \implies 24W_Y = 5 \implies W_Y = \frac{5}{24}

.

Determining the exact weight of Class Y.

5

Calculate the weight percentage for Class Z.

W_Z = 1 - 3\left(\frac{5}{24}\right) = 1 - \frac{15}{24} = \frac{9}{24} = \frac{3}{8} = 0.375 \text{ or } 37.5\%

.

Evaluating the target component weight requested by the question.

Key Concept

Weighted Average in Combined Financial Assets

Question 882Question →

A merchant blends two types of coffee beans: Type A, which costs $\$12$ per kilogram and contains $15\%$ caffeine by weight, and Type B, which costs $\$18$ per kilogram and contains $25\%$ caffeine by weight. The merchant mixes Type A and Type B to create a $60$ -kilogram batch of Blend C. If the total cost of the $60$ -kilogram batch of Blend C is $\$936$ , what is the percentage of caffeine by weight in Blend C?

Show answer & explanation

Answer: 21

Answer

The percentage of caffeine by weight in Blend C is 21%.

Solving the linear system for total mass (

A + B = 60

) and total cost (

12A + 18B = 936

) reveals that Blend C consists of 24 kg of Type A beans and 36 kg of Type B beans. Type A contributes

24 \times 0.15 = 3.6

kg of caffeine, and Type B contributes

36 \times 0.25 = 9.0

kg of caffeine. The total caffeine mass of 12.6 kg out of 60 kg total mass equals a concentration of

12.6 / 60 = 21\%

.

Step-by-Step Solution

1

Formulate linear system for bean masses based on total weight and cost

A + B = 60

and

12A + 18B = 936

, solving to

A = 24

kg and

B = 36

kg

The weight of each constituent bean type must be determined before total caffeine content can be evaluated.

2

Calculate caffeine mass from each component

Type A provides

3.6

kg (

24 \times 0.15

) and Type B provides

9.0

kg (

36 \times 0.25

)

Solute mass is found by multiplying total component weight by its percentage concentration.

3

Sum caffeine masses to find total caffeine in the mixture

Total caffeine =

3.6 + 9.0 = 12.6

kg

The total amount of caffeine present in Blend C is the combined sum from both bean types.

4

Determine caffeine concentration percentage of Blend C

\frac{12.6}{60} \times 100 = 21\%

Concentration is calculated as the ratio of total solute mass to total solution mass expressed as a percentage.

Key Concept

Weighted concentration combined with multi-variable cost systems

Question 883Question →

A financial firm's investment committee consists of 8 senior analysts and 4 junior analysts. If a project review panel of 3 members is chosen at random without replacement from this committee, what is the probability that the panel contains at least one junior analyst?

\frac{14}{55}

\frac{28}{55}

\frac{19}{27}

\frac{41}{55}

\frac{47}{55}

Show answer & explanation

Answer: $\frac{41}{55}$

Answer

The probability that the panel contains at least one junior analyst is

\frac{41}{55}

.

To find the probability of selecting at least one junior analyst, it is most efficient to use the complementary probability rule:

P(\text{at least one junior}) = 1 - P(\text{no junior})

. The total number of ways to choose 3 panel members from 12 committee members is

\binom{12}{3} = 220

. The number of ways to choose 3 senior analysts from 8 is

\binom{8}{3} = 56

. Thus, the probability of selecting no junior analysts is

\frac{56}{220} = \frac{14}{55}

. Subtracting this from 1 gives

1 - \frac{14}{55} = \frac{41}{55}

.

Step-by-Step Solution

1

Calculate total possible outcomes for choosing 3 members out of 12.

\binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220

total combinations.

Selection order does not matter, so combinations

nCr

are used.

2

Calculate the number of unfavorable outcomes where zero junior analysts are chosen (all 3 selected are senior analysts).

\binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56

ways.

All 3 members must be selected exclusively from the 8 senior analysts.

3

Find the probability of selecting zero junior analysts.

P(\text{no junior analysts}) = \frac{56}{220} = \frac{14}{55}

.

Divide the unfavorable outcomes by the total outcomes.

4

Apply the complementary probability principle:

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

.

P(\text{at least 1 junior}) = 1 - \frac{14}{55} = \frac{41}{55}

.

The event 'at least one junior analyst' is the logical complement of 'no junior analysts'.

Key Concept

Complementary Probability

Estimated Time:2m 0s

Question 884Question →

A manufacturing plant produces electronic components across three assembly lines: Line 1, Line 2, and Line 3. Line 1 accounts for $40\%$ of the plant's total production and has a defect rate of $2.0\%$ . Line 2 accounts for $35\%$ of total production and has a defect rate of $4.0\%$ . Line 3 produces the remaining components. If the overall defect rate for the entire manufacturing plant is $3.1\%$ , what is the defect rate of Line 3?

3.0\%

3.2\%

3.3\%

3.6\%

4.1\%

Show answer & explanation

Answer: $3.6\%$

Answer

3.6\%

The correct answer is

3.6\%

. Since Line 1 accounts for

40\%

of production at a

2.0\%

defect rate (

0.8\%

overall contribution) and Line 2 accounts for

35\%

at a

4.0\%

defect rate (

1.4\%

overall contribution), their combined contribution is

2.2\%

. Line 3 represents the remaining

25\%

(

0.25

) of production. Setting up

2.2\% + 0.25 \times d = 3.1\%

gives

0.25 d = 0.9\%

, yielding

d = 3.6\%

.

Step-by-Step Solution

1

Determine the proportion of total production for Line 3

The proportion for Line 3 is

100\% - (40\% + 35\%) = 25\%

(or

0.25

).

The sum of the production proportions of all three lines must equal

100\%

of total production.

2

Set up the weighted average equation for the overall defect rate

0.40(2.0\%) + 0.35(4.0\%) + 0.25(d\%) = 3.1\%

, where

d\%

represents the defect rate of Line 3.

The overall weighted average defect rate is the sum of each component's defect rate multiplied by its relative production weight.

3

Calculate the defect contributions from Line 1 and Line 2

Contribution of Line 1 =

0.40 \times 2.0\% = 0.8\%

; Contribution of Line 2 =

0.35 \times 4.0\% = 1.4\%

. Combined contribution =

0.8\% + 1.4\% = 2.2\%

.

Multiplying individual weights by their defect rates yields their weighted contribution to the overall plant defect rate.

4

Solve for the defect rate of Line 3 (

d

)

2.2\% + 0.25d\% = 3.1\% \implies 0.25d\% = 0.9\% \implies d = \frac{0.9}{0.25} = 3.6

. Thus, the defect rate of Line 3 is

3.6\%

.

Isolating the variable

d

gives the exact required defect rate for Line 3.

Key Concept

Weighted Average in Combined Sets

Estimated Time:2m 0s

Question 885Question →

If the roots of the quadratic equation $x^2 - (k - 2)x + (k + 1) = 0$ have a sum that is equal to twice their product, what is the value of $k$ ?

Show answer & explanation

Answer: -4

Answer

The value of

k

is

-4

.

By Vieta's formulas, the sum of the roots of

x^2 - (k-2)x + (k+1) = 0

is

k-2

and the product of the roots is

k+1

. Equating the sum to twice the product gives

k - 2 = 2(k + 1)

, which simplifies to

k - 2 = 2k + 2

and yields

k = -4

. Evaluating the discriminant with

k = -4

gives

\Delta = (-6)^2 - 4(-3) = 48 > 0

, ensuring real roots exist.

Step-by-Step Solution

1

Apply Vieta's formulas to express the sum and product of the roots in terms of

k

.

Sum of roots

= -\frac{b}{a} = k - 2

, and product of roots

= \frac{c}{a} = k + 1

.

For any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

and the product of the roots is

\frac{c}{a}

.

2

Set up an equation based on the condition that the sum of the roots equals twice their product.

k - 2 = 2(k + 1)

This directly translates the verbal condition given in the problem.

3

Solve the linear equation for

k

.

k - 2 = 2k + 2 \implies -4 = k

Subtracting

k

and 2 from both sides isolates

k

.

4

Check the discriminant to ensure real roots exist when

k = -4

.

\Delta = b^2 - 4ac = (-6)^2 - 4(1)(-3) = 36 + 12 = 48 > 0

A positive discriminant guarantees that the quadratic equation has two distinct real roots.

Key Concept

Vieta's Formulas for Quadratic Equations

Question 886Question →

A glass bowl contains 6 red candies and $N$ green candies, where $N$ is a positive integer. If two candies are selected at random one after another without replacement, the probability that both candies selected are red is $\frac{1}{3}$ . What is the value of $N$ ?

Show answer & explanation

Answer: 4

Answer

4

The correct value of

N

is 4. Because the selection is performed without replacement, the probability of drawing the first red candy is

\frac{6}{N+6}

and the second red candy is

\frac{5}{N+5}

. Setting their product equal to

\frac{1}{3}

yields

\frac{30}{(N+6)(N+5)} = \frac{1}{3}

, which expands to

N^2 + 11N - 60 = 0

. Factoring gives

(N-4)(N+15) = 0

, yielding

N = 4

as the only positive solution.

Step-by-Step Solution

1

Define total outcomes and event probabilities for dependent selection

Initial total candies =

N + 6

. Probability of first red =

\frac{6}{N+6}

. Probability of second red given first is red =

\frac{5}{N+5}

.

Since selection is without replacement, the total count and red count both decrease by 1 after the first draw, making the events dependent.

2

Set up the probability multiplication equation

P(\text{both red}) = \frac{6}{N+6} \times \frac{5}{N+5} = \frac{30}{(N+6)(N+5)} = \frac{1}{3}

.

The probability of joint dependent events

A

and

B

occurring is

P(A \cap B) = P(A) \times P(B|A)

.

3

Solve the equation for

N

(N+6)(N+5) = 90 \Rightarrow N^2 + 11N - 60 = 0 \Rightarrow (N-4)(N+15) = 0

.

Cross-multiplying gives a quadratic equation in terms of

N

.

4

Select the valid positive integer root

N = 4

.

The count of candies must be a positive number, so

N = -15

is discarded.

Key Concept

Probability of Dependent Events (Sampling without Replacement)

Question 887Question →

A cyclist and a runner start simultaneously from Point $A$ and travel along a straight path toward Point $B$ . The cyclist travels at a constant speed, and the runner travels at a lower constant speed. When the cyclist reaches Point $B$ , she immediately turns around and travels back toward Point $A$ at her original speed. She meets the runner at a location $12$ miles from Point $B$ .

If the cyclist had instead increased her speed by $20\%$ and the runner had decreased his speed by $10\%$ , the cyclist would have met the runner on her return trip at a location $20$ miles from Point $B$ .

What is the distance, in miles, between Point $A$ and Point $B$ ?

Show answer & explanation

Answer: 60

Answer

The distance between Point A and Point B is 60 miles.

The correct answer of 60 miles is obtained by setting up the ratio of distances traveled by the cyclist and the runner for both scenarios. Since time elapsed until they meet is the same for both people in each scenario, the ratio of distances equals the ratio of speeds. Setting the scaled speed ratio (4/3 times the original ratio) equal to the new distance ratio (D + 20)/(D - 20) yields the quadratic equation D^2 - 56D - 240 = 0, whose unique positive solution is D = 60.

Step-by-Step Solution

1

Set up the relative distance ratio for the first meeting condition.

v_c / v_r = (D + 12) / (D - 12)

Because time is equal for both participants, distance traveled is directly proportional to speed.

2

Adjust the speed ratio based on the given percentage changes.

v_c' / v_r' = (1.20 / 0.90) * (v_c / v_r) = (4/3) * (D + 12) / (D - 12)

Increasing speed by 20% scales it by 1.20, and decreasing speed by 10% scales it by 0.90.

3

Equate the adjusted speed ratio to the distance ratio for the second meeting condition.

(D + 20) / (D - 20) = 4(D + 12) / [3(D - 12)]

In the second scenario, the cyclist covers D + 20 miles while the runner covers D - 20 miles.

4

Expand and solve the resulting quadratic equation for D.

D^2 - 56D - 240 = 0, yielding D = 60 miles.

Cross-multiplying and simplifying yields a quadratic equation with positive root D = 60.

Key Concept

Distance-Speed Proportionality for Simultaneous Travel and Relative Motion

Estimated Time:3m 0s

Question 888Question →

A specialty tea manufacturer blends three varieties of tea leaves—Darjeeling, Assam, and Ceylon. Initially, the weights of Darjeeling, Assam, and Ceylon in a batch are in the ratio $3 : 4 : 5$ , respectively. After $20$ kilograms of Ceylon tea leaves are removed from the batch and replaced with $20$ kilograms of Darjeeling tea leaves, the new ratio of Darjeeling to Assam to Ceylon becomes $5 : 4 : 3$ . What was the total weight, in kilograms, of the initial tea batch?

Show answer & explanation

Answer: 120

Answer

The initial total weight of the tea batch was 120 kg.

By assigning

x

as the multiplier for the initial ratio

3 : 4 : 5

, the initial total weight is

12x

. The swap of

20

kg changes Darjeeling to

3x + 20

and Ceylon to

5x - 20

, while Assam remains

4x

. Setting

\frac{3x + 20}{4x} = \frac{5}{4}

gives

x = 10

, making the initial total weight

12 \times 10 = 120

kg.

Step-by-Step Solution

1

Define initial quantities using a common ratio multiplier

x

.

Darjeeling =

3x

, Assam =

4x

, Ceylon =

5x

. Total weight =

12x

.

Expressing quantities in terms of a single variable

x

allows setting up algebraic equations based on the changed quantities.

2

Adjust quantities after the exchange of tea leaves.

New Darjeeling =

3x + 20

, New Assam =

4x

, New Ceylon =

5x - 20

.

Replacing

20

kg of Ceylon with

20

kg of Darjeeling increases Darjeeling by

20

kg and decreases Ceylon by

20

kg while leaving Assam unchanged.

3

Set up a proportion equation comparing the new ratio of Darjeeling to Assam.

\frac{3x + 20}{4x} = \frac{5}{4}

The new ratio of Darjeeling to Assam is given as

5 : 4

.

4

Solve the linear equation for

x

.

x = 10

Cross-multiplying yields

12x + 80 = 20x

, which simplifies to

8x = 80

.

5

Compute the total initial weight.

12 \times 10 = 120

kg

The initial total weight is

12x

, so substituting

x = 10

gives the final answer of

120

kg.

Key Concept

Setting up algebraic equations for 3-part ratios under quantity addition/subtraction.

Estimated Time:1m 30s

Question 889Question →

Seven distinct paintings—3 landscapes and 4 portraits, one of which is a portrait of the founder—are to be displayed side-by-side in a single row along a gallery wall. If all 3 landscapes must be displayed together as a contiguous block, and the portrait of the founder cannot be placed at either end of the 7-painting row, in how many different linear arrangements can the 7 paintings be displayed?

72

288

432

576

720

Show answer & explanation

Answer: 432

Answer

432

The correct answer is 432. Treating the 3 landscape paintings as a single block leaves 5 items (the landscape block, the founder's portrait, and 3 other portraits) to arrange. There are 5! = 120 total arrangements of these 5 items. The founder's portrait occupies an end slot of the row if it is placed in either the 1st position (4! = 24 ways) or the 5th position (4! = 24 ways) among the 5 items. Subtracting these 48 invalid arrangements gives 120 - 48 = 72 valid block placements. Finally, accounting for the 3! = 6 internal arrangements of the landscapes inside their block yields 72 × 6 = 432 total linear arrangements.

Step-by-Step Solution

1

Group the 3 landscape paintings into a single block unit.

We now have 5 items to arrange linearly: the 1 landscape block, the founder's portrait, and the 3 other portraits.

Grouping elements that must remain together simplifies the arrangement problem into smaller independent choices.

2

Calculate the total number of linear arrangements of these 5 items without restrictions on the founder's portrait.

5! = 120 arrangements.

5 distinct objects can be ordered in 5 factorial ways.

3

Determine the number of arrangements where the founder's portrait is placed at either end of the row.

2 × 4! = 48 arrangements.

The founder's portrait is at an end of the 7-painting row if and only if it is in position 1 (first item) or position 5 (last item) among the 5 items. Fixing it at position 1 leaves 4! = 24 ways for the remaining items, and fixing it at position 5 gives another 4! = 24 ways.

4

Subtract the invalid end-position arrangements from the total 5-item arrangements.

120 - 48 = 72 valid arrangements of the 5 items.

Using complementary counting isolates the cases where the founder's portrait is not at either end.

5

Multiply by the number of internal arrangements of the 3 landscape paintings within their block.

72 × 3! = 72 × 6 = 432.

The 3 distinct landscapes within the single block can be ordered internally in 3! = 6 ways for each overall arrangement.

Key Concept

Permutations with Block Constraints and Complementary Restriction Counting

Question 890Question →

An oceanographic research station uses two deep-sea water sampling pumps, Pump $A$ and Pump $B$ , to fill a designated storage tank with seawater. Working alone at its constant rate, Pump $A$ can fill the tank in $10$ hours. Working alone at its constant rate, Pump $B$ can fill the tank in $15$ hours. If Pump $A$ begins filling the empty tank alone and operates for $4$ hours before shutting down, after which Pump $B$ operates alone to finish filling the tank, how many total hours does it take to fill the tank?

Show answer & explanation

Answer: 13

Answer

The total time required to fill the storage tank is 13 hours.

Pump A completes 2/5 of the tank during its 4 hours of operation. The remaining 3/5 of the tank is filled by Pump B in 9 hours. Summing the 4 hours from Pump A and the 9 hours from Pump B gives a total time of 13 hours.

Step-by-Step Solution

1

Calculate individual work rates

Pump A rate = 1/10 tank/hour; Pump B rate = 1/15 tank/hour

Work rate is equal to the fraction of the job completed per unit of time (1 divided by total time).

2

Calculate work completed during the first stage

Work done by Pump A = 4 * (1/10) = 2/5 of the tank

Work completed equals work rate multiplied by operating time.

3

Determine the remaining work

Remaining work = 1 - 2/5 = 3/5 of the tank

The entire job is represented by 1 whole unit.

4

Calculate time required for the second stage

Time for Pump B = (3/5) / (1/15) = 9 hours

Time taken equals remaining work divided by the rate of Pump B.

5

Sum the time for both stages

Total time = 4 + 9 = 13 hours

The question asks for the total elapsed time from the start of filling to completion.

Key Concept

Work Rate and Combined Work

Estimated Time:2m 0s

Question 891Question →

If $x$ is a real number, what is the sum of all valid real solutions to the equation $\sqrt{3x - 2} - \sqrt{x + 3} = 1$ ?

1

3

4

6

7

Show answer & explanation

Answer: 6

Answer

The sum of all valid real solutions is 6.

To solve the radical equation, we isolate

\sqrt{3x - 2}

to get

\sqrt{3x - 2} = 1 + \sqrt{x + 3}

. Squaring both sides yields

3x - 2 = x + 4 + 2\sqrt{x + 3}

, which simplifies to

x - 3 = \sqrt{x + 3}

. Squaring both sides a second time produces the quadratic

x^2 - 7x + 6 = 0

, giving candidate solutions

x = 6

and

x = 1

. Checking in the original equation reveals that

x = 6

yields

\sqrt{16} - \sqrt{9} = 1

, which is true, whereas

x = 1

yields

\sqrt{1} - \sqrt{4} = -1 \neq 1

, making

x = 1

extraneous. The only valid solution is 6, so the sum of all valid solutions is 6.

Step-by-Step Solution

1

Isolate one radical expression on one side of the equation.

\sqrt{3x - 2} = 1 + \sqrt{x + 3}

Isolating a radical allows for clean squaring of both sides without creating cross-terms with multiple roots.

2

Square both sides of the equation.

3x - 2 = 1 + 2\sqrt{x + 3} + (x + 3) \implies 3x - 2 = x + 4 + 2\sqrt{x + 3}

Eliminates the radical on the left side.

3

Isolate the remaining radical term and simplify.

2x - 6 = 2\sqrt{x + 3} \implies x - 3 = \sqrt{x + 3}

Simplifies the algebraic expression prior to squaring a second time.

4

Square both sides again to obtain a quadratic equation.

(x - 3)^2 = x + 3 \implies x^2 - 6x + 9 = x + 3 \implies x^2 - 7x + 6 = 0

Eliminates all remaining radical terms.

5

Factor the quadratic equation to find algebraic candidate solutions.

(x - 6)(x - 1) = 0 \implies x = 6 \text{ or } x = 1

Identifies potential real roots.

6

Test candidate solutions in the original equation

\sqrt{3x - 2} - \sqrt{x + 3} = 1

to remove extraneous solutions.

For

x = 6

:

\sqrt{16} - \sqrt{9} = 4 - 3 = 1

(Valid). For

x = 1

:

\sqrt{1} - \sqrt{4} = 1 - 2 = -1 \neq 1

(Extraneous). Thus,

x = 6

is the sole valid solution.

Squaring equations during algebraic steps can introduce extraneous roots that do not satisfy the original radical equation.

Key Concept

Solving Radical Equations and Eliminating Extraneous Solutions

Estimated Time:2m 0s

Question 892Question →

If 3 components are randomly selected without replacement from a batch of 10 components containing exactly 2 defective components, the probability that at least one selected component is defective is equal to $\frac{8}{15}$ .

True

False

Show answer & explanation

Answer: True

Answer

True. The probability of selecting at least one defective component is indeed

\frac{8}{15}

.

The statement correctly computes the probability of selecting at least one defective component using

1 - P(\text{no defective components}) = 1 - \frac{\binom{8}{3}}{\binom{10}{3}} = 1 - \frac{7}{15} = \frac{8}{15}

.

Step-by-Step Solution

1

Identify the complementary event

The complement of 'at least one defective component' is 'zero defective components' (all 3 selected components are non-defective).

Calculating

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

avoids calculating multiple dependent individual outcomes.

2

Calculate the total combinations for choosing 3 components from 10

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

.

This establishes the sample space size.

3

Calculate the combinations for choosing 3 non-defective components from 8

\binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56

.

There are 8 non-defective components in the batch.

4

Compute

P(\text{0 defective})

and subtract from 1

P(\text{0 defective}) = \frac{56}{120} = \frac{7}{15}

. Therefore,

P(\text{at least 1 defective}) = 1 - \frac{7}{15} = \frac{8}{15}

.

Completing the complementary calculation confirms the statement is correct.

Key Concept

Complementary Probability and At-Least-One Scenarios without Replacement

Question 893Question →

A software provider offers two cloud subscription plans for enterprise clients: Tier A and Tier B. The monthly fee per user is $\$x$ for Tier A and $\$y$ for Tier B. In 2024, a corporate customer purchased 50 user licenses for Tier A and 30 user licenses for Tier B, resulting in a total monthly cost of $\$13,500$ .

In 2025, the provider adjusted its rates by increasing the Tier A monthly fee per user by $20\%$ and increasing the Tier B monthly fee per user by $10\%$ . Additionally, the customer altered its order by increasing its Tier A licenses by $10\%$ and decreasing its Tier B licenses by $20\%$ . If the customer's total monthly cost in 2025 was $\$15,180$ , what was the original monthly fee per user for Tier A in 2024?

\$125

\$150

\$175

\$200

\$225

Show answer & explanation

Answer: $\$150$

Answer

The original monthly fee per user for Tier A in 2024 was

\$150

.

The correct option correctly models the two-variable linear system. By expressing the 2024 total cost as

5x + 3y = 1,350

and the 2025 cost as

66x + 26.4y = 15,180

, solving the simultaneous equations yields

x = 150

for Tier A and

y = 200

for Tier B.

Step-by-Step Solution

1

Formulate the linear equation for the 2024 monthly cost.

50x + 30y = 13,500

, which simplifies to

5x + 3y = 1,350

.

The total cost is the sum of the products of license quantities and their respective per-user prices.

2

Determine the 2025 price per user and license quantities for both tiers.

Tier A: New price =

1.20x

, New quantity =

50 \times 1.10 = 55

.
Tier B: New price =

1.10y

, New quantity =

30 \times 0.80 = 24

.

Price and quantity changes must be applied independently as percentage increases or decreases.

3

Formulate the linear equation for the 2025 total monthly cost.

55(1.20x) + 24(1.10y) = 15,180 \implies 66x + 26.4y = 15,180

.

Multiply the updated unit price by the updated quantity for each tier.

4

Simplify the 2025 equation and solve the system of linear equations.

Multiply

66x + 26.4y = 15,180

by

5

to eliminate decimals:

330x + 132y = 75,900

, which simplifies by dividing by

6

to

55x + 22y = 12,650

.
Multiply

5x + 3y = 1,350

by

11

:

55x + 33y = 14,850

.
Subtract the two equations:

(55x + 33y) - (55x + 22y) = 14,850 - 12,650 \implies 11y = 2,200 \implies y = 200

.
Substitute

y = 200

into

5x + 3(200) = 1,350 \implies 5x = 750 \implies x = 150

.

Eliminate variable

x

to solve for

y

, then back-substitute to find

x

.

Key Concept

Multi-variable system modeling with multi-step percentage modifications

Estimated Time:2m 30s

Question 894Question →

An investment portfolio consists of three asset classes: Equities, Bonds, and Real Estate. The total monetary value of Equities to Bonds is in the ratio $3 : 2$ , and the total value of Bonds to Real Estate is in the ratio $4 : 5$ . Over the past year, Equities yielded an average return of $12\%$ . If the overall weighted average return of the entire portfolio was $10.4\%$ , and the average percentage return of Bonds was $25\%$ higher than the average percentage return of Real Estate, what was the average percentage return of Bonds?

Show answer & explanation

Answer: 10.5

Answer

The average percentage return of Bonds was 10.5%.

The correct return of Bonds is obtained by first unifying the given ratios into a continuous ratio of weights

6 : 4 : 5

for Equities, Bonds, and Real Estate respectively. Substituting these weights and the algebraic relation between Real Estate return (

r

) and Bond return (

1.25r

) into the weighted average expression yields

r = 8.4\%

. Multiplying this by

1.25

gives the bond return of

10.5\%

.

Step-by-Step Solution

1

Determine the combined ratio of portfolio weights for all three asset classes

The ratio Equities : Bonds : Real Estate is 6 : 4 : 5, giving a total weight sum of 15 units.

Since Equities to Bonds is 3:2 (or 6:4) and Bonds to Real Estate is 4:5, the common term Bonds allows combining the ratios into 6 : 4 : 5.

2

Express the unknown asset returns in terms of a single variable

Return of Real Estate =

r\%

, Return of Bonds =

1.25r\%

.

The problem states the return of Bonds is 25% higher than that of Real Estate (

r + 0.25r = 1.25r

).

3

Set up and solve the weighted average formula for the portfolio

r = 8.4\%

.

Using

\frac{6(12) + 4(1.25r) + 5r}{15} = 10.4

, we get

72 + 10r = 156

, leading to

10r = 84

.

4

Calculate the targeted return for Bonds

Return of Bonds =

10.5\%

.

Multiplying

1.25

by

8.4

yields

10.5

.

Key Concept

Combining multiple ratios to establish weighted average component weights and solving multi-variable weighted average equations

Estimated Time:2m 30s

Question 895Question →

A charity organization's steering board consists of $6$ donors and $4$ volunteers. Two board members are selected at random, one after another without replacement, to attend a national conference. What is the probability that the first member selected is a donor and the second member selected is a volunteer?

\frac{6}{25}

\frac{4}{15}

\frac{1}{3}

\frac{8}{15}

\frac{12}{25}

Show answer & explanation

Answer: $\frac{4}{15}$

Answer

The probability that the first member selected is a donor and the second member selected is a volunteer is

\frac{4}{15}

.

The correct probability is calculated by multiplying the probability of the first event by the conditional probability of the second event given that the first event occurred. The probability of choosing a donor first is

\frac{6}{10}

. Since the selection is made without replacement, there are

9

total members remaining for the second draw,

4

of whom are volunteers. Therefore, the probability of selecting a volunteer second is

\frac{4}{9}

. The overall probability is

\frac{6}{10} \times \frac{4}{9} = \frac{24}{90} = \frac{4}{15}

.

Step-by-Step Solution

1

Calculate the probability of selecting a donor on the first choice.

Since there are

6

donors out of

10

total board members,

P(\text{1st Donor}) = \frac{6}{10} = \frac{3}{5}

.

The sample space initially contains

10

members,

6

of whom are donors.

2

Calculate the conditional probability of selecting a volunteer on the second choice given that a donor was chosen first.

After one donor is selected,

9

members remain,

4

of whom are volunteers. Thus,

P(\text{2nd Volunteer} \mid \text{1st Donor}) = \frac{4}{9}

.

The selection is made without replacement, reducing both the total number of members in the pool and the sample space size.

3

Multiply the sequential probabilities for dependent events.

P(\text{1st Donor and 2nd Volunteer}) = \frac{6}{10} \times \frac{4}{9} = \frac{24}{90} = \frac{4}{15}

.

For dependent events

A

and

B

, the joint probability is

P(A \cap B) = P(A) \times P(B \mid A)

.

Key Concept

Probability of Dependent Events (Sampling Without Replacement)

Question 896Question →

If $x$ is a real number satisfying the inequality $|x - 3| + 2|x + 1| \leq 7$ , what is the minimum possible value of $x^2 - 2x$ ?

-1

0

-2

3

8

Show answer & explanation

Answer: $-1$

Answer

The minimum possible value of

x^2 - 2x

is

-1

.

The absolute value inequality

|x - 3| + 2|x + 1| \leq 7

restricts

x

to the closed interval

[-2, 2]

. The quadratic function

x^2 - 2x

can be rewritten as

(x - 1)^2 - 1

, which has a minimum value of

-1

at

x = 1

. Because

x = 1

is inside the interval

[-2, 2]

, the minimum possible value is

-1

.

Step-by-Step Solution

1

Identify the critical points of the absolute value expressions to set up case analysis.

The critical points are

x = -1

and

x = 3

, dividing the real line into three intervals:

x < -1

,

-1 \leq x \leq 3

, and

x > 3

.

Absolute value expressions

|x - 3|

and

|x + 1|

change sign at

x = 3

and

x = -1

, respectively.

2

Solve the inequality

|x - 3| + 2|x + 1| \leq 7

across all three cases.

Case 1 (

x < -1

):

3 - x - 2(x + 1) \leq 7 \implies 1 - 3x \leq 7 \implies x \geq -2

. Combining gives

-2 \leq x < -1

.
Case 2 (

-1 \leq x \leq 3

):

3 - x + 2(x + 1) \leq 7 \implies x + 5 \leq 7 \implies x \leq 2

. Combining gives

-1 \leq x \leq 2

.
Case 3 (

x > 3

):

x - 3 + 2(x + 1) \leq 7 \implies 3x - 1 \leq 7 \implies x \leq \frac{8}{3}

. No solutions since

\frac{8}{3} < 3

.
The combined solution set is

-2 \leq x \leq 2

.

Piecewise analysis determines the exact range of valid values for

x

.

3

Minimize the quadratic expression

f(x) = x^2 - 2x

over the interval

[-2, 2]

.

Completing the square gives

f(x) = (x - 1)^2 - 1

. The vertex occurs at

x = 1

. Since

x = 1 \in [-2, 2]

, the minimum value is

f(1) = -1

.

A parabola opening upward achieves its global minimum at its vertex if the vertex lies within the allowed domain.

Key Concept

Solving piecewise absolute value inequalities and finding the extreme values of a quadratic function over a bounded interval.

Estimated Time:2m 0s

Question 897Question →

A tech hub tracked $150$ start-up companies to determine their use of three cloud services: Storage ( $S$ ), Analytics ( $A$ ), and Machine Learning ( $M$ ). Every company uses at least one of the three services. Exactly $95$ companies use Storage, $80$ use Analytics, and $65$ use Machine Learning. If exactly $20$ companies use all three services, and $45$ companies use exactly two of the services, how many companies use exactly one of the three cloud services?

65

75

85

95

105

Show answer & explanation

Answer: $75$

Answer

The number of companies that use exactly one of the three cloud services is

75

.

Using the region-based formula for three overlapping sets, the total number of elements is equal to the sum of elements in exactly one set, elements in exactly two sets, elements in all three sets, and elements in none of the sets. Here, Total

= 150

, Exactly Two

= 45

, Exactly Three

= 20

, and Neither

= 0

. Substituting these values gives

150 = \text{Exactly One} + 45 + 20 + 0

, which simplifies to

\text{Exactly One} = 75

.

Step-by-Step Solution

1

State the standard 3-set overlapping sets formula in terms of exact regions.

Total

= (\text{Exactly 1}) + (\text{Exactly 2}) + (\text{Exactly 3}) + (\text{Neither})

Dividing the Venn diagram into mutually exclusive regions simplifies calculating specific counts.

2

Substitute the known values into the regional total equation.

150 = (\text{Exactly 1}) + 45 + 20 + 0

Since every company uses at least one service,

\text{Neither} = 0

.

3

Solve for the number of companies using exactly one service.

\text{Exactly 1} = 150 - 65 = 75

Subtracting the sum of companies using exactly two services (

45

) and all three services (

20

) from the total (

150

) yields

75

.

Key Concept

Three-Set Overlapping Sets Regional Decomposition

Question 898Question →

If $x$ is a real number satisfying the equation $|x - 4| = 2x - 1$ , what is the value of $|x - 5|$ ?

8

\frac{7}{3}

\frac{10}{3}

4

\frac{34}{3}

Show answer & explanation

Answer: $\frac{10}{3}$

Answer

\frac{10}{3}

Solving

|x - 4| = 2x - 1

requires

2x - 1 \geq 0

, or

x \geq \frac{1}{2}

. Splitting into cases gives

x - 4 = 2x - 1 \implies x = -3

(invalid because

-3 < \frac{1}{2}

) and

-(x - 4) = 2x - 1 \implies x = \frac{5}{3}

(valid because

\frac{5}{3} \geq \frac{1}{2}

). Substituting

x = \frac{5}{3}

into

|x - 5|

yields

|\frac{5}{3} - 5| = |-\frac{10}{3}| = \frac{10}{3}

.

Step-by-Step Solution

1

Determine the non-negativity constraint for the absolute value equation

Since absolute values are non-negative,

2x - 1 \geq 0 \implies x \geq \frac{1}{2}

.

The right-hand side of

|x - 4| = 2x - 1

must be greater than or equal to 0.

2

Split into linear cases and solve for candidate values of x

Case 1:

x - 4 = 2x - 1 \implies x = -3

.
Case 2:

-(x - 4) = 2x - 1 \implies -x + 4 = 2x - 1 \implies 3x = 5 \implies x = \frac{5}{3}

.

An absolute value equation

|A| = B

splits into

A = B

or

A = -B

.

3

Eliminate extraneous solutions

x = -3

fails the condition

x \geq \frac{1}{2}

(since

2(-3) - 1 = -7 < 0

).

x = \frac{5}{3}

satisfies

x \geq \frac{1}{2}

(since

2(\frac{5}{3}) - 1 = \frac{7}{3} > 0

). Thus,

x = \frac{5}{3}

is the sole valid solution.

Candidate solutions must satisfy the original non-negativity constraint.

4

Substitute the valid solution into

|x - 5|

|\frac{5}{3} - 5| = |-\frac{10}{3}| = \frac{10}{3}

.

Calculate the value of the requested expression using

x = \frac{5}{3}

.

Key Concept

Absolute Value Equations and Extraneous Solutions

Question 899Question →

An investment fund allocated its total initial capital among three venture portfolios: Healthcare, Clean Tech, and Artificial Intelligence. Initially, $\frac{2}{5}$ of the capital was allocated to Healthcare and $0.35$ was allocated to Clean Tech, with the remaining fraction allocated to Artificial Intelligence. During a mid-year portfolio rebalancing, the amount in Healthcare was increased by $25\%$ , the amount in Clean Tech was decreased by $20\%$ , and the amount in Artificial Intelligence remained unchanged. If the total capital across all three portfolios after rebalancing was $\$5,150,000$ , what was the initial amount, in dollars, allocated to Artificial Intelligence?

$1,250,000

$1,287,500

$1,400,000

$1,750,000

$2,000,000

Show answer & explanation

Answer: $1,250,000

Answer

The initial amount allocated to Artificial Intelligence was $1,250,000 dollars.

The correct option of

1,250,000 is obtained by first calculating the initial proportion allocated to Artificial Intelligence as

1 - 0.40 - 0.35 = 0.25

. Applying the percentage adjustments yields Healthcare at

0.50C

, Clean Tech at

0.28C

, and Artificial Intelligence at

0.25C

. The sum of these updated allocations gives

1.03C = \

5,150,000

, which implies

C = \$5,000,000

. Taking

25\%

of

\$5,000,000

yields

\$1,250,000

.

Step-by-Step Solution

1

Determine the initial decimal fraction allocated to each portfolio.

Healthcare =

\frac{2}{5} = 0.40

; Clean Tech =

0.35

; Artificial Intelligence =

1 - (0.40 + 0.35) = 0.25

.

The sum of all initial portfolio fractions must equal

1

.

2

Express the post-rebalancing portfolio values in terms of the initial total capital

C

.

Healthcare =

0.40C \times (1 + 0.25) = 0.50C

; Clean Tech =

0.35C \times (1 - 0.20) = 0.28C

; Artificial Intelligence =

0.25C

.

A

25\%

increase scales a value by

1.25

, and a

20\%

decrease scales a value by

0.80

.

3

Sum the new portfolio values to set up an equation for total capital

C

.

0.50C + 0.28C + 0.25C = 1.03C = 5,150,000

.

The sum of all rebalanced portfolios equals the given final capital of

\$5,150,000

.

4

Solve for initial total capital

C

and calculate the Artificial Intelligence allocation.

C = \frac{5,150,000}{1.03} = 5,000,000

; Artificial Intelligence =

0.25 \times 5,000,000 = 1,250,000

.

Multiplying the initial total capital by

0.25

yields the original Artificial Intelligence allocation.

Key Concept

Combining sequential percentage changes and decimal/fraction conversions to solve multi-step financial allocation word problems.

Question 900Question →

For how many positive integers $n$ is the expression $\frac{n^2 + 7n + 120}{n + 3}$ equal to an integer?

6

9

10

11

12

Show answer & explanation

Answer: 9

Answer

9 positive integers

By dividing the numerator by the denominator, the expression simplifies to

n + 4 + \frac{108}{n + 3}

. For this to yield an integer,

(n + 3)

must be a positive divisor of 108. The number 108 has prime factorization

2^2 \cdot 3^3

, giving it

(2+1)(3+1) = 12

total positive divisors. Since

n

must be a positive integer (

n \ge 1

),

n + 3

must be at least 4. Eliminating the divisors 1, 2, and 3 leaves 9 valid values for

n + 3

, which correspond to 9 unique positive integers

n

.

Step-by-Step Solution

1

Perform polynomial long division or algebraic manipulation on the numerator

\frac{n^2 + 7n + 120}{n + 3} = \frac{(n^2 + 3n) + (4n + 12) + 108}{n + 3} = \frac{n(n+3) + 4(n+3) + 108}{n + 3} = n + 4 + \frac{108}{n + 3}

Separating the expression into an integer term

(n + 4)

and a proper fractional term

\frac{108}{n + 3}

isolates the divisibility condition.

2

Determine the condition for the fractional term to be an integer

The expression is an integer if and only if

(n + 3)

is a positive integer divisor of 108.

Since

n

is a positive integer,

(n + 4)

is always an integer, so

\frac{108}{n + 3}

must also be an integer.

3

Calculate the total number of positive divisors of 108

Prime factorization:

108 = 2^2 \cdot 3^3

. Total divisors =

(2+1)(3+1) = 3 \cdot 4 = 12

.

The formula for the number of positive divisors of

p_1^{a} p_2^{b}

is

(a+1)(b+1)

.

4

Apply the constraint that

n

is a positive integer (

n \ge 1

)

Since

n \ge 1

, we have

n + 3 \ge 4

. The divisors of 108 are {1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108}. The divisors less than 4 are {1, 2, 3} (3 divisors).

If

n + 3

were 1, 2, or 3,

n

would be

-2

,

-1

, or

0

, none of which are positive integers.

5

Subtract invalid divisors from total divisors

Valid divisors =

12 - 3 = 9

.

Each divisor

d \ge 4

yields exactly one unique positive integer

n = d - 3

.

Key Concept

Algebraic Divisibility and Prime Factorization Divisor Counting

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Answer

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Common Mistakes

Answer

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