All practice questions

Question 761Question →

A research group evaluated 150 renewable energy projects. Each project utilizes at least one of three primary energy technologies: solar, wind, or hydro. Exactly 85 projects utilize solar power, 60 projects utilize wind power, and 45 projects utilize hydro power. Furthermore, 20 projects utilize both solar and wind power, 15 projects utilize both solar and hydro power, and 10 projects utilize both wind and hydro power. How many of the surveyed projects utilize all three energy technologies?

Show answer & explanation

Answer: 5

Answer

5 projects utilize all three energy technologies.

Applying the three-set inclusion-exclusion principle Total = |A| + |B| + |C| - (|A ∩ B| + |A ∩ C| + |B ∩ C|) + |A ∩ B ∩ C| + Neither gives 150 = 85 + 60 + 45 - (20 + 15 + 10) + |A ∩ B ∩ C| + 0. Simplifying yields 150 = 145 + |A ∩ B ∩ C|, which gives |A ∩ B ∩ C| = 5.

Step-by-Step Solution

1

Identify known set sizes and recall the Principle of Inclusion-Exclusion for three sets.

Total = 150, |S| = 85, |W| = 60, |H| = 45, |S ∩ W| = 20, |S ∩ H| = 15, |W ∩ H| = 10, Neither = 0.

Setting up known quantities ensures proper application of set intersection rules.

2

Substitute the set sizes into the formula: Total = |S| + |W| + |H| - (|S ∩ W| + |S ∩ H| + |W ∩ H|) + |S ∩ W ∩ H| + Neither.

150 = 85 + 60 + 45 - (20 + 15 + 10) + |S ∩ W ∩ H| + 0.

Pairwise overlaps are double-counted when individual sets are summed and must be subtracted; the central triple overlap is over-subtracted and must be added back.

3

Combine known terms and isolate the unknown triple intersection.

150 = 190 - 45 + |S ∩ W ∩ H| => 150 = 145 + |S ∩ W ∩ H| => |S ∩ W ∩ H| = 5.

Subtracting 145 from 150 yields the exact count of projects belonging to all three sets.

Key Concept

Three-Set Inclusion-Exclusion Principle

Estimated Time:2m 0s

Question 762Question →

A boutique perfumery blends three essential oils—Jasmine, Sandalwood, and Bergamot—in an initial volume ratio of $4 : 3 : 2$ , respectively. To modify the fragrance profile, the perfumer adds $20\text{ milliliters}$ of Sandalwood and $40\text{ milliliters}$ of Bergamot to the mixture, leaving the amount of Jasmine unchanged. If the resulting volume ratio of Sandalwood to Bergamot is $5 : 6$ , what was the total initial volume, in milliliters, of the fragrance batch?

Show answer & explanation

Answer: 90

Answer

The total initial volume of the fragrance batch was 90 milliliters.

By representing the initial amounts of Jasmine, Sandalwood, and Bergamot as

4x

,

3x

, and

2x

respectively, the total initial volume is

9x

. Setting up the proportion

\frac{3x + 20}{2x + 40} = \frac{5}{6}

yields

x = 10

. Substituting

x = 10

into

9x

gives the initial volume of 90 milliliters.

Step-by-Step Solution

1

Define variables for the initial quantities using the given ratio.

Jasmine volume =

4x

, Sandalwood volume =

3x

, Bergamot volume =

2x

, where

x

is a positive multiplier. Total initial volume =

4x + 3x + 2x = 9x

.

Representing ratio parts algebraically allows setting up equations for the modified amounts.

2

Formulate an equation based on the new ratio after adding essential oils.

New Sandalwood volume =

3x + 20

, New Bergamot volume =

2x + 40

. Equating their ratio to

5 : 6

gives

\frac{3x + 20}{2x + 40} = \frac{5}{6}

.

The problem specifies the new relationship between Sandalwood and Bergamot.

3

Solve the proportion for

x

.

Cross-multiplying gives

6(3x + 20) = 5(2x + 40) \implies 18x + 120 = 10x + 200 \implies 8x = 80 \implies x = 10

.

Finding the multiplier

x

allows us to calculate the exact initial quantities.

4

Compute the total initial volume.

Total initial volume =

9x = 9 \times 10 = 90

milliliters.

The question asks for the total initial volume of the batch.

Key Concept

Solving multi-step ratio problems involving additions to individual components by establishing an algebraic multiplier.

Question 763Question →

For a certain constant $k$ , the quadratic equation $x^2 - 12x + k = 0$ has two real roots, $r_1$ and $r_2$ . If $r_1^2 + r_2^2 = 94$ , what is the value of $k$ ?

Show answer & explanation

Answer: 25

Answer

25

Applying Vieta's formulas gives

r_1 + r_2 = 12

and

r_1 r_2 = k

. Substituting these into the identity

(r_1 + r_2)^2 = r_1^2 + r_2^2 + 2r_1 r_2

yields

12^2 = 94 + 2k

. Solving

144 = 94 + 2k

gives

2k = 50

, so

k = 25

.

Step-by-Step Solution

1

Apply Vieta's formulas to the given quadratic equation

x^2 - 12x + k = 0

.

The sum of the roots is

r_1 + r_2 = 12

and the product of the roots is

r_1 r_2 = k

.

For a quadratic equation

ax^2 + bx + c = 0

, the sum of roots is

-b/a

and product is

c/a

.

2

Expand the square of the sum of the roots

(r_1 + r_2)^2

.

(r_1 + r_2)^2 = r_1^2 + r_2^2 + 2r_1 r_2

This algebraic identity connects the sum of roots, sum of squared roots, and product of roots.

3

Substitute

r_1 + r_2 = 12

,

r_1^2 + r_2^2 = 94

, and

r_1 r_2 = k

into the identity.

12^2 = 94 + 2k \implies 144 = 94 + 2k

Replacing terms with known numerical values creates a linear equation in

k

.

4

Isolate and solve for

k

.

2k = 144 - 94 = 50 \implies k = 25

Basic algebraic manipulation yields the exact value of

k

.

Key Concept

Vieta's Formulas and Symmetric Polynomial Identities

Estimated Time:1m 30s

Question 764Question →

A pharmaceutical laboratory prepares a liquid vaccine solution by blending three ingredients: Active Compound X, Active Compound Y, and Distilled Water in the volume ratio of $3 : 4 : 8$ , respectively. To meet updated formulation guidelines, a chemist adds $12$ liters of Active Compound Y and $36$ liters of Distilled Water to the solution, leaving the quantity of Active Compound X unchanged. If the new ratio of Active Compound X to Distilled Water in the resulting solution is $1 : 4$ , what was the total volume, in liters, of the solution before any ingredients were added?

Show answer & explanation

Answer: 135

Answer

The total volume of the solution before any ingredients were added was 135 liters.

Representing the initial volumes of Active Compound X, Active Compound Y, and Distilled Water as

3x

,

4x

, and

8x

, the initial total volume is

15x

liters. After adding

36

liters of Distilled Water while keeping Compound X at

3x

liters, the ratio of X to Water becomes

\frac{3x}{8x + 36} = \frac{1}{4}

. Solving

12x = 8x + 36

gives

4x = 36

, so

x = 9

. Multiplying

15

by

9

gives the initial total volume of

135

liters.

Step-by-Step Solution

1

Define the initial component quantities using a common ratio multiplier

The initial volumes of Compound X, Compound Y, and Distilled Water are

3x

,

4x

, and

8x

liters, giving an initial total volume of

15x

liters.

A ratio of

3 : 4 : 8

means the actual quantities are integer multiples of a common constant

x

.

2

Set up the ratio equation reflecting the additions

The equation comparing the unchanged Compound X to the updated volume of Distilled Water is

\frac{3x}{8x + 36} = \frac{1}{4}

.

No Compound X was added, so its volume stays

3x

, whereas

36

liters were added to the initial

8x

liters of Distilled Water.

3

Solve the algebraic equation for the multiplier

x

4(3x) = 1(8x + 36) \implies 12x = 8x + 36 \implies 4x = 36 \implies x = 9

.

Cross-multiplying isolates the terms with

x

and allows solving for the multiplier.

4

Calculate the target initial total volume

15 \times 9 = 135

liters.

Substituting

x = 9

into the initial total volume expression

15x

yields the final answer.

Key Concept

Solving multi-step ratio word problems by setting up unknown multiplier equations based on partial component alterations

Question 765Question →

A container initially holds $100$ liters of an acid solution that is $40\%$ acid by volume. First, $x$ liters of the solution are drained and replaced with an equal volume of pure water. After the mixture is thoroughly stirred, $x$ liters of the new solution are drained and replaced with an equal volume of pure acid. If the final solution is $45.6\%$ acid by volume, what is the value of $x$ ?

Show answer & explanation

Answer: 20

Answer

The value of

x

is

20

liters.

The solution requires tracking the amount of pure acid through two distinct replacement operations. Initially, there are

40

liters of acid in

100

liters of solution. Draining

x

liters removes

0.4x

liters of acid, leaving

(40 - 0.4x)

liters of acid after adding

x

liters of pure water. In the second step, draining

x

liters removes a fraction

\frac{x}{100}

of the remaining acid, leaving

(40 - 0.4x)(1 - \frac{x}{100})

liters of acid. Adding

x

liters of pure acid yields a total acid volume of

(40 - 0.4x)(1 - \frac{x}{100}) + x = 45.6

liters. Expanding and solving the resulting quadratic equation

x^2 + 50x - 1400 = 0

gives

x = 20

(since

x > 0

).

Step-by-Step Solution

1

Determine initial volume of solute (pure acid).

Initial acid volume =

40\% \times 100 = 40

liters.

Establishing the starting amount of pure acid in the

100

-liter container.

2

Model the acid amount after the first replacement (with water).

Acid volume after first replacement =

40\left(1 - \frac{x}{100}\right)

liters.

Draining

x

liters removes

\frac{x}{100}

of the total acid, and adding water adds zero acid.

3

Model the acid amount after the second replacement (with pure acid).

Final acid volume =

40\left(1 - \frac{x}{100}\right)\left(1 - \frac{x}{100}\right) + x

liters.

Draining

x

liters of the new solution removes

\frac{x}{100}

of its acid, and replacing with pure acid adds

x

liters of acid.

4

Set up and solve the quadratic equation given final acid volume of

45.6

liters.

x^2 + 50x - 1400 = 0 \implies (x+70)(x-20) = 0 \implies x = 20

.

Discarding the negative root (

x = -70

) because volume must be positive.

Key Concept

Multi-stage sequential mixture removal and replacement

Question 766Question →

A oceanographic research institute uses three autonomous submersibles��Submersible A, Submersible B, and Submersible C—to perform high-resolution seabed mapping. Working together at their respective constant rates, Submersible A and Submersible B can complete a full mapping mission in $15$ hours. Working together at their respective constant rates, Submersible B and Submersible C can complete the same mapping mission in $24$ hours. During a specialized operation, Submersible A works alone for $6$ hours, after which it is recalled. Submersible B then works alone for $10$ hours, after which Submersible C joins Submersible B, and both work together for an additional $12$ hours to complete the remaining portion of the mission. How many hours would it take Submersible A to complete the entire seabed mapping mission working alone at its constant rate?

18

20

24

30

36

Show answer & explanation

Answer: 24

Answer

24 hours

The option stating 24 is correct because setting up the system of work rates

r_A + r_B = \frac{1}{15}

and

r_B + r_C = \frac{1}{24}

, along with the multi-stage work equation

6r_A + 10r_B + 12(r_B + r_C) = 1

, allows us to eliminate

r_C

and solve for

r_A = \frac{1}{24}

. Taking the reciprocal gives 24 hours.

Step-by-Step Solution

1

Define rate variables and set up combined rate equations.

Let

r_A

,

r_B

, and

r_C

be the hourly rates of Submersibles A, B, and C respectively (in fraction of mission per hour). We are given: (1)

r_A + r_B = \frac{1}{15}

and (2)

r_B + r_C = \frac{1}{24}

.

Work rates are additive reciprocals of completion times for constant-rate work problems.

2

Express total mission work completed in the multi-stage operation.

6r_A + 10r_B + 12(r_B + r_C) = 1

, which simplifies to

6r_A + 22r_B + 12r_C = 1

.

Total work equals the sum of work done across all sequential operational phases.

3

Substitute equation (2) into the multi-stage work equation.

Since

r_C = \frac{1}{24} - r_B

, we get

6r_A + 22r_B + 12\left(\frac{1}{24} - r_B\right) = 1 \implies 6r_A + 10r_B + \frac{1}{2} = 1 \implies 6r_A + 10r_B = \frac{1}{2}

.

Substituting

r_C

eliminates one variable, reducing the system to two equations in

r_A

and

r_B

.

4

Solve the two-variable system for

r_A

.

Multiply equation (1) by

10

to get

10r_A + 10r_B = \frac{10}{15} = \frac{2}{3}

. Subtract

6r_A + 10r_B = \frac{1}{2}

from this to get

4r_A = \frac{2}{3} - \frac{1}{2} = \frac{1}{6} \implies r_A = \frac{1}{24}

.

Eliminating

r_B

isolates Submersible A's rate.

5

Calculate the time taken by Submersible A working alone.

Time

= \frac{1}{r_A} = \frac{1}{1/24} = 24

hours.

Completion time is the reciprocal of the hourly work rate.

Key Concept

System of Work Rate Equations for Multi-Stage Combined Work

Estimated Time:2m 30s

Question 767Question →

A container holds $4$ red tokens and $6$ blue tokens. If two tokens are selected at random one after another without replacement, what is the probability that both selected tokens are red?

\frac{4}{25}

\frac{8}{45}

\frac{2}{15}

\frac{2}{5}

\frac{1}{3}

Show answer & explanation

Answer: $\frac{2}{15}$

Answer

\frac{2}{15}

The option stating

\frac{2}{15}

is correct because selecting without replacement makes the second event dependent on the first. The first probability is

\frac{4}{10}

and the second probability is

\frac{3}{9}

. Multiplying these gives

\frac{4}{10} \times \frac{3}{9} = \frac{12}{90} = \frac{2}{15}

.

Step-by-Step Solution

1

Calculate the probability of selecting a red token on the first draw.

P(\text{First Red}) = \frac{4}{10} = \frac{2}{5}

There are

4

red tokens out of a total of

10

tokens.

2

Calculate the conditional probability of selecting a red token on the second draw given the first was red.

P(\text{Second Red} \mid \text{First Red}) = \frac{3}{9} = \frac{1}{3}

Because the selection is without replacement,

1

red token and

1

total token have been removed, leaving

3

red tokens out of

9

remaining tokens.

3

Multiply the sequential probabilities to find the combined probability.

P(\text{Both Red}) = \frac{2}{5} \times \frac{1}{3} = \frac{2}{15}

The probability of both dependent events occurring sequentially is the product of their individual conditional probabilities.

Key Concept

Probability of Dependent Events Without Replacement

Question 768Question →

A barista has $40$ ounces of a flavored drink mixture that is $30\%$ syrup by volume. How many ounces of pure water must be added to reduce the syrup concentration to $24\%$ by volume?

Show answer & explanation

Answer: 10

Answer

10 ounces of pure water must be added.

Adding 10 ounces of pure water increases the total volume of the mixture from 40 ounces to 50 ounces while keeping the volume of pure syrup constant at 12 ounces. The resulting concentration is 12 / 50 = 0.24, or 24% by volume.

Step-by-Step Solution

1

Calculate the volume of pure syrup present in the initial mixture

12

ounces of syrup

The initial 40-ounce mixture contains 30% syrup by volume (

40 \times 0.30 = 12

).

2

Set up the concentration equation after adding

x

ounces of pure water

\frac{12}{40 + x} = 0.24

Adding pure water increases the total volume to

40 + x

ounces without changing the amount of pure syrup.

3

Solve the linear equation for

x

x = 10

Multiplying both sides by

40 + x

yields

12 = 9.6 + 0.24x

, which simplifies to

2.4 = 0.24x

, giving

x = 10

.

Key Concept

Dilution of a mixture by adding pure solvent

Question 769Question →

The function

f

is defined for all real numbers by

f(x) = \begin{cases} 3x - 1 & \text{if } x \le 4 \\ x^2 - k & \text{if } x > 4 \end{cases}

where

k

is a constant. If

f(f(3)) = 14

, what is the value of

k

?

Show answer & explanation

Answer: 50

Answer

50

To solve for

k

, evaluate the composite function

f(f(3))

step-by-step. First, find

f(3)

. Since

3 \le 4

, we use the first branch of the definition:

f(3) = 3(3) - 1 = 8

. Next, evaluate

f(8)

. Since

8 > 4

, we use the second branch of the definition:

f(8) = 8^2 - k = 64 - k

. Given that

f(f(3)) = 14

, we set

64 - k = 14

, which yields

k = 50

.

Step-by-Step Solution

1

Evaluate the inner function value f(3)

f(3) = 8

Because 3 is less than or equal to 4, the definition f(x) = 3x - 1 applies, yielding f(3) = 3(3) - 1 = 8.

2

Express f(f(3)) in terms of k

f(8) = 64 - k

Since the inner value is 8 and 8 > 4, the piecewise condition specifies using f(x) = x^2 - k, giving f(8) = 8^2 - k = 64 - k.

3

Solve the linear equation for k

k = 50

Setting the calculated expression equal to the given value 14 gives 64 - k = 14, which simplifies to k = 50.

Key Concept

Piecewise Function Evaluation and Parameter Resolution

Estimated Time:1m 30s

Question 770Question →

For all non-zero real numbers $a$ and $b$ , the binary operator $\star$ is defined by $a \star b = \frac{a^2 + b}{a}$ . The function $f$ is defined by $f(x) = x \star 16$ for all $x \neq 0$ . If $k$ is a positive real number such that $f(f(k)) = 10$ , what is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is

4

.

By definition of the custom operator,

f(x) = x + \frac{16}{x}

. Evaluating

f(4) = 4 + \frac{16}{4} = 8

and then

f(f(4)) = f(8) = 8 + \frac{16}{8} = 10

confirms that

k = 4

satisfies the condition.

Step-by-Step Solution

1

Express

f(x)

by applying the custom operator definition

f(x) = x \star 16 = \frac{x^2 + 16}{x} = x + \frac{16}{x}

Using

a = x

and

b = 16

in

a \star b = \frac{a^2 + b}{a}

simplifies the function formula.

2

Set

m = f(k)

and solve

f(m) = 10

for

m

m + \frac{16}{m} = 10 \implies m^2 - 10m + 16 = 0 \implies (m - 2)(m - 8) = 0

, yielding

m = 2

or

m = 8

Replacing the inner function evaluation with an intermediate variable creates a manageable quadratic equation.

3

Solve

f(k) = 2

and

f(k) = 8

for positive real values of

k

For

f(k) = 2

:

k^2 - 2k + 16 = 0

has discriminant

(-2)^2 - 4(1)(16) = -60 < 0

, giving no real solutions. For

f(k) = 8

:

k^2 - 8k + 16 = 0 \implies (k - 4)^2 = 0 \implies k = 4

.

Only

m = 8

produces a real value for

k

because

k + \frac{16}{k} \geq 8

for all positive real numbers

k

.

Key Concept

Custom Binary Operators and Nested Function Evaluation

Question 771Question →

In a corporate workshop of $80$ executives, $52$ executives attended the Leadership module and $40$ executives attended the Negotiation module. If $18$ executives attended neither module, how many executives attended both modules?

Show answer & explanation

Answer: 30

Answer

30 executives attended both modules.

Applying the formula

\text{Total} = \text{Group A} + \text{Group B} - \text{Both} + \text{Neither}

yields

80 = 52 + 40 - \text{Both} + 18

. Simplifying gives

80 = 110 - \text{Both}

, so the number of executives who attended both modules is

30

.

Step-by-Step Solution

1

Determine the number of executives who attended at least one of the two modules.

Executives attending at least one module =

80 - 18 = 62

.

Subtracting those who attended neither module from the total gives the union of the two sets.

2

Apply the inclusion-exclusion principle to find the intersection of the two sets.

\text{Both} = \text{Leadership} + \text{Negotiation} - \text{At least one} = 52 + 40 - 62 = 30

.

Summing the participants of each module double-counts those who attended both, so subtracting the number attending at least one module isolates the overlap.

Key Concept

Two-Set Overlapping Sets Formula (Inclusion-Exclusion Principle)

Estimated Time:1m 0s

Question 772Question →

At a software firm, an audit of 200 developers evaluated proficiency in three programming languages: Python, Java, and C++.

- 120 developers are proficient in Python.
- 105 developers are proficient in Java.
- 95 developers are proficient in C++.
- 15 developers are proficient in all three languages.
- 10 developers are proficient in none of the three languages.
- The ratio of the number of developers proficient in BOTH Python and Java ONLY to the number of developers proficient in BOTH Python and C++ ONLY is $3 : 2$ .
- The number of developers proficient in ONLY Java is equal to the number of developers proficient in ONLY C++.

How many developers are proficient in ONLY Python?

Show answer & explanation

Answer: 55

Answer

55 developers are proficient in ONLY Python.

By decomposing the 3-set Venn diagram into 7 mutually exclusive regions, the given ratio

d : e = 3 : 2

gives

d = 3k

and

e = 2k

. Comparing the totals for Java (

105

) and C++ (

95

) under the condition that Java-only equals C++-only shows that

k = 10

. Thus,

30

developers are in Python and Java only,

20

are in Python and C++ only, and

15

are in all three. Subtracting these three regions from the total

120

Python developers yields

120 - 30 - 20 - 15 = 55

proficient in Python only.

Step-by-Step Solution

1

Assign variables to the Venn diagram regions and incorporate given ratios.

Represent

d = |P \cap J \text{ only}| = 3k

and

e = |P \cap C \text{ only}| = 2k

, with

g = 15

and

\text{None} = 10

.

Establishing explicit variables for non-overlapping regions simplifies system solving.

2

Set up set total equations for

|J|

and

|C|

to solve for

k

.

|J| \implies b + 3k + f = 90

and

|C| \implies b + 2k + f = 80

. Subtracting yields

k = 10

.

Since

b = c

(Java only = C++ only), subtracting the two set equations eliminates

b

and

f

, directly giving

k

.

3

Calculate the region values

d

and

e

, then solve for

a = |P \text{ only}|

.

d = 30

,

e = 20

. Then

a = 120 - (30 + 20 + 15) = 55

.

Subtracting all other regions of set

P

from the total proficient in Python isolates those proficient ONLY in Python.

Key Concept

Three-Set Overlapping Sets and Region Decomposition

Question 773Question →

Vessel A contains $40$ kilograms of an organic fertilizer mixture that is $60\%$ Nitrogen by weight. Vessel B contains $60$ kilograms of a fertilizer mixture that is $20\%$ Nitrogen by weight. First, $x$ kilograms of the mixture are removed from Vessel A and transferred into Vessel B, where the contents are thoroughly mixed. Then, $x$ kilograms of the newly formed mixture in Vessel B are transferred back into Vessel A. If the final mixture in Vessel A is $50\%$ Nitrogen by weight, what is the value of $x$ ?

10

12

15

16

20

Show answer & explanation

Answer: $12$

Answer

The value of

x

is

12

.

The correct answer of 12 kg accounts for the dynamic concentration change in Vessel B after the initial transfer. When

x = 12

kg is moved from Vessel A to Vessel B, Vessel B contains 72 kg total mixture with 19.2 kg Nitrogen, giving a concentration of

26.67\%

(or

\frac{4}{15}

). Transferring 12 kg of this mixture back to Vessel A adds

12 \times \frac{4}{15} = 3.2

kg Nitrogen to the 16.8 kg Nitrogen remaining in Vessel A, resulting in exactly 20 kg of Nitrogen in a 40 kg total mixture (

50\%

concentration).

Step-by-Step Solution

1

Calculate the initial mass of Nitrogen in each vessel.

Vessel A initially has

0.60 \times 40 = 24

kg of Nitrogen. Vessel B initially has

0.20 \times 60 = 12

kg of Nitrogen.

Tracking exact solute quantities is required for setting up the mixture conservation equation.

2

Determine the amount of Nitrogen in both vessels after transferring

x

kg from Vessel A to Vessel B.

Vessel A retains

(40 - x)

kg of solution containing

(24 - 0.60x)

kg of Nitrogen. Vessel B now has

(60 + x)

kg of total mixture containing

(12 + 0.60x)

kg of Nitrogen.

The solution removed from Vessel A carries Nitrogen at a concentration of 60%.

3

Express the concentration of Nitrogen in Vessel B prior to the second transfer.

The concentration in Vessel B is

C_B = \frac{12 + 0.60x}{60 + x}

.

Concentration equals total mass of Nitrogen divided by total mass of the mixture in Vessel B.

4

Formulate the equation for the final Nitrogen mass in Vessel A after returning

x

kg from Vessel B.

Final Nitrogen in Vessel A:

(24 - 0.60x) + x \cdot \left(\frac{12 + 0.60x}{60 + x}\right) = 0.50 \times 40 = 20

.

The final volume of Vessel A is restored to 40 kg with a target concentration of 50%.

5

Solve the algebraic equation for

x

.

4 - 0.60x + \frac{12x + 0.60x^2}{60 + x} = 0 \implies (4 - 0.60x)(60 + x) + 12x + 0.60x^2 = 0 \implies 240 + 4x - 36x - 0.60x^2 + 12x + 0.60x^2 = 0 \implies 240 - 20x = 0 \implies x = 12

.

Multiplying through by

(60 + x)

cancels the non-linear

x^2

terms, yielding a linear relation.

Key Concept

Two-Stage Transfer and Dilution

Estimated Time:2m 30s

Question 774Question →

A regional logistics company maintains a fleet consisting of delivery vans, medium trucks, and heavy cargo trucks in the initial ratio of $3 : 4 : 5$ , respectively. After the company retires 4 delivery vans, purchases 12 medium trucks, and purchases 6 heavy cargo trucks, the ratio of delivery vans to heavy cargo trucks becomes $1 : 2$ . What was the total number of vehicles in the fleet initially?

120

144

168

182

196

Show answer & explanation

Answer: 168

Answer

168

The initial ratio of

3 : 4 : 5

implies that the number of vans, medium trucks, and heavy cargo trucks can be represented as

3x

,

4x

, and

5x

for some positive multiplier

x

. The initial total fleet is

3x + 4x + 5x = 12x

. Modifying the counts according to the problem gives

3x - 4

vans and

5x + 6

heavy trucks. Equating their ratio to

1/2

produces

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

. Cross-multiplying gives

6x - 8 = 5x + 6

, which simplifies to

x = 14

. Multiplying

14

by the initial sum of ratio units (

12

) yields the correct initial total of

168

.

Step-by-Step Solution

1

Define initial quantities using a common multiplier

x

Delivery vans

= 3x

, Medium trucks

= 4x

, Heavy cargo trucks

= 5x

. Initial total fleet

= 3x + 4x + 5x = 12x

.

Expressing each vehicle count in terms of

x

preserves the given initial ratio of

3 : 4 : 5

.

2

Formulate the equation based on the updated vehicle counts and new ratio

New number of vans

= 3x - 4

. New number of heavy cargo trucks

= 5x + 6

. Equation:

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

.

The problem states that after retiring 4 vans and adding 6 heavy cargo trucks, the ratio of vans to heavy cargo trucks becomes

1 : 2

.

3

Solve the algebraic equation for

x

2(3x - 4) = 1(5x + 6) \implies 6x - 8 = 5x + 6 \implies x = 14

.

Cross-multiplying eliminates the fraction and isolates

x

.

4

Calculate the initial total number of vehicles

Initial total

= 12x = 12 \times 14 = 168

.

Substituting

x = 14

into the initial total expression

12x

yields the total fleet size prior to any modifications.

Key Concept

Multi-part ratio scaling and structural algebraic modeling under quantity changes

Question 775Question →

If $x$ is a real number such that $x^3 - 3x^2 = 4x - 12$ , what is the sum of all possible values of $x$ ?

-3

0

2

3

5

Show answer & explanation

Answer: $3$

Answer

The sum of all possible values of

x

is 3.

Moving all terms to one side gives

x^3 - 3x^2 - 4x + 12 = 0

. Grouping the terms yields

x^2(x - 3) - 4(x - 3) = 0

, which factors completely into

(x - 2)(x + 2)(x - 3) = 0

. The real solutions are

x = 2

,

x = -2

, and

x = 3

. Adding these values together yields

2 + (-2) + 3 = 3

.

Step-by-Step Solution

1

Rearrange the equation to set it equal to zero.

x^3 - 3x^2 - 4x + 12 = 0

Grouping terms requires having all terms on one side of the equation.

2

Factor by grouping terms in pairs.

x^2(x - 3) - 4(x - 3) = 0 \Rightarrow (x^2 - 4)(x - 3) = 0

Extracting the common factor

(x - 3)

allows reducing the cubic polynomial into a linear factor and a quadratic factor.

3

Factor the difference of squares and set each factor to zero to find all real roots.

(x - 2)(x + 2)(x - 3) = 0 \Rightarrow x = 2, x = -2, x = 3

By the zero-product property, the expression equals zero when any factor is equal to zero.

4

Calculate the sum of all distinct real solutions.

2 + (-2) + 3 = 3

The question asks for the sum of all possible real values of

x

.

Key Concept

Polynomial Factoring by Grouping and Zero-Product Property

Estimated Time:1m 30s

Question 776Question →

A retailer purchases a jacket for $80. The retailer marks up the purchase price by 25% to establish a list price. During a clearance sale, the jacket is sold at a 10% discount off the list price. What is the retailer's net profit on the sale of the jacket?

$8

$10

$12

$18

$20

Show answer & explanation

Answer: $10

Answer

The retailer's net profit is $10.

To find the net profit, first determine the list price by adding a 25% markup to the cost price of

80, which gives

100. Then apply a 10% discount to the list price of

100, yielding a selling price of

90. Subtracting the initial cost of

80 from

90 gives a net profit of $10.

Step-by-Step Solution

1

Calculate the list price by applying the 25% markup to the cost price.

\text{List Price} = \$80 + (0.25 \times \$80) = \$80 + \$20 = \$100

Markup is calculated as a percentage of the original cost price.

2

Calculate the selling price after applying the 10% discount to the list price.

\text{Selling Price} = \$100 - (0.10 \times \$100) = \$100 - \$10 = \$90

Discounts are calculated based on the list price, not the original cost price.

3

Calculate the net profit by subtracting the cost price from the selling price.

\text{Net Profit} = \text{Selling Price} - \text{Cost Price} = \$90 - \$80 = \$10

Profit is the difference between the final selling price and the initial purchase cost.

Key Concept

Profit, Loss, and Markup with Successive Percentages

Question 777Question →

At a research foundation, annual grant funding was initially allocated among three divisions—Artificial Intelligence, Biotechnology, and Renewable Energy—in the ratio $7 : 4 : 3$ , respectively. At mid-year, the foundation redistributed the funding by transferring $\$10$ million from the Artificial Intelligence division to the Renewable Energy division, while the Biotechnology division's funding remained unchanged. After this transfer, the ratio of Artificial Intelligence funding to Renewable Energy funding became $3 : 2$ . What was the total annual grant funding, in millions of dollars, allocated across all three divisions?

Show answer & explanation

Answer: 140

Answer

The total annual grant funding allocated across all three divisions was

140

million dollars.

By setting the initial funding amounts to

7x

,

4x

, and

3x

million dollars, the total funding is

14x

. The transfer alters Artificial Intelligence funding to

7x - 10

and Renewable Energy funding to

3x + 10

. Setting their ratio to

\frac{3}{2}

yields the equation

2(7x - 10) = 3(3x + 10)

, which simplifies to

5x = 50

, giving

x = 10

. Thus, total funding is

14 \times 10 = 140

million dollars.

Step-by-Step Solution

1

Define variables for the initial allocation based on the ratio

7 : 4 : 3

.

Artificial Intelligence funding

= 7x

, Biotechnology funding

= 4x

, Renewable Energy funding

= 3x

, and total funding

= 7x + 4x + 3x = 14x

.

Ratios represent relative parts of a whole multiplier

x

.

2

Model the redistribution of funds.

New Artificial Intelligence funding

= 7x - 10

; New Renewable Energy funding

= 3x + 10

.

Transferring

\$10

million decreases Artificial Intelligence funding by

10

and increases Renewable Energy funding by

10

.

3

Equate the new ratio to

3 : 2

and solve for

x

.

\frac{7x - 10}{3x + 10} = \frac{3}{2} \implies 14x - 20 = 9x + 30 \implies 5x = 50 \implies x = 10

.

Cross-multiplication converts ratio relationships into a solvable linear equation.

4

Calculate the total annual grant funding.

Total funding

= 14x = 14 \times 10 = 140

million dollars.

Substituting

x = 10

into the total funding expression

14x

gives the final answer.

Key Concept

Algebraic setup of multi-part ratio redistribution problems

Question 778Question →

What is the sum of all distinct real solutions to the equation $(x^2 - 4)^2 = 9(x - 2)^2$ ?

-4

-2

0

1

3

Show answer & explanation

Answer: $-2$

Answer

The sum of all distinct real solutions is

-2

.

Factoring

(x^2 - 4)^2 = 9(x - 2)^2

gives

(x - 2)^2(x + 2)^2 - 9(x - 2)^2 = 0

. Factoring out

(x - 2)^2

yields

(x - 2)^2[(x + 2)^2 - 9] = 0

, which simplifies to

(x - 2)^2(x - 1)(x + 5) = 0

. The distinct real solutions are

x = 2

,

x = 1

, and

x = -5

. Adding these values together gives

2 + 1 + (-5) = -2

.

Step-by-Step Solution

1

Rewrite

x^2 - 4

as

(x - 2)(x + 2)

and substitute into the left side

[(x - 2)(x + 2)]^2 = (x - 2)^2(x + 2)^2

To identify common factors on both sides of the equation

2

Rearrange the equation and factor out

(x - 2)^2

(x - 2)^2(x + 2)^2 - 9(x - 2)^2 = 0 \implies (x - 2)^2[(x + 2)^2 - 9] = 0

Factoring out common terms avoids losing roots by division

3

Factor the inner quadratic expression using difference of squares

(x + 2)^2 - 9 = (x + 2 - 3)(x + 2 + 3) = (x - 1)(x + 5)

To break down the remaining quadratic into linear factors

4

Identify all distinct real roots and sum them

Roots are

x = 2

,

x = 1

, and

x = -5

. Sum:

2 + 1 + (-5) = -2

To find the sum of all unique real solutions

Key Concept

Polynomial Factoring and Root Preservation

Estimated Time:2m 0s

Question 779Question →

In a group of $100$ students, $60$ students study Spanish and $40$ students study French. Among the $60$ students studying Spanish, $15$ also study French. If a student who studies Spanish is selected at random, what is the probability that the selected student also studies French?

Show answer & explanation

Answer: 0.25

Answer

The probability that a randomly selected Spanish-studying student also studies French is 0.25.

Since the student is chosen from the group of

60

Spanish-studying students, the sample space is restricted to

60

. Within this subset,

15

students study French. The conditional probability is therefore

\frac{15}{60} = 0.25

.

Step-by-Step Solution

1

Identify the total number of outcomes in the restricted sample space.

The sample space is restricted to students studying Spanish:

n(\text{Spanish}) = 60

.

The question specifies that the student is chosen from those who study Spanish.

2

Identify the number of favorable outcomes within this restricted sample space.

The number of students studying both Spanish and French is

n(\text{Spanish} \cap \text{French}) = 15

.

We need the count of students who satisfy both the given condition and the target event.

3

Calculate the conditional probability.

P(\text{French} | \text{Spanish}) = \frac{15}{60} = 0.25

.

Conditional probability is calculated by dividing the intersection count by the given condition's total count.

Key Concept

Conditional Probability:

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

Question 780Question →

A quality control inspector tests items from a batch containing $n$ high-grade items and $4$ low-grade items, where $n > 1$ . Two items are selected sequentially. If the first item selected is high-grade, it is kept out of the batch for testing, and additionally $1$ low-grade item is also removed from the batch before the second selection. If the probability of selecting two high-grade items in succession is $\frac{3}{8}$ , what is the value of $n$ ?

Show answer & explanation

Answer: 6

Answer

The initial number of high-grade items

n

is

6

.

Because the outcome of the first draw alters both the count of high-grade items and the count of low-grade items, the two draws are dependent events. The probability of drawing a high-grade item on the first draw is

\frac{n}{n+4}

. Following this successful outcome, the remaining pool consists of

n-1

high-grade items and

3

low-grade items, yielding a total of

n+2

items. The probability of a second high-grade selection is therefore

\frac{n-1}{n+2}

. Multiplying these probabilities gives

\frac{n(n-1)}{(n+4)(n+2)} = \frac{3}{8}

. Solving the resulting quadratic equation

5n^2 - 26n - 24 = 0

gives

n = 6

(since

n

must be a positive integer).

Step-by-Step Solution

1

Calculate the probability of selecting a high-grade item on the first draw.

P(H_1) = \frac{n}{n+4}

The initial batch contains

n

high-grade items out of a total of

n+4

items.

2

Determine the conditional sample space for the second draw after event

H_1

occurs.

High-grade items remaining =

n-1

; low-grade items remaining =

4-1=3

; total remaining items =

(n-1)+3 = n+2

.

Drawing a high-grade item removes

1

high-grade item and

1

additional low-grade item from the batch.

3

Calculate the conditional probability of selecting a high-grade item on the second draw.

P(H_2 | H_1) = \frac{n-1}{n+2}

There are now

n-1

high-grade items out of

n+2

total remaining items.

4

Set up and solve the algebraic probability equation.

\frac{n(n-1)}{(n+4)(n+2)} = \frac{3}{8} \implies 8n^2 - 8n = 3(n^2 + 6n + 8) \implies 5n^2 - 26n - 24 = 0 \implies (5n+4)(n-6) = 0 \implies n = 6

.

Because

n > 1

is required, the negative fraction root

-\frac{4}{5}

is discarded.

Key Concept

Probability of dependent events with non-standard sample space modifications

Answer

Step-by-Step Solution

Key Concept

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

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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

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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

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